Laser beam interaction with materials for microscale applications

Transcription

1 Worceser Polyechnc Insue Dgal WPI Docoral Dsseraons (All Dsseraons All Years) Elecronc Theses and Dsseraons 5-- Laser beam neracon wh maerals for mcroscale applcaons Krysof A. Nowaows Worceser Polyechnc Insue Follow hs and addonal wors a: hps://dgalcommons.wp.edu/ed-dsseraons Reposory Caon Nowaows K. A. (5). Laser beam neracon wh maerals for mcroscale applcaons. Rereved from hps://dgalcommons.wp.edu/ed-dsseraons/46 Ths dsseraon s brough o you for free and open access by Dgal WPI. I has been acceped for ncluson n Docoral Dsseraons (All Dsseraons All Years) by an auhored admnsraor of Dgal WPI. For more nformaon please conac wp-ed@wp.edu.

2 Laser beam neracon wh maerals for mcroscale applcaons A Dsseraon submed o he Faculy of he Worceser Polyechnc Insue n paral fulfllmen of he requremens for he Degree of Docor of Phlosophy n Mechancal Engneerng by Krysof A. Nowaows November 5 Approved: Prof. Rysard J. Prypunewc Maor Advsor Prof. Cosme Furlong Member Dsseraon Commee Prof. Rchard D. Ssson Jr. Member Dsseraon Commee Dr. Thomas F. Marns Draper Laboraory Cambrdge MA Member Dsseraon Commee Prof. John M. Sullvan Jr. Graduae Commee Represenave

3 Ths Dsseraon s dedcaed o my Wfe and Chldren

4 Copyrgh 5 by NEST NanoEngneerng Scence and Technology CHSLT Cener for Holographc Sudes and Laser mcro-mechatroncs Mechancal Engneerng Deparmen Worceser Polyechnc Insue Worceser MA 69-8 All rghs reserved 3

5 Nomenclaure a s spo se f he collmaed beam exng he laser a coeffcens of he leas-square f funcon b he focused beam se of he lease beam on he fber face c specfc hea or speed of lgh or he desred fber fll facor c (T ) specfc hea c p specfc hea a consan pressure c pl specfc hea a consan pressure for lqud sae c pv specfc hea a consan pressure for vapor sae c s velocy of sound n he sold d mnmum fber opc dameer da dfferenal area d mn se of he mnmum spo d he nfnesmal me sep dx spaal ncremens n x drecon dy spaal ncremens n y drecon d spaal ncremens n drecon e cc eccenrcy erf error funcon erfc complemenary error funcon f laser pulse frequency or focal lengh or fracon of elecron energy fcf focusng correcon facor fs femoseconds ffp focal plane poson fps frames per second g gravaonal consan or he gravaonal acceleraon g e degeneracy facors for elecrons g degeneracy facors for ons g degeneracy facors for neural aoms h mel hcness or specfc enhalpy or Planc s consan h c convecve hea ransfer coeffcen h ch convecve hea ransfer coeffcen nsde he hole h pl plasma plume-affeced focusng radus h r radave hea ransfer coeffcen h rh radave hea ransfer coeffcen nsde he hole h he value of nodal enhalpy a me number of phase seps ) ) ) he un vecors along each of he Caresan coordnae drecons erfc complemenary error funcon hermal conducvy or magnary par of he refracve ndex or 4

6 exncon coeffcen (T ) hermal conducvy ar hermal conducvy of he ar surroundng he worpece B Bolmann s consan g he hermal conducvy of he asss gas l hermal conducvy n lqud sae s hermal conducvy n sold sae v hermal conducvy n vaporaon sae l hermal dffuson lengh or mode number subscrp used n represenaon of he hea ransfer coeffcens refer o he sample s surface wh surface normal ponng n he negave (or lef) drecon of he coordnae axes l d dffuson lengh l s peneraon sn deph l h hermal peneraon deph lbwp laser beam was poson m average mass of an evaporaon aom or mass of an elecron or he absolue value of he magnfcaon of he magng sysem m e mass of mel eecon or elecron mass m m he mass of meled meal ha was eeced m s mass of melng of sold meal m v mass of evaporaon n real par of he refracve ndex or densy of a wealy oned plasma or neger ndcang number of orbs n he aom or he order of he leas-squares f polynomal n ndex of refracon n ndex of refracon ns nanosecond n a he elecron number densy n ag aom number densy of gas g n ec or n ec crcal elecron densy n e elecron densy or plasma densy n g flow velocy of he asss gas n densy of sngly oned aoms n oal gas densy or he neural on aom densy nsde he plasma n Zg Z-charged on number densy of gas g n complex ndex of refracon p permeer of he worpece or pressure or mode number or poson facor or he ransform varable p(τ) he normaled beam power ps pcoseconds p n pressure componen eners he fne elemen volume p ou pressure componen exs he fne elemen volume 5

7 p b p c p e p eff p p r p s p sa (T s ) p(τ) ps q q c q n q r q rad r r r b r blur r c r g r m r pl s dsp e e l max p pulse u u o v v c v d v de v dm he pressure whn he bubble he asss gas pressure a he nole ex he exernal pressure exered on he maeral by evaporang aoms he effecve sac gas pressure he asss gas pressure nsde he nole evaporaon recol pressure sauraed vapor pressure sauraed vapor pressure as a funcon of emperaure normaled oal beam power as a funcon of normaled me no he pulse pcosecond hea flux or lens shape facor convecon hea flux he magnude of he hea flux n he n-drecon radaon hea flux he magnude of radaon flux radal coordnae or drecon along he mel surface or subscrp used n represenaon of he hea ransfer coeffcens refer o he sample s surface wh surface normal ponng n he posve (or rgh) drecon of he coordnae axes he radal dsance a whch he surface emperaure s equal o he bolng pon he radus of crcal bubble he blur nroduced because of dffracon effecs correlaon lengh he asss gas densy he mel puddle radus half axs wdh of he plasma plume he ransform varable me mel dsplacemen me he elecron coolng me he me for mel dsplacemen from he mel puddle he lace heang me he maxmum me laser pulse duraon he oal me of he power meer measuremen dummy varable or nernal energy weldng speed lnear vaporaon rae or collson frequency elecron-aom/on-phonon collson frequency drllng velocy drllng velocy due o eecon of he mel mel surface velocy deermned by mel eecon 6

8 v dv drllng velocy due o evaporaon of he mel v dvs velocy a whch he surface of he condensed phase recedes due o loss of maeral by vaporaon v e mel-eecon velocy v l specfc volumes for lqud sae v m mel velocy v r radal mel flow velocy averaged over he mel layer hcness v v evaporaon velocy or specfc volumes for vapor sae v s speed of sound n he sold w dameer of he core of fber opc cable w beam radus wsp a record of he number of vapored nodes above he node () xy Caresan coordnaes drecon normal o he mel surface or he spaal Caresan coordnae axal coordnae of defocusng beam () eq he equlbrum dsance f he locaon of he focal plane F laser maeral neracon area pl half axs lengh of he plasma plume hresh deph hreshold A surface area or surface absorpvy or opc absorbance A amb an amben pressure dependen coeffcen A eff he effecve area of flow enerng he hole A l he area of he ncden laser beam Ar F laser maeral neracon area B he e plae dsance B he evaporaon consan C arbrary consan CAZ chemcal affeced one CW connuous wave CP crcal pon CPA chrped pulse amplfcaon D dameer of he lens or beam dameer on he focusng lens or mcroscopc dsplacemen or arbrary consan DOF deph of focus D_ave average dameer of he hole D_enrance enrance spo dameer on he op surface D F dameer of he laser beam n he focal plane of he focusng lens D L laser beam se when propagaes o he fron sde of he focus obecve lens D_m mnmum hole dameer mnmum beam dameer ha can be acheved D mn 7

9 D n D_ex D DH E E E ζ E EM EMW E a E dsp E dr E bλ E f E E m E phonon E req E s F F h FDM FO G HAZ H f H v H v I I() I A I (xy) I abs I (τ) I mach I R (xy) I s he beam se a locaon Z n ex spo dameer on he boom surface dameer of laser beam on he lens or dameer of he beam was dgal holography energy or modulus of elascy or elecrc feld or phasor amplude of he EM feld maxmum elecrc feld nensy on axs energy of elecrons a consdered regon energy of phonons a consdered regon elecromagnec feld elecromagnec wave an acvaon energy evaporaon per aom he dsplacemen hreshold energy drllng hreshold energy blacbody monochromac emssve power an elecron a he fnal connuum sae onaon poenal for he neural aoms n he gas a he nal connuum sae he melng hreshold energy mean energy of phonon energy requred for mcromachnng process elecrc feld nensy he focal lengh of he las focusng opcal lens on he pah of he laser beam pea load or focal number of focusng opc or facor ha depends on he characersc of he fber fluence breadown hreshold fne dfference model Fber opc collecon of me and space erms of fne dfference equaon hea affeced one he laen hea of fuson he laen hea of vaporaon s he laen hea vaporaon a absolue ero nensy he nensy of he ncden radaon a a gven dsance no he absorbng medum from he rradaed surface numercal approxmaon of laser beam nensy of he normaled area s laser radaon nensy a he maeral surface absorbed laser nensy s he laser beam emporal characersc he opmal power densy s he spaal laser beam characersc fber opc hreshold rradance 8

10 I v K K(nqp) L L r LASER LSC LSD LSR LST LTE M M a M M exp M roe M MASER MAZ MEF MEMS N NA N a N G N u N Ob OP P PCM P c P e P L P max P lcond P lconv P levp P ln P lrad P L P( ) hreshold nensy he aberraon of he fber opcs he facor dependen on he refracve ndex he characersc lengh he characersc lengh lgh amplfcaon by smulaed emsson of radaon laser suppored combuson laser suppored deonaon laser suppored radaon laser-sonc-echnque local hermodynamc equlbrum mass of a neural aom or expanson facor of he beam expander or he aomc mass aomc mass qualy of he laser beam he expulson fracon of maeral removed n laser drllng he relave aomc mass of he meal elemen he oal mass of maeral n laser drllng one mcrowave amplfcaon of smulaed emsson of radaon mechancal hea one mel eecon fracon mcro-elecronc number of laser pulse or number of aoms or he number of free elecrons per un volume numercal aperure of he fber Avogadro s number he number of aoms evaporaed per un me per un area Nussel number he number densy of elecrons whch ransfer energy o d he permeer of he worpece opcal pah laser power or dpole momen per un volume phase conugaed mrrors he power loss from elasc collsons he Pecle number he power loss from nelasc collsons maxmum oal beam power durng a laser pulse oupu power per un lengh of he molen layer due o conducon oupu power per un lengh of he molen layer due o convecon power per un lengh spen for he evaporaon npu power per un lengh of he molen layer oupu power per un lengh of he molen layer due o radaon power loss from nelasc collsons normaled oal power of he laser beam 9

11 P consan power durng a laser pulse or he amben pressure Pr Prand number of he asss gas P rp he reacve power generaed due o oxdaon P(T) sauraed vapor pressure defned by Clausus-Clapeyron equaon Q he rae of laser energy absorbed per un volume n he rradaed medum or a volumerc erm accounng for he nernal generaon of hea Q c convecve coolng rae per un area x y Q cond conv rad he hea ransfer due o conducon convecon and radaon o he amben regons n he x y and drecon R reflecvy or radus of wavefron curvaure or unversal gas consan or he recol force R a Ralegh number R d elecron dffuson rae Re he Reynolds number a e ex R elecron producon rae R oe he absolue reacon rae R r elecron recombnaon rae R x laser beam radus along he x-axs S dsance from lens x S scalng facors descrbng he lengh of he elemen n he x drecon y S scalng facors descrbng he lengh of he elemen n he y drecon S scalng facors descrbng he lengh of he elemen n he drecon SOTA sae-of-he-ar SST sanless seel T emperaure or he power ransmsson TAS hermal analyss sysem TEM ransverse elecromagnec modes of he laser beam T a amben emperaure or room emperaure T b bolng emperaure T c crcal emperaure T e plasma emperaure or elecron emperaure T l emperaure n lqud sae T m melng emperaure T plmax maxmum plasma emperaure T s surface emperaure of he worpece or emperaure n sold sae T v vaporaon emperaure T nal emperaure or he pea emperaure a me he pea emperaure us pror o rradaon T

12 wall T emperaure of wall surface grd pon () SS T seady-sae emperaure T* he average emperaure n he mel layer E dscree uns of energy or phonon energy E pulse average energy delvered per pulse E oal he oal change n energy of he meer P(r) he dfference beween he local equlbrum vapor pressure and he amospherc pressure S v vaporaon enropy T change n emperaure me ncremen f dslocaon of he focal plane U energy of evaporaon per aom or hermal dsplacemen volume of he meled maeral or recesson velocy a coeffcen of he order of magnude of he sound velocy Z average onc charge n he plasma or he charge number Z he beam was locaon α e coeffcen of hermal expanson β pulse parameer or he coeffcen of volumerc hermal expanson δ molen layer hcness or he nerval separang he wo values of funcon u δ m mel-layer hcness a he edge of he laser spo T he emperaure graden a he surface along he normal (- axs) ε emssvy or componen n oal sran vecor or permvy ε he elemenary charge or he permvy of free space ε he average elecron energy ξ dmensonless parameer o express he poson coordnae x ζ geomercal coeffcen γ he nemac vscosy or pulse parameer θ dvergence of a laser beam or he half-angle laser dvergence or aomc consan hermal dffusvy m hermal dffusvy of he mel meal s hermal dffusvy of he sold meal v hermal dffusvy of he vapored meal λ wavelengh µ bul absorpon coeffcen or he dynamc vscosy of he gas µ m he mel vscosy µ s bul absorpon coeffcen n solds µ v bul absorpon coeffcen n vapor η s he fracon of recondensng aoms

13 Θ a angular faul (half angle) of he beam θ ac real beam dvergence ν mel vscosy υ phonon frequency ρ densy σ Sefan-Bolmann consan or sress or roo-mean-square surface roughness or surface enson of he lqud-vapor nerface ψ dmensonless parameer o express he poson coordnae y τ hermal me consan or normaled me relave o he pulse lengh τ l laser pulse wdh τ e laser pulse wdh φ opcal phase or angle of ncdence φ geomercal coeffcen φ preexponenal facor ω laser beam radus a he surface of a worpece or he laser frequency angular frequency ω laser beam radus where he nensy s reduced from s maxmum value a he beam cener by a facor of /e or he laser beam radus a he beam was ω e elecron collson frequency ω he elecron-aom collson frequency of gas g eag ω lens he laser beam radus measured a he lens ω p plasma frequency - he nverse Laplace ransform Φ he absorbed laser power densy or dvergence of he laser beam Φ() he emperaure of he lace se aoms n d Θ(ζ) he emperaure of he elecrons when hey arrve a d graden operaor

14 Acnowledgmen Frs of all I would le o han Professor Rysard J. Prypunewc for gvng me he opporuny o wor wh Hm a he Cener for Holographc Sudes and Laser mcromechatroncs (CHSLT) and for supporng and advsng me hroughou my graduae sudes. Suppor came n many ways and forms always apprecaed and remembered from CHSLT s members. I would le o han Professor Cosme Furlong for hs help and undersandng specal hans goes o Peer Hef Dr We Han Adam Klempner Ryan Marns and Shvananda P. Mar. I would le o express my graude o my famly for her love and encouragemen and for supporng me n many ways hroughou my academc career. Fnally a very specal hans o my wfe Ela who has been wh me every sep of he way for her love suppor and paence. I han you and love you. 3

15 Table of conens Copyrgh 3 Nomenclaure 4 Acnowledgmen 3 Table of Conens 4 Ls of Fgures 8 Ls of ables 7 Dsseraon summary 8. Leraure revew 33.. Mechansms for lqud formaon and mel eecon durng laser mcrodrllng 5.. Normal evaporaon Homogeneous bolng Heerogeneous bolng Phase exploson Subsurface heang 6. Modelng of laser neracon wh maerals 6.. Inroducon 6.. Hgh power laser sysem Ineracon of elecromagnec radaon wh maer Phonon consderaon Lace wave Laser-nduced opcal breadown Avalanche onaon Mulphoon onaon Analyss of laser energy Tme consderaon Spaal consderaon Frequency consderaon Amplude consderaon Non lnear opcal effecs Smulaed Raman scaerng Smulaed Brlloun scaerng Second harmonc generaon 89 4

16 Opcal Kerr effec Physcal laser beam characerscs Wavelengh consderaon Coherence consderaon Mode/dameer consderaon Polaraon consderaon Focal spo se consderaon Dffracon lmed spo se Deph of focus consderaon Qualy of laser beam Sphercal aberraon Thermal lensng.5. Theorecal consderaon of laser absorpon Hea dffuson and pulse lengh consderaons Opcal funcons Effec of wavelengh.5.4. Effec of emperaure.5.5. Effec of surface flms Effec of angle of ncdence Effec of maerals and surface roughness Effec of hgh aspec rao drllng Effec of polaraon 7.6. Plasma consderaon Plasma absorpon consderaons Analyss of he physcal phenomena of laser mcromachnng Maeral removal mechansm and energy ranspor n mulple phases aporaon from rradaed surface Recol forces Energy balance n he molen layer and effec of asss gas pressure Forced convecon coolng by asss gas Thermal energy balance concep Expulson consderaon Threshold of laser mcromachnng Consderaon of he maerals Mahemacal formulaon of he model Boundary condons 3. Soluons of he governng equaon Analycal mehods Analycal soluon of hea ransfer equaon wh spaal dependen laser pulse heang Analycal soluon of hea ransfer equaon 5

17 wh me dependen Gaussan laser pulse heang Analycal soluon of hea ransfer equaon wh me dependen Gaussan laser pulse heang wh convecve boundary condons Analycal soluon of governng hea ransfer equaon consderng ransfer evaporave case Analycal soluon of hea ransfer equaon wh nec heory approach Elecron-phonon analycal soluon Comparson of Fourer and nec heory Fne dfference mehods Fne dfference formulae Explc formulaon of governng equaon Sably of explc soluons Model geomery General elemen equaon Generaled elemen Compuer program consderaons Phase ranson and hole formaon Absorpon coeffcen consderaons Enhalpy consderaon FEM - TAS soluons FDM soluons Expermenal nvesgaons Nd:YAG laser sysem Fber opc aachmen Fber opc opcal consderaon Characeraon of he laser beam Average power and energy measuremens of laser beam Measuremens of emporal dsrbuon Measuremens of spaal rradance dsrbuon Transverse mode of laser beam Sably of laser beam dsrbuon Expermenal nvesgaons of laser mcrodrllng Couplng of laser energy wh he arge maeral Maeral removal ess Invesgaon of laser mcrodrllng usng hgh speed flmng Expermenal examples of he laser mcrodrllng Effec of laser energy Effec of laser beam was poson (lbwp) Correlaon beween compuaonal and expermenal nvesgaons 36 6

18 5.. Effec of crcal emperaure and absorpon coeffcen Observaons and recommendaons Conclusons Fuure wor 377 References 379 Appendx A. Compuer program 39 Appendx B. Holes n slcon 44 Appendx C. Holes n 34 sanless seel 46 7

19 Ls of fgures Fg... A pressure-emperaure dagram showng: () he bnodal () he spnodal. The area I s he measable regon (Bulgaova and Bulgaov b). 54 Fg... Schemac of laser beam mpngng on a worpece. 65 Fg... Hgh power laser sysem used n hs sudy. 67 Fg..3. The elecrc and magnec fled vecors of EM radaon. 7 Fg..4. Phonon vbraons. 73 Fg..5. Phonon wave. 75 Fg..6. Schemac of elecron avalanche by collsonal mpac onaon. 77 Fg..7. The mulphoon onaon process. The bound elecron s oned by smulaneously absorbng m phoons. 78 Fg..8. Energy flow n laser-maeral neracon. 8 Fg..9. Inerrelaonshp beween laser nensy beam se and pulse duraon (Columba 5). 83 Fg... Represenave TEM modes (Columba 5). 93 Fg... Focus paern of parallel lgh (Columba 5). 95 Fg... Laserbeam focusng ono he sample. 98 Fg..3. Measuremen of beam properes. 99 Fg..4. Reflecvy as a funcon of wavelengh for dfferen meals (Han 5). 4 Fg..5. Reflecvy as a funcon of emperaure for.64 µm radaon (Han 5). 4 Fg..6. Ablaon rae as a funcon of sample hcness a dfferen wavelenghs. Fg..7. Average ablaon rae vs laser fluence. Fg..8. Ionaon vs. emperaure. 3 Fg..9. A surface flm acng as an nerface couplng an-reflecon coang. 3 Fg... Absorpon as a funcon of hcness of an oxde flm on seel for.64 µm radaon. 4 Fg... Reflecvy of seel o polared.64 µm radaon (Columba 5)). 6 Fg... Average ablaon rae n sanless seel sample as a funcon of aspec rao. 7 8

20 Fg..3. Plasma necs for LSC and LSD wave propagaons (Boulmer 993; Herger 986). Fg..4. araon of he elecron densy and emperaure as a funcon of he amben pressure for an ncden laser power of.5 W. Shaded area: elecron densy calculaed from he Saha-Egger equaon usng he measured emperaure (erwaerde e al. 995). 7 Fg..5. Examples of plasma phoographs under dfferen pressure condons for a laser power of.5 W. A drople of lqud meal s always vsble a he arge surface (erwaerde e al. 995). 8 Fg..6. Number denses of aoms elecrons and ons n an ron plasma p= bar (Bec el al. 995). 3 Fg..7. Opcal properes of varous plasmas; p= bar as a funcon of emperaure: absorpon coeffcen b) he real refracve ndex (Bec el al. 995). 33 Fg..8. A sech of he plasma plume (Bec el al. 995). 34 Fg..9. Represenave examples of plasma phoograph under normal pressure condons for a Nd:YAG laser power of 3.75 W rradang he 34 SST sample. 36 Fg..3. Conours of he plasma plume aen by a CCD camera a varous Ar-gas flow raes (Bec el al. 995). 36 Fg..3. The plasma-affeced focusng radus relave o he undsurbed focusng radus and absorpon whn he plasma plume dependen on he F- number of he focusng opcs and he poson of he plasma plume: p = bar wh focusng opcs: (a) F = 4 (b) F = 6 (c) F = 8 (d) F = and (e) F = (Bec el al. 995). 38 Fg..3. Dslocaon of he focal plane Aq and plasma-affeced focusng radus relave o he undsurbed focusng radus n he effecve focal plane dependen on plasma emperaure and on he sheldng gas conen: p = bar F = 7 conen of sheldng He gas: (a) (b) 4% (c) 6% and (d) 8% (Bec el al. 995). 38 Fg..33. The plasma-affeced focusng radus relave o he undsurbed focusng radus and absorpon whn he plasma plume dependen on he se and poson of he plume; p = bar T plmax = K TEM oo mode F = 7 wh He conen 8%. For: (a) pl = mm r pl =.5 mm; (b) pl =4 mm r pl = mm; (c) pl =6 mm r pl =.5 mm; (d) pl =8 mm r pl = mm; and (e) pl = mm r pl =.5 mm (Bec el al. 995). 39 Fg..34. The plasma-affeced focusng radus relave o he undsurbed focusng radus and absorpon whn he plasma plume dependen on he se and poson of he plume; p = bar T plmax = K r pl =.5 mm 9

21 TEM oo mode F = 7 wh He conen: (a) (b) 4% (c) 6%; (d) 8%; and (e) 9% (Bec el al. 995). 4 Fg..35. Weldng deph for ron and alumnum dependen on focal poson and laser power; ω o =.4 mm u o = m/mn CO laser K =. and F = 7 (Bec el al. 995). 4 Fg..36. The average absorpon coeffcen n he plasma laser sources versus deph of a Gaussan beam usng focusng radus of mm on 34 SST (Solana e al. 999). 48 Fg..37. Schemac of energy ranspor n mulple phases n laser drllng he physcal model of mel removal from he neracon one he cross secon n he x plane s shown. 5 Fg..38. The recol force as a funcon of pea surface emperaure and he surface enson force a he melng pon (Basu and DebRoy 99). 6 Fg..39. Pea emperaure vs me for a sanless seel sample rradaed by sngle pulses of (a).5 and (b). ms duraons (Basu and DebRoy 99). 6 Fg..4. Dmensonless acceleraon of lqud meal as a funcon of he dfference beween he pea surface emperaure and he crcal emperaure for expulson T=T s - T c (Basu and DebRoy 99). 6 Fg..4. The emperaure dependence of he mel surface of ron on he absorbed laser nensy for a laser beam radus of ω = :9 x - cm (Sema and Masunawa 997). 67 Fg..4. Dependence of he drllng velocy v d and s componens v dm and v dv on he absorbed laser nensy for he case of ron and a laser beam radus of ω = :9 x - cm (Sema and Masunawa 997). 67 Fg..43. The dependence of he mel velocy of ron on he absorbed laser nensy for a laser beam radus of ω = :9 x - cm (Sema and Masunawa 997). 68 Fg..44. Schemac of effecve gas flow area enerng hole defned by he laser beam dameer ω and he cylndrcal area of radal loss flow (Fere and Ward 986). 7 Fg..45. Approxmaon of he forced convecon coolng of he mel surface by he asss gas a he hole boom (Sema e al 999). 7 Fg..46. Bloc dagram for he man processes of maeral removal and her muual nfluences durng laser drllng. 79 Fg..47. Amoun of ablaed maer for dfferen laser rradaons agans absorbed laser energy (Boulmer-Leborgne e al. 993) nrogen 337 nm;

22 XeCI. 38 nm; Nd:YAG. 355 Nd:YAG 53nm: Nd:YAG 64 nm; ArF 93 nm. 83 Fg..48. Comparson of predced and expermenal maeral removal rae (Zhang and Faghr 999). 84 Fg..49. Mel dsplacemen me dsp and poron of laser pulse remanng afer begnnng of surface melng τ m as a funcon of pulse energy (Sema el a. 3). 85 Fg..5. Thermophyscal properes of 34 sanless seel. 89 Fg..5. Thermophyscal properes of sngle crysal slcon (EMIS 988). 9 Fg..5. Laserbeam mpngng on a fne se sample. 9 Fg..53. Calculaed emperaure of ron surface a he beam axs for he cases whou (op curves) and wh (boom curves) mel flow for he dfferen maxmum absorbed nensy values and dfferen laser pulse duraons: 3 µs (.5 MW cm ) 7 µs ( MW cm ) and 5 µs (5 MW cm ) (Sema e al. 999). 6 Fg. 3.. Sem-nfne meal slab. Fg. 3.. Temperaure dsrbuon nsde a maeral. 5 Fg Temperaure dsrbuon on he surface of he maeral. 6 Fg Paramerc sudy of hermal properes. 7 Fg Temperaure dsrbuon of he maeral wh respec o me. 8 Fg Temperaure graden dsrbuon nsde he maeral wh respec o me. 9 Fg Temperaure graden dsrbuon nsde he maeral for dfferen mes. Fg Equlbrum dsance. Fg Temperaure dsrubuon of laser rradaed maeral wh he lowes values of hermo-physcal parameers a) emperaure as a funcon of deph and me b) empearure or melng of he maeral wh deph calculaed usng Eq.. 3 Fg. 3.. Temperaure dsrubuon of laser rradaed maeral wh he hghes values of hermo-physcal parameers a) emperaure as a funcon of deph and me b) emperaure or melng of he maeral wh deph. 3 Fg. 3.. Rae of coolng of he worpece afer he end of laser pulse. 4 Fg. 3.. a) Temperaure dsrbuon nsde he maeral (-axs) wh respec o me: b) emperaure vs. deph of he worpece a few dfferen me form he begnng of he pulse wh he hghes values of hermophyscal parameers calculaed usng Eq.. 8

23 Fg a) Temperaure dsrbuon nsde he maeral (-axs) wh respec o me: b) emperaure vs. deph of he worpece a few dfferen me form he begnng of he pulse wh he lowes values of hermophyscal parameers calculaed usng Eq.. 8 Fg a) Temperaure dsrbuon nsde he maeral (-axs) wh respec o me: b) emperaure vs. deph of he worpece a few dfferen me form he begnng of he pulse wh he hghes values of hermophyscal parameers calculaed usng Eq Fg a) Temperaure dsrbuon nsde he maeral (-axs) for few dfferen mes; b) ransen dsrubuon of he worpece a few dfferen dephs wh he hghes values of hermo-physcal parameers calculaed usng Eq Fg Temperaure dsrbuon nsde he maeral (-axs) wh respec o me wh he lowes values of hermo-physcal parameers calculaed usng Eq Fg a) Temperaure dsrbuon nsde he maeral (-axs) for few dfferen mes; b) ransen dsrubuon of he worpece a few dfferen dephs wh he lowes values of hermo-physcal parameers calculaed usng Eq Fg Temperaure dsrbuon of he worpece wh consderaon of ransfer evaporave case calculaed usng Eq Fg Elecron movemen a he meal surface vcny (= s he surface). 57 Fg. 3.. Temperaure closed form soluon of he elecron nec heory approach wh respec o deph of he sample and me. 65 Fg. 3.. dt/d predced from he Fourer and nec heory along he -axs for wo pulse lenghs: 6-9 s and 6 - s (Ylbas ). 66 Fg. 3.. General hree-dmensonal fne dfference grd. 7 Fg Three-dmensonal fne dfference subdvson of a worpece. 76 Fg The four ypcal nodal elemens of a worpece. 78 Fg Hole formaon. 88 Fg Temperaure dsrbuon n an alumnum plae. 93 Fg D model for TAS calculaons used n hs Dsseraon. 97 Fg Temperaure dsrbuon of a ypcal cross-secon of he worpece n -drecon usng TAS. 98 Fg Temperaure dsrbuon of a ypcal cross-secon of he worpece n x-y-drecon usng TAS (op surface). 99

24 Fg Flow char for he FDM process. 3 Fg Example of a D emperaure dsrbuon around he hole cross secon profle could be obaned from he compuer program a any me durng or afer he end of he laser pulse of he worpece n he axal cross secon plane of he laser beam. 3 Fg Example of emperaure dsrbuon around he hole profle could be obaned from he compuer program a any me durng or afer he end of he laser pulse of he worpece: a) D and b) 3D surface emperaure dsrbuon aen from he boom of he worpece. 33 Fg Example of emperaure dsrbuon around he hole profle could be obaned from he compuer program a any me durng or afer he end of he laser pulse of he worpece: a) D and b) 3D surface emperaure dsrbuon aen from he mddle of he worpece. 33 Fg. 4.. Laser sysem used n hs sudy. 35 Fg. 4.. Fber opc connecor: a) phoograph b) schemac. 36 Fg Toal nernal reflecon n an opcal fber. 37 Fg µm end of he FO cable (lef pcure) wll be conneced o he focusng head as llusraed on he rgh pcure wh he arrow. 39 Fg µm end of he FO cable conneced o FO head. 3 Fg FO laser beam delvery subsysem o wo FO cables for laser mcromachnng. 3 Fg Coheren Labmaser Power Deecor Model LM. 36 Fg Evoluon of he emperaure of a arge by a recangular pulse (Boulmer- Leborgne e al. 993). 37 Fg Dfferen pulse shapes from TEA-CO laser: A and B relae o wo dfferen gas mxure concenraons n he laser (Boulmer-Leborgne e al. 993). 37 Fg. 4.. Surface emperaure evoluon resulng from TEA-CO laser pulses for T arge: A and B relae o wo dfferen gas mxure concenraons n he laser (Boulmer-Leborgne e al. 993). 38 Fg. 4.. Seup for measuremen of he emporal laser beam characerscs. 39 Fg. 4.. Typcal measured normaled emporal characerscs of he Nd-YAG laser beam whch shows normaled laser beam nensy p(τ) vs. normaled me τ. 3 Fg Temporal laser beam characerscs for dfferen pulse lenghs. 3 3

25 Fg Gaussan spaal laser beam dsrbuon measured by beam nensy profler. 3 Fg Seup for measuremen of he spaal laser beam characerscs. 33 Fg Dgal opcal power meer used for expermens. 34 Fg Spaal laser beam dsrbuon measured a close proxmy of he focal plane poson. 34 Fg Spaal laser beam dsrbuon measured a he hgher gan of he amplfer a perpheres of he laser beam. 39 Fg Mahemacal spaal laser beam represenaon of a beam focused a.4 mm above focal plane poson (lbwp). 33 Fg. 4.. Typcal burn paerns creaed bynd:yag laser a) wh and b) whou aperure n laser cavy (Nowaows 99). 33 Fg. 4.. Analyng he spaal beam dsrbuon usng blac burn paper and as a resul deermnng mnmum beam was poson. 334 Fg D and D laser beam represenaon of a spaal laser beam dsrbuon: a) obaned by scanng he beam wh he phoo deecor b) blac paper burn of hs same beam a hs same focal locaon of he focusng lens (abou 83.6 µm). 335 Fg Sably of he Nd:YAG laser beam. 336 Fg Schemac of laser drlled hole profle. 337 Fg Laser seup for mcrodrllng. 339 Fg Laboraory seup used durng laser drllng of 34 sanless seel samples. 343 Fg Parcles of 34 sanless seel eeced from he drlled hole where open glass ube placed co-axal wh he laser beam durng mcrodrllng process. 344 Fg Maeral removed durng laser drllng of 34 sanless seel samples. 344 Fg CCD camera used n he expermens. 347 Fg Sequenal frames from flmng of laser mcrodrllng process: vaporaon and lqud eecon process plasma formaon random parcles eecon and vaporaon sponaneous vaporaon a he end of he pulse for wo dfferen pulses: a) p =3.5 ms b) p =4.5 ms. 348 Fg Mcrograph of ypcal shape and characersc dmensons of he laser mcrodrlled hole cross secon of hole profle n 76 µm hc 34 sanless seel obaned wh pulsed laser beam ransmed hrough a sngle-mode fber pulse wdh p =.9 ms energy E=7.5 J laser beam 4

26 dameer ω =33 µm and power P=4. W: a) SEM mage of he hole enrance b) cross secon of he hole along -axs. 35 Fg Mcrographs of a ypcal hole profle drlled n 76 µm hc 34 sanless seel obaned wh Nd:YAG pulsed laser beam ransmed hrough a sngle-mode fber pulse wdh p =.8 ms energy E=7 J laser beam dameer ω =3 µm and power P=. W: (a) mage of he ex of hole (b) opcal mage of he cross secon (c) mage of he enrance of hole. 35 Fg Mcrographs of a ypcal hole profle drlled n 3 µm hc sngle crysal slcon obaned wh sngle Nd:YAG pulse: pulse wdh p =.5 ms energy E=.5 J laser beam dameer ω =58 µm: (a) mage of he ex of he hole (b) opcal mage of he cross secon n axal drecon (c) mage of he enrance of he hole. 35 Fg Effec of laser energy on dmensons of laser drlled mcroholes n sanless seel 34 shees: N = and cons volage = Fg The laser beam focused on he worpece and cross secon of he laser penerang he worpece. 353 Fg Change of radus of he beam as poson of focal plane wh deph changes wh when lbwp =.88 µm. 353 Fg Ray of beam racng analyss descrbng sphercal abberraon (Karnas e al. 5). 355 Fg Typcal funcon of dameers of D_enrance D_md and D_ex for varous lbwp mpngng on 34 sanless seel worpece one pulse and energy E=7.5 J. 356 Fg Typcal funcon of dameers of D_enrance D_hole and D_ex for varous lbwp mpngng on sngle crysal slcon worpece one pulse and energy E=.5 J. 357 Fg Expermenal resuls of mcrodrllng n 34 sanless seel worpece. Effec of laser beam was poson on hole profles. Holes were drlled wh a beam dameer of 59 µm. J energy. The was was moved vercally by 89 µm beween holes. 358 Fg Expermenal resuls of mcrodrllng n 34 sanless seel worpece. Effec of laser beam was poson on hole profles. Holes were drlled wh a beam dameer of 59 µm 3. J energy. The was was moved vercally by 89 µm beween holes. 359 Fg Effec of laser beam was poson on expermenal hole profles drlled n 34 sanless seel. Holes were drlled wh a 59 µm.8 J pulse usng he 5 mm lens. The was was moved vercally by 89 µm beween holes

27 Fg Effec of laser beam was poson on expermenal hole profles drlled n 34 sanless seel. Holes were drlled wh a 59 µm 3. J pulse usng he 5 mm lens. The was was moved vercally by 89 µm beween holes. 36 Fg Effec of laser beam was poson on expermenal hole profles drlled n sngle crysal slcon. Holes were drlled wh a 59 µm dameer beam wh.8 J (SET A) and.5 J (SETB) pulse usng he mm lens. The was was moved vercally by 89 µm beween holes. 36 Fg. 5.. Comparson of compuer smulaed resuls wh expermenal daa of cross-secons of a mcrodrlled holes n 34 sanless seel maeral wh a sngle pulse of Nd:YAG measured and focused above he op surface a) lbwp = 7 µm b) 89 µm. 36 Fg. 5.. Cross secon of a mcro machned sample of a 34 sanless seel maeral wh a sngle pulse E=3. J focused o:a) =5 µm below he op surface b) lbwp =7 µm c) 9 µm. 36 Fg Compuer smulaed cross secon of a mcrodrlled sample of a sngle crysal slcon maeral wh a sngle pulse of Nd:YAG measured laser beam focused below he op surface a) lbwp = 7 µm b) lbwp = 3 µm c) 46 µm d) 8 µm. 363 Fg Compuer smulaed cross secon of a mcrodrlled sample of a sngle crysal slcon maeral wh a sngle pulse of Nd:YAG measured laser beam focused below he op surface a) lbwp = - µm b) 3 µm c) 4 µm d) 7 µm. 364 Fg Effec of crcal emperaure on he hole profle nsde he 34 sanles seel wh T c = 39 K and T c = 67 K. 365 Fg Effec of absopon ceoffcen µ on he hole profle nsde he 34 sanles seel wh µ = 6.7 µ=33.5 µ=67. µ=7 and T c = 67 K. 366 Fg. C.. Top vew of represenave holes mcromachned n he 34 sanless seel. 46 Fg. C.. Effec of laser energy on dmensons of laser mcroholes n he 34 sanless seel.76 mm shees. 46 6

28 Ls of ables Table. Complex refracve ndex and reflecon coeffcen for some maeral's opcal properes (.6 µm radaon). Table. Expermenally deermned emperaures and elecron denses. 5 Table 3. Aomc consans for varous aoms and ons and values for A and θ. 6 Table 4 Crcal elecron densy for varous laser wavelenghs (Boulmer-Leborgne e al. 993) n ec =N ec. 44 Table 5. Thermophyscal properes of O asss gas and gas nole parameers (Sema and Masunawa 997). 77 Table 6. Properes of he maerals used n hs Dsseraon (EMIS 988). 89 Table 7. Typcal burn paerns produced by laser: a) paper was above he laser s beam was and b) paper was below laser s beam was. 33 Table B.. Represenave holes mcromachned n sngle crysal slcon laser parameers: one pulse p =.7 ms volage=9 energy E=.7 J and lbwp separaon = 8.48 µm. 44 Table B.. Represenave holes mcromachned n he sngle crysal slcon laser parameers: one pulse p =.7 ms volage=9 energy E=.7 J and lbwp separaon = 8.48 µm. 449 Table B.3. Summary of characersc dmensons for he represenave holes mcromachned n he sngle crysal slcon laser parameers: one pulse p =.5 ms volage=5 energy E=.75 J and lbwp separaon=9.4µm (ref. mages of Table B.). 458 Table B.4. Summary of characersc dmensons for he mcromachned holes n he sngle crysal slcon laser parameers: one pulse p =.7 ms volage=9 energy E=.74 J lbwp separaon = 8.48 µm (ref mages of Table B.). 459 Table C.. Represenave holes mcromachned n he 34 sanless seel laser parameers are summared n Table C.. 46 Table C.. Represenave mcromachned holes n 34 sanless seel

29 Dsseraon summary The obecve of hs Dsseraon s o sudy laser beam neracon wh maerals for mcroscale applcaons. Laser mcromachnng s essenal n oday s advanced manufacurng of e.g. prned crcu boards and elecronc componens especally laser drllng (Duley 985; Treusch and Herger 986 Chryssolours 99; El-Baahgy 997; Solana e al. ). Demand for crcu mnauraon and small MEMS componens and her pacagng (Cheng e al. 998) have creaed he need for smaller holes and mcrovas smaller lands narrower lnes and spaces gher regsraon and smaller and more conrollable spoweld han ever before. All ha requres more accurae and conrolled producon process (Bruggemann 996) smaller aper of he mcroholes and more sable and conrolled laser mcromachnng process han currenly avalable. Therefore much more aenon mus focused on he laser parameers ha deermne crcal specfcaons such as he accuracy of he hole se shape and aper angle. Accordng o Karnas e al. 3 Thermal managemen of mcrodrllng process has become ncreasngly mporan and needs o be carefully consdered. In bref curren requremens n laser mcromachnng are for he laser sources wh beer energy sably hgher average power whch em a shor wavelenghs wh shor pulse duraons. Alhough laser mcromachnng has become wdely used n mcroelecroncs and pacagng ndusry a full undersandng of varous phenomena nvolved s sll a maer of rals and speculaons. Laser mcromachnng s a very complex process whch nvolves varably n shape and properes of eyhole welds (Chryssolours 99; 8

30 Marley ; Jn and L 3) or laser drlled holes (Baeh e al. ; Low e al. ) due o he very complex phenomena ncludng bu no lmed o hea ransfer flud flow plasma effec lace-phonon vbraon and meallurgcal problems (Bulgaova and Bulgaov a). Lasly sudy of laser beam neracon wh maerals enables us o explore he physcal processes nvolved beer undersand he mcromachnng processes and maybe elmnae some unwaned sde effecs as well as ncrease effcency of he mcromachnng processes. Deermnaon of process parameers n laser mcromachnng s done mosly by ral and error. Ths Dsseraon aemps o reduce he expermenal me and cos assocaed wh esablshng process parameers n laser mcromachnng. The research for hs Dsseraon has been done a he Cener for Holographc Sudes and Laser mcro-mechatroncs (CHSLT) laboraores of Mechancal Deparmen a Worceser Polyechnc Insue n Worceser Massachuses. To avod duplcaons and so called renvenng he wheel a collecon of curren sae-of-he ar (SOTA) wor and pror sudy done n he feld of laser mcromachnng has been evaluaed frs (Duley 985; Chrsensen and Tllac 3). The nvesgaons performed n hs Dsseraon are based on analycal compuaonal and expermenal soluons (ACES) mehodology (Prypunewc 993). More specfcally he sudes wll be focused on he developmen of equaons governng neracon of he laser beam wh maerals where analycal soluon of hese equaons 9

31 based on Fourer hea conducon usng he Laplace negral ransform (Ylbas and Shua 999) mehod wll be uled. Soluons obaned by fne dfference mehod (FDM) and expermenal demonsraon of laser mcromachnng wll be presened and correlaon of he analycal compuaonal and expermenal resuls wll be deermned (Prypunewc 998). Some of he ass o be performed wll nclude measuremens of emperaure dsrbuon n he worpece characeraon of he hermal deformaons and nvesgaon of shape and profle of mcrodrlled holes. Furhermore sysemac nvesgaons wll be performed o sudy he effecs of specfc laser parameers on he resuls of laser mcromachnng processes and opmaon s proposed for he specfc applcaons. Expermenal and heorecal nvesgaons were performed o sudy properes of he laser beam mpngng on he surface of he maeral under consderaon. The emporal and spaal characersc of he laser beam are presened. To reduce se of mcroholes researchers have sared usng shor laser pulses down o -9 s (followng opcal Ralegh law) (Dausnger e al. 3). As he pulse lengh reduces furher (< -9 ) and nensy ncreases (> 3 W/m ) he nec heory predcons (Ylbas e al. ) devae consderably from he Fourer heory resuls bascally because hese assumpons do no accoun for he effecs ha elecron movemens n he solds have n he laser mcromachnng processes usng ulrashor pulses. Consequenly new models of he laser neracon wh maerals based on revsed assumpons and approxmaons are needed. The modelng of laser mcrodrllng 3

32 process s essenal for beer undersandng of he physcal phenomena occurrng durng laser beam neracon wh maerals (Wang and Chen 3). Absorpon of laser energy aes place hrough phoon neracon wh bound and free elecrons n he maeral srucure whch rases hem o he hgher energy levels. Energy converson aes place hrough varous collson processes nvolvng elecrons lace phonons oned mpures and defec srucures. Ths Dsseraon examnes pulsed laser heang process by consderng boh Fourer conducon and elecron-phonon nec heory approaches. More specfcally hs Dsseraon focuses on developmen of equaons governng neracon of a Gaussan laser beam wh maerals where analycal soluon of hese equaons based on boh he Fourer hea conducon heory and nec heory approach usng a Laplace negral ransformed mehod are uled (Ylbas e al. ). A comparson of he analycal soluon of he Fourer heory and closed form soluon of he nec heory approach s also presened n hs Dsseraon. Sudy n hs Dsseraon s concenraed on one of he mos popular laser processes used n he mcroelecroncs ndusry whch s laser mcrodrllng. Furher more he expulson of molen maeral from he hole has an mporan role n mcrodrllng effcency. The samples ha are consdered n hs Dsseraon are made ou of wo maerals frs very popular n MEMS echnology slcon (Cyrynowc e al. 3) and second beng used n MEMS devces and elecronc ndusry alloy 34 sanless seel (Cheng e al. 998; News 4). 3

33 The hope s ha hs research wll faclae mprovemens and opmaons of SOTA laser mcromachnng echnques. Complee resuls are presened n he Dsseraon. The conrbuons of he Dsseraon nclude:. Invesgaon of he effecs of laser operang parameers on he profles and dmensons of he laser mcrodrlled holes.. Calculaons of he emperaure profles durng laser mcromachnng usng program developed for he purpose of hs Dsseraon usng fne dfference mehod. The program has capably of handlng 3D slab of nonlnear maeral properes emperaure dependen coeffcen of energy absorpon wh prelmnary calculaons of he flud dynamc phenomena nvolved n laser mcrodrllng. The program uses expermenally obaned spaal and emporal laser beam characerscs. 3. Expermenal nvesgaon of laser beam mcromachnng usng a fas camera (8 frames per second) o record laser beam neracon wh maerals especally o measure speed of lqud expulson durng mcrodrllng and plasma formaon. 4. Consderaons of hermal and laser beam condons and her effec on laser mcromachnng processes. 5. Inal nvesgaon of he effec of ulrasonc operang parameers on he resuls of profles and dmensons of he laser mcrodrlled holes whch wll provde gudelnes and wll advance currenly exsng laser mcrodrllng processes. 3

34 6. Correlaon beween compuaonal and expermenal resuls of he mcrodrlled holes; explanaon and mprovemens of he model based on he mechansms of laser mcromachnng process. The nvesgaons presened n he Dsseraon show praccaly of laser mcromachnng process and explan some problems occurrng durng se ups of hose process. Ths nowledge should mprove qualy of mcrodlled holes and manufacurng processes. They wll also provde gudelnes whch wll help o explan and wll provde beer undersandng of he exsng laser mcromachnng process. Resuls of he Dsseraon wll faclae mprovemens and opmaon of he SOTA laser mcromachnng processes.. Leraure revew Whle here have been a sgnfcan number of papers publshed on he subec of neracon of laser energy wh maerals only a few of hese papers presen unque nformaon. Some of he references ncluded heren are of hsorcal value whle ohers are que comprehensve n addressng he mechansms of neracon of laser energy wh maerals. A dscusson of he fndngs from seleced relevan leraure follows. Because he prmary obecve of hs chaper s o summare he exsng leraure relang o laser mcrodrllng phenomena le a normal evaporaon lqud drople eecon and he hydrodynamc effecs wll be of neres and wll be dscussed. Generally he enre feld 33

35 of laser beam neracon wh maerals can be dvded no wo groups: () resonan and () nonresonan neracon. The resonan group ncludes selecve excaon processes n sysems wh phoon dscree specra mulphoon onaon of aoms and molecules mulphoon phooelecron emsson and nonlnear opcal phenomena n solds and lquds. as dscussed laer. The followng parameers affec hese excaon processes: coherence polaraon and specral characerscs of laser radaon. In he nonresonan neracons laser-nduced hermo-chemcal and hermophyscal phenomena can be relaed o each oher as well as o laser breadown of gases and solds and laser plasma generaon. In he Dsseraon he man sudy concenraes on he nonresonan laser beam neracon wh maerals. For hs group phenomena le coherence and specral characersc are no very mporan. Prncpal effecs are conneced o he heang of maerals and mporan facors are: he laser nensy pulse duraon and focusng condons. We consder laser heang melng and vaporaon of sold maerals and relaed nsably of hese processes. We also consder he dynamcs of vapor plume expanson wh an emphass on hydrodynamc nsables of vapor expanson. Many heorecal analycal and expermenal wors wh he lasers have been developed and publshed. Summary of he mehods used and resuls obaned by seleced researchers n chronologcal order are presened below. The begnnng of laser s hsory sared n 96 based on Ensen s heory of lgh emsson and concep of smulaon emsson (Thompson 3). In 97 Ensen 34

36 proposed he concep of "phoon" (Ensen 97). We can say ha lgh s composed of ndvdual parcles called phoons whch posses a dscree amoun of energy or quana. Ensen (97) also predced ha when here exss he populaon nverson beween he upper and lower energy levels among he aom sysems would be possble o reale amplfed smulaed radaon.e. laser lgh. Quanum Mechancs was developed o explan hese new phenomena snce 9 (Kel 987). 95 Townes he nvenor of he MASER mcrowave amplfcaon of smulaed emsson of radaon a Columba Unversy frs devce based on smulaed emsson was awarded Nobel pre n 964 (Towns ) Maman nvened frs worng LASER based on Ruby rod as he lasng medum May 6 h 96 Hughes Research Laboraores New Yor (Maman 96) Ansmov developed a heorecal model o predc a emperaure profle n he maeral on whch he laser beam s mpngng (Ansmov e al. 97). He consdered emperaure pressure and densy dsconnues across he Knudsen layer n vacuum o solve analycally hea ransfer equaons he Knudsen dsconnuy s modeled by a Mo-Smh ype soluons Pae and Gaglano (97) developed a heorecal model o predc a emperaure profle assumng he laser beam of crcular cross secon and unform nensy. They found ha he absorpvy along wh beam nensy were he mos mporan facors durng he laser drllng process Wagner (974) presened a quanave model whch predcs he deph and shape of a hole drlled n alumna ceramc by a ruby laser. Expermenal and 35

37 heorecal resuls ndcaed ha he drllng mechansm for hs applcaon s no surface absorpon or conducon bu one n whch laser energy s absorbed hroughou he bul of he ceramc. Wagner (974) based on expermenal evdence developed a mahemacal model ncludng maeral removal due o pressure caused by laser radaon. The shape and deph of holes drlled n ceramc have been raher accuraely predced from he measured beam energy densy dsrbuon. He showed ha he radaon pressure of he focused beam plays an mporan role n he removal of molen maeral from heaed one Andrews and Ahey (975) developed a useful framewor of he drllng phenomena. They developed a model for complee analyss of he drllng speed as onedmensonal. They predced a connues growh of a hole deph wh me. 975 Spars (975) developed a heory of laser heang of meals and derved exac analycal soluons. He calculaed he ransen and seady-sae emperaure rse of a laser rradaed meals. Spars had evdence for plasma formaon and hs heorecal wor suggesed ha he emperaure rse caused by ordnary heang of solds by laser s orders of magnude oo small o gne he plasma bu ha hermally solaed surface mperfecons could cause gnon. 976 von Allmen (976) esablshed an analycal model ang no accoun lqud expulson. The developmen of models ncorporang mel eecon really began wh von Allmen s equaons whch allow calculang drllng velocy and drllng effcency as funcons of he absorbed laser beam nensy. Equaons are based on a D 36

38 seady-sae pson-le effec of recol pressure drvng molen maeral away from he ablaon fron. 977 Anhony and Clne (977) sudy hermal analyss of laser heang and melng of maerals. They saed ha durng laser surface melng and alloyng emperaure gradens on he mel surface (neracon lne of he sold-lqud nerface) generae surface enson gradens ha sweep lqud away from laser beam mpac lne. The resulng flow of lqud creaes a depresson of he lqud surface beneah he beam and rdgng of he lqud surface elsewhere. As he beam passes o oher areas of he surface hs dsoron of he lqud surface s froen n creang a roughened rppled surface. They show ha rpplng from surface-enson gradens can be avoded f durng surface melng he laser beam velocy exceeds a crcal velocy. Expermens were presened for sanless seel Maumder and Seen (979) wored on 3D hea ransfer model for connuous wave laser maeral processng. They developed 3D hea ransfer model wh movng Gaussan hea source usng fne dfference mehod. In her wor he sysem s consdered o be n a quas-seady-sae condon afer he eyhole naon me whch was reaed as nsananeous eo (98) dscussed laser-nduced formaon of submcron holes n hn flms. The qualave conceps of mel surface evaporaon and of mel moon under he acon of he reacve vapor pressure of surface enson forces and of adheson was shown. The basc equaons descrbng he removal process were solved numercally. I 37

39 was shown ha evaporaon or lqud phase moon plays a prncpal role n he flm removal mechansm Ursu e al. (986) wored on hgh nensy laser radaon of meallc surfaces. An analyss on he role of he perodc srucures nduced by laser radaon on meallc surfaces was gven a grea deal of aenon. One mporan aspec was he nfluence of meallc perodc srucures on he opcal characerscs of he surface. Dscusson was presened based on expermenal and heorecal resuls. They concluded ha a ceran condons of meallc surface he absorpvy of a meal surface could decrease wh emperaure Herger (986) demonsraed laser-nduced plasma by he opcal feedbac conrol by on-lne dagnoscs. He also presened 3D hea flow calculaons for dfferen energy couplng. Herger nvesgaed he vapor breadown and resulng plasma formaon by hgh-speed phoography n he framng and srea mode of operaon. He concluded ha he plasma asssed ranspor of processng maeral as vapor or mel have o be adaped o characersc me consan of he nvolved physcal process and o he hermo-physcal properes of he processng maerals n order o yeld maxmum hea npu and effcen machnng. He concluded ha hgh effcency of maerals processng could be acheved only n a narrow nensy range beween he hresholds of plasma generaon and plasma sheldng whch can be nfluenced by gas conrollng he plasma properes. Herger el a. (986) mproved laser effcency by applyng n lne he pulse separaon process n a way ha plasma absorpon has 38

40 dmnshed beween wo successve pulses and he pulse halfwdh has o be adused so no subsanal plasma absorpon durng he neracon me wll occur Treusch and Herger (986) suded precson laser drllng. They calculaed hreshold nensy for Gaussan laser beam and mel deposon a he hole enrance durng drllng. They observed ha accurae adusmens of absorbed laser power and hea losses are mporan o conrol he process of maeral drllng as well as he laser nduced plasma. For a reproducble maeral removal by lasers he followng parameers are crcal: pulse power pulse duraon duy cycle of pulses nensy dsrbuon and nensy me slope proecon of he opcs and hermal lensng of he laser medum Chan and Maumder (987) developed a D seady-sae analycal model for damage by vaporaon and lqud expulson due o laser maeral neracon. They found ha he radaon plays an nsgnfcan role n deermnng he surface emperaure rae of maeral removal n boh vapor and lqud form. Resuls were compared for condons boh wh and whou radave hea loss. The vaporaon process produces a recol pressure ha pushes he vapor away from he arge and expels he lqud so maeral s removed n boh vapor and lqud phases. The oal maeral removal rae s equal o he sum of vaporaon and lqud expulson raes. The speed of he sold-lqud or lqud-vapor nerface s a drec measure of he maeral removal rae. They predced ha he lqud expulson rae goes hrough a maxmum as he power densy vares. A lower power he vaporaon emperaure s slow. Consequenly he expulson rae s low because of low recol pressure. Wh low nenses producng a low lqud eecon 39

41 rae due o a lower rae of vaporaon and hence recol pressure. A hgh power he vaporaon emperaure s hgh. ery hgh nensy hnnng he molen layer and agan reducng he rae of lqud expulson. Resuls ndcaed pea n he rae of maeral removal by mel eecon as a funcon of power densy. A low beam power he domnan form of maeral removal s lqud expulson a hgh power vaporaon was he domnan form of maerals removal. The vapor phase s assumed opcally hn so s absorpon of he laser beam s neglgble. Close-form analycal soluons were obaned for hree dfferen maerals: alumnum superalloy and anum Armon e al. (989a) measured and analyed alumnum plaes drlled wh a CO laser beam and he followng was analyed: peneraon me he spaal nensy profle of he beam he emperaure-dependen absorpvy evaporave maeral removal hea conducon whn he worpece craer radus meled one usng heorecal approach. Assumpon for absorpon of lqud meal aenuaon by meal vapor and hea exchange wh vapor were made. 3D me dependen formulaon for meal drllng wh a laser beam was presened and D soluon for peneraon rae was he upper lm soluon ha led o raher realsc predcon n small aspec rao cases a shor pulses Fne and Smon (989) dscussed n he D seady-sae analycal model properes of seady evaporaon of meals durng rradaon of meal surface by a hgh radaon laser beam. They dsngushed hree regmes: Knudsen layer adacen o meal surface condensaon regme and hydrodynamc regme. Esmaes of emperaure 4

42 densy average velocy degree of condensaon and decrease of pressure along he e were presened. 989 Posacoglu e al. (989) nvesgaed naural frequences of oscllaon of weld pool when n s molen sae: hese frequences were suded by a combnaon of analycal and numercal mehods nner and ouer boundares of he weld pool as well as he fne deph of he pool and he dependence of he surface enson on emperaure were consdered Kar and Maumder (99) developed a D model for maeral damage durng laser melng and vaporaon: Cran Ncholson mehod was used. The problem was formulaed by usng energy conservaon equaon (he Sefan condon) a he phase nerfaces. The effec of curvaure of sold-lqud and lqud-vapor nerfaces was aen no accoun and solved numercally. The effec of laser power me number of pulses and laser beam dameer on deph and radus of he craer durng laser rradaon were also examned. Lnear relaonshp beween he maxmum craer deph and he laser nensy was derved phenomenologcally and verfed numercally Armon e al. (99) employed a me-dependen axsymerc hea conducon model and used he enhalpy mehod o deermne he ransen cavy shapes and peneraon veloces Chryssolours (99) suggesed ha for laser power denses below hreshold value (e.g. for copper abou 3 J/cm ) he surface for mos meals shows a hgh reflecvy o beam energy and no maeral removal occurs. The surface reflecvy s me dependen snce he slope of he hole wall changes rapdly wh me. Durng hs 4

43 sage mos of he ncden laser energy s refleced from he eroson fron as me progresses absorpvy ncreases and reaches he maxmum because he slope of he hole does no change much. For energy greaer hen hreshold drllng occurs bu effcency may have been lowered by wo phenomena. Frs s a plasma formaon and he second laser suppored deonaon (LSD) waves Kar e al. (99) nroduced a D model for laser-nduced maeral damage and effecs of asss gas and mulple reflecons nsde he cavy were suded Olson and Swope (99) demonsraed laser drllng wh focused Gaussan beams. A compuaonal model for drllng holes wh Gaussan laser beams was presened and compared wh expermenal resuls. The beam dvergence near he focus was aen no accoun and was shown o srongly affec he hole profle. The model correcly represens expermenal observaons ncludng movng of he beam was from he arge surface. Dependence of he hole profle on beam dvergence was demonsraed as well as he opcal condons needed o drll cylndrcal or concal holes. 995 Ho e al. (995) furher developed Kar and Maumder s model consderng effecs of gas dynamcs and Knudsen layer dsconnuy durng he ablaon process. These models assume D hea ransfer n arge maeral recognng ha he machnng deph s much smaller han he dameer of hole whch s reasonable for relavely large holes (a few hundred mcromeers n dameer). The effecs of beam profles and cavy profles were no consdered. These facors are mporan when he se of he hole s comparable o he drllng deph. 4

44 995 Toarev e al. (995) have proposed an analycal hermal model of ablaon by U lasers. The approxmaon used o descrbe he absorpon of he ncden beam n excmer laser nduced ablaon plasma for he case of sngle-phoon absorpon. The changng (ncreasng or decreasng) slope of he curve of eches deph versus logarhmc fluence Modes (996) developed a ransen 3D hea conducon model for maeral volume beng machned. The model assumed ha vaporaon occurs n a sngle sep whou melng. Gas dynamcs and dsconnuy layer were no aen no accoun. Ths s no suable for laser machnng of meals on nanosecond me scale. 996 Luf e al. (996) suded mechancal and hermal dsorons n laser drllng of meals. Hole shape was nvesgaed as well as composon of maeral n he edge regon of he drll. Increasng mechancal loads on he maeral due o he hgher pressure n he drll channel was a lmng facor for accuracy of he processng. Surface of he enrance channel was relavely smooh and conaned a hgh number of submcroscopc bolng pores. Exensve maeral bulgng upon drllng was observed whch was arbued o he drec mechancal effec of he hgh pressure n he drllng channel. Ths pressure resuls from he recol of ablaed maeral and he expandng cloud of vapor and plasma. They concluded ha hea dffuson no he bul of he maeral n he drecon of drllng s comparavely lesser problem han hea dffuson no he sde walls of he hole and s unclear whch processes conrbued mos: deposon of mels plasma eroson or mulple reflecons o he laeral hea dffuson. 43

45 996 - Nedrg and Bosanoglo (996) nvesgaed ablaon of meal flms. Chronologcal order of ablaon was observed n nanoseconds me frame. Ablaon revealed hreshold behavor. Above a laser energy densy of 5-6 J/cm he rradaed flm regon was compleely evaporaed durng he laser pulse. A he begnnng of laser heang he surface evaporaon occurred and afer he energy reached s hreshold a bul evaporaon was observed. Below hs hreshold evaporaon was margnal and he flm dsnegraed manly by lqud flow. Model was developed based on Her- Knudsen-Langmur relaon of rae of ablaed aoms per area. They have also shown ha vaporaon enhalpy mus be consdered as a funcon of emperaure for correc modelng of evaporaon by shor laser pulses. The dfferenal equaons were negraed usng 4 h order Runge-Kua scheme. Predcon of shape of he drlled hole was nroduced for he hgher han hreshold energy densy successfully Ganesh el al. (996) have developed a D numercal model o predc overall maeral removal raes ha compared well wh expermenal daa. The coupled conducon hea ransfer n he sold and he advecon-dffuson hea ransfer n he lqud meal he flud dynamcs of mel expulson and he racng of sold lqud and lqud vapor nerfaces have been mahemacally modeled for he D axsymmerc case. The donor accepor cell mehod usng he volume of flud approach was used. I aes no accoun all hermo-physcal properes ncludng laen hea of vaporaon gravy and surface enson drvng forces. The novely of hs model was o rea he meled pool surface as a deformable free surface. I was found ha resoldfcaon of mel (recas formaon) occurred hroughou he pulse nerval and had sgnfcan nfluence on he 44

46 developng of hole geomery whle he effec of vaporaon maeral removal on he hole geomery was found o be small. Comparson of he smulaed resuls ndcaed he maeral removed per Joule of energy absorbed appears o be nversely proporonal o he square roo of he pea beam nensy and he drllng rae appeared o be proporonal o he square roo of he surface pressure Ylbas and Sam (997) were analyng lqud layer wh a presence of nucleae bolng mechansms. Expermenal sudy of lqud eecon mechansm and heorecal sudy of sauraed nucleae bolng were presened. In expermenal sudy hey used srea phoography. Knec heory for hea ransfer model and ranspor of energy hrough elecron-phonon (elasc) collsons were adoped. They found ha he me measured for he lqud expulson from he heaed one was dencal wh me compued correspondng o possble sauraed nucleae bolng. They calculaed he degree of superhea requred for nucleaon o occur usng he Claussus-Clapeyron equaon. The parcle velocy measured usng he srea phoography mehod agreed wh he heorecal calculaons wh % error due o measuremen error and/or he assumpon made n he analyss Sema and Masunawa (997) carred ou a heorecal analyss of he energy balance n he laser-meal neracon. The hea ransfer due o he recol-pressurenduced mel flow was aen no consderaon. Ths mel flow carres away from he neracon one a sgnfcan poron of he absorbed laser nensy (abou 7 9% a low laser nenses); hus convecon-relaed erms can neher be gnored n 45

47 calculaons of he energy balance n he neracon one nor n calculaons of he hermal feld n he vcny of he weld pool or cung fron. 998 Cheng e al. (998) developed 3D fne dfference model ha smulaes hea flow and maeral removal by volalaon. Ths model was used o deermne hole profles and hermal felds. In addon o monolhc specmens drllng of layered maerals was modeled Toarev and Kaplan (999) developed D me dependen hea conducon equaon for surface heang and phase boundary. Analycal soluon has been made for recangular pulse shape and emperaure ndependen maeral parameers. Inensy dependence of mel deph was nvesgaed. The auhors dsngushed wo melng regmes: slow and fas Sefan problem and nec boundary condons were appled a he lqud-vapor nerface Solana e al. (999) aemped o deermne me dependence of mel eecon by ncorporang wo dsnc mechansms of molen eecon. Frs occurrng early n he pulse perod as a resul of rapd pressure buld up. Second a a laer sage by a more progressve process resulng from he pressure graden assocaed wh he radal decay n beam power. Some hgh-speed phoography observaons were presened o suppor hs heory. - osey e al. () consdered a maeral removal durng laser drllng. Mos of he maeral removal durng laser drllng of blnd holes s dsplaced by mel eecon dependng on he beam power densy and ype of meal. Comparsons were made beween hole dmensons and predcons from hea flow model. A echnque was 46

48 developed o collec he eeced maerals o deermne he proporon of he hole volume removed by mel eecon. A rend was noced ha he mel eecon fracon (MEF) o rse nally as he beam power densy was rased bu hen fall. Tha fall maybe assocaed wh a reducon n he hcness of he molen layer a hgh beam powers. - Bulgaova and Bulgaov (b) uled expermenal resuls o show he ranson from normal vaporaon o explosve phase change. They deermned numercally ha surface heang could be sgnfcan for non-meal maerals. Subsurface heang could lead o explosve bolng. No expermenal daa were obaned o suppor bolng mechansm. - Solana el al. () furher ncorporaed wo dsnc mechansms of mel eecon. Inal burs of mel eecon was modeled as he flow of molen maeral once pressure gradens n he drlled hole were grea enough o overcome surface enson forces. Ths was followed by he flow drven by he radal pressure gradens ha resuled from radal nensy gradens n he ncden beam. No expermenal evdence was gven for he second ype of mel eecon on radal varyng nensy dsrbuon. - Zhang el al. () repored expermenal and numercal nvesgaon of mcromachnng of copper usng a frequency rpled Nd:YAG laser wh 5 ns pulse duraon. They have developed an axsymerc model wh laser beam dsrbuon consderaon and s couplng wh he arge maeral. Ths model used he enhalpy mehod o rac he sold-lqud nerface Sefan and nec boundary condons were appled a he lqud-vapor nerface. Dsconnuy across Knudsen layer was consdered. The Marangon effec was negleced. Expermenal resuls were presened and compared 47

49 well wh model predced resuls. The range of hermal vaporaon domnaed machnng of copper usng nanosecond me scale lasers was suded. Opmum laser nensy for mcromachnng of copper was suggesed. Xu and Wlls () proposed phase exploson as a mechansm for lqud drople formaon and dscussed he physcs of measable lquds. Ther expermenal daa showed a ump n vapor fron velocy vapor ransmsvy and ablaon deph a he hreshold laser fluence. Wh a heang rae on he order of 9 K/s or hgher he surface layer meled by laser rradaon can reach a emperaure hgher han he normal bolng pon. The vapor pressure does no buld up as fas and hus falls below he sauraon pressure a he surface emperaure resulng n a superheaed measable sae. As he emperaure of he mel approaches he hermodynamc crcal pon he lqud undergoes a phase exploson ha urns he mel no a mxure of lqud and vapor. The me requred for nucleaon n a superheaed lqud whch deermnes he me needed for phase exploson o occur was also nvesgaed from boh heorecal and expermenal vewpons. Low e al. () nroduced a hydrodynamc Physcal Modelng of Laser drllng. D hydrodynamc physcal model was creaed based on realsc maeral removal mechansms (ncludng mel vaporaon and vaporaon-nduced recol pressure). The effec of O gas asss (exohermc reacon) and forced convecve coolng were ncorporaed n hs sudy bu were proven o be nsgnfcan. 3 Sema e al. (3) defned mcrowelds as havng fuson one dmensons of < µm. The heorecal crera defnng hreshold pulse energy and beam nensy 48

50 for mel dsplacemen for weldng or drllng were proposed. They presened he resuls of numercal smulaon as dependence of hreshold pulse energy and beam nensy as funcon of laser pulse duraon and beam radus. 3 - Furher wor by Golosnoy e al. (3a) has mproved deermnaon of shape of he hole profle by nroducon of an nerface racng echnque ha accouns for changng proporon of lqud and sold phases n each cell. The beam was assumed o have a Gaussan spaal dsrbuon. Defocusng and mulple reflecon effecs were no aen no accoun. Convecve and radave hea losses from subsrae surface were smulaed va a surface hea ransfer coeffcen. The subsrae was effecvely modeled o be sem-nfne n all n-plane drecons. 4 osey and Clyne (4) used D fne dfference mahemacal model developed by Cheng e al. (998) ha smulaes hea flow and maeral removal by volalaon o predc hole profle and hermal felds. In addon o unform monolhc subsrae hey dvded a modeled maeral no a number of layers. The hea flow a he nerface was beng handled by an nerfacal hea ransfer coeffcen. They used mproved deermnaon of he hole profle whch was done by Golosnoy e al. (3b) by nroducng he nerfacng racng echnque ha aes no accoun he changng proporon of lqud and sold phases n each cell. The convecve and radave losses from he subsrae surface were smulaed va a surface hea ransfer coeffcen. 49

51 .. Mechansms for lqud formaon and mel eecon durng laser mcrodrllng The mechansms nvolved n mel eecon are raher complex and (a hs me) poorly undersood. There are wo mporan aspecs of laser drllng no fully ye explaned. The frs s he laser nduced reflecvy drop. The second and probably he mos mporan for praccal purposes n mcrodrllng s he quanave connecon beween he absorbed laser energy and he exraced volume of maeral. Because several mechansms may conrbue o he neracon of rapdly heaed worpece o correcly model ha neracon one has o now whch mechansm occurs. The energy flux of he laser source and physcal characerscs of he affeced maeral may deermne whch mechansms occurs. The exen o whch mel eecon occurs depends on boh maeral properes and laser condons. Snce mechansms may be emperaure and/or me dependen s possble ha removal of he laser maeral durng laser mcromachnng occurs n seps or smulaneously durng an energy pulse. Below s a summary of research on expulson of maeral durng laser drllng whch wll suppor he developmen of a mahemacal model n he Dsseraon. A bref descrpon of several possble mechansms found n he leraure follow. The maory of he recen leraure dealng wh lqud drople eecon has focused on phase exploson. Ansmov e al. (97) saed ha he expulsed lqud mus come from he boom of he holes and no from he walls. The expulsed lqud e forms he envelope of a cone. Ths can be seen by collecng he lqud on a hn ransparen shee n fron of he arge surface where a rng of droples s formed. In 976 von Allmen delvered consderable expermenal evdence ha for expulson of meal n 5

52 lqud form exss a mechansm nvolvng radal lqud movemen caused by he evaporaon pressure. Drec measuremens of meal drllng effcency by a Nd:YAG laser over a broad nensy regon for meal drllng have shown ha opmal resuls can be expeced only n a que narrow nensy regon for mos meals vares from 5 o 5 MW/cm. For lower nenses hea conducon and reflecon losses are domnan. For hgher nenses effecs such as beam defocusng n he vapor cloud or even nduced ar-breadown severely degrade he drllng effcency and reproducbly of he drllng process. The shape of he drlled holes showng a regular and nearly polshed surface so von Allmen (976) suggesed a raher lamnar flow of lqud. Conrary o von Allmen a seres of explosons were proposed by Dabby and Pae (97) and laer by Moello and Kelly (995). Korner e al. (996) dscussed dfferen regmes of laser drllng. For example hey have saed As pulse nensy ncreases he laser-maeral neracon regme can change from hreshold o saonary evaporaon and hen o mel eecon. The proporon of removed maeral ncreases wh he ncrease of laser nensy. The concep s ha here s an nermedae range of pulse energes for a gven pulse duraon over whch mel eecon s mos pronounced. However Korner e al. (996) showed very lle relable expermenal daa n hs area. Low e al. () explo he expresson beween mel eecon and pulse nensy. Grad and Mona (998) conrolled mel eecon by emporally shapng pulsed or pulse rans. Murhy el al. (994) repored usng srea phoography mehod veloces of mel eeced droples on he order of m/s for drllng. Sam and Ylbas (997) clamed ypcal eecon veloces of abou 3 m/s wh eecon veloces ncreasng wh pulse wdh. They 5

53 also repored wo ypes of mel eecons an nal burs of slow parcles arbued o radal pressure graden expellng maeral. Then afer abou percen of he pulse wdh ( µs) fner faser movng parcles followed by larger slower-movng parcles. The emporal varaon n mel eecon descrbed by Rodden e al. () corresponds wh he second ype of behavor repored by Sam and Ylbas (997) who suggesed explanaon of hs nd ype of mel eecon as a nucleae bolng occurrng n he molen layer. They dd no repor any mcrosrucure evdence hough. Luf el a. (996) repored submcroscopc bolng pores. Laer wor proved ha nucleae bolng would occur abou µs afer he sar of he pulse machng well wh he observed sar me of he second ype of mel eecon. They repored ha usng shorer pulses (pulses wh hgher nensy) causes he naure of mel movemen o change from connuous lqud flow o aomaon wh nhbon of recas layer near he hole enrance and ulmaely o suppresson of eecon. Luf e al. (996) also noced ha ncreased pulse nensy ncreases he pressure generaed; hence ncrease he flow velocy of molen maeral. Murhy el al. (994) have explaned varaon n drllng velocy pror o reachng seady-sae condons n erms of mel eecon. Afer he nal burs of mel eecon here s an approxmaely exponenal decay n drllng velocy from abou m/s as he molen layer forms and hcens unl an equlbrum s reached beween he absorbed nensy and energy loss va maeral expulson resulng n a seady-sae drllng velocy of.-.5 m/s afer -5 µs. The nal drllng velocy would be herefore hgher hen ha predced by seady-sae models. Accordng o osey e al. (4) for mel eecon o occur a molen layer mus form and he 5

54 pressure gradens acng on he surface vaporaon mus be suffcenly large o overcome surface enson forces and expel he molen maeral from he hole. Snce he laen hea of vaporaon does no need o be absorbed when mel occurs hs s very effcen way of removng maeral. The energy requred o remove maeral va mel eecon s abou one quarer of he energy needed o vapore he same volume. Chrsensen and Tllac (3) developed he frs model for he mechansm of lqud drople formaon and eecon as well as predcon of he eeced drople rae se and velocy. They were he frs o consder heerogeneous bolng and homogeneous bolng and phase exploson mechansms... Normal evaporaon Normal evaporaon or heerogeneous evaporaon s he escape of molecules or aoms from a lqud surface (.e. regon of hgh concenraon of maer) o he amben gas n conac wh he lqud surface (.e. regon of low concenraon of maer). Normal evaporaon occurs whenever vapor pressure n he amben gas s less han he sauraon pressure of he lqud a he lqud emperaure (Xu and Wlls ). Chrsensen and Tllac (3) saed: As lqud emperaure ncreases so he sauraon pressure of he lqud and he rae of evaporaon also ncreases wh he emperaure; Evaporaon from he lqud surface causes a decrease n lqud emperaure. They deermned mass of evaporaed maeral as a funcon of sauraed pressure a he lqud 53

55 surface emperaure wh ad of he Claussus-Claperon equaon. Smlar deermnaons of mass removal mechansms for normal evaporaon s uled n he Dsseraon..3. Homogeneous bolng For laser beam neracon wh maerals when he lqud s heaed slowly durng phase change process he process s shown n Fg.. as bnodal. However f he hea rae s fas hen he lqud may become superheaed.e. he lqud emperaure can exceed he bolng emperaure. A superheaed lqud s n measable sae. If he emperaure connues o ncrease he spnodal (Fg...) s reached and he lqud becomes unsable and relaxes o a lqud-vapor mxure. A process ha prevens a lqud from reachng he spnodal s called homogenous bolng (or nucleaon). I consss of he sponaneous creaon of vapor nucle whn he lqud whou he help of preexsng nucleaon ses. Fg... A pressure-emperaure dagram showng: () he bnodal () he spnodal. The area I s he measable regon (Bulgaova and Bulgaov b). 54

56 A ypcal pt phase dagram of a subsance n he neghborhood of he crcal pon (CP) s shown n Fg... The lne of equlbrum for he sysem lqud vapor (he bnodal) orgnaes a he rple pon A (a pon of coexsence of sold lqud and vapor) and ends a he CP. To provde a clearer nsgh no he mechansms of phase exploson and s possble manfesaons we recall here he bascs of he heores of measable lquds and crcal sae of maer (Xu and Wlls Chrsensen and Tllac 3). () As he CP s approached he flucuaons of he maer parameers are ncreasng. The densy and he enropy hereof undergo he greaes flucuaons as compared o he oher parameers (p T). The maer aes on a fne-graned srucure scaerng lgh (he opalescence phenomenon). Smplfyng he pcure one may say ha he crcal sae s a gas of droples whose characersc se r c (or a correlaon lengh) ncreases when approachng he CP. () If he lqud beng heaed has enough me o relax o a defne equlbrum sae (far from he CP he relaxaon me s normally ns (Chrsensen and Tllac 3) he maer s sable and s sae follows he bnodal. In oher words f he sysem s heaed hrough he sequence of he equlbrum saes one may use he heory of equlbrum hermodynamcs and he Clausus Clapeyron equaon s sll vald. However wh ncrease of he correlaon lengh he flucuaon heory should be used nsead of he classcal approach. (3) The rapdly heaed sysem may undergo superheang. In oher words he emperaure of he lqud becomes hgher han ha of bolng under he gven pressure. If 55

57 so he sysem shfs from he bnodal no he regon I of he measable saes (Fg..) and as he heang rae ncreases approaches he spnodal. Ths leads o a decrease of he lfeme of he sysem and o an uncerany of s hermodynamc parameers near he spnodal. The sysem sees for equlbrum ha resuls n s reurn o he bnodal hrough explosve bolng (sharp ncrease of homogeneous nucleaon). (4) The rae of homogeneous nucleaon ncreases dramacally wh superheang. The average me for formaon of he crcal vapor nucleus (a vapor sphere ha wll grow raher han decay) can drop by 3 4 orders of magnude wh superheang by ºC whch s condoned by a fas decrease of boh he crcal nucleus se and he free energy for formaon of a sable nucleus. A he same me he rae of he vapor sphere growh ncreases drascally. The ncreasng nucleaon prevens he lqud from approachng he spnodal resulng n decay of he hghly superheaed lqud no a mxure of gas and droples (phase exploson or explosve bolng). I maybe sad ha he lqud s orn no droples by growng he numerous gas bubbles. (5) One of he possble mechansms for formaon of a new phase s a rse of he densy gradens nsde a small volume of superheaed lqud (densy flucuaons) ha leads o he loss of sably and o he appearance of a vapor bubble. If he measable regon s close o he CP (T.9 T c ) he densy graden necessary for he nucleaon due o flucuaons decreases. Ths s why he closer maer approaches he CP he more possble s superheang and phase exploson. (6) For a gven superheang he crcal nucleus se s much greaer for meals han for organc lquds and waer. As a resul sponaneous nucleaon n lqud meals s 56

58 no obaned even for large superheang. On fas heang he meals can be heaed very close o he crcal emperaure and only hen does phase exploson occur. (7) Meals loose her meallc properes near he CP. Parcularly he elecrc conducvy drops because of he dsrupon of he conducvy one due o ncreasng densy flucuaons. A varey of mehods were proposed o esmae he hermodynamc crcal emperaure of maerals. Mos of hem are based on he relaon beween he crcal and bolng emperaures whch was frs revealed by Guldberg for a number of subsances (Bulgaova and Bulgaov b). The leraure on he esmaons of T c s revewed n (Red and Sherwood 966) where dfferen modfcaons of he Guldberg law and some oher mehods are gven. A generalaon of he Guldberg law for norganc lquds was proposed as T c =T b /θ where θ s deermned by he vaporaon enropy S v as θ=ab/ S v wh he coeffcens a and b dependng on he polary of he subsance molecules. Normally he θ value s n he range for onc compounds and.5.5 for meals..4. Heerogeneous bolng In many of he papers ha address phase change mechansms of lqud drople eecon heerogeneous bolng has been negleced. Evaporaon and bolng each nvolves lqud-vapor phase change a he lqudvapor nerface. Bolng ncludes he developmen of vapor bubbles a dscree locaons 57

59 below he lqud surface whereas n evaporaon he vapor escapes from he nerface beween he lqud and he amben gas (Kandlar e al. 999). Once bolng occurs he emperaure of he lqud remans approxmaely consan because a commencemen of bolng energy ha s added o he heaed meled maeral s prmarly used for he lqudvapor phase change and no for rsng he lqud emperaure. The presence of movng vapor bubbles slows down he formaon of emperaure graden. Heerogeneous bolng occurs when vapor bubbles are formed below he surface a a nucleaon se. A nucleaon se may conss of: a) a scrach or p a a sold-lqud nerface n whch gas or vapor s rapped or b) foregn gas bubbles exsng n he lqud. When he emperaure exceeds he sauraon emperaure of he lqud vapor bubble formaon and growh wll occur. Moello and Kelly (995) sugges ha he nucleaon se densy may be oo low for sgnfcan heerogeneous bolng o occur. Wh only a small number of nucleaon ses he nucleaon and growh of vapor bubbles would no have he ably o produce suffcen amoun of lqud droples observed durng ablaon. Cracun e al. () suggesed ha heerogeneous bolng mgh occur when a hgh fluence laser beam causes a hc lqud pool. The auhors argue ha neglecng heerogeneous bolng based on a small nucleaon se densy may be n error. I s suggesed ha hs error s a resul of he consan nucleaon se densy proposed by prevous auhors. I s proposed ha nucleaon se densy should be calculaed as a funcon of superhea whch would ncrease he number of nucleaon ses. Three papers are referenced by Cracun e al. (998). The auhors sugges ha a bubble mus reach a crcal se before can burs 58

60 and he lqud layer ha confnes he bubble mus also reach a crcal se. They mgh provde nsgh no drople se for heerogeneous bolng or explosve bolng. Cracun e al. () quesoned he explosve bolng heory on he bass ha small lqud droples eeced no a plume wh superheaed emperaures would have me o evaporae herefore no droples would be found on wness plaes. Overall he proposals by Cracun e al. () are neresng and offer an alernave vew of wha s ang place bu hey offered no heorecal bass..5. Phase exploson Much of he evdence for phase exploson caused by rapd laser heang s based upon an observable ump n ablaon rae when he laser fluence reaches a hreshold value. For hese researchers he hreshold fluence mars a ranson from normal vaporaon o phase exploson. Bulgaova and Bulgaov (a) gve an excellen heorecal and expermenal handlng of hs subec. They developed a heorecal model for normal evaporaon as a funcon of he laser fluence n D form. These observaons sugges a ranson from normal vaporaon o a more vgorous mass removal mechansm. The lqud droples have been explcly maged as brgh spars; droples have also been observed by placng a collecon plae n he ablaon plume. The obvous change n ablaon rae s arbued o explosve bolng. Smlar expermens were performed n hs Dsseraon o esablsh droples se and whch mass removal mechansm occurs durng laser beam neracon wh maer. 59

61 Xu and Wlls () repored smlar resuls for laser ablaon of ncel. Whle he same basc resuls are obaned (.e. normal evaporaon occurs for laser fluences below he hreshold value whle explosve occurs above he hreshold value) some mporan pons are ncluded n her reamen. They poned ou ha he properes of a maeral as approaches he spnodal changed drascally. These changes n maeral properes could affec he absorpon of laser rradaon he elecrcal conducvy and he formaon and eecon of lqud droples. The dffcul as of obanng properes such as surface enson and densy near he spnodal may be requred n order o accuraely predc he eeced drople se. In order for a nucleaon o occur a vapor embryo mus exs and grow o a crcal se. Once crcal se s acheved he bubble wll connue o grow n order o mnme free energy. The me lag for nucleaon s he me necessary for an embryo o grow o he crcal se. Xu and Wlls () uled expermenal daa ha show a ump n vapor fron velocy vapor ransmsvy and ablaon deph a he hreshold laser fluence. Sauraon of hese values occurs for laser fluence above he hreshold. The ump n hese values suggess a ranson from normal vaporaon o some oher form of ablaon namely explosve bolng. The sauraon of he vapor fron velocy also ndcaes a sauraon of he surface emperaure of he arge. One may wonder wha happens o he addonal energy resulng from ncreased laser fluence f a sgnfcan ncrease n surface emperaure does no occur. Sauraon of he surface emperaure s arbued o he decrease n maeral absorpvy as he spnodal s approached. Ths decrease n absorpvy resuls n laser energy penerang 6

62 deeper no he maeral. An addonal reason s ha once nucleaon s naed energy added o he sysem s used for nucleaon and growh of vapor bubbles..6. Subsurface heang Subsurface heang may occur when a hn layer of lqud s heaed volumercally whle he evaporaon of aoms or molecules a he lqud vapor nerface removes hea from he lqud surface and by dong decreases he surface emperaure. The combnaon of evaporaon and volumerc heang can resul n he exsence of emperaure maxmum below he surface of he lqud. Dabby and Pae (97) were among he frs o propose subsurface heang as he mechansm of lqud drople eecon. Moello and Kelly (995) poned ou ha he assumpon of a consan vaporaon emperaure a he lqud surface was ncorrec because napproprae boundary condons were used. Moello and Kelly (995) and Yoo el a. () concluded ha subsurface heang would no be sgnfcan and showed ha explosve bolng (or phase exploson) remaned he prmary hermal mechansm of laser neracon wh meals a hgh fluences. However Bulgaova and Bulgaov (a) deermned numercally ha subsurface heang could be sgnfcan for nonmeal maerals. As dscussed earler subsurface heang could lead o resuls smlar o explosve bolng. Whle hese suggesons were made no expermenal daa were obaned o verfy he exsence or effec of subsurface heang n meals. Based on laes fndngs n ransparen or semransparen (non-meal) maerals le sngle crysal 6

63 slcon he subsurface heang process s man phenomenon occurrng durng laser beam neracon and wll be faclaed n he Dsseraon.. Modelng of laser neracon wh maerals A complee undersandng of laser neracon wh maerals s sll a maer of rals and adusmens. The real physcal processes of laser beam neracon (drllng cung or weldng) wh maerals are very complex. Problem of laser neracon wh maerals presens many dffcules boh from modelng as well as from expermenal sdes. One would expec a reasonable descrpon of he man phenomena occurrng durng laser neracon bu hs s complcaed because many of physcal processes equally conrbue o he developmen of conservaon equaons producng draw bac because of a grea complexy of he equaons o be solved. In mos nsances hs leads o formulaon of a model needed o be solved numercally. A lac of pernen expermenal daa o compare wh forces one o smplfy some equaons and use prevous analycal and compuaonal wor done n hs fled... Inroducon Laser mcromachnng s based on he neracon of laser lgh wh maer. As a resul of a complex process small amouns (dependng on he process used) of maeral can be removed from he surface of a worpece. Two dfferen phenomena may be 6

64 recogned: pyrolhc and phoolhc processes. In boh cases shor pulses are appled n order o remove maeral n a conrolled way. Pyrolhc process s based on breang of he chemcal bonds by phoon energy. I s appled mosly n polymers by use of ulravole lasers characered by wavelenghs of 57 o 35 nm. Phoon energy s convered drecly n breang chemcal bonds and here s very lle hermal neracon wh he worpece self. Phoolhc processes are based on a rapd hermal cycle heang melng and parly evaporaon of he heaed volume. In general one can say ha laser s focused on he maeral surface and s parly absorbed. The frs sep n laser mcromachnng s laser absorpon. The absorpvy depends on he maeral surface srucure maeral defecs power densy and wavelengh. For non-u wavelenghs he absorpon mechansms are dfferen for absorbng maerals such as meals and semconducors and ransparen delecrc maerals such as glasses and plascs. For opaque maerals nonlnear absorpon can become domnan a ulrashor pulse wdhs wh hgh nensy. For ransparen maerals absorpon has o come from nonlnear processes hrough laser-nduced opcal breadown. Ths process s descrbed n more deals n Secon.3.3. The absorbed energy ransfers no bul maeral by conducon. The laser energy absorbed by he maeral sars heang melng he worpece and hen vaporaon occurs. The vapor flux generaes a recol pressure on he evaporang surface. There also exss a large emperaure graden a he lqud-vapor nerface due o spaal dsrbuon of he laser beam energy whch generaes a hermo-capllary force. The recol pressure 63

65 and he hermo-capllary force ogeher provde he drvng force for he lqud eecon. Thus he maeral s removed from he worpece boh n vapor and lqud phases. Dependng on he laser nensy he lqud-vapor nerface emperaure may rse far beyond he normal bolng pon. However here exss a maxmum emperaure called a crcal emperaure ha a lqud maeral can aan. Crcal emperaure s he emperaure below whch a gas can be lquefed by applyng pressure and above whch no amoun of pressure s suffcen o brng abou lquefacon. A he crcal emperaure he dsncon beween lqud and gas vanshes. The hgh vaporaon rae causes a shoc wave and hgh vapor pressure a he lqud surface whch consderably ncreases he bolng emperaure. Fnally he maeral s removed as a vapor by he expulson of mel as a resul of hgh pressure and by an explosve bolng of he superheaed lqud afer he end of he laser pulse. In meals a rm of resoldfed maeral caused by laser mcromachnng s clearly seen. In plascs however he process s dfferen. The maeral s removed by breang he chemcal bonds of he macromolecules and s dspersed no gas or small parcles. Modelng of mcromachnng process requres muldscplnary approach (Dowden ) nvolvng hea ransfer flud mechancs phase ransformaon physcs as well as he racng of he lqud-vapor or sold-lqud nerfaces. Research effors can be dvded n wo groups. One group concenraes on he energy ranspor ang no accoun hea ransfer n he maerals and laen hea of vaporaon (Allmen 976; Andrews and Ahey 975). The oher group concenraes on necs of he vaporaon process (Ansmov e al. 97; Kngh 979). 64

66 There are many laser-maeral neracon models bu none of hem has ye consdered all of he relevan process physcs. Ths Dsseraon s anoher aemp o develop such a model. Process of modelng of laser mpngng on a 3D worpece Fg.. can be descrbed n fve seps:. Defne he laser energy;. Characere he laser energy couplng wh he arge maeral; 3. Fnd necessary properes of he maerals; 4. Analye and smplfy he physcal phenomena o develop he governng equaons; 5. Seup boundary condons wre he governng equaons and develop a compuer program. Fg... Schemac of laser beam mpngng on a worpece. In case of mcrodrllng he basc quanes le he drllng rae he vapored maeral he hole dameer or hea affeced one (HAZ) can be drecly measured very accuraely bu he dffcul par s o descrbe he laser beam neracon nsde he hole. A prelmnary heorecal model of laser drllng process can be formulaed n wo dfferen approaches: () n erms of absorbed laser nensy and drllng velocy and () 65

67 n erms of absorbed energy and negraed deph of hole. In mos publcaons researchers employ he frs approach. The laer approach s of lesser neres because depends on he specfc emporal pulse shape. However f emporal envelope s varyng oo rapdly hen he nensy vs. velocy concep canno be uled. In hs Dsseraon he second concep s of neres. The laser beam nensy absorpon and redsrbuon nsde he hole can be measured only ndrecly as par of many neracng phenomena such as: flud moon hole nsables plasma formaon and mulple reflecons on he hole wall... Hgh power laser sysem For expermenal sudy for hs Dsseraon a Nd:YAG class 4 pulsed laser model KLS 6 manufacured by Lasag Indusral Lasers Corporaon was used Fg... The characersc parameer values whch should never be exceeded are: laser s raed a a maxmum average oupu power of W wh a sngle pulse capably up o 3 J volage up o 37 and wavelengh λ =.64 µm. The pulse duraon s varable beween. o ms (Lasag 997). Lasers have been dscussed snce Ensen proposed her exsence n hs heory of smulaed emssons n 97. Ye hey dd no come no exsence unl lae 5 s. Today lasers have evolved o be a powerful ool and a he lead of echnology fron n varous applcaons. Lasers have revoluoned he word of mechancal machnng pracces and have nroduced new capables no mcromachnng. 66

68 Fg... Hgh power laser sysem used n hs sudy. The word LASER sands for Lgh Amplfcaon by Smulaed Emsson of Radaon. The laser energy source s nroduced no a specfed mass of a sold gas or lqud maeral dependng on he ype of laser. Wh he Nd:YAG - Neodymum Yrum Alumnum Garne sold-sae laser he YAG maeral s he hos maeral ha conans a small fracon of neodymum he acve elemen. The subsuon of he yrum ons wh neodymum ons s called dopng and ypcally he dopng percenage s abou -.5 %. The YAG crysal s an deal hos for he lasng maeral Nd 3 beng physcally hard sable opcally soropc and has good hermal conducvy ha perm laser operaon a hgh average power levels. Neodymum s an excellen lasng maeral as produces he hghes level of powers of any dopng elemens. The dmensons of he laser rods are seleced for power and opcal qualy wh he maxmum rod se lmed 67

69 o abou 5 mm dameer and mm n lengh for reasons of crysal qualy and hermal managemen (Lasag 997). The acve medum requres exernal energy npu or pump source n order for lasng o occur. The Nd:YAG s sold sae laser meanng ha he medum s sold crysal and uses lgh energy as he pump source comng from flash lamps. The source of he pumpng energy s a plasma flash lamp (elecrcal dscharge n a quar ube flled wh ner gas). The generaon of a laserbeam s essenally a hree-sep process ha occurs almos nsananeously:. The pump source provdes energy o he medum excng he laser medum aoms such ha elecrons held whn he aoms are elevaed emporarly o hgher energy saes. The elecrons held n hs exced sae canno reman here ndefnely and drop down o a lower energy level. In hs process he elecrons loose he excess energy ganed from he pump energy by emng a phoon. Ths s called sponaneous emsson and he phoons produced by hs mehod are he seed for laser generaon.. The phoons emed by sponaneous emsson sre oher elecrons n he hgher energy saes. The ncomng phoons releasng he elecrons from he exced saes o a lower energy level creang anoher phoon. These wo phoons are coheren meanng hey are n phase of he same wavelengh and ravelng n he same drecon hs s called smulaed emsson. 3. The phoons are emed n all drecons however some ravel along he laser medum o sre he resonaor mrrors (locaed a boh ends of he medum) o be 68

70 refleced bac hrough he medum. The resonaor mrrors defne he preferenal amplfcaon drecon for smulaed emsson. In order for he amplfcaon o occur here mus be a greaer percenage of aoms n he exced sae han n he lower energy levels. Ths populaon nverson of more aoms n he exced sae leads o he condons requred for laser generaon. The phoon emssons are rapped beween wo reflecve mrrors one beng semransparen and he oher % reflecve defnng he opcal resonaor. When an adequae level of energy s creaed and sored whn he leasng maeral a collmaed beam of lgh wll expel hrough he sem-ransparen mrror. Ths phenomenon s descrbed as a laserbeam. Desgn of he resonaor has a sgnfcan mpac on he equaly and he spaal power dsrbuon of he emed laserbeam. The mos commonly used resonaor desgn s composed of wo sphercal or fla mrrors facng each oher wh he medum beween hem. The beam propagaon properes are deermned by he curvaure of he reflecve mrrors and he dsance hese mrrors are apar. As he laser absorbs he pump energy he laser rod heas up. If he frequency of he pump energy exceeds he hermal relaxaon me of he crysal he emperaure of he crysal ncreases. Ths ncludes emperaure gradens n he laser rod crysal ha gve rse o hermal lensng (Brons 3) whereby he crysal acs as a lens o dffrac he laser whch reduces power so he coolng s requred. The coolng s provded for boh he flash lamp and he laser rod by floodng he enre cavy wh flowng waer. 69

71 When hs laserbeam s focused o a small spo hen we can cu mel and burn hrough any maerals (ncludng damonds). Ths neracon of laser energy wh varous maerals s a subec of hs Dsseraon..3. Ineracon of elecromagnec radaon wh maer The physcal processes n laser maeral neracon are mporan for undersandng he capables and lmaons of laser mcromachnng processes. Laser radaon s an elecromagnec (EM) radaon and can be represened as an elecrc vecor feld and magnec feld Fg..3. Fg..3. The elecrc and magnec fled vecors of EM radaon. When EM wave radaon (laserbeam) sres a surface (ar/sold nerface) undergoes a reflecon and ransmsson. Some radaon s ransmed some refleced and some absorbed. As passes hrough a new medum s absorbed accordng o Beer Lamber s law 7

72 I = I e µ () where he absorpon coeffcen µ depends on he medum wavelengh of he radaon and he nensy I emperaure and he plasma formaon above he arge. Secon Chaper 4.3. s dedcaed o esablsh he value of µ. When EM wave passes hrough a small elascally bound charged parcle he parcle wll be se n moon by he elecrc force from he elecrc feld E. The force s very small and s ncapable of vbrang an aomc nucleus. Ths process of phoons beng absorbed by elecrons s nown as he Inverse Bremssrahlung Effec. Bremssrahlung effec s he emsson of phoons from exced elecrons. As he elecron vbraes wll eher reradae n all drecons or be resraned by he lace phonon. Before consderng he wave equaon any furher one has o defne wo prevously menoned phenomena: plasma and phonon..3.. Phonon consderaon Consderng he regular lace of aoms n a unform sold maeral one would expec here o be energy assocaed wh he vbraons of hese aoms. However hey are ed ogeher wh bonds so hey canno vbrae ndependenly. The vbraons ae he form of collecve modes whch propagae hrough he maeral. Such propagang lace vbraons can be consdered o be sound waves and her propagaon speed s he speed of sound n he maeral. The vbraonal energes of molecules (e.g. a daomc molecule he lowes vbraonal ransons of daomc molecules approxmae he 7

73 quanum harmonc oscllaor and can be used o mply he bond force consans for small oscllaons) are quanes reaed as quanum harmonc oscllaors. Quanum harmonc oscllaors have equally spaced energy levels wh separaon E so he oscllaors can accep or lose energy only n dscree uns of energy hυ. where h s Plan consan and υ s phonon frequency. Evdence on he behavor of vbraonal energy n solds s ha he collecve vbraonal modes can accep energy only n dscree amouns and hese quana of energy have been labeled phonons. Le he phoons of EM energy hey obey Bose-Ensen sascs. Consderng a sold o be a perodc array of mass pons here are consrans on boh he mnmum and maxmum wavelengh assocaed wh a vbraonal mode Fg..4. By assocang phonon energy h ν s h ν s n E = h υ = = () λ L wh he modes and summng over he modes Debye was able o fnd an expresson for he energy as a funcon of emperaure and derve an expresson for he specfc hea of he sold. In hs expresson v s s he speed of sound n he sold. 7

74 Fg..4. Phonon vbraons. Debye s approach was o consder a sphere n n-space whch would gve he same number of aoms. Tang n as a vecor and allowng he componens o ae only posve values gves /8 of he sphere volume. N 4 = π n 3 max (3) n 3 max = N. (4) π A phonon s a quaned mode of vbraon occurrng n a rgd crysal lace such as he aomc lace of a sold. The sudy of phonons s an mporan par of sold sae physcs because hey conrbue o many of he physcal properes of maerals such as hermal and elecrcal conducvy. For example he propagaon of phonons s responsble for he conducon of hea n solds. There are wo mporan properes of phonons. Frs phonons are bosons and second each phonon s a "collecve mode" caused by he moon of every aom n he lace. 73

75 The speed of propagaon of a phonon whch s also he speed of sound n he lace s gven by he slope of he dsperson relaon υ/. A low values of (.e. long wavelenghs) he dsperson relaon s almos lnear and he speed of sound s ndependen of he phonon frequency υ. As a resul paces of phonons wh dfferen wavelenghs can propagae for large dsances across he lace whou breang apar. Ths s he reason ha sound propagaes hrough solds whou sgnfcan dsoron. I should be noed ha he physcs of sound n ar s dfferen from he physcs of sound n solds alhough boh are densy waves. Ths s because sound waves n ar propagae n a gas of randomly movng molecules raher han a regular crysal lace..3.. Lace wave Due o he connecons beween aoms he dsplacemen of one or more aoms from her equlbrum posons wll gve rse o a se of vbraon waves propagang hrough he lace. One such wave s shown n Fg..5. The amplude of he wave s gven by he dsplacemens of he aoms from her equlbrum posons. The wavelengh λ s mared n Fg

76 Fg..5. Phonon wave. In real solds here are wo ypes of phonons: "acousc" phonons and "opcal" phonons. "Acousc phonons" have frequences ha become small a he long wavelenghs and correspond o sound waves n he lace. "Opcal phonons" whch arse n crysals ha have more han one aom n he un cell always have some mnmum frequency of vbraon even when her wavelengh s large. They are called "opcal" because n onc crysals (le sodum chlorde) hey are exced very easly by lgh (n fac by nfrared radaon e.g. Nd:YAG). Ths s because hey correspond o a mode of vbraon where posve and negave ons a adacen lace ses swng agans each oher creang a me-varyng elecrcal dpole momen Laser-nduced opcal breadown Laser-nduced breadown s a process where a normally ransparen maeral s frs ransformed no absorbng plasma by he srong laser pulse. Subsequen absorpon by he plasma of he laser energy causes heang ha leads o rreversble damage o he 75

77 hos maeral. The nonlnear processes ha cause breadown are avalanche onaon and mulphoon onaon Avalanche onaon The avalanche onaon process s llusraed n Fg..6. In a ransparen delecrc maeral he bound valence elecrons have an onaon poenal or bandgap greaer han he laser phoon energy. The bound elecrons do no absorb he laser lgh a low nenses. However n all real maerals here are always some free or conducon elecrons presen and hey are he seed elecrons for avalanche onaon. These seed elecrons can come from meallc mpures as well as from hermal or lnear opcal onaon of shallow energy levels of nclusons. A free elecron when smply wgglng n he oscllang laser feld does no gan energy when averaged over an opcal cycle hus does no absorb he laser energy. However he free elecron can absorb laser energy when colldes wh he bound elecrons and he lace hrough dephasng. Ths s he Joule heang process also nown as nverse Bremssrahlung. The seed elecron can be acceleraed enough ha s nec energy exceeds he onaon poenal of he bound elecron. Therefore he nex collson wh a bound elecron wll resul n an onaon even f he free elecron ransfers nearly all s energy o he bound elecron resulng n wo free elecrons wh low nec energes. Ths s called mpac onaon. 76

78 Ths process wll repea self leadng o an avalanche where he free-elecron densy grows exponenally from he very low seed elecron densy. When enough bound elecrons are oned by hs avalanche process a plasma wh a crcal densy s produced and he ransparen maeral s broen down and becomes absorbng (he absorpon by he nal seed elecrons s neglgble due o he very low densy). Ths crcal densy s cusomarly aen o be cm for nanosecond or longer pulses snce hs s consdered he densy a whch sgnfcan opcal absorpon occurs for rreversble damage o ae place. For shorer pulse duraons he real plasma crcal densy for he laser wavelengh defned as (n cgs uns) where s he elecron mass and s he laser frequency may be a more approprae choce. Ths s he plasma densy a whch he plasma oscllaon frequency equals he laser frequency and he ransparen maeral becomes oally opaque. For vsble and near-ir wavelenghs cm. Lasernduced breadown s accompaned by emsson of acousc waves and opcal plasma radaon (a spar) from he focus. Fg..6. Schemac of elecron avalanche by collsonal mpac onaon. Secondary free elecrons are generaed durng collsons wh elecrons whose nec energy s greaer han he bound elecrons bndng energy. 77

79 Mulphoon onaon When he laser feld srengh s very hgh as n he case of ulrashor-pulse laser maer neracon bound elecrons of he ransparen maeral can be drecly oned hrough mulphoon absorpon. Ths process s schemacally shown n Fg..7. A bound elecron can be lfed from s bound energy level or valence band o he free energy level or conducon band by smulaneously absorbng phoons n he laser pulse such ha where s he energy of he phoon and s onaon poenal or bandgap. Fg..7. The mulphoon onaon process. The bound elecron s oned by smulaneously absorbng m phoons. Ths s called mulphoon onaon. Only a very hgh feld srengh s mulphoon onaon sgnfcan. Therefore for long pulse wdhs where he feld srengh a breadown s lower he mulphoon onaon conrbuon s neglgble and laser-nduced breadown s domnaed by avalanche onaon. However a ulrashor pulsewdhs mulphoon onaon plays an mporan role: deermnes he breadown hreshold behavor. The laser-nduced breadown process aes me o buld up and depends on he laser feld srengh. I exhbs a hreshold behavor: a a gven laser pulsewdh only when he laser feld srengh exceeds a ceran hreshold can he plasma densy grow o 78

80 he crcal value where rreversble breadown aes place. The hreshold s cusomarly expressed as a laser fluence hreshold (n un of energy per un area) as a funcon of pulsewdh..4. Analyss of laser energy Developng an clear overall physcal model frs s mporan o explo and beer undersand laser energy under consderaon. Analyng laser energy one can dsngush four arbues: me spaal dsrbuon frequency and amplude. Analyss of energy phenomena s very helpful for our research and modelng..4.. Tme consderaon Temporal modulaon of energy feld has been proven o be an effecve way o mprove machnng qualy. Ths Secon manly llusraes he me scale effecs connuous/dscree effecs and he relavy effecs of me arbue. The emporal dsrbuons of energy felds a dfferen me scales can grealy affec he neracon beween energy and maeral. Hea Affeced Zone (HAZ) for pulsed laser processng s usually smaller han ha of connues wave (CW) laser processng. More srng phenomena were found when pulsed lasers progressed from ns scale ( -9 second) o under ps ( - second). For fs ( -5 second) scale laser processng changes ha are even more radcal happened. HAZ decreased o almos ero a dfferen 79

81 wavelenghs for he same maeral. These were arbued o he ulra-shor laser pulse duraon. One can brefly explan hs as follows. When laser beam acs on a maeral laser energy s frs absorbed by elecrons. The absorbed energy propagaes hrough he elecron subsysem and hen s ransferred o he lace. In hs way laser energy s ransferred o he amben arge maeral Fg..8. One can dsngush hree characersc me scales: e - he elecron coolng me whch s n he order of ps; l - he lace heang me; and p - he duraon of laser pulse. e and l are proporonal o her hea capacy dvded by a same consan and he hea capacy of elecron s much less han ha of lace so e << l e and l are maeral dependen p me duraon of laser pulse one can choose o accordng o he applcaon. For dfferen ranges of p we can dsngush hree cases o occur. Fg..8. Energy flow n laser-maeral neracon. Case one: p > ns >>l>>e. In hs case elecron absorbed laser energy has enough me o be ransferred o lace elecron and lace can reach hermal equlbrum he man energy loss s he hea conducon no he sold arge. Maeral s frs meled when he beam s srong enough evaporaon occurs from lqud sae. The 8

82 exsence of melng layer maes precse maeral removal usng laser pulses above nanosecond very complcaed. Case wo: p s n fs range and << e << l laser pulse duraon p n hs case s shorer han he elecron coolng me. Elecrons are heaed nsanly hen n abou ps elecrons ransfer her energy o her posve lace ons. When hs energy nensy s hgh enough whch s ofen rue for ulra-fas pulsed lasers hose ons ge energy hgh enough o brea he bondng of lace srucure hey brea off nsanly whou havng me o ransfer her energy o her neghborng lace ons hus drec sold-vapor ranson occurs. Hea conducon no he arge can be negleced hea affeced one s grealy reduced. For melng-free ablaon o be possble wo condons mus be me: ulra-shor pulse duraon and hgh enough pulse energy. Case hree: l >> p >> e p s of ps me scale. Ths s a ransonal suaon melng layer exss n laser ablaon. The emporal dsrbuons of energy felds can be connuous or dscree boh have her advanages and dsadvanages relave o her applcaons. An mporan me duraon effec s ha when he energy feld nensy and energy pulse duraon reach ceran exen somehng unusual may happen. One should also be aware of he relavy of me effecs. Relave o very fas evens such as very shor energy pulses he even varyng a normal speed can be reaed as sac. Below a correspondng me scale obecs usually demonsrae very abnormal properes. For example he propery of lqud can change for dfferen me scales below - second waer can no be reaed as a Newonan flud anymore. 8

83 .4.. Spaal consderaon Spaal dsrbuon of energy feld s drecly relaed o energy acng area acng locaon and relave poson beween ool/energy sources and pars. Laser beam can be ransmed over long dsances wh very small dvergence laser beam spo se can vary from µm o mm. Hghly focused beams can ac locally wh hgh nensy whch are used for precson maeral removal defocused laser beams are used for surface reang laser formng ec. In laser machnng he relave poson beween laser source and pars s mporan. Opcal fbers have been used n communcaon for a long me now hey are used o carry hgh power laser beams n manufacurng hey add valuable flexbly n laser energy ransmsson. I's he refracve ndex dfference n he fber sysem ha maes low loss lgh energy ransmsson feasble. Spaal adusmens of energy felds are wdely used o acheve varous obecves. Telescope or mcroscope s represenave example. Why are pulse duraon me and focus spo se so mporan n laser processng? Pea power of he laserbeam equals pulse energy dvded by pulse duraon me nensy s he area average of pea laser power. When he neracon beween energy feld and arge s no connuous energy nensy s usually he decdng facor. Le he absolue energy n a laser pulse be. J fxed. Le pulse lasng me be d seconds le pulse repeon rae be f= H (/f > d). Le he beam focused spo se be D cm n dameer. If pulse repeon rae can vary n he range of -4 H hen average power s f*e =.~4 W. One can vary pulse lengh and acng area compue he pea power 8

84 and nensy. One sees clearly from Fg..9 pea nensy of a fs pulse wh D= µm s W/cm whle he nensy of a µs pulse wh D= mm s 7 W/cm. Fg..9. Inerrelaonshp beween laser nensy beam se and pulse duraon (Columba 5) Frequency consderaon Frequency s he characersc frequency of energy feld whch s dfferen from Pulse Repeon Rae. Rado wave mcrowave sound wave mechancal vbraon lace vbraon gas molecule collson ec. all have her respecve characersc frequences. Thermal radaon has specral dsrbuon many oher energy forms or maeral properes are also relaed o frequences. Lasers usually have very narrow specral wdh whle oher energy forms may have very broad and complex frequency dsrbuons. Characersc frequency of some energy felds s no obvous such as gravaon feld medum feld hermal energy sress energy vacuum feld flud pressure feld ec. 83

85 The characersc frequency of energy feld s mporan because maerals may responded very dfferenly o energy felds a dfferen frequences. U laser ablaon of organc polymers s very dfferen from nfrared or vsble laser ablaon because her ablaon mechansms are very dfferen. The nfrared and vsble laser ablaon s manly phoo-hermal degradaon whle U laser ablaon nvolves drec phoo-chemcal dssocaon. The emssvy of copper vares wh wavelengh ha s he man reason why lasers a dfferen wavelenghs are used o process dfferen maerals Amplude consderaon Amplude or magnude s a drec measure of energy feld nensy decdng wha amplude s proper or opmal s a challenge. For energy feld mehod one s neresed n effecve ways of amplude modulaon and he drec or ndrec amplude effecs of varous energy felds. In opcs opcal flers polarers aenuaors beam expandng and focusng sysems are used o modulae laser nensy and spaal dsrbuon. Wh her help one can mach he laser power oupu o very dfferen applcaons n he same me whou dsurbng he laser source. There s a ceran value for he neracon beween maeral and energy feld below whch he neracon mechansm may be of smple lnear relaonshp bu beyond hs value he mechansm becomes nonlnear and a exreme condons abnormal phenomena may happen. Ths ndcaes ha some nd of ranson mechansm beween frequency and amplude exss or one can say he effec of amplude adusmen s o a 84

86 ceran exen equvalen o he effec of frequency adusmen. An example s selffocusng of laser lgh. When laser nensy s hgh enough meda refracve ndex changes he hgher he nensy he bgger he refracve ndex. Snce he cener of a Gaussan laserbeam has hgher nensy han he rms when he beam passes hrough opcal medum cenral area has bgger refracve ndex hs s equvalen o a new laser source whose wavelengh has been reduced o abou /5 of he orgnal. Dffracon free focus spo se of he laserbeam s proporonal o lgh wavelengh. For crcular beams he focal spo se s ω mn f λ =.44 (5) D L where f s he lens focus lengh λ s he lgh wavelengh D L s he unfocused beam dameer. Ths s he upper lm of laser processng precson a a specfc frequency. Theorecally laser should be free of lmaons. The deal laser should produce an nense perfecly collmaed beam of lgh ha could be focused o a very small spo se. Tha ny nense spo of lgh could be used o cu any shape n any maerals. Unforunaely such a well-focused hgh nensy laser was hard o fnd before 99 and even presen ones are mpraccal for some laser cung. The problem can be explaned hrough he followng equaon: D F = M 4 λ π f D L (6) where D F s he dameer of he laser beam n he focal plane of he focusng lens; M he beam qualy facor λ s wavelengh of he laser lgh f s he focal lengh of he focusng lens D L s he dameer of he collmaed laser beam on he focusng lens. 85

87 Analyng Eq. 6 f we wan o have small dameer of he laser beam D F hen we should mae small he rao f/d L. or mae f small and D L very large. However one should also consder he laser s Ralegh lengh R L R L f = DF (7) D L whch s he dsance above and below he focal plane where dameer of he beam has ncreased by and he beam nensy has dropped by. In pracce one can cu very hc maerals whch are as hc as Ralegh lenghs. To cu hcer maerals one would need large f/d L rao so he only parameers lef s he wavelengh and beam qualy M o be adused o reduce he spo dameer D F for he purpose of cung very small obecs wh very small laserbeam as a cung ool. Ulrafas laser oscllaors generae pulse energy nally a nj (. e. -9 J) scale amplfcaon o mj level s needed n mcromachnng bu one nows when pulse lasng me s very small pea energy nensy goes up far beyond he safe operaon range of normal opcal amplfcaon sysems. Ulrashor pulsed lasers successfully solved hs dffculy usng Chrped Pulse Amplfcaon (CPA) and Pulse Compresson echnques. The energy of ulrafas lasers are hghly concenraed n me doman bu her frequency dsrbuons are much broader han normal laser sysems. Lgh a dfferen frequences ravel a dfferen speeds hrough opcal medums. Ths s a bad hng a frs glance bu conrary o one s nuon hs forms he base for he fnal soluon. In free space dfferen componens n a broad band laser pulse ravel a nearly same speed. When normal opcal componens are n he opcal pah long wavelengh 86

88 lgh componens ravel hrough he medum faser han shor wavelengh componens hus pulse lasng me s sreched and energy nensy s lowered. Specal opcal devces such as chrped mrrors or specal prsm pars are used o compensae he pulse spreadng hey allow shor lgh componens pass hrough faser han long wavelengh componens so hey compress pulse lasng me Non lnear opcal effecs A new excng area of physcs has been opened up by he laser snce he focused beam can generae huge EM felds affecng he aomc dpoles. A normal level of radaon (several W/m ) he dpoles respond n one o one rao wh he drvng force lnearly. Thus we have lnear effecs of reflecon refracon scaerng and absorpon all of whch occur a hs same frequency. Tha s frequency of he lgh s no alered by he process. However n 96 Peer Franen and ohers a he Unversy of Mchgan focused a hgh powered ruby laser (red lgh) ono a quar crysal and generaed ulra vole lgh mxed wh he ransmed lgh. Ths was he brh of he new era of nonlnear opcs. A hgh levels of radaon (several MW/m ) he dpoles no longer respond lnearly bu exhb a varey of harmonc oscllaons. a such effecs s possble o mx he frequency of lgh waves. Ths was agans all he prncpals of he super posonng of waves nown a ha me lgh heory. 87

89 Today many elecro-opc devces depend on non-lnear opcal effecs. Those effecs nclude: second harmonc generaon Pocel s elecro-opc effec he Kerr elecro-opc effec hrd harmonc generaon general four wave mxng smulaed Brlloun scaerng smulaed Roman scaerng opcal Kerr effec phase conugaon self focusng self phase modulaon and wo phoon absorpon: onaon and emsson Smulaed Raman scaerng When low nensy lgh s ransmed hrough a ransparen maeral a small fracon s convered no lgh a longer wavelenghs wh a frequency shf correspondng o opcal phonon frequency n he maeral. Ths process s called Raman scaerng. A hgher nenses Raman scaerng becomes smulaed and from he sponaneous scaerng a new lgh beam can be bul up. Under favorable condons he new beam can become more nense han he remanng orgnal beam. The amplfcaon s equally hgh n boh drecons: forward and bacward. Ths may lead o suaons when a large fracon of a new beam can be redreced owards he source of he orgnal beam raher han owards he arge. Ths could be a problem especally when fber opc cable s used. Then a deecon echnques le LIDAR could be employed o preven damage of he sysem. 88

90 Smulaed Brlloun scaerng The same process aes place wh he acouscal phonons. However correspondng frequency shf s much smaller. Acouscal phonons are based on sound wave propagaon and he frequency shf exss only for wave n he bacward drecon. A hgh nenses he Brlloun effec becomes a smulaed process and he wave may ge much more nense han he orgnal beam. Almos he enre beam s refleced bac owards he laser source Second harmonc generaon In a case of non-lnear neracon he non-lnear radaon self couples he energy from one beam o anoher. Ths would no be possble n a vacuum. In he second harmonc generaon he non-lnear polaraon wave moves hrough he srucure a one end a one velocy and he prmary refraced wave a anoher. For hem o nerac consrucvely he phase veloces of wo waves mus mach. Ths can be done by usng brefrngen crysals such as Lhum nobae (LNbO 3 ) whose refracve ndex depends on he drecon and polaraon of propagang lgh. If a polared lgh wave passes hrough a brefrngen crysal a us he rgh angle he phase veloces of he nduced polaraon wave and he second harmonc wave can be made equal. Ths s done n envronmen wh mananng he rgh emperaure and correc angle of ncdence. Example can be frequency doublng Nd:YAG lasers where laser beam s shone no 89

91 LNbO 3 crysal held n a emperaure conrolled enclosure a correc angle and an IR beam emerges wh some 3% convered o green lgh beam Opcal Kerr effec The hrd order non lnear polaraon effec can cause a change n he refracve ndex of he maeral subec o hgh nensy radaon. One of he sranges effecs usng hs Kerr effec s opcal phase conugaon. In one form called degenerae four wave mxng wo beams converge n maeral and se up a form of a grang whn he maeral. The hrd wave couples nonlnearly wh he ohers o form a phase conugaed wave. Ths prncple s appled o phase conugaed mrrors PCM. PCMs reurn he lgh o he source any dsoron beween he source and he PCM s auomacally compensaed because of he phase reversal. Self focusng fbers are also a possbly usng hs effec Physcal laser beam characerscs The energy from a laser s n he form of a beam of EM radaon. Apar from power has he properes of wavelengh coherence mode/dameer and polaraon. Le us brefly dscussed hose as follows. 9

92 Wavelengh consderaon Wavelengh depends on he ransons ang place by smulaed emsson. If one wshes o acheve a very shor pulse of lgh for example of a femo second (-5s.e. a beam of lgh abou.3 um long) s no possble whou frs mang a laser wh a broader waveband as s requred by he Fourer seres whch defnes such a shor pulse waveform Coherence consderaon The smulaed emsson phenomenon means ha he radaon s generang self and n consequence a connuous waveform s possble wh low order mode beams. The lengh of he connuous waveran may be many meers long. Ths long coherence lengh allows some exraordnary nerference effecs wh laser lgh (e.g. specle nerferomery holography and Doppler velocy measuremens) Mode/dameer consderaon A laser cavy s an opcal oscllaor. When s oscllang here wll be sandng EMW se up whn he cavy and defned by cavy geomery. Modes are he sandng oscllang EM waves whch are defned by he cavy geomery. If he cavy s of closed form.e. boh he mrrors and sdewalls are reflecve here wll be large amouns of longudnal modes oscllang nsde he cavy a ypcal value can be 9 modes for a 9

93 He:Ne laser. People had hough closed form could mprove he oupu power bu urns ou ha he oupu beam can no be well focused for closed caves wh so many modes. So open oscllaors are used whose laeral walls are no reflecve. Tha s lgh ncden on hs par s absorbed. Ths can reduce he possble longudnal modes. When hese modes oscllae hey nerfere wh each oher formng he ransverse sandng wave paern on any ransverse nersecon plane. Ths mechansm decdes he Transverse Elecromagnec Modes (TEM) of he laser beam whch s he wave paern on he oupu aperure plane. We use he sgn TEM pq o specfy a TEM mode where p s he number of radal ero felds q s he number of angular ero felds q s he number of longudnal felds and we usually use TEM pq o specfy a unambguous TEM mode whou he hrd ndex. Represenave TEM paerns are shown Fg... Clearly he mode paern affecs dsrbuon of he oupu beam energy whch affecs he machnng process. Then wha s he dameer of a laser beam? Usually hs dameer s defned as he dsance whn whch /e of he oal power exss. The hgher he order of he mode he more dffcul s o focus he beam o a fne spo snce he beam of hgher order s no from a vrual pon bu from paerns as hose n Fg.. Columba 5. 9

94 Fg... Represenave TEM modes (Columba 5). Alhough assumng Gaussan beam profle s reasonable f he laser oupus pulses a TEM mode wll cause nolerable error f one assumes he laser beam propagae as perfec Gaussan beam he error can easly go beyond 5%-%. Bu s usually very dffcul o drecly measure he focused beam especally for cases when he focused spo se s below mcrons. One soluon s o combne expermenal measuremen wh opcal calculaons o overcome hs dffculy Polaraon consderaon When elecrc vecors of ravelng EMW are lneup hus he beam s polared. Many modern laser sysems do no have a fold n a cavy wll produces randomly polared beams. In hs case he plane of polaraon of he beam changes wh me and machnng qualy may reflec. To avod hs s necessary o nroduce no he cavy a fold mrror of some form. Ousde he cavy such a fold would mae no noceable 93

95 dfference bu nsde he cavy s a dfferen maer snce he cavy s an amplfer and hence he leas loss roue s he one beng amplfed n preference o he oher n fac almos o s oal excluson. Polared beams have a dreconal effec n ceran process for example cung due o he reflecvy effecs Focal spo se consderaon Focal spo se deermnes he maxmum energy densy ha can be acheved when he laser beam power s se so he focal spo se s very mporan for maeral processng. In order o adus he beam o gude o he worpece and shape here are many devces. These devces wll be dscussed ogeher wh he basc heory of her desgn n Secon. In nearly all of hem he smple laws of geomerc opcs are suffcen o undersand how hey wor. However o calculae he precse spo se and deph of focus one needs o refer o Gaussan opcs and dffracon heory Dffracon lmed spo se A beam of fne dameer s focused by a lens ono a plane as shown n Fg... When a beam of fne dameer D s focused by a lens ono a plane he ndvdual pars of he beam srng he lens can be maged o be pon radaors of new wavefron. The lgh rays passng hrough a lens wll converge on he focal plane and nerfere wh each oher hus consrucve and desrucve superposon wll ae place and lgh energy s 94

96 dsrbued as shown n Fg... The cenral maxmum conans abou 86% of he oal power. The focusng dameer s measured beween he pons where he nensy has fallen o /e of he cenral pea value. φ f Fg... Focus paern of parallel lgh (Columba 5). For a crcular beam wh a plane wavefron he dffracon lmed beam dameer whch s he smalles focal dameer s gven by Eq. 5 and he smalles possble focal spo se n hs case s (Columba 5) f λ ω =.44 ( p l ) (8) D where f s he lens focal lengh D s he beam dameer a he lens λ s wavelengh of he lgh p l are he mode numbers. From Eq. 8 we can clearly see he nfluence of modes on he focal propery. There are oher facors ha affec focal spo se such as sphercal aberraon and hermal lens effecs. Mos lenses are made wh a sphercal shape bu hey canno be of perfec shape because here exss sphercal aberraon. Lenses n laser sysems ransm or reflec hgh power laser radaon. However laser power varaons can cause shape 95

97 changes of he lenses so he focal pon wll change when he radaon power changes hus affecng he focal spo se. Accordng o Lasag manufacurer of Nd-YAG laser sysem he spo dameer of laser beam focused ono he worpece Fg.. can be approxmaed as (Lasag 997) f θ ω (9) M where f s he focal lengh of he lens θ s he dvergence of he laser beam before he expander and M s he expanson facor of he beam expander. For laser mcromachnng small focal spo se s preferable. Based on he prevous dscusson one can learn ha he focal dameer s relaed o beam mode wavelengh beam dameer and focus Deph of focus consderaon The laser lgh s frs converged a he lens focal plane and hen dverged o a wder beam dameer agan. The deph of focus (DOF) s he dsance over whch he focused beam has abou he same nensy. Ths s also called beam was Fg... I s defned as he dsance over whch he focal spo se changes ±5%. The equaon for DOF s (Lasag 997) DOF 8 λ = π f D =.44 λ f D () 96

98 where λ s he wavelengh f s he lens focal lengh and D s he unfocussed beam dameer. Usually longer deph of focus s preferred because equal energy densy along he beam s preferred when usng he laser o process maerals. Poson of he focal plane s nown o have an effec on fnal shape of he hole as well as degree of peneraon. As he laser focusng spo moves up and down he laser maeral neracon area Ar F vares oo accordng o expresson (Sema and Masunawa 997) Ar F = nω ω / ω f D () where ω s he laser beam radus a he beam was equals o ω whch s he mnmum radus of he laser beam. In our analycal and numercal calculaons Eq. was proved o be he mos accurae and s used n hs Dsseraon. Whle calculang he drlled hole profles defocusng of he laser beam effecs should be aen no accoun. As a rule he dvergence of he laser beam would ncrease he radal hea flux on he hole walls by decreasng energy a he enrance of laserbeam of he evolvng hole. However mulple reflecons nsde he hole wll accumulae he energy a he boom of he drlled hole whch s descrbed n deal laer. 97

99 Fg... Laserbeam focusng ono he sample Qualy of laser beam The concep of M s mporan o descrbe acual propagaon of laser beams. M s a beam qualy ha measures he dfference beween he acual beam and he Gaussan beam. In order o fnd ou he M of a laser sysem we need frs measure he spo se along he laser opcal axs. Edge mehod s used f beam proflomeer s no avalable. Edge mehod uses a nfe-edge o bloc he laser beam a powermeer measures he power afer he blocng of he nfe-edge by recordng he 86% and 4% locaon of he full power. Subracng he wo values one ges he beam spo se a ha pon. Ths mehod measures he /e radus of he laser beam. Because he laser spo se s raher small he relave measuremen error for such dmenson usng edge mehod can be large. Usually a beam expander or a collmaor s used o expand and parallel he beam he spo se ou of 98

100 he collmaor s several mllmeers. For such dmenson nfe-edge mehod can easly reduce he relave measuremen error o less han %. As llusraed by Fg..3 sx measuremens a hree dfferen dsances from he collmaor are aen Z n D n where n= 3. The dsance from any chosen pon along he opcal axs and he spo se a ha locaon are recorded. Fg..3. Measuremen of beam properes. The beam se a locaon Z n sasfes he followng equaon: D n 4M λ Z n Z = D n = 3 π D () where D n s he beam se a locaon Z n D s he beam was Z s he beam was locaon λ s he wavelengh M s he beam qualy parameer whch s unnown. In hs relaon λ s nown D n and Z n can be measured M D and Z are unnowns. Tang he measured daa no Eq. 6 we ge hree hghly nonlnear equaons for hree unnowns. One can solve hese equaons usng e.g. MahCAD. Knowng M one can calculae he beam dvergence focused spo se and deph of focus (DOF) as follows: 99

101 D mn 4 fm λ = π D L (3) Af θ Gaussan = (4) πd M λ D3 D θ ac = θ nf ny = (5) π D ( Z 3 Z ) Dmn DOF = ±.8λ (6) M λ where D L s he laser beam se when propagaes o he fron sde of he focus obecve lens f s he focus lengh D mn s he mnmum beam dameer ha can be acheved. θ ac s he real beam dvergence. I can be verfed ha he equaons are solved wh an error less han -. In praccal cases one should frs measure he pulse energy or average power of he laser beam hen measure he beam spo se along he opcal axs. Usng Eqs 3 and 4 one can calculae he beam was and beam was locaon and one can fnd M of he beam hen can calculae he oher ndexes n Eq. 6. Knowng DOF one can calculae M value as / ( Dmn ) M =.8 (7) DOF λ where D mn s he mnmum beam dameer ha can be acheved. Knowng dameer a any locaon one can deermne he nensy of he laser beam a ha locaon.

102 Sphercal aberraon There are wo reasons why a lens wll no focus o a heorecal pon. Frs s he dffracon lmed problem dscussed n Secon.4.7. and he second s ha lens s no of a perfec shape. Mos lenses are made wh a sphercal shaped snce hs can be accuraely manufacure whou oo much cos and algnmen of he beam s no so crcal as wh a perfec aspherc shape. The ne resul s ha he ouer ray enerng he lens s brough o a shorer axal focal pon han he rays nearer he cener of he lens. Ths leaves a blur n he focal pon locaon. The plane of bes geomerc focus s a lle shor of he plane wavefron (paraxal pon). The se of he mnmum spo d mn s gven by d mn 3 DL = K( n q p) S a = Θ as (8) f where Θ a s angular faul (half angle) S s dsance from lens D L s dameer of op ha beam mode on lens f s he focal lengh of he lens K(nqp) s he facor dependen on he refracve ndex n q s he lens shape and p s lens poson 3 n n K ( n q p) = q 4( n ) pq (3n )( n ) p (9) 8n( n ) n n where q s he lens shape facor = (r r )/(r -r ) r r are he rad of curvaure of he wo faces of he lens and p s poson facor = f/s.

103 Thermal lensng In opcal elemens whch ransm or reflec hgh power radaon here wll be some heang of he componen whch wll aler s refracve ndex and shape (Brons 3). As he power or he absorpon changes so wll he focal pon and spo se. The wo man elemens usually concern are he oupu coupler and he focusng lens. The beam gudance mrrors could also be of concerned f adequae coolng s no suppled (waer coolng or dry ar coolng). Thermal lensng s manly caused by he rse n emperaure ncreasng he refracve ndex (dn/dt) and hus shorenng he focal lengh. A lesser effec s he hermal dsoron (dl/dt). The focal lengh shf for hn lenses can be calculaed o be A P F dn F = () π D dt L where A s he opc absorbance s he hermal conducvy T s he emperaure P s ncden power of he laser beam. Uneven heang causes furher complcaons. The approxmae Gaussan power dsrbuon of he ncden beam heas he mddle more han he edge causng a radal emperaure graden (usually he edge of he lens s beng cooled). A ypcal emperaure dfference would be 4 degrees for a 5 W beam of 38 mm dameer passng hrough an opc wh.% absorbance. A hs me le us menon anoher problem. Durng laser mcromachnng from he fac ha he absorpon s on he surface of he lens here s also gong o be a emperaure graden n he deph drecon. The hcer he opc he more bowed wll be

104 he nernal soherms. Such an aberraons wll affec he M value ransverse mode and on he spaal dsrbuon of he laser beam whch wll be dscussed and verfed expermenally n Secon Theorecal consderaon of laser absorpon Ths chaper descrbes dfferen effecs of he laser beam physcal and opcal properes on he maeral s absorpon of he elecromagnec wave. The absorpon of he laser energy aes place hrough phoon neracon wh bound and free elecrons n he maeral srucure whch rases hem o he hgher energy levels. Energy converson aes place hrough varous collson processes nvolvng elecrons lace phonons oned mpures and defec srucures. If he surface beng machned reflecs oo much lgh energy he absorbed energy s decreased he operaon effcency s lowered and he refleced lgh may do harm o he opcal sysems. Reflecon and absorpon of laser beams s closely relaed o laser mcromachnng. The value of absorpon and reflecon s relaed by Reflecvy = - Absorpvy - Transmsvy (for ransparen maerals). In meals he radaon s predomnanly absorbed by free elecrons n an elecron gas. These free elecrons are free o oscllae and reradae whou dsurbng he sold aomc srucure. As a EM wave-fron arrves a a surface of he arge hen all he free elecrons n he surface vbrae n phase generang an elecrc fled 8 ou of phase wh he ncomng beam creang elecron gas. Ths elecron gas whn he 3

105 meal srucure means ha he radaon s unable o penerae meals o any sgnfcan deph only one o wo aomc dameers (or free pahs) hus meals are opaque and hey appear shny. Accordng o Fg..4 reflecvy decreases as wavelengh becomes shorer whle absorpon ncreases when phoon energy ncreases. Fg..4. Reflecvy as a funcon of wavelengh for dfferen meals (Han 5). Fg..5. Reflecvy as a funcon of emperaure for.64 µm radaon (Han 5). 4

106 If suffcen energy s absorbed hen he vbraon becomes so nense ha he molecular bondng s sreched so far ha s no longer capable of exhbng mechancal srengh and he maeral s sad o have meled. On furher heang he bondng s furher loosened due o he srong molecular vbraons and he maeral s sad o have evaporaed. The vapor s sll capable of absorbng he radaon bu only slghly snce wll only have bound elecrons. The excepon occurs f he gas s suffcenly ho so ha elecrons are shaen free and he gas s hen sad o be a plasma..5.. Hea dffuson and pulse lengh consderaons For absorbng maerals such as meals and semconducors here s a large densy of free elecrons and valence elecrons wh an onaon poenal less han he phoon energy. For long pulses lnear absorpon domnaes and he maeral s heaed hrough Joule heang (Inverse Bremssrahlung effec) (Lu e al. 997). When melng or vaporaon emperaure s reached he maeral s consdered broen down and damaged. The breadown s also accompaned by acousc waves and opcal radaon. The rae of heang s deermned by he rae of laser energy absorpon and he rae of energy loss from he focus manly hrough hermal conducon away from he focus. The rae of laser energy absorpon s approxmaely consan before he breadown. The energy s deposed n a surface layer whose hcness s gven by he absorpon or peneraon sn deph (Ansmov and Khohlov 995) l s = () µ 5

107 where µ s he absorpon coeffcen. Anoher characersc lengh s he hea dffuson lengh durng he laser pulse whch gves he hea peneraon deph due o hermal conducon. Ths dffuson lengh s gven by ld = τ () l where s he hermal dffusvy and τ l s he laser pulse wdh. For long pulses l d >l s and he volume of he maeral heaed by he laser pulse hence he emperaure s deermned by he hea dffuson lengh durng he laser pulse. Therefore for long pulses he fluence breadown hreshold vares wh laser pulse wdh as Fh τ. However l as he laser pulse wdh decreases o a value τ e such ha l = τ < l he sn deph d e s deermnes he heaed volume durng he laser pulse no he hea peneraon deph (Lu 997). The breadown hreshold becomes ndependen of he pulse wdh (Ansmov and Khohlov 995). The above argumen also apples o ransparen maerals once sgnfcan absorpon due o plasma generaon occurs and has been used o explan he observed τ l scalng of he breadown hreshold. For long pulses where he hea dffuson lengh s larger han he sn deph a large volume s heaed and meled he ablaon and maeral removal s accomplshed hrough mel expulson drven by he vapor pressure and he recol of he lgh pressure. Ths s a very unsable process snce he flud dynamcs of he flud phase and he drvng vapor condons are que complcaed. In hole drllng and cung applcaons he resoldfcaon of he mel afer he ablaon can lead o very rregular shapes n he 6

108 holes and cus. The localed energy heas he maeral very qucly pas he lqud phase o he vapor phase wh hgh nec energy (way above he vaporaon emperaure). The maeral removal s by drec vaporaon away from he surface (no vacuum or ar). Mos of hs s accomplshed afer he laser pulse rradaon s fnshed. The maeral s sll heaed by he hea dffuson over a longer me scale. However he resulng mel layer hcness wll be small because mos of he heaed maeral reaches vaporaon emperaure and here s rapd coolng due o he seep emperaure graden. The heang of he maeral by hea dffuson s furher reduced by he fac ha a large amoun of he absorbed laser energy s carred away by he drec vaporaon. Because lle lqud s nvolved ablaon and maeral removal become hghly precse n conras o he long pulse case. There s a grea dversy n absorpvy and reflecance daa found n he leraure. Indeed hese values are bul properes bu n pracce meal surfaces usually show lower reflecance han he bul due o conamnaon (absorbens oxde layers) or macroscopc defecs (flaes ps craers ec). Opcal behavor s almos compleely domnaed by surface effecs. Surface defecs and mpures have a grea mporance n he laser-maeral couplng because hey decrease he laser nensy hreshold needed o nae vaporaon of he surface. The naon mechansm s probably assocaed wh he presence of surface defecs such as flaes whch are hermally uncoupled from he bul maeral. Before a flae vapores an elecron densy can be as hgh as 3 creaed by hermonc emsson. More deals on hs subec are n Secon.6. 7

109 .5.. Opcal funcons The sandard opcal funcons are he complex refracve ndex n ' = n = ε /. (3) where n and are he refracve ndex and he exncon coeffcen respecvely and ε denoes he permvy. The surface absorpvy A s 4π A =. (4) λ The fundamenal response of a sold o an EM feld s descrbed by D = ε E = E 4π P. (5) where D E and P are he mcroscopc dsplacemen elecrc feld and he dpole momen per un volume respecvely. Reflecvy R for normal angles of ncdence from delecrc or meal surface may be calculaed from he real par of refracve ndex n and he exncon coeffcen ( n) R =. (6) ( n) For opaque maerals such as a meal surface absorpvy A can be compued usng he followng formula 4n A = R = (7) ( n) Some values of heses consans are gven n Table. 8

110 The varaon of amplude A m wh deph s gven by Beer-Lamber s Law for a wavelengh λ n vacuum π µ Am = Am exp. (8) λ Smple analycal represenaon for ε such as he Drude expresson for free carrers (EMIS 988) ( 35. e ) ε =. (9) E( E 8. e ) or he Sellmeer expresson for dsperson n regons of ransparency perm effcen calculaon of opcal funcons o supplemen abulaed values. Energy E o wavelengh λ converson s done accordng o h c E ( λ ) = = e ( mcron). (3) n where λ=55 nm h=6.6676x -34 J s c= x cm/s e=.689x -9 J and ordnary refracve ndex of dry ar n=.77 a 5 C 76 mmhg (CRC Handboo 983) so for Nd:YAG laser beam E(λ=.64)=.65 e. Based on hs value n and he real par of refracve ndex and he exncon coeffcen for sngle crysal slcon are as follows: n=3.55 and =8.9x -5 and surface absorpvy A=/cm. Inensy s proporonal o he square of amplude and hence he varaon of nensy wh deph s gven by: 4π µ I = I exp. (3) λ 9

111 Please noe ha he opcal properes are funcons of radaon wavelengh and vary wh emperaure. Table. Complex refracve ndex and reflecon coeffcen for some maeral's opcal properes (.6 µm radaon). Maerals n R Al Cu Fe Mo N Pb T W Zn Sn Glass Effec of wavelengh The shorer he wavelengh he more energec he phoons are. Phoons wh shorer wavelenghs are easer o be absorbed by he maerals han phoons wh longer wavelenghs. Thus reflecvy R normally decreases as wavelengh becomes shorer whle absorpon ncreases when phoon energy ncreases Fg..4. As he sample hcness approaches mm he ablaon rae becomes srongly wavelengh dependen Fg..6c. A smlar laser fluence U laser beam provdes almos an order of magnude faser drllng han an IR beam.

112 Fg..6. Ablaon rae as a funcon of sample hcness a dfferen wavelenghs. Ths rend s summared n Fg..7. The characersc sample hcness a whch average ablaon rae sars decreasng s greaer for shoren wavelengh. Apparenly n he hcer samples beams of dfferen wavelengh are aenuaed o dfferen degree nsde he hole. Alhough ablaon raes a he surface are smlar for all hree wavelenghs Fg..6a. Aenuaon of he beam owards he ex of he hole and subsequen reducon n maeral removal n hc samples lead o reducon of average ablaon rae Fg..7. Fg..7. Average ablaon rae vs laser fluence.

113 One possble mechansm responsble for such drasc wavelengh dependence s nverse Bremssrahlung effec. Smple esmae of plasma absorpon usng formula 3 / β.37 λ n e T. (3) = e 9 3 Assumng plasma densy n e =.4 cm and plasma emperauret e = K leads o exncon lenghs: L(64) =.5 mm a 64 nm L(535) = 4 mm a 53 nm and L(355)=3 mm a 355 nm wavelengh. Addonally one has o accoun for an avalanchele ncrease of n T for longer wavelengh due o radaon heang of plasma. More on e e plasma absorpon can be found n Secon Effec of emperaure As he emperaure of he srucure rses here s an ncrease n he phonon populaon causng more phonon-elecron energy exchanges. Elecrons are more lely o nerac wh he srucure raher han wh he ncden phoons. Thus a fall n he reflecvy and an ncrease n he absorpvy wh a rse n emperaure should be observed Fg..8. Deal consderaons on he relaonshp beween absorpon and emperaure s descrbed n Secon.6..

114 Fg..8. Ionaon vs. emperaure Effec of surface flms The reflecvy s essenally a surface phenomenon so surface flms may have a large effec. For nerference couplng he flm mus be around [(n)/4]λ o have any effec where n s any neger. The absorpon varaon for CO radaon by a surface oxde flm s shown n Fg..9 and Fg... One form of hese surface flms may be plasma provded ha plasma s n hermal conac wh he surface. Fg..9. A surface flm acng as an nerface couplng an-reflecon coang. 3

115 Fg... Absorpon as a funcon of hcness of an oxde flm on seel for.64 µm radaon Effec of angle of ncdence A full heorecal analyss of he reflecvy allowng for angle of ncdence shows a varaon boh wh angle and wh he plane of he elecrc vecor he plane of polaraon. If he plane of polaraon s n he plane of ncdence he ray s sad o be a p ray (parallel); f he ray has s plane of polaraon a rgh angles o he plane of ncdence s sad o be an s ray (senrech = perpendcular) (Seen 3). The reflecves for hese wo rays refleced from perfecly fla surface are gven by n cosϕ R p = (33) n cosϕ ( cosϕ) ( cosϕ) n R s = (34) n where φ s angle of ncdence and oher parameers are as prevously defned. 4

116 A varaon of he reflecvy wh angle of ncdence s shown n Fg... A ceran angles he surface elecrons may be consraned from vbrang. Oherwse elecrons would have o leave he surface and hey would be unable o do ha (collecve vbraonal modes) whou dsurbng he marx.e. absorbng he phoon. Thus f he elecrc vecor s n he plane of ncdence he vbraon of he elecron s nclned o nerfere wh he surface and absorpon s hus hgh. Whle f he plane s a rgh angles o he plane of ncdence hen he vbraon can proceed whou reference o he surface and reflecon s preferred. There s parcular angle he Brewser angle a whch he angle of reflecon s a rgh angles o he angle of refracon. When hs occurs s mpossble for he elecrc vecor n he plane of ncdence o be refleced snce here s no componen a rgh angles o self. Thus he refleced ray wll have an elecrc vecor only n he plane a rgh angles o he plane of ncdence. A hs angle he angle of refracon = 9 whch s he angle of ncdence and hence by Snell s law he refracve ndex n=an(brewser angle. Any beam whch has only or prncpally one plane for he elecrc vecor s called a polared beam. 5

117 Fg... Reflecvy of seel o polared.64 µm radaon (Columba 5)) Effec of maerals and surface roughness Roughness has a large effec on absorpvy due o he mulple reflecons n he undulaons. There may also be some smulaed absorpon due o beam nerference wh sdeways refleced beams. Provded ha roughness s less han he beam wavelengh he radaon wll no suffer hese evens and hence wll perceve he surface as fla. Though he refleced phase fron formed from he Huygens waveles wll no longer be he same as he ncden beam and wll spread n all drecons as a dffuse reflecon. I s neresng o noe ha should no be possble o see he pon of ncdence of a red He:Ne beam on a mrror surface f he mrror s perfec. 6

118 .5.8. Effec of hgh aspec rao drllng Fg.. llusraes how he hgh aspec rao of a hole for he hcer maerals becomes a lmng facor. Ths relaonshp s shown comparng he average ablaon rae n.46 mm hc sanless seel shee a dfferen wavelenghs and laser spo dameers. The laser fluence a he surface was mananed consan. For each gven wavelengh hgher aspec rao led o a sgnfcan n lower ablaon rae. Fg... Average ablaon rae n sanless seel sample as a funcon of aspec rao. Besdes evden aspec-rao dependence even more pronounced s he dependence of ablaon rae on wavelengh. A shorer wavelengh allows he drllng of hgher aspec rao holes a a hgher speed for he sample hcness exceedng mm.5.9. Effec of polaraon Any beam whch has only or prncpally one plane for he elecrc vecor s called a polared beam. Mos lasers produced beams whch are polared due o he 7

119 naure of he amplfyng process whn he cavy whch wll favor one plane. Any plane wll be favored n random manner unless he cavy has foldng mrrors n whch case he elecrc vecor whch s a rgh angles o he plane of ncdence on he folded mrrors wll be favored because ha s he one sufferng he leas loss. Lgh s a ransverse EMW normally he drecon of he elecrc feld of lgh beam s randomly dsrbued no drecon s domnan we say such lgh s no polared. If he lgh elecrc feld s no evenly dsrbued some drecons domnae we say he lgh s polared. The plane of he polared lgh elecrc feld s called he plane of polaraon. Presen laser sysems generae polared lgh. I s realed by usng fold mrrors nsde he cavy. The fold mrrors selec he drecon of he resonan lgh snce only lgh wh he preferred drecon can buld up n he cavy he oupu laser s hghly polared. If he lgh s no properly polared machnng qualy may change wh me because laser polaraon plane changes wh me and he plane drecon affecs he machnng qualy. Polared beams have a dreconal effec n machnng due o reflecvy effec hence n maeral machnng lasers are usually desgned frs o generae hghly polared beam hen hey are adused o he requred drecon changes usng depolarers. The beam ou of a depolarer s a "crcularly polared" beam. To summare one can say o avod he negave effecs of polaraon one les he laser generae hghly polared lgh frs hen use depolarers o change he beam no he desred form. 8

120 .6. Plasma consderaon The role of plasma durng laser neracon wh maerals has been debaed for over wo decades. Some researchers repored a consderable loss of ransmed laser energy hrough he plasma ono he arge whereas ohers repored enhanced couplng due o re-radaon from plasma. Some repor ha plasma can aler he laser beam propagaon pah and affec he energy dsrbuon ransmed o he surface or bul of he arge. Defnely plasma s one of he mos mporan aspecs of laser neracon wh maerals and sll s no fully undersood. Plasma s a fourh sae of maer well nown o physcss oher hree saes of maer beng sold lqud and gas. Plasma s a loosely bound maer of hghly charged aoms and elecrons conanng so much energy ha he forces ha hold he maeral ogeher are obleraed. Lasers can produce hs sae of maer because hey pac so many parcles of lgh called phoons durng small me nerval ha when hey nerac wh he aoms n he surface of he maeral hey srp as many as 5 elecrons off he aom. Accordng o Duley (999) n laser mcromachnng a plume s usually produced as he resul of eecon of maeral from he mcromachned area. The vapored maeral s eeced and moves hrough he ncden beam where may be heaed o emperaures n excess of he vaporaon emperaure. Ths heang arses n large par hrough collsons wh energec elecrons. Under ceran condons he overall effec s o produce a rapd ncrease n he level of onaon whn he plume wh he formaon of he plasma. 9

121 The opcal couplng of a meal s domnaed by he conducon elecrons. Only elecrons n saes close o he Ferm level referred o as free elecrons conrbue o he opcal properes. There are wo man mechansms for elecron generaon. The frs called mulphoon onaon nvolves smulaneous absorpon by an elecron of a suffcen number of phoons o be eeced from he conducon band. The second mechansm s he absorpon of laser radaon by free elecrons: he hermonc process. The mulphoon absorpon process s more effecve a low laser wavelenghs because phoons hen have a larger energy (few e) and onaon can be acheved wh he smulaneous absorpon of only few phoons. Thus he mulphoon absorpon process s mporan only a laser wavelenghs < l µm. For CO laser rradaon (.6 µm) he smulaneous absorpon of a few ens of phoons (. e) s needed o eec an elecron from a meallc conducon band. Plasma formaon requres vaporaon of he maeral surface as a frs sep. When laser radaon s absorbed a he surface of he arge lgh energy s ransformed no hea and he surface emperaure ncreases. Smulaneously hea conducon aes place no he neror of he arge hus ncreasng he hcness of he heaed layer. Whle s undersood ha evaporaon usually occurs from a lqud he sold-lqud ranson s generally negleced snce he laen hea of fuson s a small fracon of he laen hea of vaporaon of maeral a he evaporaon emperaure. Plasma s formed for hgher laser nensy. If he plasma densy s low laser energy can sll be ransmed o he maeral whou obvous absorpon f he plasma densy s hgh enough absorpon by he plasma should be consdered. Ths absorpon

122 s relaed o he elecron densy n he plasma. When he plasma densy reaches ceran value hgh absorpon and reflecon happens. Then laser energy canno be effecvely ransmed o he arge maeral hus laser and maeral neracon s decoupled. When he plasma expands dsspaes and rarefes hen laser energy can reach he surface agan anoher cycle of plasma generaon decouplng and dsspaon happens. In hs process shoc wave s generaed so he hghes laser nensy s opmum for laser machnng. Laser machnng wors n he range below srong plasma generaon. Laser shoc waves can be used for maeral processng s called Laser Shoc Processng whch s used o mprove surface hardness and sress dsrbuon. The evoluon of he plasma follows hree maor pahs dependng on rradance spo se and amben gas condons. One can dsngush he hree maor ypes of laser absorpon waves commonly nown as () laser suppored combuson (LSC) wave () laser suppored deonaon (LSD) wave () laser suppored radaon (LSR) wave (for I > GW cm - ). elocy pressure and he effec of radal expanson on he subsequen plasma evoluon are he prmary characerscs of a wave. LSC waves occur a low rradance ( 4-7 W/cm - ) he precursor shoc s separaed from he absorpon plasma - laser absorpon one. The fron edge of he plasma (laser absorpon one) propagaes no he shoced gas ( 5 cm/s). Freshly shoced gas ngesed by he LSC wave rapdly heas up. Durng expanson manans he pressure ha drves he shoc wave. Ths process propagaes layer by layer. The LSC wave velocy s lower han ha of he shoc wave;

123 hs perms he laser energy o be channeled no heang a small mass o very hgh emperaure raher han a large mass o low emperaure (Roo 989). The plasma says confned o he vapor and surroundng amben gas. LSD waves occur a an nermedae rradance ( 7-9 W/cm ). The precursor shoc s suffcenly srong for he shoced gas o be ho enough o begn absorbng he laser radaon whou requrng addonal heang by energy ranspor from he plasma. The laser absorpon one follows drecly behnd he shoc wave and moves a he same velocy ( 5-6 cm/s). The LSD wave s ypcally observed o form ahead of a arge before has even sared o evaporae. Breadown plasmas end o be srongly absorbng. Afer an LSD wave s gned begns o ravel away from he surface along he laser beam and evenually degeneraes no a plasma column (Roo 989) a reheang fron moves owards he surface. The plasma necs wave propagaon for LSC schemacally llusraed n he upper par of Fg..3 and n he lower par of he pcure for LSD wave propagaon. Fg..3. Plasma necs for LSC and LSD wave propagaons (Boulmer 993; Herger 986).

124 Laser beams have been used as he hgh-flux energy source for a wde range of ndusral maerals processng such as laser weldng cung drllng ec. The neracon of he laser beam wh he evaporaed meal vapor ofen leads o he paral onaon of meal vapor and hus nduces plasma nsde he eyhole (referred o hereafer as eyhole plasma) and above he worpece surface (referred o hereafer as plasma plume). As a good energy ransfer medum he plasma nsde he eyhole or near he eyhole op wll help he energy ranspor from he laser beam o he worpece. On he oher hand he plasma plume ha emanaes from he worpece surface may hea he laser head and affec he mcromachnng process hrough laser absorpon scaerng and refracon. In order o proec he laser head from hermal damage a sheldng gas s usually neced coaxally wh he laser beam. Ineracon beween he plasma plume laser beam and sheldng gas nduces a complcaed pcure of flud flow and hea/mass ransfer and a par of he ncden laser beam power wll be absorbed by he plasma plume hrough an nverse Bremssrahlung mechansm. When he absorbed laser energy s grea enough o balance he energy loss due o conducon convecon and radaon from he plasma plume he plume can be mananed n a saonary form wh respec o he laserbeam. The spaal dsrbuons of gas emperaure and elecron number densy n he plasma plume are always non-unform. The exsence of elecron densy gradens whn he plasma plume wll cause refracon and defocusng of he laser beam reduce he power densy of he laser beam arrvng a he worpece surface and hus affec he laser mcromachnng process. Somemes undesrable unseadness and nermence of 3

125 he laser mcromachnng process may occur a hgh laser beam nenses. In order o suppress he unfavorable effec of he plasma plume on he mcromachnng process a laerally neced asssng gas s ofen employed. The laser-nduced plasma plume has been he subec of numerous sudes (Herger 986) summared some research resuls up o he mddle of he 98s on laser-nduced plasma plume characerscs n laser weldng. Expermenal resuls were repored abou he plasma plume emperaures bu he elecron (or plasma) emperaures measured by varous auhors dffered sgnfcanly. For example he repored maxmum values of he measured emperaures for he plasma plume encounered n connuous-wave CO laser weldng of seel worpece were from around 5 K o over K and correspondng elecron denses vared n a wde range from o 4 m -3 (DebRoy e al. 99) Table. They showed ha he laser nensy a he lgh spo was reduced manly due o defocusng and only o a mnor degree due o absorpon n he plasma plume. Expermenal nvesgaons (e.g. Bec e al. 995) show a hghly frequen modulaon of he plasma emperaure. Snce he plasma-affeced focusng dameer depends on hs emperaure obvously also flucuaes durng processng. Whenever he plasma emperaure vares whn he emperaure range 6- K he focus dameer can emporarly change by a facor of wo even a a sheldng gas conen of 9% He. Ths mgh be one cause for process nsables. To ncrease process sably herefore he emperaure-dependence of he focusng dameer should be suppressed as much as possble. Ths can be acheved by applyng approprae sheldng mxures o reduce he emperaure-dependence of he plasma's opcal properes. 4

126 Locaon of he plasma n laser mcromachnng s me and nensy dependen. Under ceran condons he plasma appears o evolve from nsde he eyhole no he amben medum. Table. Expermenally deermned emperaures and elecron denses. A a relavely low laser nensy he plasma remans aached o he enrance of he hole and s no mporan n aenuang ncden laser radaon. A hgher laser nensy he plasma separaes from he surface bu sll s relavely sable. The properes of he laser nduced plasma are descrbed by applyng he heory for a plasma n local hermodynamc equlbrum (a LTE plasma). For such a LTE plasma he degree of onaon can be calculaed accordng o he Saha-Egger equaon (erwaerde e al. 995) n n e n = g g g e ( πm T ) e b 3 h e 3 E exp bt e (35) where n e n and n are respecvely he elecron on and neural aom denses and T e s he elecron emperaure g e g and g are he degeneracy facors for elecrons 5

127 ons and neural aoms respecvely E s he onaon poenal for he neural aoms n he gas and m e s he mass of he elecron b E are respecvely he Bolmann consan and he onaon poenal of he aom. Moreover because he plasma s neural and he aoms are a mos sngly oned hen n = n Equaon 35 can be e rewren as n e 3 4 e [ exp( θ / T )] = An T. (36) e Numercal values for A and θ for several elemenal gases are gven n Table 3. Equaon can be used o esmae he nal concenraon of elecrons ha are he seed elecrons ha nae plasma heang va nverse Bremssrahlung. Table 3. Aomc consans for varous aoms and ons and values for A and θ. Elemen g g A (cm -3/ K -3/4 ) θ ( K) He Ar Al Fe Zn One can assume ha he plasma s an deal gas wh s pressure n equlbrum wh he amben amosphere p. Because he neural on aom densy n o nsde he plasma plume s raher hgh all he presen parcles have he same emperaure T e hen he equaon of sae for hs vapor s p = ( n n n ) T. (37) o e e 6

128 Snce he plasma s wealy oned ( n < n ) one can neglec conrbuon of he charged parcles o he pressure. Then subsung Eq. 37 no he Saha-Egger Eq 35 he elecron densy can be relaed o he square roo of he pressure and n case of an on gas gves he relaon / n ( ) =.6 ( ) ( ) exp. ( ) o cm x p Torr Te K (38) Te K Equaon 38 can be compared wh he measured elecron densy. araon of he elecron densy accordng o Eq. 38 versus he pressure s dsplays Fg..4 where he expermenally observed elecron emperaure T e measured by specroscopy has been used. The spread of he daa resuls from ang no accoun he expermenal unceranes n T e. The raher good agreemen observed beween he expermenally measured elecron densy n e and he compued one (maxmum dfference of abou -3) s e o remarable gven he srong sensvy of n e wh T e n Eq. 38 a 5% varaon of T e nduces a 4% varaon of n e. Fg..4. araon of he elecron densy and emperaure as a funcon of he amben pressure for an ncden laser power of.5 W. Shaded area: elecron densy calculaed from he Saha-Egger equaon usng he measured emperaure (erwaerde e al. 995). 7

129 erwaerde e al. (995) performed expermenal sudy of connuous CO laser weldng a sub-amospherc pressures and meallc vapor eeced by he eyhole for dfferen helum pressures Fg..5. Ths lumnosy decreases when he helum pressure s reduced and seems o dsappear beween and 6 Torr. A smlar evoluon was observed earler by Araa e al. 984 and Kabasawa e al. 99. Thus he elecron densy s prncpally conrolled by he amben pressure. An mporan sgnfcance of hs sudy s ha a low pressure he perurbng effecs of he plasma plume (he lensng effec and absorpon by nverse Bremssrahlung are compleely suppressed by he one order of magnude reducon of he elecron densy. Fg..5. Examples of plasma phoographs under dfferen pressure condons for a laser power of.5 W. A drople of lqud meal s always vsble a he arge surface (erwaerde e al. 995). Moreover accordng o expermenal nvesgaons showng ha expanson veloces of he plasma ousde he eyhole are abou - m/s one can asser ha hs flow s subsonc (erwaerde e al. 995). Therefore when he amben pressure s 8

130 reduced he elecron densy of he plasma nsde he eyhole follows he same varaon as ha of he plasma plume. Ths plasma reducon nsde he eyhole has srong consequences for he laser energy deposon. These are readly observed n he meallographc sudes descrbed furher n whch one analyes he energy deposon and deermnes a local absorpon coeffcen. The absorbed power s he same a every pressure bu he meled profles are dfferen. Hence he energy deposon mode of he laser beam nsde he eyhole s dfferen a hgh pressure han a low pressure. The energy deposon mode can be undersood qualavely from he evoluon of plasma wh decreasng pressure. Densy and emperaure of he plasma decrease when he pressure s reduced and he plasma dsappears a low pressures. Therefore he absorpon nsde he eyhole becomes que dfferen. A amospherc pressure and for low weldng speeds he man mechansms of energy absorpon nsde he eyhole are nverse Bremssrahlung and Fresnel absorpon along he eyhole wall. The energy deposon can herefore be descrbed o he frs approxmaon by a bul absorpon coeffcen µ. By analyng weld seam profles and usng a hermal model of he eyhole µ s deermned o range from.4 o.7 cm - for such weldng condons. A consequence of hs hgh absorpon coeffcen s ha much more laser energy s deposed near he op of he eyhole han a s boom; he energy deposon law follows a quas-nverse exponenal dependence. In conras when he pressure s reduced a smlar dervaon of he absorpon coeffcen gves a value of abou.4-.5 cm - for a 5 Torr pressure expermens. The plasma beng almos enrely suppressed he absorpon nsde he 9

131 eyhole s only due o he Fresnel absorpon and he oal local absorpon coeffcen becomes much smaller. Propagaon of an EMW s descrbed by he wave equaon (Bec el al. 995) ( n ) E = (39) where E s phasor amplude of he EM feld s propagaon vecor of he opcal feld and he opcal properes defnng he propagaon meda are represened by he complex ndex of refracon n. For plasmas n whch elecrons ha are no bound o he aomc shell domnae he neracon wh he EMW he ndex of refracon can be calculaed from he dsperson relaon n ω p = (4) ω ( ω ω ) p e where ω p s plasma frequency ω e s elecron collson frequency and ω s he laser frequency. The dsperson relaon descrbes he close connecon beween he laser frequency and he plasma properes represened by he plasma ω p and he elecron collson frequency. Usng a ray racng mehod (Rocsroh e al. 987; Ducharme e al. 994) descrbed he effec of plasma plume on propagang laser beam. These auhors showed ha properes of he beam are modfed no only by absorpon bu also by refracon due o he spaally varyng opcal properes whn he plasma plume. The maor drawbac of he ray-racng mehod used by hose auhors s ha s no applcable o descrbe he beam propagaon n he focal area. Oher auhors (Bec el al. 995) performed numercal and expermenal analyss for a focused laser beam condons. 3

132 They have suded nensy dsrbuon n he focal plane and s exac poson dependen on he plasma plume and also wh regard o he properes of he focusng opcs (especally s F-number). Opcal properes of he plasma were derved n erms of her dependences on plasma pressure and emperaure as well as on sheldng gas composon and laser wavelengh. In he calculaons properes of he plasma plume (spaal emperaure dsrbuon and sheldng gas conen) were pre-se and paramercally modfed correspondng o expermenal resuls documened n he leraure. The oal number densy of all parcles whn he plasma depends on emperaure and pressure and s gven for a mxure of dfferen parcles by Dalon s law p n n = e a g n Z g bt ε (4) where n a s elecron number densy n ag s aom number densy of gas g and n Zg s Z- charged on number densy of gas g. Snce he LTE plasma s quas-neural he number denses of elecrons and ons are relaed by n (4) e = Zn Z g where Z s he charge number. The elecron densy herefore can be drecly calculaed by combnng Eqs 35 4 and 4. As Fg..6 demonsraes he elecron densy n frs srongly rses wh ncreasng emperaure due o he onse of onaon. Afer compleely reachng he frs onaon level he elecron number densy drops accordng o he expanson of he gas wh hgher emperaures whenever he plasma pressure s ep consan. The begnnng hgher order onaon canno compensae hs rarefacon 3

133 and he maxmum elecron densy of he plasma herefore occurs durng he frs onaon. The laser-plasma neracon no only depends on he number of free elecrons ha defne he plasma frequency ω p / e n e ω = p (43) ε me bu also on he energy ransfer beween elecrons and ons and aoms expressed n erms of he elecron collson frequency ω e = ωea g ωe Z g (44) g Accordng o (Tannenbaum 973) he elecron-aom collson frequency s gven by 8(π ) 3/ / ω ea g = ( btc ) na g Da g. (45) 3 m e Fg..6. Number denses of aoms elecrons and ons n an ron plasma p= bar (Bec el al. 995). 3

134 To calculae he collson frequency beween elecrons and ons for ceran gas g (Mchener and Kruger 973) proposed he relaon Z e nz g ω e. g = ln Λ (46) 3/ 3ε m (π T ) usng he Coulomb logarhm e b e ( ) ln ln εbte Λ = 3 e ne 3/ (47) where m e s he elecron mass and ε s he elemenary charge. Knowng he elecron densy and collson frequency he opcal properes of he plasma can be calculaed by he dsperson relaon gven by Eq. 4. Resulng absorpon coeffcen and he real par of he complex refracon ndex are presened n Fg..7. Relaed o he elecron densy he absorpon coeffcen as well as he real refracve ndex exhb maxmal and mnmal values a her frs complee onaon. a) b) Fg..7. Opcal properes of varous plasmas; p= bar as a funcon of emperaure: absorpon coeffcen b) he real refracve ndex (Bec el al. 995). 33

135 To derve he absorpon and refracon of he laser beam whn he plasma plume emperaure dsrbuon and sheldng gas conen whn he plume have o be exacly nown properes ha are exremely dffcul o oban heorecally. In hs Dsseraon herefore hese properes are prese for calculaon. They are paramercally vared o demonsrae he possble effecs of he plasma plume upon laser focusng. The plasma plume s assumed o be ellpcally shaped Fg..8 wh a Gaussan-le emperaure dsrbuon (Bec e al. 995) accordng o ( r ) ( ). max exp r pl T r = T p (48) e pl pl where T pl/max s he maxmum emperaure n he cener of he plasma plume s he axal coordnae n drecon of beam propagaon and r pl s he half axs wdh of he plasma plume. Fg..8. A sech of he plasma plume (Bec el al. 995). 34

136 Plasma s n local hermodynamc equlbrum (LTE). LTE s he assumpon ha all parcles have a Maxwellan velocy dsrbuon and ha collson processes domnae he rae equaons such ha Bolmann sascs apply. To sasfy hese condons elecron denses of 6 cm -3 are suffcen (Ready 97; Dowden e al. 994). Snce formaon of he plasma plume s no he subec of hs sudy no aemp has been made o solve he plume formaon mahemacally. More deals on hs subec has been presened by Dowden e al. (994). Insead he obecve of hs Secon s o demonsrae he effecs of a plasma plume expermenally observed for example by a vdeo sysems laser beam focusng and plasma formaon. I has o be poned ou ha demonsraed calculaons and expermenal resuls do no gve exac numbers for he beam propagaon and he focal dameer Bec e al The sudes cover a wde range of suaons han he presened daa demonsrae he magnude of laser absorpon and defocusng of he laser beam and her dependence on locaon he shape and he emperaure of he plasma as well as on he characerscs of he opcs used. In hs Dsseraon o demonsrae he effecs of a plasma plume n he neracon of laser energy wh maer expermens were conduced usng a fas CCD camera o monor plasma. The hgh-speed vdeo camera (characered by a ypcal recordng speed of 8 frames/s) recorded spaal exen of he plasma plume. A represenave pcure from he camera recordng plasma of he 34 sanless seel (SST) samples s dsplayed n Fg

137 Fg..9. Represenave examples of plasma phoograph under normal pressure condons for a Nd:YAG laser power of 3.75 W rradang he 34 SST sample. Temperaure dsrbuon whn he plasma plume and s locaon and shape are herefore srcly pre-se for he calculaon of he beam propagaon. To derve he effecs of he plasma plume sysemacally properes of he plume were paramercally vared whn he ypcal range gven by he expermenalss Fg..3. The sheldng gas s assumed o be homogeneously dsrbued whn he plume. Fg..3. Conours of he plasma plume aen by a CCD camera a varous Ar-gas flow raes (Bec el al. 995). 36

138 The effec of he plasma plume on he focal radus h pl and subsequenly on he focus nensy s no only dependen on he plasma properes bu also on he F-number of he focusng opcs as s demonsraed n Fg..3. I becomes evden ha he smaller he F-number of he focusng opcs he sronger he deeroraon of refracve effecs of he laser-nduced plasma s. Snce he focusng radus produced by a small F- number opcs s very sensve o plasma formaon usng small F-number opcs o ensure hgh focal nenses presupposes an approprae plasma conrol. The plasma lens no only ncreases he beam radus bu also changes locaon of he effecve focal plane whch ndcaes poson of he mnmal beam radus of he focused beam. Ths dslocaon s demonsraed n Fg..3. Whenever he meal vapor plasma s no hnned by sheldng gases he dspersng effec of he plasma plume may exceed he focusng effec of he lens. In hs case he plasma plume prevens furher focusng of he beam and he poson of he mnmal beam radus s locaed whn he plasma plume. The effecve focal poson herefore s above he worpece ndcaed by a posve dsplacemen f n Fg..3. As long as he dspersng effec of plasma s less han he focusng effec of he lens he effecve focal plane s ousde he plasma plume and shfed n he drecon of he beam propagaon. Increasng he sheldng gas conen whn he plume reduces he dslocaon of he focal plane. However even a sheldng gas conen of 6% He may lead o a dslocaon of as much as mm. For accurae predcons of he hole profle he defocusng effec of he plasma has o be compensaed by he dslocaon of he focal plane f. 37

139 Fg..3. The plasma-affeced focusng radus relave o he undsurbed focusng radus and absorpon whn he plasma plume dependen on he F-number of he focusng opcs and he poson of he plasma plume: p = bar wh focusng opcs: (a) F = 4 (b) F = 6 (c) F = 8 (d) F = and (e) F = (Bec el al. 995). Fg..3. Dslocaon of he focal plane Aq and plasma-affeced focusng radus relave o he undsurbed focusng radus n he effecve focal plane dependen on plasma emperaure and on he sheldng gas conen: p = bar F = 7 conen of sheldng He gas: (a) (b) 4% (c) 6% and (d) 8% (Bec el al. 995). 38

140 The effecs of he plasma plume se and s dsance from he surface are shown n Fg..33. The resuls elucdae defocusng and absorpon of he focused laser beam by a plasma plume leavng he worpece ndcang ha whenever he plasma plume sars o deach from he surface defocusng ncreases (Bec el a. 995). The nensy dsrbuon s manly changed by defocusng especally for small plasma plumes. Even for a sheldng gas conen of 8% of He he focusng radus can be enlarged by a facor of egh. In conras o he severe defocusng power absorpon s moderae. Fg..33. The plasma-affeced focusng radus relave o he undsurbed focusng radus and absorpon whn he plasma plume dependen on he se and poson of he plume; p = bar T plmax = K TEM oo mode F = 7 wh He conen 8%. For: (a) pl = mm r pl =.5 mm; (b) pl =4 mm r pl = mm; (c) pl =6 mm r pl =.5 mm; (d) pl =8 mm r pl = mm; and (e) pl = mm r pl =.5 mm (Bec el al. 995). The effec of plasma emperaure and sheldng gas conen s shown n Fg..34. For all sheldng gas conens dependence of he focus dameer on he plasma emperaure s noceable. 39

141 Fg..34. The plasma-affeced focusng radus relave o he undsurbed focusng radus and absorpon whn he plasma plume dependen on he se and poson of he plume; p = bar T plmax = K r pl =.5 mm TEM oo mode F = 7 wh He conen: (a) (b) 4% (c) 6%; (d) 8%; and (e) 9% (Bec el al. 995). Defocusng s maxmum a plasma emperaures of abou K a low He fracon. Snce he elecron denses drop wh hgher emperaures defocusng agan decreases for hgher plasma emperaures up o he second onaon of he ron aoms sarng a abou K. Thnnng he ron plasma by applyng He as sheldng gas lms enlargemen of he focus radus. For 8% sheldng gas conen he enlargemen facor s less han wo whn a emperaure range up o K. In conras o he defocusng absorpon s hardly affeced by he plasma emperaure (for plasma emperaures of more han K bu s manly dependen on he sheldng gas conen. As n he prevous example he assumpon of a sheldng gas conen of abou 9% leads o mnor absorpon of he laser beam less han 5%. 4

142 Snce he plasma-affeced focusng dameer depends on hs emperaure obvously also flucuaes durng processng. Whenever he plasma emperaure vares whn he emperaure range - K he focus dameer can emporarly change by a facor of wo even a a sheldng gas conen of 9% He. Ths mgh be one cause for process nsables. To ncrease process sably herefore he emperauredependence of he focusng dameer should be suppressed as much as possble. To avod emperaure-dependence of he focus dameer he lensng effec of he plume mus no be dependen on emperaure; ha s he gradens of he refracve ndex n he radal drecon have o reman consan whn he plume. Whereas pure gases lead o a focal spo se ha s srongly dependen on plasma emperaure he opmed sheldng gas mxure reduces hs dependence. Ths ensures an almos complee decouplng of he nensy dsrbuon n he focal plane from plasma emperaure flucuaons. Le defocusng he dslocaon of he focal poson also depends on he plasma emperaure and can be reduced by ulng an approprae sheldng gas mxure. For example he plasma defocusng dslocaes focal poson from he surface of he worpece o approxmaely mm below and enlarges he effecve focal radus by abou 3%. Snce he defocusng s almos consan whn he emperaure range 8 4 K hese effecs can be aen no accoun durng process opmaon. The He:Ar sheldng gas mxure s supposed o be favorable o oban a sable mcromachnng process. The expeced mcrodrlled or weld deph when he focal radus s ep consan and he focal poson s dslocaed s llusraed n Fg..35. The maxmum weldng 4

143 deph s aaned for a focal poson below he surface of he worpece by abou a quarer of he oal weldng deph for ron and by abou a sevenh of he oal weldng deph for alumnum. As he calculaons ndcae opmng he focal poson s mos mporan when weldng wh laser powers close o he mnmum laser power (Bec el a. 995). In hs case even small dslocaons of he focal plane may nerrup he deep weldng process. Fg..35. Weldng deph for ron and alumnum dependen on focal poson and laser power; ω o =.4 mm u o = m/mn CO laser K =. and F = 7 (Bec el al. 995). A plasma plume hnned by sheldng gases s consdered o be hghly ransparen. Neverheless a consderable amoun of refracon s presen enlargng he focus radus and dslocang he focal plane. Even a small plume deachng from he surface may lead o an ncrease n he focus radus ha s suffcen o nerrup he mcromachnng 4

144 process. Plasma defocusng and no plasma absorpon herefore s assumed o be he man cause of he so-called plasma sheldng observed n laser mcromachnng. Addonally defocusng by he plume depends on he plasma emperaure. Snce he plasma emperaure s frequenly modulaed he plasma-affeced focus radus and poson flucuae as well. Ths leads o process nsables whch can parally be suppressed by approprae sheldng gases. Applyng an approprae sheldng gas mxure furher reduces he dependence of plasma defocusng on he plasma emperaure and herefore furher dampens he plasma-nduced nensy flucuaons of he focused laser beam. A mxure of helum and argon n a rao of 3: l s found o be favorable of enhancng process sably..6.. Plasma absorpon consderaons Opcal absorpon by elecrons depends on he elecron densy n he plasma. The elecron densy s crcal n c when he plasma frequency ω p equals o he frequency of he ncden lgh. When n c s reached n he plasma he laser radaon s compleely absorbed before sres he arge. Crcal elecron denses n ec values for dfferen laser wavelenghs are lsed n Table 4. 43

145 Table 4 Crcal elecron densy for varous laser wavelenghs (Boulmer-Leborgne e al. 993) n ec =N ec. The absorpon and scaerng of ncden radaon by a wealy oned plasma are deermned by he elecron densy wh he refracve ndex approxmaely gven by ( ) n = n e n (49) c where n c s crcal elecron densy defned as n c meε ω = = 3.4 ω cm e 3 ( ) (5) where ε s he permvy of free space and ω s he angular frequency. For CO laser radaon ω =.78 rad/sec and n c = cm. When n e < nc he refracve ndex s real and radaon can propagae hrough he plasma; s only when n = n ha e c radaon s prevened from enerng he plasma. Treusch (986) descrbed a srong dependence of absorpon coeffcen for nverse Bremssrahlung as vcω p µ = (5) c( ω v ) p c 44

146 where v c s elecron-aom/on-phonon collson frequency c speed of lgh whch decreases wh ncreasng frequency ω p resulng n lowerng he hreshold nensy wh ncreasng wavelengh. The absorpon and dsspaon of ncden laser radaon by he laser plasma can be obaned from he followng me-dependen equaons: dn d e = R R R (5) d r dε ne = µ I PC d P L (53) where R R d and R r are he elecron producon dffuson and recombnaon raes respecvely; ε s he average elecron energy µ s he absorpon coeffcen for laser radaon I s he laser nensy P c s he power loss from elasc collsons and P L s he power loss from nelasc collsons. Condon for rapd plasma heang s ha he rae of energy npu µ I exceeds he power loss due o elasc collsons. Ths requremen can be wren as (Myamoo and Araa 984). me µ I = ne vε (54) M where M s he mass of a neural aom and v s he collson frequency. The mean elecron energy wll be some fracon of E. Wh ε.e neme µ I ~ ve. (55) 5M Solana and Ocana (997) wored on consrucng a full 3D weld eyhole geomery. In her calculaons of ablaon losses and evaporaon effecs hey have 45

147 ncluded energy-absorpon mechansms called Fresnel and nverse Bremssrahlung. For proper evaluaon of he collson energy aenuaon n he plasma was necessary for hem o deermne he dependence of absorpon coeffcen µ on he emperaure nsde he eyhole. An expresson for absorpon coeffcen n plasma due o nverse Bremssrahlung (Kaplan 994) s wh µ ( T ) = 3 6πξ m 3 ω chω ω e 6 e n n / hω x exp.5ln( Λ T e me πt ) / (56) Λ = πε m T e 3 e ω me p / Z Z Z / (57) where all varables were defned earler. The plasma s descrbed as a funcon of emperaure wh equal densy for elecrons and ons so he equaon for pressure s Z p = ne T (58) Z whch ogeher wh Saha-Egger equaon provded ha he plasma s a amospherc pressure gves he value of he elecronc densy as a funcon of he emperaure whch n urn s necessary o oban an expresson for he absorpon coeffcen dependen only on he emperaure. The nverse Bremssrahlung absorpon coeffcen (assumng LTE condons Myamoo 986) s 46

148 6 n n Z e m w e π e µ ( m ) = exp g m chw m T e T (59) ε e π e where Z s he average onc charge n he plasma h s Planc s consan c s he speed of lgh and g s he quanum mechancal Gaun facor whch ypcally s.3 o.6 for laser weldng plasmas a.6 µm. For CO or YAG laser radaon and plasma wh T e ~ 8 3 o K exp( hw / Te ) ~ hw/ Te he Eq. 59 becomes 6 nen Z e g µ ( m ) =. 3 3 (6) 6 3mε ( ) ( ) cw π mete For wealy oned plasma wh Z= g =.5 m=. and 4 w =.78 rad/sec Eq. 6 smplfes o ne ( m ) ~ µ (6) T 3 e where ne s gven n meers -3 and T e s gven n K. Therefore wh 3 3 n e = m and Te 3 = 6. K he nal absorpon coeffcen can be calculaed o be µ = 7 m. There are wo man effecs resulng from a plasma whch srongly neracs wh he laser beam. The frs s a reducon n lgh energy ncden on he opaque arge maeral. The second s he hermalaon of he ncden lgh energy wh he assocaed effecs on he pressure and densy of he gas phase. In addon he hermaled gas becomes a source for blacbody radaon bu hs should be small compared o he energes of he ncden laser radaon. 47

149 Solana e al. (999) and Solana and Ocana (997) show varaon n he average absorpon coeffcen wh deph and he emperaure. He observed correlaon beween absorpon and he power densy of he laserbeam ang degree of onaon no accoun. Hs resuls were n a good agreemen wh oher researchers (erwaerde e al. 995; Kaplan 994 Duley 999). The average absorpon coeffcen n he plasma laser sources versus deph for a Gaussan beam s shown n Fg..36. Fg..36. The average absorpon coeffcen n he plasma laser sources versus deph of a Gaussan beam usng focusng radus of mm on 34 SST (Solana e al. 999). The laserbeam energy can be convered o hea energy whch s spread hroughou he enre cavy raher han beng delvered us o he sample surface. The absorpon coeffcen reachng maxmum value relavely close o he eyhole walls. The ncreasng emperaure owards he cener of he eyhole ends o decrease he value of absorpon coeffcen because he elecron densy s decreasng 48

150 here so absorpon coeffcen s hgher a he walls of he eyhole and lower n he cener of and s a funcon of emperaure. Wh ncreasng deph n he eyhole he emperaure graden decreases and hus he leadng effec on he absorpon coeffcen as one moves owards he cener of he eyhole s no he elecron densy bu raher he effec of ncreasng onaon. Absorpon coeffcen s decreasng wh deph of he eyhole along he axs of he sample. Durng he Nd:YAG laser energy neracon wh few samples he formaon of a plasma was examned. I was evden ha for he energy densy hgher han 3x W/m plasma s presen and he surface absorpon coeffcen as well as he bul absorpon coeffcen s a funcon of emperaure and resuls n deph of propagang eyhole as saed before. Dscusson on developng hs funcon s shown n Secon Analyss of he physcal phenomena of laser mcromachnng The mechansms nvolved n laser mcromachnng especally mel eecon phenomena are raher complex and poorly undersood. There are wo mporan aspecs of laser neracon wh maerals no fully ye explaned. The frs s he laser-nduced reflecvy drop. The second and probably he mos mporan n mcromachnng s he quanave connecon beween he absorbed laser energy and he exraced volume of maeral durng laser drllng. 49

151 The exen o whch mel eecon occurs depends on boh maeral properes and laser condons. Expulsons of maeral durng laser drllng were nvesgaed n many publcaons; a summary of our leraure revew s shown n Chaper. To consruc a mahemacal descrpon correspondng o he presened physcal model shown n Fg.. he followng assumpons were used: (a) he lqud s ncompressble; (b) he hermo physcal parameers of maerals depend on emperaure (c) he mel layer s a hn boundary layer and (d) he nfluence of surface enson and gravy on he mel flow are assumed o be nsgnfcan compared o he effec of evaporaon nduced recol pressure..7.. Maeral removal mechansm and energy ranspor n mulple phases In laser mcromachnng process hea ransfer and flud mechancs convecon as well as dffuson of he energy feld should be consdered. Thus he hea conducon equaon should be replaced by he more general energy equaon. Then he hea ransfer analyss and he flud moon analyss of he process become coupled. Combnng he boundary condons and nal condons a sysem of equaons can be solved. In he programmng he non-lneary of he propery are consdered usng eher ceran lnearaon approxmaons or usng eraons. A physcal model of he melng fron a he nal sage of drllng s shown n Fg..37. The laser nduced pressure graden a he evaporaon fron pushed maeral radally ouward away from he axs of he drllng beam. The dsplaced maeral can 5

152 hen eher flow along he walls and ou of he craer or as observed wh worpeces ha deform durng ablaon remans rapped whn. Eher way as long as he dsplaced maeral does no obsruc he cenral par of he beam he drllng velocy a he cener of he drlled hole ncreases. Fg..37. Schemac of energy ranspor n mulple phases n laser drllng he physcal model of mel removal from he neracon one he cross secon n he x plane s shown. Evaporaon from he surface molen maeral causes mel surface shf no he maeral wh velocy v dv whch s dreced normal o he mel surface. Temperaure of he mel surface s dfferen n dfferen locaons. Therefore he evaporaon rae and he relaed recol pressure on he mel surface vary along he surface. In parcular he 5

153 values of pressure a he oppose sdes of he compuaonal fne elemen p n and p ou are dfferen. Ths pressure dfference causes he mel flow Fg..37. Because he mel flow acceleraes or deceleraes dependng on he relave values of p n and p ou and he cross secons of he fne elemen volume a he flow enrance and ex are dfferen he quanes of mel enerng and leavng he compuaonal volume are no generally he same. Therefore he mel hcness δ Fg..37 decreases or ncreases resulng n he moon of he mel surface wh velocy v dm whch le v dv s normal o he mel surface. Noe ha he evaporaon componen of he mel surface velocy v dv s always dreced no he sample whle he componen of he mel surface velocy due o mel flow v dm can be dreced eher no he sample or n he oppose drecon. The resulng velocy of he mel surface v d he drllng velocy s he sum of he wo componens v = v v. (6) d dm dv The laser beam s absorbed n he near surface layer whn a maeral hcness much smaller han he characersc dmensons of he sample. Thus he hea source can be approxmaed as a surface hea source. As dscussed above he surface of he mel moves no he sample. The hea source moves ogeher wh he surface. Ths causes melng of he addonal sold meal and he melng fron propagaes no he sample. Smulaneously he mel flow ransfers hea along he sold lqud nerface. Thus boh dsplacemen of he hea source and he moon of he mel affec he emperaure feld. Assumng ncompressble lqud n he mel layer ndependen hermo-physcal properes of he emperaure for he worpece maeral and surface hea source for he 5

154 laser power he equaons of conservaon of momenum and energy can be wren as (Sema e al. 999) dv = (63) v ρ gradv = grad p µ m (64) T gradt = c ρ T (65) where ρ c T and are he velocy densy specfc hea emperaure and hermal conducvy of sold or lqud maeral µ m s he mel vscosy and p s pressure. The boundary condon a he lqud-vapor nerface s defned n Eq. 63 and for he classc Sefan boundary condon s appled o he sold-lqud boundary (melng fron = m ) s defned n Eq. 66. The lqud pool emperaure dsrbuon s nfluenced by he surface enson graden drven convecon. I s nown from recen heorecal and expermenal research ha when a lqud meal s heaed by an nense laser beam propagaon of srong convecon currens n he lqud pool manly drven by Marangon force and o a much lesser exen by buoyancy force s nsuffcen o elmnae he commonly presen srong emperaure graden whn he lqud. In laser processng of meals and alloys he pea emperaure reached a he surface s very hgh and exceeds he bolng pon. For example von Allmen deermned molen pool emperaures n excess of bolng pon for laser reamen of copper. Baanov e al. (973) ndcaed ha emperaures on he surface of a laser rradaed maeral can be hgher han he bolng pon. Paul and DebRoy; and Zachara e al. (999) have repored heorecally calculaed emperaures 53

155 close o he bolng pon for laser weldng. Khan and DebRoy (999) arrved a he same concluson from an analyss of he expermenally deermned values of relave vaporaon raes of varous alloyng elemens. Chan and Maumder (987) have also repored emperaures greaer han he bolng pon durng laser rradaon of alumnum anum and a superalloy. Theorecal calculaons of he vaporaon raes by Kngh (979) and Ansmov e al. (97) are based on he premse ha he lqud pool surface emperaures are hgher han he bolng pon. Thus durng laser rradaon he equlbrum pressures a he pool surface are ofen hgher han he amospherc pressure and he excess pressure provdes a drvng force for lqud expulson. Several nvesgaors have repored expermenal resuls of meal expulson durng laser rradaon. Chun and Rose (97) rradaed an alumnum arge wh o 3 J laser pulses of duraon.5 o. ms and observed ha as much as 9% of he maeral los was removed from he molen pool as lqud. Durng laser scrbng of ceramcs wh ruby laser pulses Wagner (974) observed ha a concal molen regon was formed and all lqud was expelled. As shown n Fg..53 calculaons may predc surface emperaures n excess of he maeral s bolng emperaure T v ; whch s approxmaely 335 ºK for he ron. Ths s n conras o mos oher models of laser weldng and cung or drllng whch assume ha he mel surface emperaure canno exceed he bolng pon. Ths assumpon orgnaes from belef ha evaporaon s a hreshold process and ha volumerc evaporaon (bolng) can ae place n he case of laser maeral neracons. 54

156 Anoher consequence of hs belef s ha propagaon of he evaporaon fron can be smulaed n erms of he problem of Sefan. However evaporaon s no a hreshold process. I can occur a any emperaure exceedng absolue ero (Ansmov and Khohlov 995). In addon heorecal sudes (Afanasev and Krohn 967) have shown ha volumerc evaporaon canno ae place durng laser maeral neracons under ypcal ndusral condons; only surface evaporaon can occur under hese condons. Thus he surface emperaure can sgnfcanly exceed he bolng pon f he absorbed laser beam nensy s hgh. The velocy of he evaporaon fron s deermned by exponenal equaon Eq. 64 (Ansmov and Khohlov 995) whch s applcable a leas for mel surface emperaures no exceedng he crcal emperaure. Therefore due o he exponenal dependence on he surface emperaure he approxmaon of evaporaon fron velocy n erms of he problem of Sefan may resuls n sgnfcan errors aporaon from rradaed surface Model used n hs Dsseraon o oban he emperaure and pressure a lqud layer surface s an ougrowh of an earler models presened by Ansmov e al. (97) or on Allmen (976). I s assumed ha he surface emperaure T s = T v and ha surface properes are mos nfluenced by he properes of vapor n he Knudsen layer where parcles leave he meled surface parcpae n enough collsons so Maxwellan ype 55

157 dsrbuon funcons can be appled. The deph of hs regon s of he order of several mean free pahs (sub-mcron regon). Please bear n mnd ha our consderaons are focused so far on he ablaon processes ha occur before mel sars o flow. The vaporaon model suggesed n hs Dsseraon ncludes wo dfferen mechansms: surface and volume vaporaon. Under equlbrum condons he rae of vaporaon s equal o rae of condensaon.e. he rae a whch parcles leave he surface of he lqud s equal o he rae a whch parcles collde wh and sc o he surface. By compung he equlbrum rae a whch gaseous parcles collde wh surface and correcng he fracon whch scs s possble o fnd he rae a whch parcles eec from he surface of he condensed phase as a funcon of emperaure. The rae of collsons per area of parcles wh he mel s found by negrang he parcle veloces over her Maxwellan velocy dsrbuon. Evaluang he negral and usng Her-Knudsen- Langmur relaon (Nedrg and Bosanoglo 996) relaon of rae of ablaed aoms per area s P( T ) n& e = (66) πm T B s where P(T) s sauraed vapor pressure defned by Clausus-Claperon equaon. Treang he vapor as an deal gas he equlbrum vapor pressure a mel surface P(T) can be relaed o he vaporaon emperaure by ( ) exp H v P T = P (67) R Tv Tv 56

158 where R s unversal gas consan and P s he amben pressure assocaes wh he reference vaporaon emperaure T v. Expresson n Eq. 67 esmaes he rae of vaporaon for a sngle componen maeral. If one deals wh an alloy le 34 sanless seel whch s a mul-componen maeral Eq. 67 gves paral pressures separaely for each componen and he oal pressure would be he sum over all of he componens. The velocy v dvs a whch he surface of he condensed phase recedes due o loss of maeral by vaporaon s relaed o molar evaporaon flux by (Nedrg and Bosanoglo 996) δ s = v dvs = ( η)n& e M ρ m (68) where η s he fracon of recondensng aoms and for mos meal s assumed o be 8% The volume evaporaon s assumed o occur by homogeneous nucleaon (see Chaper ). The formaon of crcal nucle (of volumerc veloces) s gven by (Nedrg and Bosanoglo 996) as 3 ρ 6 m σ H vm πσ n& = exp 3 ( ) en (69) M πm BTs BTs pb pe where H v s he enhalpy of vaporaon and σ s he surface enson of he lqud-vapor nerface p e P(T)/ p b s are he exernal pressure exered on he maeral by evaporang aoms and he pressure whn he bubble. The number of aoms whn a crcal bubble s deermne by 57

159 N b 4 3 Pb πrb = 3 (7 T B s where r b s he radus of crcal bubble defne by r b σ = δ vdv N b. (7) p p b e The change of he layer hcness δ due o volume evaporaon s proporonal o he nucleaon rae he number of aoms and he layer hcness ha s δ v = v dv v = M ρ δ vdv Nb. (7) The facor n Eq. 7 s an esmaed correcon whch aes no accoun bursng of bubbles when mcro debrs and small droples are expelled. Toal change of he layer hcness s sum of wo expressons gven by Eq. 68 and Eq Recol forces Absorpon of he laser beam resuls n heang and melng of he sold meal ncreases surface emperaure T s of he mel and produces recol pressure and consequen eecon of he mel from he neracon one. Mel eecon and evaporaon resul n a decrease n he mel hcness and melng of a new poron of he sold meal. The man dffculy n he applcaon of he mel eecon model o sudy he crcal condons for he naon of lqud meal expulson s ha he reardng effec of 58

160 surface enson s no consdered. Thus a precondon for he applcaon of he model s ha he vapor recol force mus be sgnfcanly hgher han he surface enson force. Basu and DebRoy (99) esablshed ha he pea emperaure mus exceed a crcal emperaure for lqud meal expulson o occur. Furhermore hey demonsraed ha he ranson from melng o expulson can be undersood from he fundamenal prncples of ranspor phenomena. Accordng o Basu and DebRoy (99) value of he maxmum weld pool surface emperaure and us before he naon of lqud meal expulson can be deermned from he followng creron: lqud expulson aes place when he vapor recol force overcomes he surface enson force of lqud meal a he perphery of eyhole pool. The recol force R s calculaed from he relaon r = R π r P( r) dr (73) where r s he radal dsance a whch he surface emperaure s equal o he bolng pon and P(r) s he dfference beween he local equlbrum vapor pressure and he amospherc pressure and s a funcon of radal dsance from he beam axs. The calculaed values of he surface enson force a he perphery of he weld pool πr σ where σ s he surface enson a he melng pon and he recol force for varous values of T for ron are shown n Fg..38. I s observed ha he recol force s a srong funcon of he pea emperaure. The value of T o for whch he recol and surface enson forces are equal s he crcal emperaure for lqud expulson. Lqud expulson from sanless seel can ae place when he emperaure a he cener of he pool T o reach a crcal emperaure T c whch s 338 ºK 59

161 Fg..38. The recol force as a funcon of pea surface emperaure and he surface enson force a he melng pon (Basu and DebRoy 99). Durng laser rradaon he rse and decay of he pea emperaure a he surface of a meal can be deermned from (Basu and DebRoy 99) o be T T = I for p πρ c p (74) T p T = I for p π ρ c p π ρ c p. (75) where To and T are he pea emperaure a me and us pror o rradaon respecvely I s he local nensy of absorbed laser radaon ρ and c p are he hermal conducvy densy and specfc hea of he meal respecvely. The emporal emperaure profle calculaed from Eqs 74 and 75 neglecs he effecs of phase change 6

162 varable maeral properes and weld pool convecve hea ranspor bu allows rough esmaon of he changng pea emperaure on he surface. The emporal surface emperaure profles for sanless seel rradaed by sngle pulses of lengh.5 and. ms are presened n Fg..39 (Basu and DebRoy 99). I s observed ha for a pulse lengh of.5 ms he pea emperaure reached s sgnfcanly lower han he mnmum pea emperaure of 338 K requred for lqud expulson o occur. However for a pulse lengh of. ms he pea emperaure exceeds he crcal emperaure. The resuls are conssen wh he expermenal daa. I should be noed ha alhough here s good agreemen beween he predced and he observed condon for expulson he equaons employed are no exac and he compued emperaures provde dependable rends bu no rgorously compued emperaure daa. Fg..39. Pea emperaure vs me for a sanless seel sample rradaed by sngle pulses of (a).5 and (b). ms duraons (Basu and DebRoy 99). Even for a small ncrease n emperaure above he crcal emperaure for meal expulson Fg..4 he lqud experences a large acceleraon and s mmedaely 6

163 expelled. Ths concluson s suppored by he observaon ha afer commencemen of he expulson of lqud meal no resdual lqud meal s observed n he cavy. When he pea emperaure a he cener of he pool exceeds he crcal emperaure T c lqud expulson aes place. When molen meal s expelled from he weld pool he laser beam ges defocused resulng n a decrease n he hea flux. As a resul he pea emperaure may drop below he crcal value and no furher expulson may ae place. Thus he exen of melng a he me when he crcal emperaure s reached deermnes he amoun of lqud meal expelled. Fg..4. Dmensonless acceleraon of lqud meal as a funcon of he dfference beween he pea surface emperaure and he crcal emperaure for expulson T=T s - T c (Basu and DebRoy 99). Applcaon of he hn boundary layer assumpon allows he pressure across he molen layer o be approxmaed by he evaporaon recol pressure p r appled o he mel surface. The evaporaon-nduced recol pressure generaes mel flow wh he domnan drecon beng ha n whch he recol-pressure graden s he hghes. Assumng ha a seady sae has been reached and he dsrbuon of he recol pressure along he shor axs of he spo x- or y-axs s close o unform hen he mel flow s a 6

164 one-dmensonal flow and we can use Bernoull s equaon o relae he mel velocy v m a he edge of he laser spo o he value of he recol pressure p r (Sema and Masunawa 997) p r ρ m ν m =. (76) The recol pressure s proporonal o he sauraed vapor pressure p s (Sema and Masunawa 997) whch n urn depends on he mel s surface emperaure T s. Applcaon of he hn boundary layer assumpon allows he pressure across he molen layer o be approxmaed by he evaporaon recol pressure p r appled o he mel surface. Usng he resuls presened n Ansmov e al. 97 one can show ha he evaporaon recol pressure a he surface s relaed o he surface emperaure T s accordng o he equaon U exp p = r AambB Ts (77) Ts where A amb s an amben pressure dependen coeffcen and B s he evaporaon consan and he parameer U s he energy of evaporaon per aom defned as U M H v = (78) N a B where: H v s he laen hea of vaporaon (per un mass) B s Bolmann s consan M s he aomc mass and N a s Avogadro s number. Accordng o he calculaons performed by Ansmov and Khohlov (995) he coeffcen A amb depends on he amben pressure and s value vares from.55 for evaporaon n vacuum o uny for he case of evaporaon under a hgh amben pressure. Calculaons (Ansmov and 63

165 Khohlov 995) also have shown ha for praccal values of he amben pressure he coeffcen A amb s close o s mnmal value of.55. The approxmaon of he hn boundary layer allows one o model he mel flow usng quas-one-dmensonal equaons n conservave form represened by a modfcaon of he S. enan equaons for ncompressble open-channel flow for he case of mass source (heang) due o melng and mass sn (coolng) due o evaporaon ( δ ) ( r vr δ ) = v r r dv v m (79) ( vrδ ) r rδ v r r pδ p δ δ v = ρ ρ r r r r µ v ρ δ m r ( v v ) v dv m r (8) where he r drecon s along he mel surface he drecon s normal o he mel surface δ s he mel hcness v r s he radal mel flow velocy averaged over he mel layer hcness and µ m s he mel vscosy. The second erm on he rgh-hand sde of Eq. 8 represens deceleraon/acceleraon due o mass addon/loss deermned by he combned effec of melng and evaporaon. The hrd erm on he rgh-hand sde of Eq. 8 represens he effec of mel vscosy. Examnng Eq. 79 one can easly see ha he second erm on he lef-hand sde and he frs erm on he rgh-hand sde represen he componens of mel surface velocy deermned by mel eecon v dm and mel evaporaon v dv respecvely. Thus he velocy of he mel surface or he drllng velocy v d s gven by v d ( rvrδ ) = v r r dv = v dm v dv. (8) 64

166 In formulang problem of neracon of laser beam wh maerals one can assume radal symmery ogeher wh an ncompressble molen subsrae eeced from he rradaed spo as he surface reaches an elevaed emperaure. Tang no consderaon he radal moon of he mel wh an average velocy v r deermned by Eqs 79 and 8 and normal o he mel surface moon of he mel-vapor nerface wh velocy v d deermned by Eq. 8 one can wre he hea ransfer equaons for he lqud and sold phases n he followng form (Sema e al. 999): T v r T r v d T = l ρ c l l T r T T r r (8) T v d T = s ρ c s s T r T T r r (83) where he subscrps l and s refer o he lqud and sold phases and ρ c T and and he densy specfc hea emperaure and hea conducvy respecvely. If he seady-sae regme of evaporaon and mel eecon s reached hen boh he melng fron and he mel-vapor nerface propagae nsde he maeral along he axs wh consan drllng velocy v d for he maeral removal sage. In hs case followng he conservaon of mass he melng rae of he sold meal equals he sum of he vaporaon and mel eecon raes (Sema and Masunawa 997) dm d s dmm dmv = (84) d d where m s s he mass of meal meled m v s he mass of meled meal ha was vapored m m s he mass of meled meal ha was eeced. The lef-hand sde of Eq. 84 s he melng rae of he sold and he frs and second erms on he rgh-hand sde of he 65

167 66 equaon represen he vaporaon and mel eecon raes respecvely. Rewrng Eq. 84 for a hole dameer equvalen o he laser spo dameer of ω gves dv m m m m d s v v v = ρ ω π ρ δ ω π ρ ω π (85) where ρ s and ρ m are he denses of he sold and lqud phases correspondngly δ m s he mel-layer hcness a he edge of he laser spo v m s he mel-eecon velocy along he shor spo axs (he x axs) a he edge of he spo and v dv s he evaporaon fron s velocy dreced along he. Calculang he drllng velocy v d from Eq. 85 gves. dv s m m m s m d v v v = ρ ρ ω δ ρ ρ (86) Afer assumpon ha durng seady-sae fron propagaon wh velocy v d he mel hcness s approxmaed as / d m m v δ (87) where m s he hermal dffusvy of he mel meal hen by subsuon Eq. 87 no Eq. 86 and solvng he quadrac equaon and ang only he posve soluon gves. 8.5 = m m s m dv s m dv s m d v a v v v ω ρ ρ ρ ρ ρ ρ (88) In analycal calculaons he value of he evaporaon fron velocy v dv n Eq. 88 can be deermned by he mel s surface emperaure T s (. e. for ron he emperaure dsrbuon of mel surface s shown n Fg..4) and s descrbed by Eq. 64.

168 Fg..4. The emperaure dependence of he mel surface of ron on he absorbed laser nensy for a laser beam radus of ω = :9 x - cm (Sema and Masunawa 997). In analycal calculaons value of drllng velocy and mel eecon velocy can be esmaed by ang values from he graphs shown n Fg..4 and Fg..43. Fg..4. Dependence of he drllng velocy v d and s componens v dm and v dv on he absorbed laser nensy for he case of ron and a laser beam radus of ω = :9 x - cm (Sema and Masunawa 997). 67

169 Fg..43. The dependence of he mel velocy of ron on he absorbed laser nensy for a laser beam radus of ω = :9 x - cm (Sema and Masunawa 997) Energy balance n he molen layer and effec of asss gas pressure The effec of he pressure nduced on he mel surface when an asss gas e s used durng laser drllng s consdered. I can be shown ha for frconless and adabac (senropc) gas flow he oal pressure along a sreamlne s consan and equals o he sum of he sac and dynamc pressures (Fere and Ward 986). The oal pressure s ofen called he sagnaon pressure snce s he sac pressure of he gas f s velocy was senropcally reduced o ero. Snce he hole boom s assumed o be perpendcular o he gas axs Fg..44 and by assumng unform asss gas pressure whn he laser beam and ero ousde he beam he dynamc gas pressure can be negleced. A he nole ex he gas has acceleraed up o he local speed of sound leadng o he crcal sae due o adabac expanson of he γ ( γ ) p c = p γ (89) 68

170 where he asss gas pressure a he nole ex p c s deermned by he asss gas pressure nsde he nole p. For daomc gases such as O he specfc hea rao γ=.4. For smplcy he crcal sae s assumed o be found a he laser-drlled hole enrance bu wh a reduced pressure due o he radal expanson of he gas ouwards from he hole. Esmae of he reduced pressure s made by consderng he geomercal areas of he gas flow enerng he hole cavy and he gas flow ha flows radally ouwards as shown n Fg..44 expressed as A = π (9) eff ω A = ω d nπ (9) where A eff s he effecve area of flow enerng he hole defned by he laser beam radus ω snce ero asss gas pressure ousde he beam was assumed A ω s he cylndrcal area where he radal loss of gas pressure flows as defned by he nole ex dameer d n and nole-worpece dsance n. As a resul of hese wo flows he asss gas pressure p c a he nole ex s reduced o p eff as follows Aeff p eff = pc = A A eff ω f ( p ) (9) where p eff s he effecve sac gas pressure of he asss gas acng n he same drecon as he recol pressure on he mel surface and f(p ) sands for he funconal dependence on he nner gas pressure nsde he nole. The pressure of he asss gas e can be consdered by addng he effecve sac gas pressure o he lef-hand sde of Eq. 9 yeldng ρ mvm pr peff =. (93) 69

171 Clearly can be seen from Eq. 93 ha dependng on he magnude of p eff he use of an asss gas e wll provde an addonal momenum and wll ncrease he mel eec velocy o allow faser melng of a new poron of he sold meal. The followng consders he energy balance n he molen layer exposed o he laser beam and O asss gas when seady-sae mel eecon from he neracon one (.e. hole boom) s reached. In he coordnae sysem relaed o he melng fron he sold s movng owards he mel fron wh velocy v dv mels and hen he mel s parally eeced by he recol pressure radally wh velocy v m and parally evaporaed from he mel surface. The npu power of he molen layer can be gven as n P = I π ω P (94) l abs rp where I abs s he absorbed laser nensy and ω s he laser beam radus (he area of he ncden laser beam A l ) and P rp s he reacve power generaed due o oxdaon when an O asss gas s employed. Fg..44. Schemac of effecve gas flow area enerng hole defned by he laser beam dameer ω and he cylndrcal area of radal loss flow (Fere and Ward 986). 7

172 Forced convecon coolng by asss gas. Many researchers have found ha coolng by he asss or shroud gas can have a sgnfcan effec n laser maerals processng especally for low laser nensy processes such as hgh power dode laser removal of mullayer chlornaed rubber and dode laser weldng of hermocouples. From wor sudyng he laser cung of meals was generally found ha he coolng provded was reasonably small excep for very large flow raes. However lle wor has been conduced o nvesgae he possble effecs of asss gas coolng durng laser drllng. As an approxmaon he case of forced convecon perpendcular o a plae has been consdered as shown n Fg..45. The hea ransfer by forced convecon beween he mel surface a he hole boom and he gas flowng perpendcular o he mel surface s smulaed. The consdered area s defned by he laser beam spo se whereby he hea ransfer beween he hole walls and he asss gas s assumed neglgble. Ths s a reasonable approxmaon snce he man concern s o nvesgae he coolng rae of he mel surface whch drecly affecs he drllng velocy. Frs he Reynolds number for he asss gas flow can be expressed as Re v g ω ρ g = (95) µ where r g s he asss gas densy n g s flow velocy of he asss gas µ s he dynamc vscosy of he gas and he wdh of he mel surface s approxmaed by he laser beam spo se ω. The convecve coolng rae per un area Q c beween he mpngng asss gas and he mel surface s Q = h T T ) (96) c c ( s 7

173 where T s and T are he mel surface and asss gas emperaures respecvely h c s he hea ransfer coeffcen whch can be deermned from h c g n 3 = C Re Pr / c c (97) ω where g and Pr are he hermal conducvy and Prand number of he asss gas respecvely C c and n c are he consans for forced convecon perpendcular o he mel surface plae. Fg..45. Approxmaon of he forced convecon coolng of he mel surface by he asss gas a he hole boom (Sema e al 999). Combnng Eqs 95 o 97 he power los due o coolng by he asss gas P coolng whn he mel surface area can be esmaed o be P coolng n c cc g v ω ρ g g 3 = Pr / ( Ts T ) πω (98) µ I s mporan o menon ha due o he complexy n he evaporaon from surfaces he above forced convecon coolng analyss s reaed as a frs approxmaon. For example f subsanal evaporaon s acheved a Knudsen layer may exs and he coolng effec may be affeced (Sema e al 999). 7

174 An exohermc reacon s generaed when he asss gas conans O whereby he O would chemcally reac wh he mel surface o generae an addonal hea (Adams 97). Snce O asss gas s ofen used n mos praccal laser drllng processes s herefore mporan o consder he conrbuon of he exohermc reacon n he model. Durng he drllng process he elemens whn he meal or alloy would be oxded and hus formng varous oxde phases. In general he oal chemcal equaon for he reacon beween a meal Me and oxygen gas O o form an oxde Me a O b may be wren as ame(b/)o -> Me a O b. (99) Usng X-ray dffracon (XRD) or we chemcal analyses (Ivarson e al. 99) s possble o denfy he prncpal oxde phases formed by analyng he spaer or eeced maeral. Consequenly n order o quanfy addonal hermal npu he hea generaed o ransform each meal elemen o her relevan oxde phases especally n alloys needs o be consdered separaely. Fundamenally he rae of reacon s deermned by he drllng velocy or cung speed as conrols he rae a whch he maeral s beng cu or drlled and hereby subecng he mel o oxdaon. Usng he drllng velocy v d descrbed n Eq. 86 he absolue reacon rae R oe (mol/s) for a parcular meal elemen n consderaon s R oe vdπω ρm = () M r oe where M roe s he relave aomc mass of he meal elemen n consderaon. The numeraor on he rgh-hand sde of Eq. s he rae of mass removal. Consequenly 73

175 he hea generaed due o he oxdaon of a parcular elemen can be gven as (Ivarson e al. 99) P r oe vdπω ρ m = M r oe m oe H ox () where m oe s he esmaed percenage mass of he elemen n consderaon ha s oxded H ox s he reacon hea of oxdaon o form he parcular oxde. For low carbon seel he preferred oxdaon s Fe/O ->FeO and H ox =58.75 J/mol (Barn and Knace 973). The oal hea generaed due o he oxdaon of dfferen elemens for a parcular meal or alloy n consderaon can be descrbed as follows P rp = v πω ρ m d m = M r oe m oe H ox () where m oe s he number of sgnfcan oxde phases ha were formed for a parcular meal or alloy. The oupu power of he molen layer s composed of a conducon erm P cond a convecon erm P conv a vaporaon erm P vap a radaon erm P rad as well as he coolng erm when an asss gas s used as follows: P ou l = P P P P. (3) cond conv rad coolng I has been shown ha he radaon relaed energy ransfer from he mel layer s relavely small and can be negleced (Sema and Masunawa 997). The conducon erm can be esmaed from he sum of forward and radal componens.e. Pcond s forwardtπω s radaltπ ω (4) 74

176 75 where s s he hermal conducvy of he sold meal and he gradens of emperaure are aen a he mel fron along forward and perpendcularly radal o he moon of he rereang sold meal. The forward componen can be gven as ( ) / w v T T c w v T T w T d m s s d s m s forward s π ρ π π = = (5) where s s he hermal dffusvy of he sold meal. The radal componen can be esmaed as (Sema and Masunawa 997) ( ) ω π ω ρ π ω ω π ω s d s m d m s s d m d s m s radal s v v T T c v v T T T. (6) Snce he forward componen s used o pre-hea he sold meal only he radal componen s consdered as losses of energy from he mel layer. Knowng he mel hcness δ m he power spen for convecon can hus be expressed as follows: ( ) ( ) * * d m e m m m m e m m m conv v v L T c v L T c P ρ π ω δ ρ π ω π = (7) where T* s he average emperaure n he mel layer defned as ( ) * m s m T T T T = α (8) where T m s he melng emperaure T s s he surface emperaure of he mel and α s a consan smaller han uny. By ang no accoun of he vaporaon fron velocy v dv he power used for vaporaon s gven as π ω ρ v dv m vap H v P = (9) where he velocy of he evaporaon fron v dv s deermned by Eq. 64 and H v s he laen hea of evaporaon.

177 76 By combnng Eqs. 94 o 9 and equang he absorbed laser nensy I abs o he lef-hand sde of he Eq. 94 gves ( ) ( ) v dv m s d s m d m s s n g g g c m ox oe oe r m d d m s s d m m m m m abs H v w v v T T c T T v w c H m M v v T T c w v v H T c I c ρ ρ µ ρ ω ρ πω ρ ρ = = 3 / * ) ( Pr ) ( () As can be seen from he forgong consderaons drllng velocy and s componens such as vaporaon fron velocy and recol pressure as well as he absorbed laser nensy and s fracons expended for heang melng and conducon and vaporaon losses are expressed hrough Eqs These equaons mplcly depend on he hermophyscal properes of he meal he surface emperaure of he mel as well as he geomercal properes of he laser beam. In addon he possble effecs of usng O asss gas have been consdered. The momenum provded by he asss gas e whch prncpally affecs he mel eec velocy v m s esmaed hrough Eqs. 9 and 93. Usng hermophyscal properes colleced n Table 5 he forced convecon coolng by he asss gas on he mel surface can be esmaed by Eq. 97 and relaed o he energy balance n Eq.. The addonal energy generaed due o he exohermc reacon s expressed n Eq. and fnally consdered n he energy balance n Eq..

178 Sema and Masunawa (997 reached smlar conclusons defnng absorbed laser energy. They also expressed he drllng velocy v d and he mel eecon velocy v m as a funcon of he surface emperaure T s of he mel usng Eq.. Table 5. Thermophyscal properes of O asss gas and gas nole parameers (Sema and Masunawa 997). Equaon gves an mplc dependence of he melng fron velocy and s componen absorbed nensy and he hermo-physcal parameers of he worpece as he funcon of he melng surface emperaure. Therefore by selecng a proper range of he melng surface emperaure he range of he recol pressure and he melng fron can be calculaed..7.. Thermal energy balance concep The energy balance whn he neracon one beween he laser beam and worpece s he ey pon n modelng laser maeral processng especally for laser 77

179 machnng on meals where losses due o hgh reflecance and greaer hermal conducvy are sgnfcan. The laser parameers have o be balanced n meal drllng o he maeral properes le absorpvy and hermal dffusvy. The drllng process s nfluenced by mel expulson absorpon and refracon of laser radaon n he expandng vapor plasma. A bloc dagram for varous processes of maeral removal and her muual nfluence for laser drllng s shown n Fg..46. The energy balance whn he neracon one beween he laser beam and maer s he ey pon n modelng laser eyhole weldng and cung of meals. The error n esmang he relave values of varous componens n he balance equaon can change no only he quanave bu also he qualave characer of he numercal resuls n for example calculaon of he hermal feld nduced n he sample durng laser reamen. Anoher mporan aspec closely relaed o he energy balance problem s he mechansm of he propagaon of he beam-surface nerface no a maeral. Is mporance s deermned by he fac ha correc undersandng of how he drllng he weldng or cung fron advances no he meal bul wll allow one o answer such val quesons as wha s he expeced peneraon deph and where s he orgn of nsables affecng he qualy of he process. 78

180 Transmsson o meal surface Laser beam Plasma defocusng (sheldng) Absorpon by meal surface Reflecon loss Hgh pressure vapor/plasma Meal surface heang aporaon of meal Maeral removed as vapor Thermal dffuson Lquefacon of meal Meal removed as lqud Fg..46. Bloc dagram for he man processes of maeral removal and her muual nfluences durng laser drllng. The hermal balance concep concenraes on couplng beween ncden laser radaon and reversble hermal response of a localed regon on he meal. Ths process s bes descrbed as a consan pressure hea addon phenomenon durng whch a maeral s beng heaed a consan pressure followng a hermal response curve. Durng hs phenomenon emperaure vs. me relaonshp s characered by four prncpal ones. The frs one represens he amoun of consan pressure hea needed o brng he worpece from room emperaure of sold o molng emperaure T m. The second one represens he addon of hea needed o brng he worpece compleely from sold o molen phase. Ths s sohermal hea addon process called fuson. The hrd crcal one s a consan pressure hea addon whch brngs he molen maeral up o s 79

181 bolng emperaure T v. A he bolng pon he maeral undergoes anoher sohermal ransformaon called vaporaon. A hs pon he maeral reaches a gaseous or vapor sae. The oal hea appled o brng he worpece from room o vaporaon emperaure can be wren as Q oal = M c T T ) M H M c ( T T ) M H () p ( m a f p v m v where M s oal mass of maeral n laser drllng one (mllgrams) c p s specfc hea of he maeral H f s laen hea of fuson and H v laen hea of vaporaon Expulson consderaon In laser mcrodrllng maeral s removed as a mxure of mel and vapor. The proporons of whch depend on maeral properes and laser beam nensy. Large amoun of removed maerals s advanageous from drllng effcency pon of vew bu on he oher hand can creae more nsably n he hole formaon. The hcer he layer of mel he less s defned he surface geomery he more he hole geomery flucuaes. The hcness of a layer molen durng a laser pulse neracon wh meals can be calculaed and can be approxmaed as δ = () p where δ s molen layer hcness s he hermal dffusvy p s pulse duraon (> ps o be vald). Analyng Eq. he bes way o ncrease drllng accuracy s by smply shorenng he molen layer hcness by shorenng he laser pulse me. 8

182 The molen maeral s eeced from he hole by he pressure graden developed durng he vaporaon of he poron of he maeral. As a resul more mass s removed compared o vaporaon alone. The hermal balance model addresses expulson concep drecly n he hea ransfer equaons. Snce he eeced maeral has o leave n molen form our assessmen accouns for all maeral eeced o leave when localed drllng one reaches he molen sae. In order for mel dsplacemen o occur he evaporaon recol pressure mus accelerae mel creaed anywhere n he fuson one o a velocy suffcen for o reach he edge of he mel pool durng he poron of laser pulse remanng afer he melng has occurred. Thus an approxmae creron for he esablshmen of mel dsplacemen can be formulaed (Sema el a. 3) e rm = p m (3) v m where e s he me for mel dsplacemen from he mel puddle r m s he mel puddle radus v m s he mel velocy averaged over he laser pulse duraon p and m s he me o he naon of surface melng. Please noe ha f condon n Eq. 3 s sasfed hen boh eyhole weldng and drllng can ae place. The ranson from weldng o drllng requres applcaon of anoher creron whch descrbes wheher surface enson a he mel edge s capable of reanng mel dsplaced from he beam rradaed area and formng a weld pool. Consderng he reflecvy and expulson parameers he orgnal hermal balance equaon (gven by Eq. ) was modfed o he frs order approxmaon no 8

183 ( R) Q oal = M c p ( M ( T exp m T ) M H a ) M H v f ( M exp ) M c p ( T v T ) m (4) where R s reflecvy and M exp s expulson fracon of maeral removed. Boulmer-Leborgne e al. (993) deermned he mass of ablaed maerals m (Cu) for dfferen rradaons and her resuls Fg..47 are n raher good accordance wh he formula proposed by (Boulmer-Leborgne e al. 993) Φ 3/ 4 m = λ (5) 4 3 where Φ s he absorbed laser power densy (W/cm ). I s clear from Fg..47 ha shor laser wavelenghs favor he ablaon process. The average maeral removal rae n he laser drllng process can be obaned by he followng fomula (Zhang and Faghr 999): MR = πρ p s( r ) rdr p (6) wll be compared wh expermanal daa and analyss dsscussed n he expermenal par n Chaper 4. 8

184 Fg..47. Amoun of ablaed maer for dfferen laser rradaons agans absorbed laser energy (Boulmer-Leborgne e al. 993) nrogen 337 nm; XeCI. 38 nm; Nd:YAG. 355 Nd:YAG 53nm: Nd:YAG 64 nm; ArF 93 nm. The comparson of calculaed maeral removal rae for he cases wh and whou conducon hea loss and ha of expermenal daa s shown n Fg..48 (Zhang and Faghr 999). The expermenal maeral removal rae was obaned by scalng mcrographs of sngle sho drlled holes for pulse on me of 7 µs and radus of.54 mm a he Pra and Whney drllng facly (formarly) a Norh Haven CT. I can be seen ha he predced maerals removal rae wh conducon hea loss s slghly lower han ha whou conducon hea loss bu he dfference s less han %. Ths suggess ha he overall effec of conducon hea loss on he maeral removal raes s no sgncan. As can be seen from Fg..48 he maeral removal rae predced by he Zhang and Faghr s (999) model s hgher han ha of expermenal daa for mos cases. The possble cause of he over predcon may nclude uncerany of absorpvy and possble paral laser beam blocage due o plasma whch s no aen no accoun n he 83

185 model. Consderng hese complcaed phenomena he agreemen beween calculaed resuls and he expermenal daa s raher good. Fg..48. Comparson of predced and expermenal maeral removal rae (Zhang and Faghr 999). Smlar expermens were done by osey e al. () and eo e al. (98). More nformaon on hs subec and own expermenal evdence n hs feld can be found n hs Dsseraon n he Secon Threshold of laser mcromachnng The meal surface s heaed by he absorbed laser power up o he melng pon and afer a phase change fnally o he vaporaon emperaure. The refleced radaon from he meal surface can nfluence he laser oupu va opcal feedbac. The hreshold nensy I v for maeral removal s deermned by he losses due o surface reflecon and 84

186 hea conducon no he maeral. The hreshold s a funcon of beam radus ω a he meal surface he processng me p he surface absorpon coeffcen A and he hermal dffusvy of he meal. For a Gaussan nensy dsrbuon he hreshold nensy I v s gven by (Treusch and Herger 986) Tv π I v = (7) Aω arcg 8 / p ω where T v s he vaporaon emperaure and s hermal conducvy of he worpece maeral. A schemac represenaon of he dependences of mel dsplacemen me and melng me as a funcon of pulse energy s shown n Fg..49. Obvously an ncrease n pulse energy resuls n a decrease of he dsplacemen me because for a gven pulse duraon a hgher pulse energy corresponds o a hgher beam nensy a hgher surface emperaure a hgher evaporaon recol pressure and consequenly a hgher mel dsplacemen velocy. Fg..49. Mel dsplacemen me dsp and poron of laser pulse remanng afer begnnng of surface melng τ m as a funcon of pulse energy (Sema el a. 3). 85

187 Sema el a. (3) concluded ha he dependence of dsplacemen me dsp (presened n Fg..49) versus pulse energes exss under he assumpon ha mel always exss on he surface. Of course for energes lower han he melng hreshold E m he par of he curve represenng dsplacemen me has no physcal meanng and s represened by a dashed lne. For pulse energes greaer han he melng hreshold he me when mel exss a he surface τ m ncreases. The mel exsence me ncreases from ero when he pulse energy s equal o he hreshold value E m and asympocally approaches he pulse duraon τ. The dsplacemen hreshold E dsp ndcaes he laser pulse energy when he dsplacemen me becomes equal o he mel exsence me. One can nroduce a drllng hreshold E dr correspondng o a condon when he velocy of mel dsplaced from he mel one s hgh enough so ha he dynamc pressure of he mel flow exceeds he (resranng) surface enson pressure a he mel pool edge. For pulse energes exceedng E dr drllng aes place. For pulse energes hgher han he dsplacemen hreshold E dsp bu less han he drllng hreshold E dr eyhole weldng aes place. Lasly for pulse energes beween he melng hreshold E m and he dsplacemen hreshold E dsp so-called conducon lmed weldng aes place. The poson of he focal plane s nown o have an effec on fnal shape of he hole as well as degree of peneraon. As he laser focusng spo moves up and down he laser maeral neracon area F vares accordng o expresson (Sema e al. 3) λ = ω L F (8) π ω 86

188 where λ L s he laser lgh wavelengh and ω he laser beam radus a he beam was. The ncden power densy s manly deermned by he focus-surface poson as s shown n Eq. 8. The opmal power densy I mach s denfyng expermenally. In laser mcrodrllng usng fber opc cable nowng he machnng hreshold s mos crcal for drllng o occur. Maeral s beng drlled wh a consan focal poson Eq. 8 and he removal sops when he deph hresh s obaned Consderaon of he maerals In hs Dsseraon here were wo vasly dfferen maerals consdered: 34 sanless seel and sngle crysal slcon. In he elecronc ndusry where componens are ofen processed n a clean room envronmen dscharges of meal vapors are no accepable. Durng laser mcrodrllng especally durng mcroweldng evaporaon of alloyng elemens needs o be mnmed. Ths s anoher reason o perform a qualave undersandng of evaporaon maerals (34 sanless seel) under consderaon n laser processng. Loss of alloyng elemens can resul n sgnfcan changes n he mcrosrucure and degradaon of mechancal properes. He e al. (3) and Moon e al. (3) nvesgaed he change n properes of alumnum alloy before and afer weldng usng CO laser wh He gas sheld. They found ha he ensle properes of he welds were nferor o he base meal manly because of magnesum depleon loss of sran hardened srucure and porosy. They found ha he hardness of weld meal was lower 87

189 han he base meal due o magnesum vaporaon. The loss of hardness was conrbued o a reducon n he sold srenghenng effec of magnesum. Durng laser mcromachnng he man consuens of sanless seel vapor are ron manganese chromum and ncel. In hs Dsseraon all properes of he maeral under nvesgaon 34 sanless seel was acqured eher from manufacurer of he maeral or from he leraure and are summared n Table 6. Based on daa obaned from seel s manufacures he 34 sanless seel samples had he followng composon:.83w%mn 8.w%Cr 8.w%N.39w%S.4w%C.3w% P.w% S.6w% N and balance Fe. In calculaon of he laser mcromachnng commonly used maeral properes are: (for energy balance analyss) densy hea capacy specfc hea rao hea conducvy hea dffusvy laen hea melng pon vaporaon pon; (for sress and momenum analyss): vscosy modulus of elascy shears modulus Poson s rao sress-sran consuve relaon. Typcal basc maeral properes of sanless seel and sngle crysal slcon are summared n he Table 6 Fg..5 Fg..5 and were used n calculaons n hs Dsseraon. 88

190 Table 6. Properes of he maerals used n hs Dsseraon (EMIS 988). Nomenclaure Sanless seel Sngle crysal Uns 34 slcon Densy ρ g m Laen hea of vaporaon H v J/g Laen hea of fuson H f J/g Thermal conducvy a 3 C Wm K 5 C Wm K Specfc hea c p.5.7 J g K Melng emperaure T m K aporng emperaure T v 9 68 K Crcal emperaure T c K Thermal conducvy W/m-K Thermal conducvy vs. emperaure 34 SS W/m-K Temperaure K Hea capacy J/g-K 9 Hea capacy vs. emperaure 34 SS J/g-K Temperaure K Thermal dffusvy m/s Thermal dffusvy vs. emperaure 34 SS m /s 7.5E-6 7.E-6 6.5E-6 6.E-6 5.5E-6 5.E-6 4.5E-6 4.E-6 3.5E-6 3.E Temperaure K Densy g/m3 Densy vs. emperaure 34 SS g/m Temperaure K Fg..5. Thermophyscal properes of 34 sanless seel. 89

191 Thermal conducvy W/m-K Thermal conducvy vs. emperaure slcon W/m-K Temperaure K Hea capacy J/g-K Hea capacy vs. emperaure slcon J/g-K Temperaure K Thermal dffusvy m/s Thermal dffusvy vs. emperaure of slcon m /s Densy g/m Densy vs. emperaure of slcon g/m Temperaure K Temperaure K Fg..5. Thermophyscal properes of sngle crysal slcon (EMIS 988) Mahemacal formulaon of he model The mahemacal model mus be derved such ha he parameers are easly handled. Once he model s verfed by expermens can smulae he process and provde nformaon such as hea-affeced one ransen emperaure dsrbuon and coolng raes. Therefore he model can reduce he expermenaon by deermnng he effecs of parcular parameers beforehand. 9

192 Ths secon presens he formulaon of he general governng equaon usng he concep of he dvergence of a ranspor nensy as he ne accumulaon rae of energy per un volume of he medum under consderaon. Predcon of hermal effecs produced by laserbeam scannng he surface of an absorbng sample requres ha hreedmensonal hea ransfer equaon be solved subec o fne se condons of a sample. The geomery consdered n hs Dsseraon s llusraed n Fg..5 whch s fne n dmensons slab rradaed by a laserbeam mpngng on s surface subec o convecve and radave losses. Nex Secon.7.6 consders he resulng boundary condons. Fg..5. Laserbeam mpngng on a fne se sample. Defnng q n as he magnude of he hea flux n he n-drecon (wh n beng x y ) one may nroduce he vecor sum q of he dreconal fluxes ) ) ) q = q q q (9) x y ) ) ) where ( ) are he un vecors along each of he Caresan coordnae drecons. 9

193 Havng defned a vecor represenaon of he energy ranspor nvolved one may wre he expresson for he ne accumulaon rae of hermal energy per un volume as u q Q = () where Q s a volumerc erm accounng for he nernal generaon of hea. The rgh hand sde of Eq. represens he oal rae of change of he elemen s specfc nernal energy. Thus Eq. s a general saemen of he frs law of hermodynamcs. Usng hermodynamc argumens Eq. can be expended furher. If one consders sample under consderaon o be a homogenous connuum composon wh more han one phase han one can unquely deermne he sae of any propery of he connuum usng wo ndependen properes specfc energy and specfc enhalpy one can wre (Nowa 99) du dh um = dv cvdt () v T hm = dp c pdt () p m m T where he specfc heas a consan volume and pressure were defned respecvely as c c c p u = T v (3) h = T. (4) p For solds and ncompressble fluds he specfc volume s assumed consan. If one also neglec pressure changes n he gven process Eqs and 4 can be smplfed o yeld 9

194 du m cvdt (5) dhm = c pdt. (6) Recallng he defnon of specfc enhalpy h = u pv (7) m m one can combne Eqs 4 and o fnd ha c = c c for dv dp. (8 p v = If pressure s allowed o vary Eq. 7 sll holds approxmaely for solds and ncompressble fluds. Tang he relaons n Eqs 5 and 7 and mulplyng by he densy of he medum resuls n expressons n erms relave o he un volume as opposed o he un mass. Usng Eq. 7 hese new volumerc erms can be expressed as du = ρ c( T ) dt (9) and dh = ρ c( T ) dt (3) h = u ρ pv. (3) Inegrang boh sdes of Eq. 9 over proper varables and dfferenang wh respec o me yelds u T = ρ c( T ) dt (3) T where T s he emperaure a he begnnng and T s he emperaure a he end of he nfnesmal me sep d. For he parcular very nfnesmal me nerval d one can assume a consan specfc hea han Eq. 3 smplfes o 93

195 u T = ρ c (33) snce he nal emperaure s consan wh respec o me. Subsung Eq. 33 no Eq. yelds he hea dffuson equaon T q Q = ρ c. (34) Soluon of Eq. 34 for he me-dependen feld T(x y ) requres he use of a consuve equaon relang he emperaure o hea flow. For conducve hear ransfer equaon hs relaon s Fourer s law of conducon. I saes ha he hea flux n a drecon n s proporonal o he emperaure graden n ha drecon. Mahemacally hs s expressed as ) T ) q n n = n (35) n where s hermal conducvy and q n s he hea flux n he n-drecon. Negave sgn s necessary o sasfy he second law of hermodynamcs. In he analyss of he Fourer hea conducon model he hea flux hrough a gven plane s consdered as beng a funcon of he spaal emperaure graden a ha plane. Ths depends upon he assumpon ha he emperaure graden remans almos consan beween wo successve and closely spaced planes. However he dsance beween hese planes s fne herefore error occurs when hgh-order erms whch are negleced become mporan a hgh power laser nenses. The hea flux hrough a gven plane depends on he elecron energy dsrbuon hrough he maeral herefore he maeral canno be consdered as a homogeneous connuum when one s analyng very shor 94

196 pulses (shorer hen a pcosecond) a nermolecular level (dsances less han. µm). Subsung Eq. 35 no Eq. 34 yelds T ( T ) Q = ρ c (36) whch s he general form of he governng dfferenal equaon for sobarc hermal conducon n a homogenous sold or ncompressble flud. In laser mcromachnng he nernal energy generaon Q s commonly hough of as he rae of laser energy absorbed per un volume n he rradaed medum. In case of meals hs absorpon occurs a begnnng n a very hn layer a he surface of he worpece and for many praccal cases can be consdered as a boundary condon o Eq. 36. Ths absorpon process s calculaed by Beer-Lamber s law v. I( ) = ε I () exp( µ ) (37) where I() s he nensy of he ncden radaon a a gven dsance no he absorbng medum from he rradaed surface ε s he surface emssvy of he medum and µ s he absorpon coeffcen of he maeral measure of he absorpon of radaon propagang hrough he medum. Usng he ermnology used n Eq. dfferenang Eq. 53 wh respec o he drecon of propagaon of he laser beam yelds he volumerc exracon rae of energy absorbed by he dfferenal elemen. Snce he accumulaon rae s negave of he exracon rae for a gven volume he necessary volumerc erm for use n Eq. 37 s gven by I ( ) Q = = ε µ I () exp( µ ). (38) 95

197 Usng he Drude-Zener heory (Ylbas 997) Eq. 38 leads o he followng expresson for Q ( x y ) Q( x y ) = A exp( β ) µ I ( x y )exp( µ ) (39) where A s he surface absorpvy β s he pulse parameer and I (x y ) s laser radaon nensy a he maeral surface ( = ). In Eq. 39 absorpon coeffcen of he maeral µ measurng he absorpon of radaon propagang hrough he medum s consdered consan a hs me bu based on a number of publcaons µ should be a funcon of emperaure and axal poson of he rradaed laser beam n respec o he maeral under consderaon. A hs me le us modfy Eq. 39 o be Q( x y ) = A exp( β ) µ ( T ) I ( x y )exp( µ ( T ) ) (4) and funcon µ(t) wll be defned laer on n hs Dsseraon. Incorporang Eq. 39 no Eq. 36 for laser pulse wh a poson-dependen nensy for hs form of pulse npu he Fourer dfferenal Eq. 36 can be rewren as T ρ c = ( T ) Q (4) where ρ s he densy of he maeral of he wor pece c (T ) s he emperaure dependen specfc hea of he maeral (T ) s he emperaure dependen hermal conducvy T T ( x y ) = s he resulng hree-dmensonal me dependen emperaure dsrbuon n he maeral s me T s he nal emperaure and are he spaal Caresan coordnaes. Q ( x y ) x y s he rae a whch hea s suppled o he sold per un me per un volume depends on he laser pulse parameers and 96

198 physcal and opcal properes of maeral rradaed. Noe ha boh A and µ are funcons of emperaure and he wavelengh of he ncden radaon. Sold or lqud evaporaes a any emperaure greaer hen K. The evaporaon raes srongly depend on he surface emperaure T s. Equaon 4 consders hea dffuson no maeral only hrough conducon. Based on expermenal and heorecal evdence evaporaon aes place so one has o consder movng elemens of vapor and lqud nsde he maeral durng laser beam neracon. Consderng hs fac he general governng dfferenal equaon allowng a phase change process can be wren as T ρ ( T ) c( T ) = ( T ) ρ( T ) c( T ) n ( T ) T Q (4) where n s he normal componen of he evaporaon fron or meled fron velocy (recesson velocy). Le us defne velocy n. From leraure (Toarev e al. 995) he nerface moves no he deph of he maeral a he speed defned as c M ) = psa ( T ) (43) ρ π T ( Ts s B s where c s recondensaon facor whch s usually aen as.8 and p sa (T s ) s sauraed vapor pressure defned from Clapeyron-Clausus equaon as ( E ) exp. = a psa Ts ϕ (44) BTs where φ s preexponenal facor and E a s an acvaon energy evaporaon per aom. 97

199 Noe ha Eq. 43 s a pressure dependen funcon whch has o be calculaed smulaneously wh emperaure dependen velocy. Thus le us derve a velocy as a funcon of emperaure and he laen hea of phase ranson whch s also emperaure dependen. The rae of change of laen hea wh emperaure can be expressed as (Ylbas e al. 996) H T = H T ( c pv c pl ) v v H v l vv T p vl T p (45 where c pv and c pl are specfc heas a consan pressure for vapor and lqud saes respecvely and v v and v l are specfc volumes for vapor and lqud saes. One has o reale ha negraon of he laen hea over he emperaure rangng from o T c s dffcul because we do no have enough nformaon abou values for he laen hea especally a exreme value of he range under consderaon. I s safe o assume (Ylbas 996) ha lle naccuracy s nvolved n ang he room emperaure laen hea as he laen hea a absolue ero because by nowng ha vv vl v v v l and T T p p. e. he specfc volume of gas v v s much graer han he condensed lqud v l and s rae of change wh emperaure a consan pressure s correspondngly greaer. Thus c p s exremely small for emperaure up o a room emperaure T a. Accordng o Maxwell s law he funcon of velocy dsrbuon of molecules can be defned as (Tabor 99) 98

200 f ( n ) d n = M π T B s Mn exp π BT s d n (46) where n s he velocy n he drecon normal o he surface and he oher parameers were defned earler. Usng vernacular erms funcon f( n )d n s a rao of number of aoms wh velocy n o n d n per un volume o he oal number of aoms per un volume. Only hose molecules whose velocy s greaer han mn obaned from equaon gven by (Tabor 99) M mn = H ( T ) (47) wll escape from he reanng poenal where mn les n he drecon. If n s he number of aoms per un volume hen he number of aoms wh veloces n o n d n per un volume s n f n ) d and he number of aoms wh hese veloces passng a ( n un area per un me s n f ( ) d. n n n Wh he assumpon made n Chaper.3 ha all he aoms for whch n mn do no reurn o her equlbrum poson are assumed o be evaporaed. If N G s he number of aoms evaporaed per un me per un area hen N G = [ n f ( n ) n ] mn = n M π T B s mn d n Mn exp π BT s n d n. (48) Afer negraon and subsuon of Eq. 47 no Eq. 48 we oban 99

201 N G = n M π T B s H ( T ) exp BT s n. (49) If aoms are equally spaced whn he lace a surface layer would conss of n /3 wh an evaporaon me n /3 N G. The average velocy of he surface n would be N = G n / 3 / 3 n n (5) n BT ( ) exp H T = πm BT. (5) Consderng Eq. 5 he general governng dfferenal equaon defned n Eq. 5 wh phase change processes could be wren as T BT H ( T ) ρ ( T ) c( T ) = ( T ) ρ( T ) c( T ) exp T Q. (5) πm BT Noe ha laen hea H(T) n Eq. 4 based on some heorecal and expermenal evdence s emperaure dependen and he fnal form of s gong o be descrbed n Secon In order o solve Eq. 4 approprae boundary condons should be appled. Secon.7.6 s dscussng formulaon of he approprae boundary condons Boundary condons Boundary condons are mporan par of a model. They nfluence he programmng and calculaon resuls grealy. Specfyng he suable boundary condons s he bass for successful compuaon. Three nds of boundary condons

202 are ypcally encounered n hea conducon analyss. These are: ) gven he boundary emperaure T ) gven he boundary hea flux q and ) a boundary hea flux balance relaon. If he boundary emperaure s gven here s no parcular dffculy n modelng one needs only o specfy he value of he boundary grds o he specfed emperaure. The magnude of hea flux due o convecon o he amben from he sample surface s expressed usng Newonan law of coolng q conv = h T T ) (53) c ( s amb where h c s convecon hea ransfer coeffcen T s s he surface emperaure and T amb s he emperaure of surroundng. In order o deermne h c he characersc lengh L of he worpece should be Area L = (54) Ob where Area s he area of he surface and Ob s he permeer of he worpece. Then he Nussel number for he horonal plae s (Bean 993) N u.5.7 Ra = (55) where he Raylegh number Ra s R gβ 3 = L ( T γ a T amb ). (56) In Eq. 56 g s he gravaonal acceleraon β s he coeffcen of volumerc hermal expanson s he hermal dffusvy γ s he nemac vscosy and oher

203 parameers are as defned for Eq. 53. The convecve hea ransfer coeffcen h c can be calculaed as ar hc = N u (57) L where ar s he hermal conducvy of he ar surroundng he worpece. Radaon o he amben s expressed usng he relaon q rad 4 4 = σε ( T T ) (58) s amb where q rad s he magnude of radaon flux ε s he emssvy of he maeral and σ s he Sefan-Bolmann consan. Noe ha convecon s a lnear funcon of emperaure whereas radaon s a nonlnear due o s dependence on he dfference of he fourh powers of he surface and amben emperaure. The magnude of convecon and conducon n he overall ranspor of hea can be evaluaed from he value of he Pecle number P e whch s defned by P e u c p Lr = (59) where u s velocy L r s he characersc lengh aen as he pool radus a he op surface of he weld pool and he oher parameers were defned earler. Hea ranspored by a combnaon of convecon and conducon mechansm s observed n he weld pool n laser mcroweldng applcaons. When Pecle number s less han he hea ranspor whn he weld pool occurs prmary by conducon. When Pecle number s much hgher han hen he convecve hea ranspor s he man mechansm of hea ransfer n he maeral.

204 If one s usng sheldng gas n laser mcromachnng hen hea ransfer coeffcen h c requred n Eq. 57 s calculaed from Maumder and Seel (979) and Gordon and Cobonpue (96) for case of a vercally mpngng e.3.35 hc = 3Re Pr g (6) B where B s he e plae dsance Re s he Reynolds number a e ex Pr s he Prand number for gas and g s hermal conducvy of he gas. Performng an energy balance on a boundary where boh convecon and radaon losses occur one can relae he flux conduced o he nerface o he convecon and radaon losses by he expresson q = q q. (6) n conv rad Subsung Eqs and 58 no 6 one obans T n s = h c ( T s T 4 4 ) σε ( T T ) (6) s where he erm on he lef-hand sde of he Eq. 6 represens he magnude of he hea conduced normal o he boundary surface. Equaon 6 should be appled o all exposed boundary of he fne sample under consderaon. Seng he rgh-hand sde of Eq. 6 o ero accommodaes nsulaed boundares. In laser mcrodrllng Eq. 6 should be appled n he regon of he newly vapored maeral (hole area) and an approprae hea ransfer coeffcen should be used. Snce he queson of modelng of he mcrodrllng phenomena s assocaed wh creang he hole formaon has no ye been solved hen o chose an approprae hea 3

205 ransfer coeffcen s an educaed guess and open o debae. More deals on he hole formaon subec s descrbed n Secon Based on Chaper n laser mcrodrllng phase ransons occur. For he case when phase change ranson aes place he boundary condon a he lqud-vapor nerface s T ρ vdvhv = ( R) I (63) T where s hermal conducvy of he sold or lqud phase s he emperaure graden a he surface along he normal (- axs) ρ s he densy of he sold or lqud phase v dv s he componen of boundary velocy due o evaporaon R s he reflecvy for he laser wavelengh and I s he nensy of he laser beam a he surface. For surface emperaures less han approxmaely half of he crcal emperaure he energy of evaporaon per aom U can be assumed o be consan. Then he componen of he boundary velocy due o evaporaon v dv was defned by Eq. 5 and also defned n smlar fashon by Nedrg and Bosonglo (996) v dv U = exp (64) Ts where s a coeffcen of he order of magnude of he sound velocy T s s he surface emperaure and U s he energy of evaporaon per aom defned as 4

206 U M H v = (65) N a B where H v s he laen hea of vaporaon (per un mass) B s Bolmann s consan M s he aomc mass and N a s Avogadro s number. elocy n defned n Eq. 5 s equal v dv. In hs Dsseraon he laen hea of vaporaon H v s gong o be defned laer as a funcon of emperaure (only for hgher emperaures) n Chaper The classc Sefan boundary condon s appled o he sold-lqud boundary (melng fron = m ) T s l ρ H f vm = s l (66) = m T = m where H f s he laen hea of fuson v m s he melng fron velocy and subscrps s and l represen sold and lqud respecvely. The Sephan boundary condon assumes an nsan ranson from sold o lqud a he melng emperaure T m and does no allow superheang a he melng fron. The approxmaon s adequae for he slow veloces of mel fron propagaon ypcal of laser weldng and drllng where he melng necs can be dsregarded. To undersand how he boundary condons were defned le us frs comprehend maeral removal process ogeher wh energy ranspor n mulple phase ransons whch s gong o be dscus n he Secon.7.. To esablsh effecs of convecve hea ransfer coeffcen due o evaporaon recol generaed mel flow Sema e al. (999) performed wo smulaons. In he frs one recol pressure and relaed mel flow were dsregarded and n he second case recol pressure and mel flow were accouned for. Those observaons hey suppored by 5

207 calculang he mel surface emperaures a he axs of he laser beam for dfferen values of absorbed nensy I presened n Fg..53. When recol pressure and mel flow were ncluded he seady-sae values of emperaure were reached faser and he maxmum emperaure were lower and he coolng raes much hgher han for he case where recol pressure and flow were negleced. Analyng Fg..53 one can calculae ha gnorng recol pressure and assocaed convecve hea ransfer resuls n -5% error n calculaons of surface emperaure n he cener of he laser beam. Fg..53. Calculaed emperaure of ron surface a he beam axs for he cases whou (op curves) and wh (boom curves) mel flow for he dfferen maxmum absorbed nensy values and dfferen laser pulse duraons: 3 µs (.5 MW cm ) 7 µs ( MW cm ) and 5 µs (5 MW cm ) (Sema e al. 999). 3. Soluons of he governng equaon Ths Chaper consders he soluon of he general governng equaon subec o s relaed boundary condons by means of exac and numercal mehods. Ths chaper consss of a dscusson of he approxmae soluon of he problem usng mahemacal 6

208 ools and s dvded no four pars. Frs four secons Secon 3.. Secon 3.. Secon 3..3 and Secon 3..4 presen an analycal mehod of he soluon of he general governng equaon usng he Fourer heory. Secon 3..5 s consderng neracon of laser energy wh maerals usng very shor laser pulses and nroduces elecron-phonon heory approach o solve he hea ransfer problem of he neracon of ulrashor pulses wh he maer. Secon 3. consss of a dscusson of he approxmae soluon of he problem usng he fne dfference mehod (FDM) and fne elemen mehod (FEM) and presens he compuer soluons developed n hs Dsseraon. 3.. Analycal mehods There have been many analycal soluon publshed for varous specal cases of he Fourer hea ransfer equaon. Secon 3.. llusraes analycal soluon for heang case whou phase change by laser defned as a sep funcon (consan heang). Nex Secon 3.. consders analycal soluon o he governng hea ransfer equaon for laser pulse wh a me-dependen Gaussan pulse heang. Secon 3..3 consders analycal soluon of hea ransfer equaon wh me dependen Gaussan laser pulse heang wh convecve boundary condons. Secon 3..4 descrbes heang analyss wh me dependen pulse nensy and where evaporaon s consdered as he exclusve phenomenon ang place durng he ablaon process. Secon 3..5 presens he heang analyss wh pulsed laser heang process by consderng boh Fourer conducon and elecron-phonon nec heory approaches. 7

209 3... Analycal soluon of hea ransfer equaon wh spaal dependen laser pulse heang Ths Secon presens analyss of he conducon heang process nroduced for a praccal Nd:YAG Gaussan laser pulse wh a poson-dependen nensy. An analycal soluon o he problem s obaned wh approprae boundary condons. When he laser nensy s raher low no phase ranson occurs and he only effec of laser absorpon s heang of he maeral. In meals he laser radaon s absorbed by free elecron and he energy ransfer n meals s also due o elecron hea conducon. The emperaure feld s descrbed by he sandard Fourer hea conducon and for laser pulse wh a poson-dependen nensy for hs form of pulse npu he Fourer dfferenal equaon s descrbed by Eq. 4. Q ( x y ) n hs equaon s defned by he Drude-Zener heory whch leads o he followng expresson (Ylbas e al. 996) Q( x y ) = A exp( β ) µ Imax ( x y )exp( µ ) (67) where A s he surface absorpvy β s he me pulse parameer µ s he absorpon coeffcen of he maeral and I (x y ) s laser radaon nensy a he maeral surface ( = ). Accordng o Spars (975) µ s ndependen of emperaure whle surface absorpon coeffcen of he maeral A = R where R s he surface reflecvy s lnear funcon of he surface emperaure A = A ( ) A T T. (68) where A s he surface absorpvy a room emperaure T and for mos of engneerng maerals he absorpon facor A s almos uny for Nd:YAG laser wavelengh. Ths 8

210 emperaure dependence of surface absorpvy resuls from he fac ha A s proporonal o he elecron-phonon collson frequency whch n urn s proporonal o he crysal lace emperaure. For meals and ceran applcaons a emperaures above he Debye emperaure can be assumed ha (T ) and c(t ) do no change dramacally wh emperaure. Therefore assumng consan specfc hea and hermal conducvy for a parcular me nerval Eq. 4 can be smplfed o T T Imax ( x y )exp( ) = c µ µ ρ (69) In many praccal cases he ransverse dmensons of laser focusng spo are large compared o he hcness of he heaed layer and hea conducon problem. Equaon 69 can be consdered one-dmensonal and can be solved usng sandard mehods Carslow and Jaeger (969). I s unnecessary o solve for he complee pulse snce he complee soluon may be obaned by summaon of he soluons for he ndvdual pars of me exponenal hen equaon s lnear. Rearrangemen of Eq. 69 gves T µ I max exp T [ µ ] = (7) where hermal dffusvy = wh boundary condons ρ c T x= = (7) T ( ) = (7) 9

211 T ( ) =. (73) Laplace ransform can be used o solve ceran ype of paral dfferenal equaons wh wo or more ndependen varables. To solve Eq. 7 le us show how o solve n deal dfferenal equaon usng Laplace ransformaon of one dmensonal hea equaon T = T. (74) The physcal model of Eq. 74 s a sem-nfne slab of meal wh a plane face on whch he orgn of he -axs s locaed wh he posve half of he axs dreced no slab. Ths suaon s llusraed n Fg. 3.. The approach wll be o ae Laplace ransform of he dependen varable T() n he hea equaon wh respec o he me as a resul of whch an ordnary dfferenal equaon wh as s ndependen varable wll be obaned for he ransformed varable ha wll hen depend on boh he Laplace ransform varables s and. Afer he ordnary dfferenal equaon has been solved for he ransformed varable he nverse Laplace - ransform ( ) wll be used o recover he me varaon and so o arrve a he requred soluon as a funcon of and. Please noe ha f he Laplace ransform s appled o he ndependen varable n he funcon of wo varables T( ) he varable wll behave le a consan. Consequenly he rules for ransformng dervaves of funcons of a sngle ndependen varable also apply o a funcon of wo ndependen varables.

212 Fg. 3.. Sem-nfne meal slab. Usng he noaon [ ] ) ( ) ( T s T = o denoe he Laplace ransform of T( ) wh respec o he me. The formula for he ransform of dervave s [ ]. ) ( ) ( ) ( T s st T = (75) To proceed furher we mus now use he condon ha a me = he maeral of he slab s a ero emperaure so T()= as a resul of whch. ) ( ) ( s T s T = (76) Nex snce s regarded as a consan one has. ) ( ) ( s T T = (77) Usng Eq. 77 when ang he Laplace ransform of he hea equaon wh respec o and usng he lneary propery of he ransform one obans ) ( ) ( = d s T d s T s (78) where now one can use an ordnary dervave wh respec o so can be consdered o be he only ndependen varable. Therefore Eq. 78 can be rewren as ) ( ) ( = s T s d s T d (79)

213 Eq. 79 has he general soluon of s s T ( s) = C exp D exp. (8) For s C = (emperaure has o be fne for > and > ) so he Laplace ransform of emperaure s seen o be gven by s T ( s) = D exp. (8) To deermne D we can use boundary condon on he plane face of he slab ha requres T()=T from whch follows ha [ ( ) ]. ransform of he soluon wh respec o he me s T T = Thus he Laplace s T s T ( s) = exp s. (8) To recover he me varaon from Eq. 8 s necessary o fnd nverse of - Laplace funcon ( ).e. [ ( s) ] T ( ) - T =. (83) Performng mahemacal manpulaons or usng reference boo wh Laplace ransforms funcons one can fnd ha T ( ) = T erfc (84) where he error funcon erf s defned as erf ( ) = π s e u du (85)

214 and he complemenary error funcon erfc s defned by Carslaw and Jaeger (969) as erfc( ) = erf ( ) (86) where s he ndependen varable and u s he dummy varable. Connung he forgong mahemacal calculaons he Laplace ransformaon of Eq. 7 wh respec o and subsuon of boundary condon Eq. 7 and yeld [ ] T I maxµ exp g T = µ s (87) s where g = and s he ransform varable wh T( s) whch has a complmenary and parcular soluon [ µ ] I maxµ exp T = C exp( g) D exp( g) ( s)( µ g ) (88) where C and D are arbrary consans. Subsuon of boundary Eq. 7 and Eq. 73 no Eq. 88 gves he followng consans: I maxµ D = and C =. g( s)( µ g ) (89) Therefore he complex soluon n he ransform plane s [ g ] exp[ µ ] I maxµ µ exp T =. ( ) ( ) ( ) (9) s g g µ g µ The problem now s o nver he soluon of Eq. 9 whch s a produc of wo s- funcons. There are wo ways o do hs. The frs s a convoluon negral mehod and he second one s o enal expanson of he funcons no paral fracons. Usng he second mehod he full soluon obaned by nverse Laplace ransformaon of Eq. 9 s 3

215 4 [ ]} ) exp( ) exp( ) exp( ) exp( 4 exp 4 ) ( max erfc erfc erfc I T = µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ π µ µ (9) where ) ( ) ( = = u du e erf erfc π defned by Eqs 85and 86 and nowng ha ) ( ) exp( ) ( = = erfc d erfc erfc τ ξ ξ (9) he fnal soluon of Eq. 9 s [ ] [ ] = erfc I erfc I I erfc I T µ µ µ µ µ µ µ µ µ µ exp exp ) exp( ) ( max max max max. (93) Equaon 93 gves he emperaure profle nsde he maeral for a gven laser beam power nensy and s shown n Fg. 3..

216 emperaure ºK deph µm me µsec Fg. 3.. Temperaure dsrbuon nsde a maeral. I should be noed ha as he me ends o nfny n Eq. 93.e. lm[ T ( )] = (94) no seady sae soluon exss for he emperaure dsrbuon. In he maory of calculaons regardng he laser heang of solds surfaces he emperaure dependence of he surface reflecvy usually s negleced. If we se A= growh of he emperaure a he surface n he cener of laserbeam wh me may be obaned by seng = no Eq. 93.e. I max µ T ( ) = exp( µ ) erfc( µ ) µ π (95) Graphcal represenaon of Eq. 95 s shown n Fg

217 emperaure ºK deph µm me µsec Fg Temperaure dsrbuon on he surface of he maeral. Dsrbuon n Fg. 3.3 represens he surface emperaure of he worpece durng he laser neracon and roughly one can esmae f here s gong o be a drllng or melng on ha surface. Wrng n MahCAD paramerc program as a funcon of hermal properes for dfferen maerals (hermal conducvy of lead =9.66 c=.5; anum =.5 c=.78 and 34 sanless seel =4.84. c=.48. Typcal resul of such a paramerc sudy s llusraed n Fg Usng very complcaed funcon programmng n MahCAD s no always easy. Thus o obaned smlar resul one can smply subsue values of hermo-physcal parameers for sold frs hen for lqud and by dsplayng hem on wo dfferen graphs compare resuls. Ths mehod was used on Eq. because equaon s very complex. 6

218 anum 34 SST lead emperaure ºK deph µm me µsec Fg Paramerc sudy of hermal properes. Examnng furher Eq. 93 assumng maxmum absorpon coeffcen le us say ha µ goes o nfny one obans he followng resul I T ) = ( max erfc. (96) Graphcal represenaon of Eq. 96 s shown n Fg

219 8 Fg Temperaure dsrbuon of he maeral wh respec o me. Dfferenaon of Eq. 93 wh respec o gves he emperaure graden nsde he maeral.e.. ) exp( ) exp( ) exp( ) ( max max max max = erfc I erfc I erfc I I T d d µ µ µ µ µ µ µ (97) I s evden ha dt( )/d Fg. 3.6 wll only be ero a he surface.e. maxmum emperaure wll occur a he surface. emperaure ºK me µsec deph µm

220 dt/d ºK/µm deph µm me µsec Fg Temperaure graden dsrbuon nsde he maeral wh respec o me. The varaon of dt/d wh dsance.e. (deph expressed n µm ) as a funcon of ree dfferen me nsances s shown n Fg As llusraed n Fg. 3.7 he slope of he curves decreases reachng he mnmum and hen ncreases o aan almos ero as he emperaure profle becomes almos asympoc wh. In hs case he behavor of dt/dx wh may be dvded no hree regons whch are ndcaed n Fg In he frs regon he hea gan due o laser rradaon domnaes he conducon losses;.e. he nernal energy ncrease s consderably hgh as compared o conducon losses. In he second regon he slope has a = mnmum value; n hs case he energy gan due o ncden laser beam balances he conducon losses.e. he nernal energy of he maeral remans almos consan. In hs case he dsance correspondng o hs pon may be defned as he equlbrum dsance () eq (Fg. 3.8) and dt/d becomes (dt/d) mn. In he hrd regon he slope ncreases o reach almos ero. In hs regon conducon 9

221 losses are domnan and he energy gan due o he exernal feld s nsgnfcan.e. he nernal energy decreases as he dsance ncreases. deph d µm dt/d ºK/µm ( 9 T_Grad) d 67 ( 9 T_Grad) d 5 ( 9 T_Grad) d I II III 3.. Fg Temperaure graden dsrbuon nsde he maeral for dfferen mes. araon of he equlbrum dsance shown n Fg. 3.8 can be dmensonled by m ( ) C( µ ) µ = (98) eq q and s expeced ha ncrease n heang me ncreases he dmensonless equlbrum dsance whch n urn ncreases he dmensonless equlbrum emperaure whch can be defned as T ( ) I max / µ eq = C( µ ) m eq. (99) where C s he consan and m s he power. In oher words he equlbrum emperaure s defned as he emperaure where dt/d s mnmum.

222 deph d. µm ºK/µm.5 dt/d ( 9 T_Grad) d 7..5 Fg Equlbrum dsance. Based on research done by Ylbas and Sam (997) he relaonshp beween equlbrum emperaure and equlbrum dsance on he logarhmc scale s lnear for all maerals and pulse lenghs. If he pulse lengh s approxmaely -9 s he analyss of he hea ransfer process usng he Fourer equaon becomes nvald. In hs case he heang process would be non-equlbrum. In sem-nfne medum as shown n Fg. 3. a T= wh no hea flow a = and wh he hea source defned n Eq. 37 soluon o hs same problem s I max T ( ) = erfc µ I max exp( µ ) µ I max exp µ I max exp µ [ µ µ ] erfc µ [ µ µ ] erfc µ. ()

223 and s llusraed n Fg Soluon n Eq. s dencal o Spars s (976) soluon for fne hcness slab l >> δ. Le us consder varaon of he emperaure dsrbuon wh respec o he dfference from he maeral properes varaons wh emperaure. Analycally one can deermne ha based on exreme maeral properes for maxmum and mnmum values of hermal properes nown from he leraure. Thermal conducvy vares beween 4.9 and 6. for amben emperaure T a and 3 for vaporaon emperaure T v hea capacy c p vares from 5 o 84 and densy ρ vares from 787 o 8 g/m 3. Dscrepances relaed o he hermo physcal properes of he maeral could be deermned by subsung he lowes and he hghes values o soluon gven by Eq.. Examnng graphs n Fg. 3.9 and Fg. 3. one can conclude ha neglecng emperaure dependences of he maeral may nvoe an esmaed error roughly abou 5%.

224 deph µm emperaure K T d T d T d T d 5 T v me µsec a) b) deph m Fg Temperaure dsrubuon of laser rradaed maeral wh he lowes values of hermo-physcal parameers a) emperaure as a funcon of deph and me b) empearure or melng of he maeral wh deph calculaed usng Eq.. d T d 5 emperaure K T d 875 T d T d 5 T v deph µm me µsec deph m a) b) Fg. 3.. Temperaure dsrubuon of laser rradaed maeral wh he hghes values of hermo-physcal parameers a) emperaure as a funcon of deph and me b) emperaure or melng of he maeral wh deph. 375 d 6 Based on Ansmov and Khohlov (995) rae of coolng afer he end of he pulse can be esmaed by Eq. and graphcally represened as n Fg

225 4. ) ( ) ( max = erfc erfc erfc erfc AI mpc p p µ µ () Fg. 3.. Rae of coolng of he worpece afer he end of laser pulse. deph µm me µsec mp ºK/µm

226 3... Analycal soluon of hea ransfer equaon wh me dependen Gaussan laser pulse heang Ths Secon presens me-unseady analyss of he conducon lmed heang process nroduced for a praccal Nd:YAG laser pulse wh a me-dependen nensy. An analycal soluon o he problem s obaned wh approprae boundary condons. The oupu from a pulsed Nd:YAG laser s descrbed by approxmang he form of he rue oupu by he subracon of wo exponenal funcons. Ths analycal form s gven by I [ exp( ) exp( )]. = I max γ β () For meals and ceran applcaons a emperaures above he Debye emperaure can be assumed ha (T ) and c(t ) do no change wh emperaure. Therefore assumng consan specfc hea and hermal conducvy for a parcular me nerval Eq. 4 can be smplfed o T exp T β max (3) [ ( γ ) ] µ I ( x y ) exp( µ ) = ρ c. I s unnecessary o solve for he complee pulse snce he complee soluon may be obaned by summaon of he soluons for he ndvdual pars of me exponenal hen equaon s lnear. Rearrangemen of Eq. 3 gves T µ I max exp( µ ) exp T [ ( β ) exp( µ ) ] = (4) wh boundary condons T x= = (5) 5

227 T ( ) = (6) T ( ) =. (7) Laplace ransformaon of Eq. 4 wh respec o and subsuon of boundary condon from Eq. 6 gves T g T = I max µ exp ( s β ) [ µ ] (8) s wh T = ( s) g = and s he ransform varable whch has a complmenary and parcular soluon: [ µ ] I maxµ exp T = Aexp( g) B exp( g) ( s β )( µ g ) (9) where A and B are arbrary consans. Subsuon of boundary Eqs 5 no Eq. 7 gves hese consans as B = I maxµ and A = herefore he complex soluon n he ransform plane g( p β )( µ g ) s [ g ] exp[ µ ] I maxµ µ exp T =. ( ) ( ) ( ) () p β g g µ g µ The problem now s o nver hs soluon whch s a produc of wo p-funcons. There are wo ways o do hs. The frs s a convoluon negral mehod and he second mehod s smpler enalng expanson of he funcons no paral fracons. Usng he second mehod he full soluon obaned by nverse Laplace ransformaon of Eq. s 6

228 7 ( ) ( ) } ) exp( ) exp( ) exp( exp ) exp( ) exp( exp exp ) exp( ) ( ) ( max erfc erfc erfc erfc I T = β µ µ α µ µ µ µ αµ β β β β β β α µ µ β µ. () Usng he relaonshp erfc(-) = -erfc() gves he soluon ( ) } )] ( exp[ exp ) exp( ) exp( exp exp ) exp( ) ( ) ( max = erfc erfc erfc erfc I T µ β µ µ µ µ αµ β β β β β β µ µ β µ () The emperaure dsrbuon calculaed n Eq. s llusraed n Fg. 3..

229 deph µm me µsec emperaure ºK a) b) TmpH d TmpH d TmpH d TmpH d 5 T v Fg. 3.. a) Temperaure dsrbuon nsde he maeral (-axs) wh respec o me: b) emperaure vs. deph of he worpece a few dfferen me form he begnng of he pulse wh he hghes values of hermo-physcal parameers calculaed usng Eq.. d 6 deph µm 8 emperaure ºK TmpH d TmpH d TmpH d TmpH d 5 T v 6 4 deph µm me µsec a) b) 5 5 d 6 deph µm Fg a) Temperaure dsrbuon nsde he maeral (-axs) wh respec o me: b) emperaure vs. deph of he worpece a few dfferen me form he begnng of he pulse wh he lowes values of hermo-physcal parameers calculaed usng Eq.. The me a whch he maxmum value of he pulse occurs can be obaned by dfferenang Eq. wh respec o for he pulse. Mahemacally speang one has 8

230 o fnd n he condon ha he frs dervave of hs funcon should be equal o ero whch gves he condon for he maxmum me max or γ ln β max = (3) γ β γ ln β β max =. (4) γ β Inseron of Eq. 4 no Eq. gves he maxmum emperaure when he maxmum pulse occurs. Solvng Eqs and 4 gves he condon ha he maxmum emperaure may concdes wh he maxmum pulse amplude.e. a hs pon β =.8 and γ / β =.596. max The me a whch he maxmum emperaure occurs s a funcon of pulse parameers provded ha he pulse rse mes are much greaer han he equlbrum me C / µ descrbed n Secon 3... The surface emperaure depends on absorpon deph n he nal sages of he pulse and ably o absorb he laser depends on he shape of he pulse Ylbas e al The nowledge of pulse parameers β and γ allows a leas heorecally o adus he pulse for maxmum desred effec based on deducons from he physcs of lasng sysem. 9

231 3..3. Analycal soluon of hea ransfer equaon wh me dependen Gaussan laser pulse heang wh convecve boundary condons Ths Secon presens me-unseady analyss of he conducon lmed heang process nroduced for a praccal Nd-YAG laser pulse wh a me-dependen nensy. An analycal soluon o he problem s obaned wh approprae boundary condons. The oupu from a pulsed Nd-YAG laser s descrbed by approxmang he form of he rue oupu by he subracon of wo exponenal funcons. Ths analycal form s gven by I max [ exp( ) exp( ) ]. = AI β γ (5) For meals and ceran applcaons a emperaures above he Debye emperaure can be assumed ha (T ) and c(t ) do no change wh emperaure. Therefore assumng consan specfc hea and hermal conducvy for a parcular me nerval Eq. 4 can be smplfed o T T ρ c = exp max µ [ ( β ) ] exp[ γ] A µ I ( x y ) exp( ). I s unnecessary o solve for he complee pulse snce he complee soluon may be obaned by summaon of he soluons for he ndvdual pars of me exponenal hen equaon s lnear. Rearrangemen of Eq. 6 gves (6) T µ I A max exp T [ β ] exp[ γ ] exp[ µ ) ] = (7) wh boundary condons T x= = h ( T ( ) T ) (8) 3

232 T ( ) = (9) T ( ) =. () Complee soluon can be obaned by subracon of soluons for he ndvdual pars of he me exponenal pulse. I should be noed ha for he soluon of a complee pulse he amben emperaure s consdered as ero (T =). Hence he hea ransfer for he half-pulse becomes T µ I A max exp T [ ( β µ ) ] =. () The soluon of Eq. can be obaned hrough he Laplace ransformaon mehod wh respec o T AI max µ exp ( p β ) [ µ ] = [ pt T ( ) ] () wh T =( p) and T()= where p he ransform varable whch has a complmenary and parcular soluon. The full soluon obaned by nverse Laplace ransformaon of Eq. s 3

233 3 ( ) ( ) ( ) ( ) ) ) exp( exp( exp exp ) exp( ) ) exp( exp( exp exp ) exp( exp exp ) exp( ) ( = erfc w erfc w w w a a w erfc w w w l erfc l erfc l erfc l erfc l a T µ β αµ µ β β µ µ β µ µ αµ µ β β β β β β µ (3) where ) ( max h I a µ µ = (4) max I a µ = (5) 3 T h a = (6) The emperaure dsrbuon calculaed n Eq. 3 s llusraed n Fg. 3.4.

234 emperaure ºK deph µm me µsec Fg a) Temperaure dsrbuon nsde he maeral (-axs) wh respec o me: b) emperaure vs. deph of he worpece a few dfferen me form he begnng of he pulse wh he hghes values of hermo-physcal parameers calculaed usng Eq emperaure ºK TY d77 TY d TY d 7 TY d47 T v emperaure ºK TY TY TY 4 T v d 6 deph µm τ 6 me µsec Fg a) Temperaure dsrbuon nsde he maeral (-axs) for few dfferen mes; b) ransen dsrubuon of he worpece a few dfferen dephs wh he hghes values of hermo-physcal parameers calculaed usng Eq

235 emperaure ºK deph µm me µsec Fg Temperaure dsrbuon nsde he maeral (-axs) wh respec o me wh he lowes values of hermo-physcal parameers calculaed usng Eq emperaure ºK TY d77 6 TY d TY d 7 TY d47 T v 4 emperaure ºK TY TY TY 4 T v d 6 deph µm τ 6 me µsec Fg a) Temperaure dsrbuon nsde he maeral (-axs) for few dfferen mes; b) ransen dsrubuon of he worpece a few dfferen dephs wh he lowes values of hermo-physcal parameers calculaed usng Eq

236 3..4. Analycal soluon of governng hea ransfer equaon consderng ransfer evaporave case Ths Secon presens analyss of he heang process. In he heang analyss evaporaon s consdered as he exclusve phenomenon ang place durng he ablaon process. Based on forgong dscusson melng whou vaporaon can be produced only n very narrow range of laser parameers. Usually boh phase ransons (melng and vaporaon) occurs almos smulaneously n sandard laser applcaons. Noe ha he laen hea of vaporaon s larger hen ha of fuson by he facor of -5 mes. Thus evaporaon plays he mos mporan par n he energy balance. To smplfy our dscusson we wll assume ha he vaporaon occurs n vacuum and he vapor does no absorb he laser energy. When he vapor does no absorb he laser lgh he process n he condensed and gaseous phases can be consdered separaely. An analycal soluon o he problem s obaned wh approprae boundary condons. In hs case plasma formaon lqud expulson and nucleae bolng are omed. Sold or lqud evaporaes a any emperaure greaer hen K. The evaporaon raes srongly depend on he emperaure T. Ths dependence can be wren n he form U n = exp (7) T where n s he normal componen of he evaporaon fron velocy and MH v U = (8) N a B and H v s he laen hea of vaporaon (per un mass). B s Bolmann s consan M s he aomc mass and s a consan whose value s of he order of he speed of sound n 35

237 he condensed phase. The Fourer dfferenal equaons allowng a phase change process can be wren as T ρ( T ) c T ( T ) dt = T ( T ) x x T ρ( T ) c( T ) T ( T ) y y Q T ( T ) (9) where ρ (T) s he densy of he maeral of he worpece c (T ) s he emperaure dependen specfc hea of he maeral (T ) s he emperaure dependen hermal conducvy s he recesson velocy and T T ( x y ) = s he resulng hreedmensonal me dependen emperaure dsrbuon n he maeral s me T s he nal emperaure and x y are he spaal Caresan coordnaes. ( x y ) Q s he rae a whch hea s suppled o he sold per un me per un volume depends on he laser pulse parameers and opcal properes of maeral rradaed and defned by Eq. 67. Spars (975) defned absorpon coeffcen of he maeral as a lnear funcon of he surface emperaure n Eq. 66. For praccal reason he absorpon facor A we assume as uny. Ths emperaure dependence of A resuls from he fac ha A s proporonal o he elecron-phonon collson frequency whch n urn s proporonal o he crysal lace emperaure. The recesson velocy of he surface can be formulaed from energy balance a he free surface of rradaed worpece. In hs case he energy flux a he free surface can be wren as 36

238 I = ρ ( T ) [ ct s H ] (3) v where I = I ) (3) ( max R and T s s he surface emperaure and H v s he laen hea of vaporaon and R s surface reflecvy. Rearrangemen of Eq. 3 yelds I = ρ ( T ) [ c( T ) T s H v ]. (3) For meals and ceran applcaons a emperaures above he Debye emperaure can be assumed ha (T ) ρ (T ) and c(t ) do no change dramacally wh emperaure. Therefore assumng consan specfc hea and hermal conducvy for a parcular me nerval Eq. 9 can be smplfed o T ρ c T T = c ρ I x y )( R)exp( β ) µ exp( µ ) max (. (33) I should be noed ha pea power nensy I max does no vary wh me. Snce he surface emperaure s me dependan he recesson velocy vares wh me. Ths resul s non-lnear form of Eq. 33 whch canno be solved analycally by Laplace ransformaon mehod. For a nown surface emperaure here s a unque value for recesson velocy so an erave mehod can be appled here o solve Eq. 33 analycally. In our case eepng he recesson velocy consan n Eq. 3 enables us o deermne he surface emperaure analycally and afer obanng he surface emperaure he recesson velocy can be calculaed usng Eq. 3. Ths procedure can be repeaed so long as he surface emperaure and recesson velocy converge o correc resuls. Rearrangemen of Eq. 33 gves 37

239 38 ) )exp( exp( T I T T = µ β µ (34) where hermal dffusvy c = ρ wh boundary condons. ) ( ) ( = = = = T T L T x ρ (35) We can apply he Laplace ransform defned as. ) ( ) ( = d e T p u p (36) Laplace ransformaon of Eq. 34 wh respec o and subsuon of boundary condon from Eq. 35 gves [ ] ) ( exp β µ µ = p I T p T T (37) wh T= ( p) and p he ransform varable whch has a complmenary (homogenous) and parcular soluon. T h T p T = (38) The characersc equaon for he homogenous soluon can be wren as = ς ς p (39) whch leads. 4 ς p ± = (4)

240 39 The homogeneous soluon from Eq. 39 yelds h e c e c T ς ς = (4 or. 4 4 = p p h e c e c e T (4) For he parcular soluon one can propose exponenal soluon. p e A T δ = (43) Subsuon of T p gven by Eq. 43 no Eq. 33 resuls n e H e A p e A e A δ δ δ δ δ δ δ = (44) where ) ( p I H δ β = (45) δ δ = p H A (46) and subsung Eqs 46 no Eq. 43 and Eq. 4 no Eq. 38 we oban. 4 4 δ δ = p H e c e c e T p p (47) Defnng δ = δ (48) subsung Eq. 48 and usng boundary Eq. 35 hen c = herefore. ) )( ( ] 4 [ p p e I c e T p = β δ δ (49)

241 4 We can calculae c by usng boundary condon and defnng I δ ϖ = (5) ) )( ( ] 4 [ ] 4 [ p L p p e e c p T p = = = ρ β δϖ δ (5) from Eq. 5 c s ) 4 ( ) 4 )( )( ( ρ β ϖ δ p p L p p p c = (5) Hence ) )( ( ) 4 )( )( ( ] 4 [ mm p T p p e p p p e T = β ϖ β ϖ δ δ (53) where. ) 4 ( ] 4 [ ρ p p e L T p mm = (54) For he purpose of easer handlng calculaons for he fuure le us defne: H H and H 3 o be ) 4 )( )( ( ] 4 [ β p p p e H p = (55) ) )( ( = = p p p p H β β β (56)

242 4. ) 4 ( ] 4 [ 3 p p e H p = (57) Consequenly ) ( T T = (58). ) ( H L H e H T = ρ ϖ ϖ µ µ (59) To oban he nverse ransformaon of funcons - H and 3 - H le us nroduce ρ = 4 s or dp ds = 4 and ) ( 4 s p = (6) herefore = c c p dp p H e H ) ( - π (6) or ) ( = c c ds s s s e e H β π (6) where. 4 c c = (63) We can use one more ransformaon by leng ϑ 4 = or ϑ d ds = 4 (64) afer lenghy algebra we oban ) ( = c c p d e e e H ϑ ϑ ϑ β ϑ π ϑ (65)

243 4 where. 4 / ˆ c c = (66) hence H e H = (67) where. ) ( p p p e H p β = (68) Smlarly 3 - H can be obaned.e H e H = (69) where ( )( ) 7 p p e H p = (7) or ( )( ) 8 7 ξ ξ = p p e H p (7) where ξ =. (7) Inroducng paral fracons and rearrangemen yelds ( ) ( ) ( ). 8 7 = ξ ξ ξ p e p e p e H p p p (73) I s noed from he Laplace nverson ha - = a erfc e ae e p a e a a p π (74)

244 43 where complmenary error funcon s defned as erf erfc ) ( ) ( = (75) and. ) ( ) ( = = u du e erf erfc π (76) Therefore 4 - = erfc e e e p e p π ξ (77). 4 - = erfc e e e p e p π ξ (78) Leng s p = ξ and usng he defnon of nverse Laplace negral ( ) - ξ p e p becomes ( ) p e erfc e e p e = π ξ (79) usng Eqs and afer smplfcaons he Laplace nverson of H 3 becomes ( ) e erfc erfc e H = π (8) Le 4 w = = 4 w 3 = w 4 = (8)

245 44 hen H 4 becomes ( ) ( ) p e w p w p w p H = (8) and afer usng paral fracon expanson H 5 becomes ( ) ( )( )( ) ( ) ( ) p e w p D w p D w p D w p D w p D H = (83) where ( )( ) 4 3 w w w w w D = (84) ( )( ) 4 3 w w w w w D = (85) ( )( ) w w w w w D = (86) ( )( ) w w w w w D = (87) ( )( ) w w w w D = (88) Afer subsung Eqs no Eq H can be rewren as

246 = w erfc e e w e D w erfc e e w e D w erfc e e w e D w erfc e e w e D w erfc e e w e D H w w w w w w w w w w π π π π π (89) However s nown from Eq. 58 ha T(x) can be wren as ] [ 8 ) ( H L H e H e T T ρ ϖ ϖ µ µ = = (9) noe ha ) ( e e H = β β (9) Subsung Eqs no Eq. 9 we ge

247 46 ( ) w w w w w w w w w w e L erfc erfc e L e e e w erfc e e w D w erfc e e w D w erfc e e w D w erfc e e w D w erfc e e w D D D D D D e e T = β µ π ρ α β ϖ π ϖ µ ) ( ] [ ] [ ) ( (9) Knowng ha D D D 3 D 3 D 5 = Eq. 9 becomes

248 47 e L erfc erfc e L e e e w erfc w e w D w e w w w erfc w e w w w erfc w e w w w e w w w erfc w e w w w erfc w e w w w e e T = π ρ α β µ β ϖ µ ϖ 4 ] [ ) (. (93) In order o generae graphcal represenaon of Eq. 93 le us defne he followng varables whch wll be laer used n MahCAD program:..... N h N = = (94) L L τ = = (95) µ τ = (96)

249 = µ or = (97) µ = β (98) µ β w = β (99) 4 = w 3 µ (3) w 4 = µ (3) D 5 = β O ( )( µ ) or D = 5. β (3) One can use MahCad program o perform calculaons so our funcon T(x) can be formulaed as T and subsung above parameers we can oban 48

250 49 ( ) [ ] ( ) ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] = d e e e I erf erf e pl erf erf erf erf erf x I T max max exp 4 4 exp exp 4 exp exp 4 ) ( 4 4 exp 4 exp 4 4 exp 4 4 exp 4 exp ) ( 4 exp exp ) ( β β µ τ π β β β β β β β β β β τ µ (33)

251 5 O E J I 4 = β (34) ( ) E 4 = (35) ( ) 4 3 = E (36) ( ) 4 4 Z O J I e E = (37) I Z e E = 4 5 β (38) E J I 6 = (39) E 7 = (3) ( ) 4 8 = E (3)

252 5 ( ) [ ] ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ]. exp 4 4 exp exp 4 exp ( Re )) ( Re ( ) ( exp exp ) ( max = d d d d d d d d d d d d d d d d d T d d d E d e e I E erf erf e L E erf E E erf E E E erf E e E erf E E erf E E I T τ τ β τ β µ τ τ π τ τ τ τ ρ τ β β β β β τ τ µ ( 3) Temperaure dsrbuon wh respec o deph of he worpece and me s graphcally shown n Fg. 3.8 where recesson velocy of vaporaon surface was esmaed o be abou 33 m/s.

253 emperaure ºK deph µm me µsec Fg Temperaure dsrbuon of he worpece wh consderaon of ransfer evaporave case calculaed usng Eq. 3. A closed form soluon llusraed n Fg. 3.8 s lmed o he surface ablaon only and does no consderng he plasma formaon and lqud expulson from rradaed worpece. The emperaure graden aans relavely lower values n he surface vcny of he subsrae maeral han ha correspondng o some deph below he surface. In hs case nernal energy gan n hs regon due o absorpon of a laser beam energy becomes more mporan as compared o dffusonal hea ransfer from he surface vcny o he bul of he subsrae maeral due o emperaure graden. As he heang perod progresses he pon of mnmum emperaure graden moves owards he sold bul of he subsrae maeral. In he regon beyond he pon of mnmum emperaure graden he dffusonal energy ranspor plays an mporan role nsde he subsrae maeral. 5

254 The large change n he power nensy occurs wh ncreasng β ; however he magnude of emperaures n he surface regon does no aler consderably. Ths s because of he rae of evaporaon whch ncreases wh ncreasng power nensy. Consequenly he convecve boundary condon a he surface suppresses he emperaure rse n he surface vcny a hgh power nenses Analycal soluon of hea ransfer equaon wh nec heory approach Ths Secon examnes pulsed laser heang process by consderng boh Fourer conducon and elecron-phonon nec heory approaches. A D nec heory approach s presened. More specfcally hs Secon s focused on developmen of equaons governng an neracon of a Gaussan laser beam wh maerals where analycal soluon of hese equaons based on boh he Fourer hea conducon heory and nec heory approach usng a Laplace negral ransformed mehod s uled. A comparson of he analycal soluon of he Fourer heory and closed form soluon of he nec heory approach s nroduced n he nex Secon of hs Dsseraon. The emperaure dsrbuon for he heaed maeral predced from he nec heory are compared wh he Fourer heory fndngs. Mos heorecal wor relang o laser beam neracon wh maerals s based on he soluon of he classcal hea conducon equaon derved from Fourer heory. I has been shown ha he Fourer heory of hea conducon s no fully applcable o shor pulsed laser heang due o he assumpons made n he heory. These assumpons 53

255 nclude: maer s assumed as connuous and homogeneous and ha he hea flux across any plane s a funcon of only he emperaure graden a ha plane. The frs assumpon s no vald for dsances less han neraomc spacng and he second s only rue f all he energy crossng he sohermal plane s accouned for. Conducon n meals occurs due o subsequen collsons beween exced elecrons and lace se aoms. The elecron moon n he subsrae s random whch means ha elecrons move from he surface o he bul as well as from he bul o he surface. Moreover he amoun of energy ransferred o lace se aoms depends on he elecron energy dsrbuon n a parcular regon. Harrngon (967) showed based on nec heory consderaons ha elecrons whn fve mes of he elecron mean free pah conrbue 98.5% of he oal energy ranspored provded ha T/ s consan over hs dsance. Consequenly aempng o generae he Fourer equaon he applcaon of q=- T/ s lmed o planes n excess of en mes he elecron mean free pah apar. In addon he absorpon deph of he meals s of he order of en elecron mean free pahs; herefore he graden T/ s no unform over he spaal ncremen ( λ λ beng he elecron mean free pah). In hs case he hgher order gradens ( 3 T/ 3 ) whch are negleced n he Fourer heang model become mporan and he valdy of he Fourer heang model comes no queson. Therefore becomes necessary o examne he laser-nduced conducon heang on a mcroscopc scale. The applcably of he Fourer equaon n laser heang s lmed o he cases n whch low power laser nenses are employed (Ylbas e al. ). Ths s due o he followng facs: 54

256 . In he analyss of he Fourer hea conducon model he hea flux hrough a gven plane s consdered as beng a funcon of he spaal emperaure graden a ha plane. Ths depends upon he assumpon ha he emperaure graden remans almos consan beween wo successve and closely spaced planes. The dsance beween hese planes s fne herefore error occurs when hgh-order erms whch are negleced become mporan a hgh power laser nenses.. The hea flux hrough a gven plane depends on he elecron energy dsrbuon hrough he maeral herefore he maeral canno be consdered as a homogeneous connuum. Consequenly a new model may be requred o be developed for heang mechansm whch s approprae o hgh power laser heang process. A model consderng a nec heory approach descrbng he ranspor of energy by elecrons whn he elecron mean free pah may be suable n hs case. The bass of hs model was nroduced by Ylbas () for one-dmensonal heang. I was shown ha he predcons made from he new model agree well wh he expermenal fndngs. He adoped he elecron moon n meals o formulae he laser pulse heang process. The heang process was esablshed based on a nec energy ransfer mechansm whch occurred durng he collsons beween exced elecrons and lace se aoms. In hs case he excess elecron energy was ransferred o lace se aoms resulng n ncreased amplude of lace se vbraons durng he collson process. In hs Secon comparsons of he elecron nec heory approach are made wh Fourer heory models for a pulse laser heang process. The emperaure feld due o 55

257 each model s predced for sep nensy as well as exponenally decayng nensy pulses. The sudy s exended o nclude he analycal soluon o he elecron nec heory approach for Gaussan nensy pulses. Elecron nec heory approach s based on elecron and phonon movemens Elecron-phonon analycal soluon The nec heory approach essenally deals wh he nec energy ransfer mechansm ha occurs when elecrons and lace aoms wh dfferen energes nerac. In order o smplfy he phenomenon some useful assumpons are made. These nclude omsson of hermonc emsson aanmen of seady space charge and mean free pah of molecules beng ndependen of emperaure. A ne flow of elecrons occurs n he subsrae by he presence of an elecron source a nfny and he elecron gves fracon of s excess energy o he lace se aoms durng an elecron phonon collson and hs fracon s assumed o be consan hroughou he successve collsons. When he elecron absorbs he ncden laser energy some excess energy of he elecron s ransferred o he lace se aoms durng elecron phonon collson process. Ths energy manfess self as an ncrease n he amplude of he aomc vbraon (phonon). As a resul neghborng aoms n he lace are forced away o new equlbrum posons and absorb some of hs exra energy n he process. A sage may be reached where evenually he lace se aoms n he localed regon around he orgnal collson se are all n equlbrum and have ncreased her vbraonal energes. 56

258 I s hs energy mechansm whch defnes he conducon process n he sold subsrae when subeced o a laser heang pulse. The amoun of energy whch elecrons from secon I ransfer o lace se aoms n he same secon Fg. 3.9 can be calculaed as follows. The number of elecrons leavng secon I s: N A d where = dx dy and N s he number densy of elecrons whch ransfer energy o d from dζ and s he average elecron velocy enerng he conrol volume n he -axs across area A a me d. A Fg Elecron movemen a he meal surface vcny (= s he surface). The energy of N aoms n one drecon due o lace vbraon s: E = N T whch can be descrbed as he phonon energy. The number of collsons whch aes place beween elecrons and phonons hrough he maeral can be assessed by he oal collson probably of elecrons. Probably of elecron ravelng a dsance where B <<λ (λ beng he mean free pah) whou mang a collson s exp- λ he probably of an elecron havng us collded n d s. d λ 57

259 In general hermal conducvy s usually defned wh respec o seady hea ransfer hrough homogenous medum n a random process. Thus he hermal conducvy n one drecon can be defned as N = Bλ. (33) 6 I has been shown ha for he emperaures hgher han he Debye emperaure hermal conducvy can be assumed consan. The number of he elecrons whch have dζ us collded n secon I s: Nζ A d. Therefore he number of elecrons λ whch have us collded n dζ durng d s: ζ dζ d N ζ A exp λ (34) λ λ ζ and whch hen ravel o secon II before colldng n d where dζ d exp λ λ λ s he oal elecrons-lace se aoms collson probably as descrbed n (Ylbas ). The negave sgn of he negral s due o a mrror mage nroduced a a surface. Ths mrror mage represens refleced elecrons from he free surface. The ne ransfer of energy durng he elecron-phonon collson hrough he enre body can be wren as E ζ d dζ exp f ζ λ λ λ [ E E ] = (35) where E ζ and E are energy of elecrons and energy of phonons a a consdered regon respecvely n he -axs and parameer f s a fracon of elecron energy. I s suggesed ha he rae of ransfer of energy beween he elecrons and he molecules wll be deermned only by he dfference n emperaure of he elecrons and he lace 58

260 vbraons. If he emperaure of he lace se aoms n d s Φ() and he emperaure of he elecrons when hey arrve a d s Θ(ζ) hen he energy ransfer o he lace se aoms n d from collsons wh elecrons n whch he elecrons gve up a fracon ``f'' of her excess energy s dζ d ζ N A d exp ζ f B [ Θ( ζ ) Φ( ) ]. Summng he λ λ λ conrbuons from all such secons o oban he energy n secon II gves Nζ f ζ B E exp = Ad d [ Θ( ζ ) Φ( ) ] dζ. (36) λ λ λ Durng elecron-phonon collson some fracon f of he elecron excess energy s ransferred o he phonon. For any nelasc collson he conservaon of energy n any secon may be wren as Elecron energy enerng he secon = elecron energy lvng he secon energy ransfer o phonons n he secon (37) Ths gves elecron energy n - elecron energy ou f = excess elecron energy n f ( EeI ) ( ) n - EeI ( E ) E ou =. (38) ei n phonon provdng ha f from he energy conservaon where (E el ) excess = (E el ) n - E phonon and E phonon = mean energy of phonon. The effecve f value over a regon suffcenly large o allow many collsons approaches uny and hs corresponds o he aanmen of 59

261 hermal equlbrum. In he case of a sngle collson f depends only upon he masses of he colldng parcles accordng o f M m = (39) ( M m) where M and m are he masses of an aom and an elecron respecvely. Subsung approprae values shows ha he f value s of he order of -4. The mos energec phonon s only. e assumng c s = 5 cm/s (velocy of sound n he sold) bu elecrons near he Ferm level have energes of several e; hence when such elecrons are scaered only a small fracon of her energy can be gven durng an elecronphonon collson. In he presen analyss f s equal o -4 and assumed as a consan over successve collsons. The change of rradance of he laser beam passng hrough a homogenous medum (meal) as a funcon of dsance s gven by di / d = µ I ( ) where µ s he absorpon coeffcen. The negave sgn ndcaes he reducon n beam rradance due o absorpon as µ s a posve quany. The absorpon of he ncden laser bam aes place n -axs. Inegrang defned di / d he nensy of he ncden beam a any plane nsde he subsrae s: = I exp( µ ) where I s he pea nensy of ncden rradance. The lm for a small secon wde a he energy absorbed s I 6

262 di d d d = ( µ I exp( µ ) ) or I = I f ( ) d d ' or I = I f ( ) (3) where f () s he absorpon funcon. Snce he mrror mage suaon s consdered Fg. 3.9 for elecron movemen a he surface he laser beam wll be absorbed n a ' d manner descrbed by: f ( ) = exp( µ ) for all. Usng Eq. 3 for nensy of d he laser beam he rae of appled exernal energy a d durng he me nerval d can be gven as µ E = I µ e A d d (3) abs he oal energy ncrease n he maeral a d durng me d s N A E E ) d = E E. (3) ( d abs The oal amoun of energy whch s absorbed n an elemen ξ area A n me d s: I A d dξ f '( ) snce all he beam energy s absorbed n he drecon. One ξ mus allow for he possbly ha elecron denses may vary hroughou he maeral and n parcular he number ravelng from dζ o d may no be he same as ha from d o dζ. Therefore he proporon of energy whch s absorbed by he elecrons whch ravel from dζ o d n d s: ζ A d dξ f '( ξ. Nζ N ζ I ) N The average energy absorbed by one elecron n dξ n a me d s: I f '( ξ ) dξ ( N N ) ζ ζ and he oal amoun absorbed by hs elecron from d o dζ s: ζ f '( ξ ) dξ I ( N N ) ζ ζ. 6

263 6 The fnal emperaure of he elecrons n d afer he collson process can be readly found from he conservaon of energy.e. Toal elecron energy afer collson = oal elecron energy n durng d change of lace se energy. The assumpon ha all drecons of ravel are equally probable gves: 6 ' N N N = = ζ ζ where N s he number of free elecrons per un volume. Therefore Φ Θ = Φ ζ ζ ξ λ ζ λ ζ λ ζ λ ζ ζ λ ζ λ ρ d d f f I d f d f c ) '( exp ) ( exp ) ( exp )) ( ( ' 3 3 (33) and ( ). ) '( )exp ( ) ( ) ( exp ) ( ) ( exp 3 3 Θ = Φ Θ ζ ζ ξ λ ζ ζ ζ λ ζ λ ζ ζ ζ λ ζ λ d d f f d f d f d (34) Equaons 33 and 34 are of neres o laser machnng. The mehod of soluon o be used n he followng analyss s he ransformaon of he smulaneous dfferenalnegral Eqs 33 and 34 usng he Fourer negral ransformaon wh respec o. The resulng ordnary dfferenal equaons may hen be handled much more convenenly. The Fourer ransformaon of a funcon f() s defned (Ylbas ) by

264 = ( ω ) F [ f ( )] exp f ( ) d = F( ω) (35) and he Fourer nverson s gven as f ( ) = F( ω)(exp( ω ) dω) = F( ω) π. (36) The Fourer ransformaon of he convoluon negral f ( ξ ) g( ζ ) dζ s he produc of he ransforms f ( ω) g( ω) and he ransform of funcon exp s λ λ. Applyng Fourer ransformaon o Eqs 33 and 34 yelds ω λ δ f ω λ ) ( ρ c Φ) = ω fθ I δ f. (37) δ ω ( The mulplcaon n he ransform doman by (ω) corresponds o a secondorder dfferenal n he real plane. Hence he nverson of Eq. 37 gves λ µ Φ Φ f ρ c = f I µ f exp( µ ). (38) If he erm λ µ f Φ ρ c s negleced for all f values Eq. 38 becomes Φ Φ ρ c = I µ exp( µ ) (39) whch s he same as a Fourer hea conducon equaon shown as Eq. 7. I s apparen ha he elecron nec heory equaons for he hea conducon process are much more general han he Fourer equaon. The new model of he conducon process s vald n regons close o he surface where an absorpon process aes place and herefore he 63

265 64 emperaure profles whch are obaned n hese regons can be expeced o be vald. One furher advanage of hs new approach s ha he problem s compleely specfed ogeher wh spaal boundary condons by he fnal equaons and ha hese equaons can be solved usng he mehod of Fourer ransformaon. The soluon of Eq. 37 n he ransform plane for he Gaussan nensy pulse s exp ) exp( ) exp( exp exp exp ) ( ) ( λ µ βλ β µ µ µ µ µ µ βλ β βλ β βλ β βλ β βλ β β βλ µ µ λ µ β ρ µ = Φ erfc erfc erfc erfc c AI (33) and gves he closed form resul of he elecron nec heory approach whch graphcally s shown n Fg. 3..

266 emperaure ºK deph µm me µsec Fg. 3.. Temperaure closed form soluon of he elecron nec heory approach wh respec o deph of he sample and me Comparson of Fourer and nec heory Comparson of analycal resuls of he Fourer relaon gven by Eq. and nec heory equaon Eq. 33 shows ha he wo are dencal when he followng wo condons are mee: µ λ and β λ. For mos maerals µ and λ are of he same order of magnude. Snce µ s of he order of -5 and λ s of he order of -8 hen β. Ths scenaro corresponds o laser mcromachnng for laser pulses wh pcosecond rse mes. Generally β and Eq. 33 reduces o exacly he analycal λ soluon obaned from he Fourer heory for he exponenal pulse provded ha >>µ λ. When β approaches ero he pulse soluon reduces o ha for a consan nensy analycal soluon of Fourer heang Eq.. The comparson of he emperaure graden Φ / predced from he Fourer and elecron nec heores 65

267 wh he dsance n he -axs for wo pulse lenghs s shown n Fg. 3.. The emperaure gradens predced from boh heores are smlar for he long pulse lengh pulse = 6x s. In general he emperaure graden decreases sharply n he surface vcny o reach s mnmum. Fourer heory (6x -9 ) Knec heory (6x -9 ) Fourer heory (6x - ) Knec heory (6x - ) deph m Fg. 3.. dt/d predced from he Fourer and nec heory along he -axs for wo pulse lenghs: 6-9 s and 6 - s (Ylbas ). As he dsance from he pon of mnmum ncreases furher nsde he subsrae Φ / ncreases gradually. The sharp decrease of Φ / n he surface vcny s due o he rapd ncrease of he emperaure n hs regon. In hs case he energy absorbed by he elecrons n he surface vcny s convered no he nernal energy gan of he subsrae hrough collson process. Ths gves rse o a sharp ncrease of he lace se emperaure. The energy balance aans among he absorbed energy nernal energy gan and he conducon process a he pon of mnmum Φ /. As he dsance ncreases beyond he pon of mnmum he gradual ncrease n Φ / reveals ha he conducon effec due o phonon relaxaon domnaes. However as he pulse lengh 66

268 reduces less han 6x s he emperaure gradens predced from boh heores pulse dffer consderably. The emperaure graden predced from he Fourer heory reduces sgnfcanly as compared o s counerpar predced from elecron nec heory. The dfference n Φ / predced from boh heores s due o he emperaure response of he maeral for a shor laser heang pulse as ndcaed earler. In summary one can deduc a followng conclusons: Temperaure profles predced from Fourer heang model wh nec heory approach for -dmensonal model were compared n hs sudy. In general emperaure profles for -dmensonal Fourer heang case and nec heory approach are very smlar. Alhough he Fourer heang model fals o predc correc emperaures for he shor pulse heang case. The equlbrum me exss for a gven maeral he balance occurs beween he nernal energy gan due o laser rradaon and he conducon losses. The analycal soluon of he elecron nec heory approach obaned for he exponenally decayng pulse reduces o a sep npu nensy soluon when β approaches ero. Moreover he closed form soluon of he elecron nec heory for he sep npu λ nensy approaches he Fourer soluon when µ. The elecron nec heory f resuls devae consderably from he Fourer heory resuls when β >>. The dfference n emperaure profles occurs because of he fac ha he elecrons n he surface vcny absorb he ncden laser energy and he exced elecrons do no mae suffcen collsons wh he lace se aoms o ransfer her excess energy n he surface regon. Thus lace se emperaure n hs regon becomes lower han he 67

269 elecron emperaure as evden from Fg. 3. n whch he elecron emperaure dsrbuon nsde he subsrae s shown. Therefore he Fourer heory fals o predc he emperaure rse n he surface vcny accuraely for heang me of pulse 6x s. 3.. Fne dfference mehods In hs Secon he soluon of governng equaon s accomplshed usng an explc fne dfference approxmaon and he correspondng boundary condons. The represenave resuls for compuaonal nvesgaons are presened and dscussed. To solve general governng equaon Eq. 4 fne dfference mehod (FDM) was used and he approxmaons o model he governng equaon were based on forward-dfference n me and cenral-dfference n space Fne dfference formulae Defnng funcon u whose value u s nown a a number of dscree pons he approxmaon of he frs dervave u a pon one can defne as u ' u = u δ (33) where δ s he nerval separang he wo values of funcon u. The expresson from Eq. 33 s nown as a forward-dfference approxmaon of he dervave of u a pon. The cenral-dfference formula can be expressed as 68

270 ' u u u = (33) δ The error resulng from a cenral dfference s smaller han from a forward or bacward dfference. Consequenly s benefcal o use cenral dfference mehod for he calculaons whenever possble. The second dervave of u a pon can be consdered as dervave of dervave of u a hs same pon or u '' u = u δ u (333) whch s consdered he cenral-dfference formula for u. Examnaon of he governng equaon shows ha he hea conducon equaon requres evaluaon of he frs-order dervave wh respec o me and he second-order dervaves wh respec o each of he coordnae drecons x y and. The me dervave s commonly evaluaed usng a forward dfference and each of he spaal dervaves s expressed usng he cenral dfference expresson. Summared a hreedmensonal model under consderaon requres creang a hree-dmensonal grd of pons a whch correspondng soluon values are deermned. Ths grd s ofen referred as a mesh and he ndvdual pons of he mesh as nodes Explc formulaon of governng equaon In order o calculae he emperaure from he governng equaon usng FDM we mus express he emperaure a node ( ) a me d explcly n erms of he 69

271 emperaures of he surroundng nodes a me. Ths approach s called an explc formulaon whch allows one o smplfy march hrough he fne dfference mesh and solve for he emperaure a succeedng me sep provded he nal emperaure of he enre grd s nown. Somemes one can call a forward-dfference formulaon snce he me dervave s expressed usng a forward dfference. The oher basc echnque nvolves solvng he emperaures n he nodal mesh a me n erms of nodal emperaure a me d s called an mplc formulaon snce he emperaure a any gven node s no explcly saed n he resulng equaon bu a se of lnear equaons nvolvng he enre nodal mesh s consdered. Boh of he above mehods have her advanages and dsadvanages. I was shown ha selecon of small ncremens n he node spacng requres he use of a small me sep o nsure a convergen soluon. The mplc mehod on he oher hand does no have any such sably requremens relang he node spacng and he me sep. (-) (-) (-) dy () d dx () () () Fg. 3.. General hree-dmensonal fne dfference grd. Consequenly much larger me ncremen can be used o solve a gven as bu wh dffcules conneced wh he soluon of he smulaneous equaons resulng from 7

272 he formulaon. The soluon approach seleced for hs Dsseraon was he explc mehod. In order o smplfy furher developmens wll be useful o expend he governng equaon Eq. 4 n convenonal noaon T ρ( T ) c T ( T ) dt T = ( T ) x x ρ( T ) c( T ) T ( T ) y y T Q T ( T ) (334) whch apples for any neror pon of he nernal nodes n he sample and all erms of he equaon were defned earler. Analyng Eq. 334 one can noce ha he frs hree erms on he rgh hand sde of he equaon are he volumerc accumulaon raes of energy due o he varaon of he hermal conducvy due o varaon of hreedmensonal emperaure feld. Noe ha hose erms depend on boh he magnude of he hermal graden and he emperaure dependence of he hermal conducvy. The forh erm of Eq. 334 expressng energy los on melng expulson of meled and vapored maeral where s drllng velocy of vapored fron of vapor normal o Knudsen layer. The las erm of he rgh hand sde of Eq. 334 s he Caresan analog of he volumerc accumulaon rae due o he equvalen nernal generaon of energy. The sum of hose erms equals he ne volumerc accumulaon rae of energy whch s wren on he lef-hand sde of Eq Subsuon of Eqs 33 and 333 no Eq. 334 yelds 7

273 7 ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) = Q T T T T y T T x T T T T T y T T T x T T T c T ρ (335) where T T T = (336) The nomenclaure n above equaons was defned earler Fg. 3.. Please noe ha he specfc hea and densy are dependen from he emperaure. One way o accoun for ha s o evaluae c = c(t ) or ρ = ρ (T ). For furher analyss s convenen o rewre Eq. 335 no a general form as ( ) T c T φ ζ ρ = (337) where he parameers ζ and φ are defned respecvely as

274 73 ( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) T T T T y T T x T T T T T y T T T x T T T = ς (338) and ( ) ( ) ( ). = y x φ (339) Sably of explc soluons The explc formulaon of governng equaon as wren n Eq. 337 s only condonally sable as was saed earler. The sably depends on he relaonshp beween maeral properes he me ncremen and he spaal ncremens dx dy and d. For sably analyss one can rewre Eq. 337 as. = T c T φ ζ ρ φ (34) For seady-sae emperaure SS T a node () Eq. 34 convergence o ero f

275 SS T ζ =. (34) φ Subsung Eq. 34 no Eq. 337 one obans a relaonshp whch mahemacally can be llusraed as T φ = ρ c SS ( T T ). (34) Analyss of Eq. 34 reveals ha suffcen condon for convergence of Eq. 337 o he seady sae emperaure s φ ρ c. (343) The sably creron presened n Eq. 34 does relae he me nerval o he gven maeral properes and o geomercal coeffcenφ. Procedure for wrng he ransen equaon for a gven ype of node solvng for he seady-sae emperaure and hen subsung bac no he orgnal equaon s general and should be repeaed for every ype of node elemen presen n he model under consderaon. Subsequen secons presen more comprehensve nformaon of hs procedure for he compuer model developed for hs Dsseraon Model geomery To faclae soluon by FDM he worpece mus be subdvded no an array of nodes n Fg The model geomery of he problem s shown n Fg

276 Nowaows (99). The orgn of he Caresan coordnae sysem s locaed a he op far lef corner of he sample. Assocaed wh each node s a volume elemen. Noe ha he elemens correspondng o he surface edge and corner nodes are smaller han a ypcal neror elemen. Ths s due o he fac ha nodes correspondng o hese elemens are placed on he boundares defnng he worpece. Noe also ha he cener of he rradang laser beam s placed n he cener of he op surface of he worpece. Each face of he model can have unque hea ransfer coeffcens. Tha possbly enables one o consder varous boundary condons. For example one can model an nsulaed boundary by seng he hea ransfer coeffcen o ero along he proper face. Analyng he geomercal posons of nodal pons wh respec o unque hea ransfer coeffcens on each sde of he sample one can dsngush four ypcal nodal elemens n order o descrbe he processes of hea ransfer. These ypcal nodal elemens are nernal nodes wall nodes edge nodes and corner nodes Fg

277 LASERBEAM hl hyl X hxl hxr Y hyr Z hr hxl sum of convecon and radaon coeffcens on he worpece surface wh surface normal ponng n he negave x-drecon hyl sum of convecon and radaon coeffcens on he worpece surface wh surface normal ponng n he negave y-drecon hl sum of convecon and radaon coeffcens on he worpece surface wh surface normal ponng n he negave -drecon hxr sum of convecon and radaon coeffcens on he worpece surface wh surface normal ponng n he posve x-drecon hyr sum of convecon and radaon coeffcens on he worpece surface wh surface normal ponng n he posve y-drecon hr sum of convecon and radaon coeffcens on he worpece surface wh surface normal ponng n he posve -drecon Fg Three-dmensonal fne dfference subdvson of a worpece. 76

278 3..5. General elemen equaon Examnaon of he model geomery llusraed n Fg. 3.3 ndcaes ha he ncluson of fne model dmensons nroduces varaons on he general dscreaon mesh. As shown hese varaons produce volume elemens correspondng o neror wall edge and corner nodes. Noe ha n addon o changng boundary condons for each elemen he defnon of neror wall edge and corner nodes placed on he regon boundares leads o elemens of varyng ses. In hs Dsseraon fne dfference equaons relang o he ypcal nodes shown n Fg. 3.4 were developed. In he frs sep o developng a general node equaon consder frs he soluon of general equaon wh a consan hermal conducvy; he effecs of varable hermal conducvy wll be mplemened no he analyss laer. Developmen of he fne dfference model for ha case of consan hermo physcal parameers was prevously developed (Nowaows 99). 77

279 Fg The four ypcal nodal elemens of a worpece. Equaons 335 hrough 337 were developed by he smple explc FDM. Because hs mehod s condonally sable n order o guaranee he convergence of he soluon a furher sably analyss for hs FDM model s requred (Nowaows 99). A presen me for hs dscusson only he resuls are mporan Generaled elemen Equaons developed by Nowaows (99) can be modfed and combned no one general formulaon. Such a general elemen has he advanage of beng easly 78

280 79 modfed o accoun for varous boundary condons whch occur when 3D modfcaon of he node mesh s ncorporaed n he process of hole formaon. Recallng our dscusson n Secon 3.. one can noce ha he governng hea ransfer equaon and s lnear boundary condons are all expressed n erms of hermal gradens a each node defned by physcal coordnaes x y. Each node can be hough of as conssng of wo erms: s erm conanng nformaon on he hea ransfer mechansms presen a he node nd erm descrbng s neghborng nodes. Those wo erms have been already defned n Eqs 338 and 339. In order o faclae hs dscusson le us frs sar wh a consderaon of an neror node equaon wh consan hermal conducvy. In hs case Eq. 337 can be rewren no a form of Eq. 337.e. ( ) T c T φ ζ ρ = (344) ( ) ( ) ( ) ( ) T T T T y T T x T T = ς (345) ( ) ( ) ( ). = y x φ (346) In a smlar manner equaons for oher ypes of nodes can be wren n he form of Eq For each case a new par of coeffcens mus be defned. For he case of a ypcal wall node Fg. 3.4 he coeffcens are

281 8 ( ) ( ) ( ) ( ) ( ) T T T x h T T y T T x T a xl = ς (347) and ( ) ( ) ( ). = x h y x xl φ (348) For ypcal edge node he coeffcens are ( ) ( ) ( ) T T h x h T y T T x T a l xl = ς (349) ( ) ( ) ( ). = h x h y x l xl φ (35) For he corner node one obans ( ) ( ) ( ) a l yl xl T h y h x h T T y T x T = ς (35) ( ) ( ) ( ). = h y h x h y x l yl xl φ (35) In Eqs 344 o 35 T a s he amben emperaure whle h x h y and h are hea ransfer coeffcens on he sample faces normal o he x y and drecons respecvely. Subscrps l and r used n represenaon of he hea ransfer coeffcens refer o he sample s surface wh surface normal ponng n he negave (or lef) and posve (or

282 rgh) drecon of he coordnae axes respecvely. Noe ha he radaon losses and hermal conducvy emperaure dependence s no ye aen no accoun. Consderng possble forms of hea ransfer ha can occur across he boundares of a ypcal elemen one can formulae a general equaon on he bass of energy balance on hose elemens ha s ' T T x y ρ c = Qcond Qconv Qrad. (353) x y S S S Proceedng furher one can oban where T ρ c. ' T x y S S S x S x y S Q Q y Q xl cond xl conv xl rad S Q Q Q xr cond xl conv xr rad = Q Q Q xl conv yl rad yl cond Q Q Q yr rad yr cond xl conv Q Q Q l rad l cond xl conv Q Q Q r rad xl conv r cond. (354) are scalng facors descrbng he lengh of he elemen n he x y and drecons respecvely whle xl Q cond xl Q conv xl Q rad Q Q xl rad xl rad xl Q rad xl Q rad xl Q rad refer o he hea ransfer due o conducon convecon and radaon o he amben regons n he x y and drecons respecvely. xl Q rad To ae full advanage of he compuer algorhm one can mae use of Boolean expressons correspondng o each possble mode of hea ransfer a every node represened by subscrps and and s neghbors he coeffcens needed o evaluae Eq. 354 can be expressed as and x y Q = ς ς ς S Q ς (355) 8

283 8. y x ς ς ς φ = (356) where ( ) ( ) = a xr hxr xl hxl h hxl xl h hxl xl x x T x h S x h S T x h S x S T x h S x S S ς (357) and [ { ] ( ) [ ]. = x h S S x h S x h S x S S h hxr hxl xr axr xl axl xr xl x x φ φ (358) I should be noed ha Eqs 357 and 358 are for he x-drecon only. For y- and - drecons one can wre smlar expressons usng superscrp y for y-drecon and for -drecon. Noe ha hose coeffcens are he Boolean expressons whch modfy he general node formula o he specfc node under consderaon. In he case where hermal conducvy s no consan wh he emperaure general expresson for coeffcen ς and φ are

284 83 [ ] { } ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( = x S S S T x h S x S x S S S T x h S x S S S x S S S S xr xl xl h hxr xr xr xl xr h hxl xl xr xl xr xl xl x x ς (359) and [ ] ( ) [ ] ( ) [ ] ( ) ( ). ) ( ) ( ) ( ) ( = x h S h S h S S x S S S S x S S S S S x S S S xr hxr xl hxl h xl xl hxr xl xr xl xr xl xr xr xl xr xl x x φ (36) Followng he forgong procedure equaons smlar o Eqs 359 and 36 can be wren for y-and -drecons. The ncluson of radaon erms no expressons for he coeffcens ς and φ s farly sraghforward when lneared he radaon boundary condons wh he defnon of emperaure-dependen effecve radaon hea ransfer coeffcen R h such ha

285 84 ( ). 3 R T h = ε σ (36) The resulng modfcaons o Eqs 359 and 36 are [ ] 4 x T S S S a x Rxr Rxl x x = ε σ ς ς (36) and [ ] x h S S S R x Rxr Rxl x x = φ φ (363) respecvely. Equaons 36 and 363 can be derved for he y- and -drecons and he fnal form of modfed parameers are defned as Q y x Q S = ς ς ς ς (364) and = y x φ φ φ φ (365) where Boolean expresson has ero value for all nodes of he mesh excep hose rradaed drecly by he laserbeam. Subsuon of Eqs 364 and 365 no Eq. 344 yelds he fnal dfference formulaon of general governng equaon subec o he specfc boundary condons. The fnal form of Eqs 364 and 365 was used n hs Dsseraon o develop compuer program o sudy neracon of he laserbeam wh maer. The compuer program oulne s gven n he flow dagram shown n Fg The effecveness of hs soluon was subec o parameers defnng operang characerscs of he laser sysem. Those characerscs were measured and calculaed durng he course of hs Dsseraon.

286 3..6. Compuer program consderaons Search for an adequae commercally avalable sofware o smulae laser beam neracon wh maer faled. The man reason was ha o effecvely model neracon of laserbeam wh maerals he laser beam s characerscs mus be measured and hen lned drecly o he compuer algorhm. Soluon of he general governng equaon (Eq. 4) was obaned usng compuer sofware developed n hs Dsseraon. The soluon procedure adoped n hs Dsseraon was an explc fne dfference formulaon (fne forward n me and cenral dfference n space) of Eq. 4 and s correspondng boundary condons descrbed n Secon.7.6. The effecveness of hs soluon was subec o parameers defnng operang characerscs of he laser beam such as: energy per pulse pulse duraon and laser beam dameer emporal and spaal laserbeam characersc. These characerscs were measured durng he course of hs Dsseraon. Soluon of Eq. 4 wll be dscussed n he subsequen secons of hs Dsseraon. The resulng se of fne dfference equaon were solved usng he algorhm oulned n Fg Borland Turbo Pascal sofware was used o wre compuer code lsed n Appendx A. Geomery of he worpece uled n he expermens ( mm by 5 mm by.76 mm) and because of he fac ha sgnfcan emperaure graden under he laser beam rradance exs only a a small laser beam spo hen he worpece wh he dmensons of.3 mm by.3 mm by.76 mm was used n he FDM compuaon (Han 85

287 999). The user frs needs o choose he npu daa fle whch conans he maeral hermal and physcal properes he nerface would calculae and oupu he maxmum me sep n order o acheve convergence more deals on hs subec n Secon The sofware used o dagnose he laser beam dsrbuon and nerface programs for 3D sages were wren n Quc Basc Secon 4.. Daa analyss le emperaure dsrbuon across he sample was done usng MahCad Grapher and Surfer sofware. All daa npu and oupu formas were developed usng ASCI formas as lsed n Appendx A Phase ranson and hole formaon In mcromachnng applcaons heang of he sample undergoes varous phase ransformaons (descrbed earler) le sold o lqud lqud o vapor and vapor o plasma phases. Compuer programs developed consder all of he phase ransformaons. The procedure s very general and may be used for mos of maerals (no only meals) and ncludes some of he mass expulson phenomena. The hole developmen n he worpece changes he sample geomery and mus herefore be raced hroughou he soluon procedures. The process for phase ransons s sraghforward. As he compuer program algorhm marches hrough he fne dfference grd a a gven me sep he prevous emperaure a node () s compared o he emperaure for naon of a gven phase ranson. If he node emperaure falls whn a prescrbed olerance he gven phase ransformaon occurs. The phase change s modeled as an sohermal 86

288 process n whch he energy requred for ranson s equal o he laen hea of he parcular ranson (melng or vaporaon). The emperaure of ha parcular node s held a he ranson emperaure poenally for mulple eraons unl has accumulaed he amoun of energy equal o he laen hea of he ranson. The emperaure of ha parcular node s held a he ranson emperaure poenally for mulple eraons unl has accumulaed he amoun of energy equal o he laen hea of he ranson. In oher words he change n elemen nernal energy s expressed as a change n enhalpy of ha elemen or mahemacally hs process s mplemened by modfyng Eq. 344 as h = h ρ ( ζ φ T ) (366) In Eq. 366 h s he value of nodal enhalpy a me and he second erm on he rgh hand sde of he equaon s he ncremenal change n nodal enhalpy occurrng as a resul of he energy ransfer durng he presen eraon of he soluon. Isohermal naure of he phase change a node () s guaraneed by evaluang Eq. 366 n place of Eq. 336 and by seng T = T. (367) Ths procedure s followed unl he varable h s equal o he laen hea of he gven ranson. A ha me he emperaure a node () s allowed o vary by evaluang Eqs 336 and Eq. 344 for subsequen eraons n me a node () unl he nex ranson. 87

289 A he end of evaporaon ranson he node s consdered o be annhlaed or a he ceran laser energy densy s gong o change no plasma and o consue n any case a regon correspondng o a hole. The hole creaon n he worpece changes he maeral geomery and mus herefore be raced hroughou he soluon procedure. For semransparen maerals for example he possbly of subsurface hole formaon mus also be consdered. The effec of hole formaon on he problem geomery s llusraed schemacally n Fg Examnaon of hs llusraon reveals few ey problems one has o face wh and hey have o be consdered n developmen of he soluon model. laserbeam Fg Hole formaon. The essenal modelng ssue s relaed o he queson of he effecs of maeral removal on beam aenuaon and how o model he hea ransfer along he sde and boom of he hole (.e. Marangon effec and Knudsen layer). Compuer code was wren (as a par of hs Dsseraon) n modular and unversal way and can suppor any value of surface convecon coeffcen ncludng a emperaure dependen funcon Appendx A. The approach aen n hs dsseraon s he followng. Under normal condons he hea s absorbed frs n a very hn Knudsen layer. When ha layer become lqud s sll beng heaed unl reaches he crcal 88

290 emperaure whch was calculaed o be abou.4 mes he bolng emperaure. Then he lqud meal exhbs delecrc behavor and becomes ransparen o he laserbeam. Tha mples ha here canno be any laser absorpon a hs pon n he superheaed layer. All of he absorpon s concenraed hen on very small regon below where he emperaure s us below he crcal emperaure. Maeral s removed a he op of he delecrc layer by evaporaon. A he sdewalls maeral s forced away by he plasma pressure on he lqud whle a he end of he pulse when he pressure suddenly drops maeral s removed by he bolng of superheaed lqud. In hs Dsseraon a coeffcen of hea ransfer beween he nodes n he hole and he hole wall has been consdered. The emperaure of he hole s wall was o be T wall and of he node n he vapored regon T h. The resulng hea loss across he hole boundary can be expressed by modfyng Eq. 4.e. q h h( h wall = h T T ) (368) where h h s he wall hea ransfer coeffcen defned by h = h h. (369) h ch rh In Eq. 369 h ch and h rh s convecve and radave hea ransfer coeffcen respecvely. If sheld gas s used or any oher nole for asssng gas n mcromachnng applcaon hen convecve hea ransfer coeffcen h ch was deermned by Gordon and Cobonque (96) 89

291 .5.33 h = gas ch 3Re Pr. B (37) where B s he e plae dsance Re s he Reynolds number a e ex Pr s he Prand number for gas and s he hermal conducvy of gas. The radave hea ransfer coeffcen h rh was deermne from s defnon q r rh ( h wall = h A T T ) (37) and he Sefan-Bolmann Eq. 59 from whch derves ( h h wall wall h = σ A T T )( T T ). (37) rh The melng phase change process was acheved by calculang wall T for wall surface grd pons and f exceeded he bolng pon of he maeral hen ha grd pon was deemed o be ransparen and he ncden power fell on he grd pon below afer sufferng some absorpon and so on hrough he enre subsrae whle he ransparen grd pons ep her hgh emperaure. In real condons ha would be he case because n he vaporaon one durng laser neracon me he hole s flled wh ho vapor and plasma Absorpon coeffcen consderaons Frs ssue s he effec of maeral removal on he aenuaon of he laserbeam a pon below he hole. Examnng Eq. 39 one sees ha he equvalen nernal power generaon due o he laserbeam s reduced exponenally wh he dsance he beam propagaes whn he maeral. As a resul Eq. 39 can be modfed o 9

292 Q( x y ) = A exp( µ ) µ I exp ( x y ) { [ µ ( µ µ ) wsp ] } s v s (373) where wsp s a record of he number of vapored nodes above he node () a me. Ths s mporan for he laserbeam decreasng he amoun of aenuaon maeral hrough whch he beam mus pass n order o reach he node (). Selecon for he values for absorpon coeffcens µ µ s and µ v s dependen from he maeral properes of he rradaed surface. To oban an accurae heorecal value for he absorpon coeffcen seems o be raher dffcul as because no only he nverse Bremssrahlung effecs and Fresnel absorpon on he walls of he hole are presen bu also neracon wh some of he lqud droples deached from he lqud layer of he drll hole occurs. These droples would acually bloc some of he ncden radaon by absorbng he ncden energy or parally reflec bac. The acual value depends on he lqud densy and vscosy as well as on he npu laser beam nensy. Is value can be nferred for dfferen maeral and hcness. The frs sep consdered n hs Dsseraon was lnear model whch could be vewed as very realsc provdng ha he predced resuls correlae well wh observed resuls. Analyng Eq. 373 and he fac ha f he power enerng he meled one undergoes exponenal decay accordng o Beer-Lamber law one can carry ou he followng analyss. The range of reasonable values for absorpon coeffcen was calculaed by consderng he fac ha a maeral (34 sanless seel) has a ypcal surface reflecvy a 9

293 .64 mm radaon of 6% he remanng 4% of ncden power from he laser s suffcen o nae a mel of maeral a whch pon he mel behaves as a blac body and absorbs % of he power. Absorpon aes place n he plasma feld due o hgh elecron densy of he ho vapor (~6 deg K). Based on expermenal daa from he leraure Maumder and Seen (98) ha he laser peneraon a 3 W no seel acheves a deph of.3 mm vrually ndependen over speed range 5-5 mm/s. A 5 W deph was.5 mm and a 7 W deph of.7 mm. To calculae ha only an absorpon coeffcen of 67 m - s less han 4% of he power avalable a he maxmum peneraon deph.e. less han he power requred o nae he melng. Thus for he 34 sanless seel sample (maeral hcness equals.76 mm) frs consderaon n hs Dsseraon was ha he nal µ was chosen o be 67 m - (.67 mm - ) a value correspondng wh ha used by ohers researchers. The precse value depends on he lqud densy and vscosy as well as on he npu laserbeam nensy. Is value can be nferred for dfferen maeral and hcness. In order o pursue a smpler way of carng ou he overall calculaons of absorpon coeffcens for lqud and vapor he average Fresnel absorpon was fed no exponenal curve so ha Fresnel and Bremssrahlung mechansms were o be descrbed approxmaely (Nedrg and Bosanoglo 996). I was found ha negraed absorpon coeffcen once has average nsde he hole n he range - m - dependng on he process parameers. If a lnear approach was assumed values of absorpon coeffcens were n he followng ranges: for sold phase µ s =67m - for lqud µ l =45 m - and for he vapor phase µ v = m -. The mplemenaon of a 9

294 lnear absorpon coeffcen reduces he compuaon me whle eeps an accepable agreemen wh expermen daa bu dd no creaed realsc hole profles when he compuer program was runnng and resuls were compared o expermenal daa. Absorpon coeffcens menoned above were very dfferen dependng on a specfc applcaon. Nedrg and Bosanoglo (996) nerpolaed he exncon coeffcen usng power funcon. T µ [ T ] = 7.7. (374) Tv I can be seen from Fg. 3.6 (Shen and Zhang ) ha he effec of he emperaure-dependen absorbance s greaes a he surface and hen decreases wh deph. Fg Temperaure dsrbuon n an alumnum plae. 93

295 Based on Eq. 374 absorpon of he radaon penerang he worpece was consdered o be dependen of emperaure and based on Fg. 3.6 s dependen on deph and laeral poson of he laserbeam. Implemenaon of a consan absorpon coeffcen would reduces he compuaon me bu wll no mae realsc hole profles. Based on he fndngs n leraure and ral and error n our FDM compuer program absorpon coeffcen was chosen as funcon of emperaure and poson (radal and axal) µ[xyt(xy)] as T ( x y ) µ [ x y T ( x y ) ] = µ ( ) (375) Tc where µ nal µ ( ) = µ fnal exp (376) and crcal emperaure T c whch s calculaed o be abou.4 mes he bolng emperaure. The accoun was also aen for he emperaure varaon whn he plasma flled eyhole and was assumed o be n he range beween vaporaon emperaure T v and crcal emperaure T c and a he op surface plasma emperaure T p o be abou mes T v. Equaon 376 was esablshed based on he fac ha he energy deposon law follows a quas-nverse exponenal dependence whch means ha for hgh absorpon coeffcen much more energy s deposed a he near enrance of he hole han a s boom. Absorpon coeffcen defned n Eq. 376 s smlar o equaon for surface enson (Nedrg and Bosanoglo 996) and s based on he fac ha absorpon of he laser energy s hgher a he walls of he eyhole cavy han n he cener whch 94

296 corresponds o he hgher emperaure n he cener of he laser beam han a he edge of he beam (he hgher emperaure he lower he absorpon coeffcen). Summarng he absorpon of he radaon penerang he sample was consdered o be no only dependen on deph and laeral poson of he laser beam bu also on emperaure of he maeral under consderaon. In he cenral poron of he laser beam he emperaure s hgher hen a he wall a a eyhole and he absorpon a he walls seems o be hgher hen n he cener of he beam. The emperaure varaon whn he plasma flled eyhole was assumed o be n he range beween vaporaon emperaure T v and crcal emperaure T c o be.4 mes vapor emperaure T v. The model proposed n hs Dsseraon could be vewed as very realsc provdng ha he calculaed resuls correlae well wh he observed resuls. The nex dffculy arsng n modelng of laser neracon wh maer s smulang he effecs of he hgh pressure resulng from subsurface vaporaon or he plasma formaon. Ceran smplfcaon on he problem formulaon was deemed an approprae rade-off for he presen me hus he assumpon of consan pressure (.e. amben pressure) model was adoped n he Dsseraon. Afer performng prelmnary expermens and some calculaons of amoun of maeral removed durng mcrodrllng a lnear funcon was adoped n he compuer code whch deermnes rao of amoun of removed maeral (vapored and eeced) o he oal maeral of he holes. 95

297 Enhalpy consderaon Frequenly enhalpy of vaporaon s assumed o be consan. The enhalpy srongly nfluences he vapor pressure whch can be evaluaed from he Clausus- Clapeyron relaonshp gven by Eq. 67. Descrbng evaporaon ranson he lqud surface layer formed whle he laser pulse moves no he maeral a he rae deermned by he quany of vapor expelled. As he emperaure of he lqud molecules ncreases he addonal energy needed o free molecules from he bndng forces decreases. The laen hea of vaporaon herefore decreases wh he emperaure unl he crcal emperaure s reached. The laen hea of vaporaon can be wren n erms of he surface emperaure as (Ylbas el a. 996; Nedrg and Bosanoglo 996) ( ) [ ( )] T x y H = v T x y H v (377) Tc where H v s he laen hea vaporaon a absolue ero. Newly defned parameer can / be nroduced now n he componen of he boundary velocy due o evaporaon v dv defned by Eq. 64 n BTs H exp v πm BT = s (378) and beng a funcon of emperaure now can be defned as v dv = BTs πm H exp v T ( x y ) Tc T B s (379) 96

298 3..7. FEM - TAS soluons To verfy own analycal soluon and also o compare hem o own numercal soluons descrbed n Secon 3..8 we employed Thermal Analyss Sofware (TAS) whch s a well nown n ndusry hermal fne modelng sofware wren by Harvard Thermal Inc. (Rosao 4). TAS s a fne elemen program. Fne elemen modelng s a process of lnng he mahemacal equaons and solvng hem smulaneously o gve a soluon o he problem under consderaon. Formulaon and neracon of laser energy wh 3D slab of maeral sared from buldng 3D model as shown n Fg. 3.7 by dvdng no some 4 erahedral elemens. Nex sep was o apply boundary condons hermo physcal properes and fnally apply he hea source. Program s very generc and here s always more han one way of programmng n TAS. The way we defned he hea source was as follows: we defned a ransen surface hea load o he cylnders we bul for he purpose of paramercally changng he se and hea load condons. In oher words he hea source vared spaally and emporally and geomercally n form of crcles appled o he op surface. Fg D model for TAS calculaons used n hs Dsseraon. 97

299 TAS conans possbly of usng emperaure dependen hermo physcal properes as well as a new ablave module o solve ablang problems. We ncorporaed no TAS emperaure dependen properes defned earler n Secon.7.4. Typcal soluon oupu of he emperaure dsrbuon across he plane perpendcular o op surface of model n -drecon s shown n Fg. 3.8 and a he op of he worpece s shown n Fg Program s very powerful and useful for paramerc sudy of dfferen emperaure depended hermo-physcal parameers and laser beam spaal and emporal varaons bu does no consder any of he phase changes of he maeral under large hea load. Tha s a reason why n order o successfully predc he profle of he drlled hole own program was wren. Fg Temperaure dsrbuon of a ypcal cross-secon of he worpece n - drecon usng TAS. 98

300 Fg Temperaure dsrbuon of a ypcal cross-secon of he worpece n x-ydrecon usng TAS (op surface) FDM soluons Usng Borland Turbo Pascal MahCAD Quc Basc Surfer and Grapher programs he fne dfference algorhm Fg. 3.3 was mplemened. A begnnng of he program he user frs needs o choose he npu daa fle maeral daa he nerface would calculae and oupu he me sep n order o acheve convergence (sably crera) as descrbed n Secon

301 Sample geomery Thermal properes Physcal properes Compuaon seup Creae FDM grd Begn User npu Error checng Esablsh sably crera Calculae properes Repea o max coun/me Yes Node nsde No Selec power dsrbuon Unform power dsrbuon Gaussan power dsrbuon Trangular power dsrbuon Measured power dsrbuon Temperaure D cross-secons Calculaed Power Updae properes Chec for he phase change Evaluae node emperaure: T = T T Temperaure 3D llusraon-surface T = T End Fg Flow char for he FDM process. 3

302 Then he user can accep ha me sep or npu anoher me sep whch s less han he calculaed maxmum me sep also npu number of eraons o sp he daa oupu o dsplay n he fle oupu n order o save some ds space. In addon here are several more opon o choose from o consder expermenally measured laserbeam characersc Gaussan beam dsrbuon square or rangular laserbeam dsrbuon. The oher opons o choose from are few dfferen emporal characerscs of he laser pulse square rangular or measured. Then he program wll compue he overall emperaure profles and oupu he emperaure convecon and radaon hea loss of any nodes on he sample and melng soherms no fles. A 3D graphc of he emperaure dsrbuon of any node a he sample can be obaned by lnng daa oupu fles drecly o MahCAD programs whch was wren for he purpose of llusrang and nerpreng he resuls of laser maeral neracon. Typcal profle of he mcrodrlled holes (usng a color palee soherms) are shown n Fg. 3.3 Fg. 3.3 and Fg Tha way one can oban a emperaure dsrbuon a any cross-secon of he sample a any me from he begnnng of he pulse and afer when he laserbeam ceased o exs. In addon wo oher programs were used: MahCAD and Grapher o demonsrae he surface emperaure dsrbuon a any surface perpendcular o mpngng laserbeam a any me from he begnnng of he laserbeam neracon example of whch s llusraed n Fg. 3.3 and Fg

303 deph µm lengh mm Fg Example of a D emperaure dsrbuon around he hole cross secon profle could be obaned from he compuer program a any me durng or afer he end of he laser pulse of he worpece n he axal cross secon plane of he laser beam. Compuer program s usng he worpece s emperaure dependen parameers whch were found n he leraure bu program needs also he oher mporan npu parameers emporal and spaal laserbeam characerscs. Those parameers were no avalable n he leraure. They were no avalable from laser manufacures eher. The use of assumed parameers ha approxmae characerscs of laserbeam leads o erroneous resuls especally for deermnng he hole profle characerscs. Therefore o effecvely model neracon of laserbeam wh maerals he laser beam s characerscs mus be measured and hen lned drecly o he compuer algorhm Secon 4.. 3

304 lengh x - µm wdh x - µm b) a) Fg Example of emperaure dsrbuon around he hole profle could be obaned from he compuer program a any me durng or afer he end of he laser pulse of he worpece: a) D and b) 3D surface emperaure dsrbuon aen from he boom of he worpece. emperaureºk emperaure ºK b) lengh x - µm wdh x - µm a) Fg Example of emperaure dsrbuon around he hole profle could be obaned from he compuer program a any me durng or afer he end of he laser pulse of he worpece: a) D and b) 3D surface emperaure dsrbuon aen from he mddle of he worpece. 33

305 4. Expermenal nvesgaons The purpose of hs chaper s o presen and dscuss some represenave resuls of he expermenal wor on neracon of pulsed laserbeam wh sanless seel and sngle crysal slcon samples. Expermenal secon sared from Chaper 4. n whch he laser sysem (ncludng fber opc aachmen) used n expermens hroughou preparaon of hs dsseraon s descrbed. Temporal spaal sably and oher characerscs of he laser sysem used n hs Dsseraon are presened n Secon 4.. Represenave resuls of laser mcrodrllng on maerals are presened and dscussed n Secon 4.3 for he effec of laser parameers on he mcrodrllng resuls. Correlaons beween compuaonal and expermenal nvesgaons for laser maeral processng are ncluded n Secon 5 follows wh Secon 6 whch conans summary of observaons done n hs Dsseraon and some recommendaons for he fuure fnshed wh conclusons and fuure wor n Secons 7 and Nd:YAG laser sysem For expermenal sudy for hs Dsseraon a Nd:YAG class 4 pulsed laser model KLS 6 manufacured by Lasag Indusral Lasers Corporaon was used Fg. 4.. The maxmum values whch should never be exceeded are: laser s raed a a maxmum average oupu power of W wh a sngle pulse capably up o 3 J and volage up o 37 and wavelengh λ =.64 µm. The pulse duraon s varable beween. o ms (Lasag 997). 34

306 The oupu energy was measured wh a bul-n power measuremen un. Pulse energy represens he energy comng ou drecly from he resonaor and s calculaed by negraon over he pulse lengh and dsplayed on a LCD dsplay. The Lasag s Nd:YAG laser sysem s also equpped wh He:Ne laser class connuous wave laser for algnmen purposes. Is maxmum oupu s.5 mw. Fg. 4.. Laser sysem used n hs sudy. When Nd:YAG laserbeam wll focused o a small spo hen one can drll cu mel and burn hrough any maerals. Ths neracon of laser energy wh varous maerals s a subec of hs Dsseraon. One of he unque feaure of Nd:YAG laser sysems s ha hey are flexble and fber cable can be aached o hem n order o ransm he laser beam o a worpece. 35

307 4... Fber opc aachmen Fber opc (FO) can be long hn srand of very pure glass abou he dameer of a human har and used o ransm lgh sgnals over long dsances. A sngle opcal fber has he followng pars: core (hn glass cener of he fber where he lgh ravels) claddng (ouer opcal maeral surroundng he core ha reflecs he lgh bac no he core) and buffer coang (plasc coang ha proecs he fber from damage and mosure) Fg. 4.a and Fg. 4.b. a) b) Fg. 4.. Fber opc connecor: a) phoograph b) schemac. The lgh n a fber-opc cable ravels hrough he core by consanly bouncng from he claddng (mrror-lned walls) a prncple called oal nernal reflecon. Because he claddng does no absorb any lgh from he core he lgh wave can ravel grea dsances. However some of he lgh sgnal degrades whn he fber mosly due o mpures n he glass. The exen ha he sgnal degrades depends on he pury of he glass and he wavelengh of he ransmed lgh (for example 85 nm = 6 o 75 percen/m; 3 nm = 5 o 6 percen/m; 55 nm s greaer han 5 percen/m). 36

308 When lgh passes from a medum wh one ndex of refracon (n ) o anoher medum wh a lower ndex of refracon (n ) bends or refracs away from an magnary lne perpendcular o he surface (normal lne). As he angle of he beam hrough n becomes greaer wh respec o he normal lne he refraced lgh hrough n bends furher away from he lne. A one parcular angle nown as he crcal angle he refraced lgh wll no go no n bu nsead wll ravel along he surface beween he wo meda (sn [crcal angle] = n /n where n and n are he ndces of refracon [n s less han n ]. If he beam hrough n s greaer han he crcal angle hen he refraced beam wll be refleced enrely bac no n whch s nown as he oal nernal reflecon llusraed n Fg. 4.3 even hough n may be ransparen. In physcs he crcal angle s descrbed wh respec o he normal lne. In fber opcs he crcal angle s descrbed wh respec o he parallel axs runnng down he mddle of he fber. n n Fg Toal nernal reflecon n an opcal fber. 37

309 The core layer of he fber has a larger refracve ndex han he ouer claddng layer deally % of he ncden laser energy s refleced up and down and s rapped nsde he core under wo assumpons. Frs he laser ncden on he fber above a crcal angle o allow oal nernal reflecon oherwse par of he lgh s refleced and par s ransmed whch means lower effcency whch s no desrable. Anoher assumpon s ha he core doesn' absorb any energy as lgh s ransmng hrough. Unforunaely all maerals absorb a poron of lgh energy or aenuae he lgh as lgh propagaes hrough. Modern opcal fber has an energy loss as small as 4dB/m. Consderng absorpon one mus eep n mnd ha he absorpon of a maeral s srongly relaed o he wavelengh. For praccal fbers he core maeral has low absorpon for he.6 mcron Nd:YAG laser bu he absorpon s much hgher a.6 mcron (CO laser) or a U wavelengh. Hgher absorpon means hgher percenage of he laser lgh beng ransformed no hermal energy. Ths s he reason why only Nd:YAG laser s suppled wh he ably of beam delvery wh fbers. Laser beam delvery hrough fbers frs found applcaon n elecommuncaons whch only use very small energy levels. In elecommuncaon he maor concern s wheher nformaon can be ransmed whou dsoron and a hgh speed. In laser machnng however he maor concern s ha wheher suffcen hgh power can ransmed o he worpece wh hgh beam qualy. Snce laser maeral processng requres very hgh rradance up o W/cm a up o 5 lowas (W) of average power he couplng of hgh power lasers no 38

310 mulmode fbers becomes an mporan ssue. To ge hgh beam qualy from he oupu of he fber however sngle-mode fbers could be he soluon. The sngle mode fber opc cable s consruced wh a cenral core ha carres he laser a claddng regon ha acs as a mrror o he laser such ha all he lgh remans n he core and an ouer meal ace o proec from lgh leaage. The core dameer of he fber can vary n dameer accordng o he requred spo se needs and npu laser power. In erms of focused spo se he core dameer drecly affecs he fnal focus spo se and herefore he pea power densy. For example a 3 mcron core fber has half he focused spo se of a 6 mcron core fber and so has four mes he pea power densy. However he 6 mcron fber shown n Fg. 4.4 has no power lmaons whereas he 3 mcron fber does. The selecon of fber s applcaon relaed. Fg µm end of he FO cable (lef pcure) wll be conneced o he focusng head as llusraed on he rgh pcure wh he arrow. Hgh precson maeral removal s needed n all mcro-machnng applcaons. Excmer lasers are wdely used for hs purpose. Excmer process s generally ablave 39

311 and reles on he shor laser wavelengh. Alhough hghly precse allows only a lmed deph of maeral (- mcrons) o be removed per pulse. Nd:YAG processng s generally hermal and allows removal dephs of a couple of hundreds of mcrons wh a sngle pulse. When a hgh beam qualy s used hs process can provde suffcen precson and feaure se for many applcaons. Nd:YAG laser machnng sysem s more flexble han excmer lasers. An addonal benef of he Nd:YAG sysem s ha lgh can be delvered o he worpece hrough opcal fbers. I s ofen easer and less expensve o process he worpece by delvery laser beam hrough he fber and focusng opcs across raher han movng eher he worpece or he laser self. Fg µm end of he FO cable conneced o FO head. Numercal Aperure (NA) for fber opcs measures he dfference beween he core refracve ndex (n ) and he claddng refracve ndex (n ): 3

312 NA = n n. (38) Couplng lgh from an ndusral Nd:YAG sysem shown n Fg. 4.4 and Fg. 4.5 no a sngle-mode fber poses praccal problems. Only par of he laser energy s coupled no he core of he fber he remander of he laser energy s coupled no he fber claddng whch can lead o hermal damage a he pon where he claddng modes are removed. In any praccal suaon he claddng of a glass opcal fber s covered by a proecve buffer layer. Even for M = only 7% of laser energy s coupled no he core. If a reflecve maeral such as a meal coang s placed n opcal conac wh he fber claddng wll absorb a small percenage of he claddng lgh on each reflecon. To maxme he amoun of energy o be coupled o FO cable algnmen of FO cable o he nernal laser opcs se Fg. 4.6 s very mporan and should be verfed ofen. Fg FO laser beam delvery subsysem o wo FO cables for laser mcromachnng. 3

313 4... Fber opc opcal consderaon A shorer fber opc cable produces lle modfcaon o he npu laserbeam. On he oher hand long and large n dameer fbers cause he oupu o be ndependen of he npu dsrbuon. Based on experence and leraure (Huner e al. 996) he fber has sgnfcanly alered he laserbeam profle and mass many properes of he source laser. The focused beam profle s subec o aberraons and magnfcaon effecs of he mpngng lens he profle on he fber oupu face. The lengh of he fber was no long enough o allow he formaon of he op-ha profle expeced for a long sep-ndex fber. Insead a poned op profle was generaed. Smaller se fbers produce less degradaon of beam qualy. The wors case beam qualy from fber opc cable can be esmaed usng he convenonal defnon of M. Ths defnon s based on a Gaussan beam and he 86% energy enclosure beam was s used and consderng he angle of dvergence one can oban he followng M π =.86r fber arcsn( NA) (38) λ where r fber s he fber radus NA s he numercal aperure of he fber. Geomercally he se of he mage of he fber face s gven by r = m r r. (38) b fber blur where m s he absolue value of he magnfcaon of he magng sysem and r blur s he blur nroduced because of dffracon effecs or aberraons n he opcal sysem and maxmum aberraon can be esmaed o be r blur =. 5 m r. fber 3

314 Clearly smaller fbers are preferred n order o ge he bes beam qualy. However he laser self mposes some consrans. The laserbeam focused spo se has o be smaller han he fber core o avod hea effecs and allow for he mechancal olerances of he fber opc connecors. The mnmum fber opc dameer d o be seleced as a funcon of M and NA s gven by Huner e al. (996) d ab λ = M K (383) c π an(arcsn( NA)) where K s he aberraon of he fber opcs and λ s wavelengh a s spo se f he collmaed beam exng he laser b s he focused beam se of he lease beam on he fber face and c s he desred fber fll facor. If a and b are.5 and c s.8 a he wavelengh of 64 nm NA=. and M K=7 he mnmum fber dameer would be 65 µm. The prmary reason for damagng he hgh power FO connecors s heang n he connecors. If he fber s npu and oupu opcs are properly desgned mos of he connecor heang comes from spurous reflecons or refracons a he fber end faces. The prmary causes are he refracve ndex dsconnuy a he fber surface (Fresnel losses) and scaerng from he surface mperfecons. The Fresnel losses can be mnmed by applyng anreflecon coangs on he fber and opmng he surface qualy. The mporance of he choces made n a laser beam delvery sysem can be seen by analyng he power levels requred for mcrodrllng or weldng wh rradance 33

315 hreshold. The laser beam power requred o exceed he hreshold rradance can be compued by π I P =.86 T ω F (384) where I s he hreshold rradance ω s he focused 86% energy enclosure radus T s he power ransmsson and F s a facor ha depends on he characersc of he fber. For sep ndex fbers F= bu for graden-ndex fbers s greaer han one. In concluson reducng se of ω reduces he oal laser power requred and ha can be done by usng a smaller core dameer fber usng graden-ndex fber or shorer focal lengh lenses. 4.. Characeraon of he laser beam Characeraon of he laser beam consss of measuremens of average laser power energy per pulse pea power per pulse and rradance dsrbuon of he laser beam as a funcon of poson across he beam (spaal dsrbuon) and me from he naon of he oupu (emporal dsrbuon). In pracce one also needs o measure he beam spo se along he opcal pah -axs and calculae he oher beam parameers le: deph of focus and beam dvergence. In mcrodrllng applcaons very mporan parameer s he ransverse mode of he laser beam whch s gong o be dscussed n hs secon also. In addon emporal sably of he laser pulse should be also checed whch s descrbed a he end of hs secon. 34

316 4... Average power and energy measuremens of laser beam Deermnaon of he average laser power and he average energy per pulse was measured wh Coheren Labmaser Power Deecor Model LM Fg. 4.7 and wh nernally bul energy meer. Expermenal seup for hs procedure s llusraed n Fg. 4.. The enre beam was focused ono he desgnaed arge n he power meer o nsure ha he full energy of he beam was delvered o he meer. For he sable laser sysem one may consder ha he oal energy measured by he power meer s equally delvered by each pulse and consequenly Eoal P oal E pulse = = (385) n n pulse pulse where E pulse s he average energy delvered per pulse E oal s he oal change n energy of he meer durng he expermenal perod n pulse s he number of pulses ncorporaed no expermen P s he average laser power and pulse s he oal me of he power meer measuremen. The Lasag s Nd-YAG laser sysem s a md-power sysem capable of delverng a maxmum average power of 3W and a maxmum energy per pulse of abou 5 J. boh for 4 msec pulse duraon. These values agree well wh manufacurer s provded daa. 35

317 Fg Coheren Labmaser Power Deecor Model LM Measuremens of emporal dsrbuon In he leraure (Boulmer-Leborgne e al. 993) he emperaure calculaons are usually made wh recangular laser pulse shapes. Evoluon of he emperaure of a arge surface rradaed by a recangular pulse s shown n Fg The vaporaon hreshold occurs as soon as T s > T vap. (T s s he surface emperaure T vap he vaporaon emperaure of he arge). Wh oher lasers he negral mus be evaluaed usng he exac emporal shape of he laser pulse. Usng dfferen pulse shapes dfferen rradaed surface emperaure evoluons are deermned resulng n dfferen laser beam-arge neracon effecs. Two dfferen pulse shapes modulaed by varyng he acve gas mxure concenraon n he laser can be obaned usng a TEA-CO laser are llusraed n Fg. 4.9 (Boulmer-Leborgne e al. 993). Evoluon of he emperaure of a anum arge surface T s rradaed by hose wo dfferen pulses A and B are shown n 36

318 Fg. 4.. The vaporaon hreshold occurs as soon as T s > T vap (T vap s he vaporaon emperaure of he arge). I can be observed ha he pulse shape (B) resuls n a longer surface heang me nduced by s long al. Thus he vapored maeral quany s larger wh hs pulse shape han wh he shor one (A) bu he vaporaon hreshold occurs laer. The quany of vapored mass depends on he laser wavelengh and on he laser energy absorbed by he sample. Fg Evoluon of he emperaure of a arge by a recangular pulse (Boulmer- Leborgne e al. 993). Fg Dfferen pulse shapes from TEA-CO laser: A and B relae o wo dfferen gas mxure concenraons n he laser (Boulmer-Leborgne e al. 993). 37

319 Fg. 4.. Surface emperaure evoluon resulng from TEA-CO laser pulses for T arge: A and B relae o wo dfferen gas mxure concenraons n he laser (Boulmer-Leborgne e al. 993). Laser beam characersc of our laser sysems were no avalable from he manufacurer and had o be measured durng hs sudy. These measuremens were made usng a phoo-dode ha was placed as s shown n Fg. 4.. Calbraon of he dode oupu o produce absolue power measuremens for a gven pulse was accomplshed by relang he dode measuremens o measuremens of he energy per pulse. Normalng he power curve relave o he maxmum oal beam power P max recorded for a gven pulse and me τ relave o he pulse lengh p one can wre P( τ ) = Pmax p( τ ) (386) where τ = (387) / p P max = p E pulse p( τ ) dτ (388) where p(τ) s he normaled beam power as a funcon of normaled me no he pulse and E pulse s average energy delvered per pulse. 38

320 Fg. 4.. Seup for measuremen of he emporal laser beam characerscs. The relave power curve can be approxmaed by a leas-squares error polynomal and evaluaon of negral of Eq. 388 can be accomplshed numercally by applyng Smpson s rule drecly o he numercal daa or by negraon of he leas-squares error polynomal n a = p( τ ) = τ (389) where a are he coeffcens of he leas-squares f and n s he order of he leas-squares f polynomal. Subsung Eq. 389 no Eq. 388 and performng he ndcaed negraon yelds 39

321 E n pulse = n a p = P( τ ) = (39) a τ Normaled nensy Normaled me τ Fg. 4.. Typcal measured normaled emporal characerscs of he Nd-YAG laser beam whch shows normaled laser beam nensy p(τ) vs. normaled me τ. Fg. 4.3 shows more examples of measured emporal characersc of Nd:YAG laser beam (Nowaows 99). 3

322 Fg Temporal laser beam characerscs for dfferen pulse lenghs Measuremens of spaal rradance dsrbuon The ncden laser beam has a spaal nensy dsrbuon whch n drllng s usually a Gaussan dsrbuon produced by a laser operang n TEM mode (M ). To measure spaal characersc of he laser beam beam nensy profler (Beamew Analyer sysem) was used (hans o Dr Marns of Draper Laboraores) was avalable. A ypcal Gaussan dsrbuon was measured on he Argon laser Fg To measure our Nd:YAG laser beam wh hs devce we encounered a couple of problems. Frs a beam nensy profler s raher large pece of equpmen and due o space lmaons nsde he enclosure box of he Nd:YAG laser sysem (laser beam s mpngng vercally on he worpece as seen on Fg. 4.5) and he second ha for nfrared wavelenghs a 3

323 specal flers were requred. Because of hose reasons our own laser beam nensy measuremen sysem was buld and used n he course of hs Dsseraon. Typcal measuremen seup of such a sysem s shown n Fg Fg Gaussan spaal laser beam dsrbuon measured by beam nensy profler. Mos mporan par of he sysem for measurng spaal bema dsrbuon was Newpor s phoo deecor model 88-F-SL wh pcowas dgal opcal power meer Model 83-C Fg. 4.6 whch s essenally hs same as ha used n emporal beam characersc excep one modfcaon. Orgnally he beam spler was used bu n he case of measurng spaal beam nensy we can no afford a rs of any dsoron comng from addonal opcs n he sysem. Dsoron could be due o he asymmerc spo formed as a resul of reflecon of an mperfecly collmaed beam a he nclned beam spler. Snce he goal of hs spaal beam characersc measuremens was o deermne nensy of he beam n he plane of ha dode han dsoron of he beam cross-secon could no be oleraed. Posonng of he deecor n he pah of he beam requred use of 3

324 aperure and proecng flers agans poenally hgh ncden flux. Consequenly a few neural densy flers were added o aenuae he ncden beam. fber opc cable y Nd- YAGlaser sysem x 8 7 beamspl er M M 6 phoo dode conroller monor PC - laser energy meer MM - nernal mrrors 6 - opcs 7 - phoo dode 8-3Dmoored sage power meer Fg Seup for measuremen of he spaal laser beam characerscs. In addon o elmnae unwaned bacground radaon whch mgh add o a nose n he readngs an nerference fler passng only a narrow band of wavelenghs of neres (64 nm) was added n fron of he deecor whch n urn aenuaed he orgnal beam even more. The resulng oal aenuaon of he laser sgnal ncden ono he phoodode was on he order of -. Ths value of aenuaon s no an exaggeraon consderng he fac ha we are dealng wh nensy of he beam n he order of W/m. The deecor was fxured on a 3D mcro compuer conrolled mcro-sage wh poson accuracy of abou.3 µm. The x-y posoner enabled scannng of he laser beam n laeral drecon of beam propagaon usng raser ype of scannng. 33

325 Fg Dgal opcal power meer used for expermens. Daa (nensy) were colleced a each pon of scannng range ( nch by nch). The nerfacng sofware for hs expermen as well oher dode measuremens descrbed were wren n Quc Basc durng hs sudy specfcally for hs as. In addon o collecng raw daa from he phoo dode anoher program was wren o presen he daa n D or 3D graphcal dsplay usng he leas square approxmaon echnque o oban he rradance dsrbuon as a funcon of boh me and poson (x-y). Fg. 4.7 shows a ypcal resul for sable quas-gaussan oupu of he laser. Fg Spaal laser beam dsrbuon measured a close proxmy of he focal plane poson. 34

326 Please ae a noe of he fac ha Fg. 4.7 represens resuls of a me-varyng spaal dsrbuon of he laser beam rradance.e. ha could be dfferen for dfferen mes durng emporal varaon of he laser pulse. A hs me for purpose of hs analyss was assumed ha he relave rradance of he laser beam s consan. Sngle ypcal funcon was used o descrbe he relave relaon beween spaal and emporal varaon of he beam. Knowledge of a me hsory of he oal laser beam power descrbed n Secon 4.. and he presen measuremen nformaon wll be enough o deermne absolue values of a spaal rradance funcon a any gven me durng course of he laser pulse. Defnng rradance as he rae of energy flow per un area one may negrae over he rradaed area o arrve a he oal rae of energy flow across he gven surface. The laser power P(τ) as a funcon of normaled me no pulse can be expressed as: P (τ ) = I( x y ) da (39) A where I(xyτ) s rradance a a pon specfed by coordnaes x and y a a gven me τ. Expressng he rradance as a produc of me-dependen funcon I (τ) and posondependen funcon I R (x y) P ( τ ) = I ( τ ) I ( x y) da (39) A R where I( x y τ ) = I ( τ ) I ( x y). (393) R The focused beam radus w s specfed as he dsance beween he cener of he beam and a pon where he nensy s reduced from s maxmum value a he bean 35

327 cener by a facor of e. To faclae for non-crcular cross-secons hs /e radus wll depend on he drecon under consderaon. The beam radus along he x-axs R x can be se o w. Defnng dmensonless parameers ξ and ψ o express he poson coordnaes x and y and he eccenrcy e cc of he laser spo n erms of w yelds ξ = x / Rx = x / w (394) y ψ = y / Ry = eccw (395) e = R / R. (396) cc y x Adopng hs convenon and usng elemen of dfferenal area da = e w dξdψ (397) cc one can rewre Eq. 39 as ' P( τ ) = ecc w I ( τ ) I ( ξ ψ ) dξdψ. (398) Soluon of Eq. 398 for I (τ) yelds ( P( τ ) I τ ) = e w I ( ξ ψ ) dξdψ cc R (399) whch s he maxmum value of he spaal rradance dsrbuon a me τ. Subsung Eq. 399 no Eq. 393 resuls n he absolue me-varyng spaal rradance dsrbuon across he laser beam.e. where E pulse I R ( ξ ψ ) n = I( ξ ψ τ ) = (4) n a ecc w p I A = a τ 36

328 I A = I R ( ξ ψ ) dξdψ (4) s numercal approxmaon for he area negral showed n Eq. 4. Lamber s law for he energy delvered o he maeral n laser mcro machnng s absorbed n he surface regon accordng o Eq. 44. Combned laserbeam characersc spaal and emporal characerscs and ulng Eq. 4 and Eq. 44 and nroducng he exponenal decay yeldng a a dsance below he rradaed surface I n E pulse I R ( ξ ψ ) aτ = µ ( ξ ψ τ ) = e (4) n a ecc ω( ) p I A = where µ s he exponenal absorpon coeffcen for he gven maeral. Approxmaon of he radus of he laserbeam a any dsance from he orgn of he sample under consderaon s gven by (Nowa 99) ω ) θ ω (43) ( = f where θ s he half-angle laser dvergence and ω s radus of laser beam a he focal dsance or a he was of he beam f he locaon of he focal plane. Beer resuls usng dfferen relaonshp for he radus of he laser beam a any dsance (Zhao and DebRoy 3) defned as 37

329 38 / ) ( ± = o f D F ω ω ω (44) was used n he Dsseraon where F s he focal lengh of he las focusng opcal lens on he pah of he laser beam us before h he worpece and D s dameer of ha laser beam on hs lens. Subsuon of Eq. 44 no Eq. 4 yelds. / ) ( ) ( n A p o f cc n R pulse e b I D F e a I E I = = ± = µ ω ω τ ψ ξ τ ψ ξ (45) The power densy a any pon (xy) he poson coordnaes defned n erms of he laserbeam dmenson a a dsance below he rradaed surface (=) a me τ and usng erms defned n Secon he ne volumerc accumulaon rae of energy due o he laserbeam s expressed as. / ) ( ) ( n A p o f cc n R pulse e a I D F e a I E A Q = = ± = µ ω ω τ ψ ξ µ τ ψ ξ (46) Equaon 46 s almos he fnal expresson whch s gong o be used n he compuer algorhm calculaons descrbed n Secon wh he excepon of defnng he absorpvy coeffcen whch remans o be performed. Examnng Fg. 4.7 more closely one can noce ha he boom of rradance dsrbuon s no smooh here are some dsurbance of nensy (small waves). In order o verfy hs we had o swch o hgher gan of measurng range n he amplfer used

330 because for hgher gan nose o sgnal rao was oo hgh. By usng hgher gan and oomng n he area of neres and collecng daa we ploed hem n Fg Fg Spaal laser beam dsrbuon measured a he hgher gan of he amplfer a perpheres of he laser beam. Analyng boh funcons of he spaal laser beam dsrbuon presened n Fg. 4.8 and Fg. 4.7 we can overlap hem ogeher and oban a full mage of he laser beam. Usng mahemacal analyss of he shape of a spaal laser beam dsrbuon and wh help of MahCAD a mahemacal approxmaon of he measured spaal laser beam dsrbuon can be defned mahemacally as γ f ( ξ ψ ) =.5cos( 3(.5 ξ) 3(.5 ψ ) e.5 (47) where γ ( ξ ψ ) ψ =.5(.7ξ ).5(.7 ) (48) whch s graphcally represened n Fg

331 Fg Mahemacal spaal laser beam represenaon of a beam focused a.4 mm above focal plane poson (lbwp). Please noe ha plo n Fg. 4.9 s normaled o maxmum rradance occurrng whn he cross-secon of he laser beam Transverse mode of laser beam One of he mos mporan parameers n laser mcrodrllng of especally delecrc maerals s he ransverse mode of he laser beam. Nd:YAG class 4 pulsed laser model KLS 6 manufacured by Lasag Indusral Lasers Corporaon used n hs Dsseraon s producng a nearly TEM mode wh use of an aperure (approxmaely 6 mm n dameer). As llusraed n Fg. 4. use of he mode elecng aperures whn he laser cavy sgnfcanly alers oupu of he laser. 33

332 The sgnfcance of changng he dameer of he laser beam before he focusng lens s manfesed n he dvergence of ha beam whch rradaes he worpece. Dvergence Φ of he laser beam of radus ω lens measured a he lens focused o a spo of radus ω by a lens of focal lengh F s defned ωlens ω ω Φ = lens (49 F F snce n mos applcaons ω lens << ω. Examnng Eq. 49 a decrease n dameer of he unfocused laser beam by a facor of n resuls n a reducon of he beam dvergence by he same facor n. Snce he dvergence of he beam governs he dameers of he laser beam s cross secon along he axs of propagaon he aper of a laser-drlled hole depends on he dvergence of he rradaed laser beam. As was menoned prevously dependence of he hole aper on laser beam dvergence s governed n par by he hermo physcal properes of he worpece. Consderng one of hem hermal conducvy one should observe a sronger correlaon beween laser beam dvergence and hole aper n delecrc maerals such as slcon han n a good conducor such as sanless seel. Fg. 4.. Typcal burn paerns creaed bynd:yag laser a) wh and b) whou aperure n laser cavy (Nowaows 99). 33

333 Addonally should be observed ha he dffracon-lmed spo se of a focused laser s mos closely approxmaed by a TEM mode laser and s gven by dff λ F ω lens =. (4) π ω lens Inspecon of Eqs 49 and 4 and observaon ha by nsallng an aperure whn he resonaor cavy leads o reducon n he dameer of he laser beam ndcaes ha he nearly TEM mode should be he preferred mode of laser operaon n mcrodrllng. Sudyng some prelmnary resuls of drlled holes we have noced ha he hole shape s no round le supposed o loo for he Gaussan ype of he laser beam. There were some dsurbances around hole and he hole was no round. Then I learned ha drlled hole was by msae processed no he beam mnmum focal spo. Tha nowledge rggered he new seres of ess performed by scannng he beam n - drecon.e. axally wh he beam are summared n Table 7. Table 7. Typcal burn paerns produced by laser: a) paper was above he laser s beam was and b) paper was below laser s beam was. Dsance from = Above was POSITIE value Above was POSITIE value Below was- NEGATIE value N/A Mn

334 Max

335 Tang an average dameer of burned spos showed n Table 7 a funcon of laser beam burned spo dameer wh respec o focal poson was bul and llusraed n Fg dameer µm wdh cener rng hgh cener rng wdh ouer rng hgh ouer rng lbwp µm Fg. 4.. Analyng he spaal beam dsrbuon usng blac burn paper and as a resul deermnng mnmum beam was poson. The burned paerns showed n Table 7 are D represenaon of a spaal laser beam characersc. In order o compare o 3D spaal laser beam characersc raser scan of a laser beam was performed agan (Fg. 4.) a hs same parcular focal poson of he beam and comparson s presened n Fg. 4. whch s 3D graphcal represenaon of D burn paper nex o he graph. 334

336 neger volage neger a) b) Fg D and D laser beam represenaon of a spaal laser beam dsrbuon: a) obaned by scanng he beam wh he phoo deecor b) blac paper burn of hs same beam a hs same focal locaon of he focusng lens (abou 83.6 µm) Sably of laser beam dsrbuon For he purpose of conrollng he process resuls and for he nowledge of he repeaably s necessary o deermne sably of he laser oupu. For he laser operang n mulmode such a measure of laser sably would no be praccal due o he random naure of laser oupu. Snce he maory of wor performed n hs Dsseraon was done usng one pulse Gaussan ype of laser he nvesgaon of laser sably was conduced n an approxmae fashon by consderng varaon of he pea value of he average beam rradance as a funcon of a seres of sngle pulses fred by he laser n a sequenal manner. The seup for measuremen of he laser sably was dencal o he seup for deermnaon of emporal laser beam dsrbuon Fg. 4.. Resuls shown n Fg. 4.3 ndcae ha he laser sysem s very sable. A he begnnng afer he laser sysem was 335

337 urn on some small flucuaons were observed for he frs few pulses. In order o se some sandard for measuremens of laser s performance for hs Dsseraon was decded o consder he oupu of he laser afer nal warm up of he sysem and afer few ral pulses nex pulse was consdered as ypcal and consdered sable. Funcon llusraed n Fg. 4.3 s a funcon of laser rradance versus volage appled o he flash lamp. As llusraed he maxmum varaon of he frs wo pea pulse rradance of he beam was no greaer han 5% of he maxmum-recorded value volage me sec Fg Sably of he Nd:YAG laser beam Expermenal nvesgaons of laser mcrodrllng Ths par of expermenal poron of hs Dsseraon deals wh a sudy of mporan parameers n he conrol of laser mcrodrllng process. In laser mcrodrllng he basc quanes le he drllng rae he amoun of vapored maeral he hole 336

338 dameer or hea affeced one can be drecly measured very accuraely bu he dffcul par s o descrbe he laser beam neracon nsde he hole and he hole profle. Laser mcrodrllng s already an esablshed mehod for producon of hgh precson feaures le va formaon especally n mcroelecroncs ndusry (Han and Prypunewc 3a 3b). Alhough holes have been made n hs Dsseraon e.g. Fg. 4.4 hey are no perfec. To mprove qualy of he laser mcromachned holes he followng ssues should be addressed and resolved:. Recas layers causng cracs;. Taper of he drlled hole may be oo large; 3. The shape of he hole s no sasfacory; 4. Hgh aspec rao s dffcul o reach; 5. Coangs around he hole end o chp off; 6. Redeposon around he hole needs pos processng. Fg Schemac of laser drlled hole profle. 337

339 In general laser drllng s governed by energy balance beween he rradaed laser energy and he conducon hea no he worpece he energy losses o he envronmen and energy requred for phase changes n he worpece. Expermens were conduced as a par of hs Dsseraon n he CHSLT laboraores a he room emperaure whou any sheldng gas wh he Nd:YAG laser sysem Fg. 4. o valdae effecs of hese characerscs. Parcular aenon was dreced o he case of drllng wo dfferen maeral 34 sanless seel and sngles crysal slcon. Thcness of he frs maeral was abou.76 mm and he second one slcon abou.3 mm. The seup used for mcrodrllng process of samples s llusraed n Fg. 4.5 whch also shows he samples were held horonally n he fxure aached o 3D x- y- posonng able wh posonng accuracy.3 mm. The resulng samples of laser drlled holes conssed of a seres of rows of holes n each worpece consdered. Parameers ha were adused durng performng he mcrodrllng operaons ncluded lens focal lengh poson of he laser beam focal pon laser energy laser pulse wdh and frequency of he laser beam per hole. Resuls of hose expermens and a dscusson of her correlaon wh exac and approxmae soluons are presened n Chaper

340 Fg Laser seup for mcrodrllng Couplng of laser energy wh he arge maeral The laser beam nensy absorpon and redsrbuon nsde he hole can be measured only ndrecly as par of many neracng phenomena such as: flud moon hole nsables plasma formaon and mulple reflecons on he hole wall. I has been proven as a par of research for hs Dsseraon ha laser beam characerscs (spaal and emporal nensy dsrbuon) as well as he varaon of maeral properes nfluence shape of he hole. When laser beam sres on he arge maeral par of he energy s refleced and par of s absorbed. The absorbed energy heas up he arge maerals. Ths absorpon and reflecon have a resonan feaure due o he mcrosrucures of maerals. When laser 339

341 beam acs on he maeral laser energy s frs absorbed by free elecrons. The absorbed energy propagaes hrough he elecron subsysem and hen s ransferred o he lace. In hs way laser energy s ransferred o he amben arge maeral. Ths process has a resonan feaure because maerals show dfferen absorpons o lasers wh dfferen wavelenghs (Han 999). Ths dependence of absorpon on wavelengh s deermned by he mcrosrucure and EM properes of he maeral. When he laser nensy s raher low no phase ranson occurs and he only effec of laser absorpon s heang of he maeral. In meals he laser radaon s absorbed by free elecrons and he energy ransfer n meals s also due o elecron hea conducon. In many praccal cases he Drude-Zener heory provdes descrpon of opcal properes of maeral whose opcal absorpon s manly due o free elecrons (Nowaows and Prypunewc 4). Sold and lqud meals are ypcal examples of such maerals. In case of machnng applcaons of a parally absorbng medum Lamber law can be appled. Ulng Eq. 46 and nroducng he exponenal decay yeldng a a dsance below he rradaed surface we oban A µ E pulse I R ( ξ ψ ) = µ Q( ξ ψ τ ) = e (4) n ± f a eccω p I A F / D o = ω where µ s he exponenal absorpon coeffcen for he gven maeral (Nowaows e al. 5) s defned by Eq Combnng Eq. 4 wh Eq. 375 he fnal defnon of he absorbed laser energy by he maeral s n a τ 34

342 34 ) ( exp exp / ) ( ) ( exp ) ( ± = = = T y x T a I D F e a I E T y x T A Q c nal fnal n A p o f cc n R pulse c nal fnal µ µ ω ω τ ψ ξ µ µ τ ψ ξ (4) Maeral removal ess In laser mcrodrllng maeral s removed as a mxure of mel and vapor Fg The proporons beween he mel and he vapor depend on maeral properes and laser beam nensy. Large amoun of removed maerals s advanageous from drllng effcency pon of vew bu on he oher hand can creae more nsably n he hole formaon. The hcer he layer of mel he less s defned he surface geomery he more he hole geomery flucuaes. The hcness of a layer molen durng a laser pulse neracon wh meals s approxmaed by Eq. 43 where δ s molen layer hcness and p s pulse duraon (>ps o be vald). Analyng Eq. 43 he bes way o ncrease drllng accuracy s by smply shorenng he molen layer hcness by shorenng he laser pulse me. The molen maeral s eeced from he hole by he pressure graden developed durng he vaporaon of he poron of he maeral. As a resul more mass s removed compared o vaporaon alone. The hermal balance model addresses hs concep drecly n he hea ransfer equaons. Snce he eeced maeral has o leave n molen

343 form our assessmens accoun for all maeral eeced o leave when localed drllng one reaches he fully molen sae. Expermens were conduced as a par of research for hs Dsseraon n he CHSLT laboraores a he room emperaure whou any sheldng gas. The arges were.76 mm hc sanless seel plaes mouned perpendcular o laser beam drecon. For drllng process of above maerals Nd:YAG laser was used Fg. 4.. Laser wavelengh s 64 nm pulse duraon was vared from. o.5 ms. Dfferen energy levels were used o explore he opmum condons for Nd:YAG laser drllng (Han e al. 4). Wh he hermal balance analycally assessed acual maeral removal ess were performed. These ess consdered neracon of 64 nm wavelengh energy wh samples of 34 sanless seel. The laser parameers and opcal ransmsson equpmen s shown schemacally n Fg Expulson consderaons were also of prncpal neres o he model. In effors o gaher nformaon on he amoun of maeral removed an expulson collecon scheme was pu ogeher. Ths plan was o measure and wegh all samples before and afer he laser drllng was perfumed consoldaed a large poron of he maeral removed n he molen sae no he glass ubes and allowed us o drecly measure. Presence of meal vapor and ny meal droples on he wall of he glass ube are shown n Fg Several droples can be seen on hs macrograph. Clearly mass loss s conrbued by vaporaon of alloyng elemens and he eecon of meal droples. Comparng amoun of vapored and eeced maeral and examnng he whole volume of he drlled hole n he sample we noced ha he volume ncreased wh he ncrease of energy delvered o he sample. Drllng ess on 34 sanless seel have 34

344 followed lnear correlaon beween maeral removal and npu pulse energy as llusraed n Fg Wh hs as a bass a hermal model wh physcal and opcal maeral characerscs and wh measured laser beam spaal and emporal characerscs was buld. Fg Laboraory seup used durng laser drllng of 34 sanless seel samples. Due o lmaons n he collecon scheme wll be dffcul o suppor he rend of expelled maeral decreasng lnearly wh ncreasng pulse energy due o he fac ha hs were us prelmnary ess and here are more ess requred o fully llusrae he occurrng phenomenon. In he scheme of he es of measurng a he enrance ex and mean dameers of he drlled holes one can calculae average volume of he holes and mass of ha maeral and compare resuls wh numercal and expermenal daa. 343

345 Fg Parcles of 34 sanless seel eeced from he drlled hole where open glass ube placed co-axal wh he laser beam durng mcrodrllng process.. wegh mg.9 maeral removed mg.8 Lnear (maeral removed mg) energy J Fg Maeral removed durng laser drllng of 34 sanless seel samples. Consderng he reflecvy and expulson parameers he orgnal hermal balance equaon was modfed. The frs order approxmaon used n hs analyss ransformed Eq. no ( R) Q oal = M c p ( M ( T exp m T ) M H a ) M c p ( T v T f m ) ( M exp ) M H v (43) where R s he reflecvy and M exp s he expulson fracon of maeral removed. Wh a hermal balance analycally assessed acual maeral removal ess were performed. 344

346 A conrol-volume based on he compuaonal approach was developed o solve he governng relaon gven by Eq. 4. Usng he enhalpy mehod o deal wh he phase change problem whle solvng Eq. 4 a 3D model and a compuer program was bul o smulae he drllng process. In he calculaons of he emperaure profles n he maeral several condons have o be aen no accoun: () he surface s movng due o evaporaon of maeral () he hea of fuson creaes a movng boundary (Sefan problem) () he lqud maeral loses s meallc properes above he crcal emperaure. Then he lqud meal exhbs delecrc behavor and becomes ransparen. A he crcal emperaure (abou.4 mes T v ) maeral absorpvy approaches ero whch mples ha here canno be any laser absorpon n he superheaed layer; all of he absorpon s concenraed n a small regon where he emperaure s us below he crcal emperaure T c. Maeral s removed a he op of he delecrc layer by evaporaon. A he sdewalls maeral s forced away by he plasma pressure on he lqud whle a he end of he pulse when he pressure suddenly drops maeral s removed by he bolng of superheaed lqud Invesgaon of laser mcrodrllng usng hgh speed flmng In Secon.6 plasma formaon was descrbed and n Secon.7..3 we defned heorecally velocy of lqud eecon as well as he velocy of vaporaon. Alhough daa are avalable and quoed n Fg..4 and Fg..43 bu o verfy hose daa and o 345

347 sudy more horoughly laser mcrodrllng process a hgh speed CCD camera was employed Fg A queson s n whch sae he maeral s removed manly. In Secon 4.3. where we calculaed how much of he removed maeral from he hole was beng removed by vaporaon vs. how much of was beng expelled n form of a lqud. Hopefully expermens wh fas rae phoography mehod wll shne on us some more lgh no he dar unnel whch s expulson of a molen maeral problem durng laser mcrodrllng. A Pxellng 6.6 MBye CCD camera Fg. 4.9a wh maxmum frame raes of 8 frames per second was used for prelmnary sudy of measuremens of expulson velocy and o sudy plasma formaon. The camera was nsalled Fg. 4.9b orhogonally o he laser beam whch was mpngng vercally on he worpece. The drllng evens were capured usng he hghes frame rae of 8 fps wh nown dmensons of vewng area. The arge was llumnaed wh He:Ne laser beam for ease of argeng he small area of neres on he worpece. The laser pulse lengh comng hrough aached fber opc cable vared beween.8 o ms wh focusng spo of.3 mm. Worpece maeral used was 34 sanless seel wh hcness of abou.76 mm. 346

348 Fg CCD camera used n he expermens. The laser mcrodrllng process can be descrbed n he followng sages: a) he maeral s beng removed as a mxure of mel and vapor n concal shape fashon Fg. 4.3a and Fg. 4.3b drven by he vapor pressure nsde he hole han he plasma s beng formed hen n he nex sage random droples of meled maeral are beng seen leavng he hole. The las sage occurs a he end of he pulse when he pressure suddenly drops and massve evaporaon aes place. 347

349 a) b) Fg Sequenal frames from flmng of laser mcrodrllng process: vaporaon and lqud eecon process plasma formaon random parcles eecon and vaporaon sponaneous vaporaon a he end of he pulse for wo dfferen pulses: a) p =3.5 ms b) p =4.5 ms. 348

350 The velocy of he mel whn he concal shee was esmaed from he hgh speed flmng o be n he range -35 m/s. The sngle droples were movng a a slower rae esmaed o be n range 5-7 m/s Expermenal examples of he laser mcrodrllng One of he goals of hs wor s o denfy he prmary cause of hole geomery varaons by undersandng of he effecs of laser parameers and he mel eecon process on hole repeaably. Pcure of a ypcal laser hole drlled n 34 sanless seel Fg. 4.3 shows effecs of srong flud moon. Recas layers of he holes are clearly seen n Fg. 4.3b and Fg. 4.3b. For lower nenses (us above he drllng hreshold) hea conducon and reflecon losses are domnan. For hgher nenses effecs such as beam defocusng n he vapor cloud or even nduced ar-breadown severely degrade he drllng effcency and reproducbly of he drllng process. Typcal laser drlled holes n slcon sngle crysal maeral s shown n Fg All ess were performed wh neracon of a sngle laser pulse duraon. The goal was o deermne relaonshp beween delvered laser energy o he sample and amoun removed from (by drec wegh measuremen and hole geomery approxmaons). 349

351 enrance TOP md ex BOTTOM a) b) Fg Mcrograph of ypcal shape and characersc dmensons of he laser mcrodrlled hole cross secon of hole profle n 76 µm hc 34 sanless seel obaned wh pulsed laser beam ransmed hrough a sngle-mode fber pulse wdh p =.9 ms energy E=7.5 J laser beam dameer ω =33 µm and power P=4. W: a) SEM mage of he hole enrance b) cross secon of he hole along -axs. a) b) c) Fg Mcrographs of a ypcal hole profle drlled n 76 µm hc 34 sanless seel obaned wh Nd:YAG pulsed laser beam ransmed hrough a sngle-mode fber pulse wdh p =.8 ms energy E=7 J laser beam dameer ω =3 µm and power P=. W: (a) mage of he ex of hole (b) opcal mage of he cross secon (c) mage of he enrance of hole. 35

352 a) b) c) Fg Mcrographs of a ypcal hole profle drlled n 3 µm hc sngle crysal slcon obaned wh sngle Nd:YAG pulse: pulse wdh p =.5 ms energy E=.5 J laser beam dameer ω =58 µm: (a) mage of he ex of he hole (b) opcal mage of he cross secon n axal drecon (c) mage of he enrance of he hole. More llusraons of mcrodrlled holes of 34 sanless seel and sngle crysal slcon can be found n Appendx B Effec of laser energy The amoun of laser energy requred for laser mcrodrllng process depends on he opcal and hermal properes of he worpece maeral. When energy delvered o he worpece exceeds he hreshold energy defned earler han he mcrodrllng process wll occur. In order o sudy he effec of laser energy on he resulan dmensons of he hole connuously ncreased laser energy was appled o he samples rangng from.6 J o 5.7 J wh a consan pulse wdh of.8 ms and he laser beam was poson (lbwp) was below he surface.e. he laser was focused /3 below he surface of he worpece. Prelmnary resuls are shown n Fg whch shows ha he power a he beam breahrough (.e. he onse of a hrough hole) was. J. For he hole enrance dameer 35

353 a he op surface desgnae as enrance as he laser energy ncreased enrance also ncreased. For he ex dameer desgnae as ex as he laser energy ncreased ex also ncreased as he power ncreased from. J o 5.7 J. For he hgher power denses he hole dmensons dd no change much. 6 enrance ex per. Mov. Avg. (ex) per. Mov. Avg. (enrance) 5 Dameer um Energy J Fg Effec of laser energy on dmensons of laser drlled mcroholes n sanless seel 34 shees: N = and cons volage = Effec of laser beam was poson (lbwp) Poson of he focal plane s nown o have an effec on fnal shape of he hole as well as degree of peneraon. As he laser focusng spo moves up and down he laser maeral neracon vares oo Fg The local beam radus as a funcon of deph can be calculaed usng he followng relaonshp (Zhao and DebRoy 3) 35

354 W ω ( ) = nω / (44) ω f D where ω he laser beam radus a he beam was poson as s llusraed n Fg The numercal values of Eq. 44 are represened graphcally n Fg Focal plane poson from now called laser beam was poson lbwp s negave when he focal plane s above he worpece and posve when s below he surface of he worpece because = poson s locaed a he op surface of he worpece Fg... laser beam focal lengh = 5 mm worpece DOF beam was =.8 mm beam focused below surface beam dameer =.3 mm maeral hcness.76 mm Fg The laser beam focused on he worpece and cross secon of he laser penerang he worpece. ω( ) race Fg Change of radus of he beam as poson of focal plane wh deph changes wh when lbwp =.88 µm. 353

355 Inensy of he laser beam a he maeral surface s one of he mos mporan processng parameers n laser mcromachnng. The maxmum rradance of he laser beam (he mos mporan parameer of laser mcromachnng) s obaned a he beam was plane Fg Beam was of he laser beam or he deph of focus (DOF) defned n Eq. s a dsance over whch he focused beam has approxmaely he same nensy and s defned as a dsance over whch he focal spo se changes by 5%. For our Nd:YAG sysem manufacurer gves us value of.8 mm for DOF (Lasag 99). In macro scale hs number can be consdered small and n hs case lbwp can be consdered as one plane (or a pon). In mcroscale applcaons s acually a funcon defned by Eq. 44 whch s mplemened by FDM n hs Dsseraon. Loong closely a ha problem here are a leas wo reasons why a lens wll no focus o a heorecal pon; one s he dffracon lmed problem dscussed n Secon.4.7. and he oher s ha a sphercal lens s no a perfec shape. Lenses are made wh more or less accuracy by dfferen manufacurers. The more deal sphercal shape of he lens s he more expensve ges. The oher lens fauls occur when mechancal and opcal axs are no correcly algned. In some cases sphercal aberraon was aen advanage of and used n posve way Karnas e al. 5. They nenonally nroduced more aberraon n he sysem Fg a frs o correc he problem laer o oban dfferen shape of he beam whch ncreased effcency of he laser drllng and n some cases qualy of drlled holes. 354

356 Summared facors ha nfluence he se of he focal spo and lbwp are he mperfecons of he opcal componens and dffracon effecs whch lm he se of he obanable focal spos. Fg Ray of beam racng analyss descrbng sphercal abberraon (Karnas e al. 5). The reason ha hs was brough up s ha nowledge s gong o be helpful laer n he dscusson secon of hs Dsseraon. Nex ssue s drecly relaed o he qualy of he ncomng beam whch can be quanfed by he dvergence of he beam. The smaller dvergence he smaller spo se one can oban. The dameer of he ncomng laser beam affecs he focal spo se. Focal spo se s nfluenced by dffracon. Hole shape s a sensve funcon of he poson of he beam was relave o he sample surface Fg Le us defne ha when he focal plane poson lbwp s se on he worpece op surface han lbwp s consdered o be ero. In he expermens he sample of sanless seel 34 shee wh he hcness of.76 mm was used he lbwp was vared beween 3. mm (above he worpece surface) and -3. mm (below he worpece surface) where he consan laser energy of 355

357 3. J and.8 msec pulse wdh and pulse frequency of H was appled. The characersc dmensons of laser drlled mcroholes were measured desgnaed as follows: for he dameers of he laser beam a he TOP of he worpece as D_enrance md secon of he hole as MID and D_ex as he beam exs he worpece Fg Holes were drlled usng sngle pulses and a plano-convex lens was used wh 5 mm focal lengh (noe: here s a huge dfference beween a plano convex lens mouned one way or he oppose way around). 6 5 D_ex D_hole Poly. (D_enrance) D_enrance Poly. (D_ex) Poly. (D_hole) 4 dameers µm lbwp µm Fg Typcal funcon of dameers of D_enrance D_md and D_ex for varous lbwp mpngng on 34 sanless seel worpece one pulse and energy E=7.5 J. 356

358 D_enrance D_ex D_Hole damer µm lbwp µm Fg Typcal funcon of dameers of D_enrance D_hole and D_ex for varous lbwp mpngng on sngle crysal slcon worpece one pulse and energy E=.5 J. The laser beam was poson lbwp was moved vercally by consan amoun of dsplacemens sep n -drecon rangng from 89 µm o µm beween holes. Laser drllng process was performed a every me sep of -mcro-sage. Afer drllng process was accomplshed holes were measured (f any) and samples were crossseconed for furher analyss Fg. 4.4 and Fg Represenave quanave and qualave resuls for cross-secon measuremens are shown n Fg. 4.4 and Fg for sngle crysal slcon and for sanless seel respecvely. Qualavely as he was s moved away from he orgnal surface n eher drecon he spo se a ha surface ncreases. Thus a he sar of drllng maeral s removed over a larger area and a a slower rae due o he reduced nensy. Expended drllng an enlarged enrance leavng less laser energy (or laser nensy) for deepenng he hole. 357

359 When he was s below he surface expermenal resuls are shown n Fg. 4.4 he spo se a he exposed surface (afer maeral s removed) decreases as drllng progresses deeper no maeral. Ths causes he hole o narrow relavely rapdly so hole walls are no very seep. If lbwp s no oo large also ends o produce narrow raher poned hole booms Fg. 4.4a. Conversely when lbwp s above he surface he spo se a he exposed surface ncreases seadly as drllng progresses nsde he maeral. Ths ends o produce holes wh relavely less seep walls and broad rounded booms Fg. 4.4a-3. When he beam was poson les above he worpece surface he beam represens a smaller arge for he eeced maer.. The calculaons found n leraure show ha he narrowes hole enrance occurs when he beam was s a he worpece s surface because he smalles se of he laser beam has a conac drecly wh he maeral surface. SET A E=. J µm Fg Expermenal resuls of mcrodrllng n 34 sanless seel worpece. Effec of laser beam was poson on hole profles. Holes were drlled wh a beam dameer of 59 µm. J energy. The was was moved vercally by 89 µm beween holes. 358

360 SET B E=3 J µm Fg Expermenal resuls of mcrodrllng n 34 sanless seel worpece. Effec of laser beam was poson on hole profles. Holes were drlled wh a beam dameer of 59 µm 3. J energy. The was was moved vercally by 89 µm beween holes. SET A a) focused above he op surface b) focused below he op surface Fg Effec of laser beam was poson on expermenal hole profles drlled n 34 sanless seel. Holes were drlled wh a 59 µm.8 J pulse usng he 5 mm lens. The was was moved vercally by 89 µm beween holes. 359

361 SET B a) a) focused above he op surface b) focused below he op surface Fg Effec of laser beam was poson on expermenal hole profles drlled n 34 sanless seel. Holes were drlled wh a 59 µm 3. J pulse usng he 5 mm lens. The was was moved vercally by 89 µm beween holes. SET A a) b) SET B a) b) a) focused above he op surface b) focused below he op surface Fg Effec of laser beam was poson on expermenal hole profles drlled n sngle crysal slcon. Holes were drlled wh a 59 µm dameer beam wh.8 J (SET A) and.5 J (SETB) pulse usng he mm lens. The was was moved vercally by 89 µm beween holes. 36

362 5. Correlaon beween compuaonal and expermenal nvesgaons The model of he laser beam expermenally deermnes he shape hrough whch he hole evolves as beam propagaes hrough he worpece. Below here s summary of expermenal resuls presened n prevous chapers. Comparson of heorecally deermned compuer smulaons of laser mcrodrllng wh expermenally obaned ones s summared n Fg. 5. o Fg Despe lmaons of he model menoned above deermnaon of profles of holes agreed well wh he expermenal daa alhough here are some regons of beer agreemen for example for nenses close o he hreshold from lqud eecon. If nensy of laser beam s below hreshold hen laser beam wll no penerae he worpece hrough bu only o a ceran deph. Despe usng a smple represenaon of beam aberraon and reamen of he maeral beng drlled calculaed hole profles agree very closely wh expermenally drlled holes. Mos mporan s ha he model correcly reproduces expermenal changes n he hole se and shape wh he beam was poson relave o he sample surface and beam dvergence whch s llusraed n Fg. 5. o Fg

363 a) deph µm T c T c b) deph µm lengh µm Fg. 5.. Comparson of compuer smulaed resuls wh expermenal daa of crosssecons of a mcrodrlled holes n 34 sanless seel maeral wh a sngle pulse of Nd:YAG measured and focused above he op surface a) lbwp = 7 µm b) 89 µm. T c a) deph µm T c b) deph µm deph µm c) lengh µm Fg. 5.. Cross secon of a mcro machned sample of a 34 sanless seel maeral wh a sngle pulse E=3. J focused o:a) =5 µm below he op surface b) lbwp =7 µm c) 9 µm. 36

364 a) b) c) deph µm deph µm deph µm deph µm T c T c T c T c d) lengh µm Fg Compuer smulaed cross secon of a mcrodrlled sample of a sngle crysal slcon maeral wh a sngle pulse of Nd:YAG measured laser beam focused below he op surface a) lbwp = 7 µm b) lbwp = 3 µm c) 46 µm d) 8 µm. 363

365 a) b) c) d) deph µm deph µm deph µm deph µm T c T c T c T c lengh µm Fg Compuer smulaed cross secon of a mcrodrlled sample of a sngle crysal slcon maeral wh a sngle pulse of Nd:YAG measured laser beam focused below he op surface a) lbwp = - µm b) 3 µm c) 4 µm d) 7 µm. 5.. Effec of crcal emperaure and absorpon coeffcen In he area where he crcal emperaure s reached here s no more absorpon of he laser energy. Meallc maeral n hs area becomes delecrc and ransparen o he beam. So he absorpon s concenrang n he places us below he area wh he crcal emperaure. Maeral s beng removed by evaporaon a he op of delecrc layer and a he sdewalls maeral s beng removed by he plasma pressure on he 364

366 exsng lqud n he eyhole. A he end of he pulse when he pressure suddenly drops maeral s beng removed by bolng of superheaed lqud. As one can noce o esablsh values for crcal emperaure s essenal for predcng profle of he hole. A good value for T c s dffcul o asceran. Frs because he slcon s a semransparen maeral. The 34 sanless seel on he oher hand s an alloy comprsed of a few meals each wh a dfferen properes. Meer () saed ha he value of he crcal emperaure T c (of abou.4 mes he bolng emperaure) s reached very fas and hen remans consan. Oher leraure s gvng dfferen numbers for T c rangng from 4 o 9 K deermned expermenally (Bec e al. 995; Bruggemann 996; Bulgaova and Bulgaov a; Chrsensen and Tllac 3 Cracun e al. ). Le us use wo dfferen numbers for T c run our compuer program and deermne how sgnfcan he value for T c s. Resuls of hs deermnaon are shown n Fg. 5.5 and Fg Expermenal pulse E=.5 J Run 6 µ =67 Tc=39 K Run 6x µ=67 Tc=69 K deph 76 µm deph µm deph µm lengh µm lengh µm lengh µm Fg Effec of crcal emperaure on he hole profle nsde he 34 sanles seel wh T c = 39 K and T c = 67 K. 365