(5) Furthermore, the third constraint implies the following equation: (6)

Transcription

1 T-Element Refactng System f Gaussan and Annula-Gaussan Beams Tansfmatn Abdallah K. Che *, Nabl I. Khachab, Mahmud K. Habb Electcal Engneeng Depatment, Cllege f Engneeng and Petleum, Kuat Unvesty, P. O. Bx 99, Safat, Kuat abdallah.che@ku.edu.k (Receved th Octbe, 7; Revsed th Decembe 7; Accepted th Decembe, 7; Publshed: 9 th Januay, ) Abstact- In ths pape, a efactng system cmpsed f t sepaate lenses s desgned t btan specfc lase adance dstbutns. Namely, the lens sufaces (nput and utput) ae deved f thee types f lase beam tansfmatn: () Gaussan t Annula-unfm; () Annula-Gaussan t Annula-unfm; and () Annula-Gaussan t Gaussan, takng nt cnsdeatn the system length, the sufaces ad f cuvatue and the ntal efactng angle f annula beams. Key Wds: Gaussan beam, Refactng system, Lens desgn.. INTRODUCTION The TEM (Gaussan) lase beam s ne f the mst lase types n cuent use f many applcatns hee a small fcusng spt s needed. Heve, thee ae many the applcatns hee unfm ntensty dstbutn s equed such as n cheent mage pcessng, patten ecgntn, and mateals pcessng tasks [-7]. Thus, t fulfl ths need, the cnvesn f Gaussan and nn-gaussan lase beams t unfm beam pfles appeas t be vey benefcal. Refactve ptcal system methd, hch eles n gemetc ptcs, s amng the methds that can make ths cnvesn [-]. Ove the yeas, seveal publcatns, descbng snglelens and t-lens efactng system, ee ppsed t acheve beam shapng such as Gaussan and annula-gaussan beam tansfmatn t unfm beams, as ell as t Bessel beam pfles [-]. In ths k, e ae ppsng a t-element efactng beam shapng systems t cnvet thee types f beams that ee nt cnsdeed befe: () Gaussan t Annula-unfm; () Annula-Gaussan t Annula-unfm; and () Annula- Gaussan t Gaussan. The mathematcal expessns that epesent the cuvatue and the asphecty f the nput and utput lenses ae deved. In addtn, dscussns f the vaus paametes that ae affectng the ppsed desgns ae pesented such as the veall length f the system, the ad f cuvatue f the sufaces, and the ntal efactng angle f annula beams.. DESIGN PROCEDURE The gemetcal ptcs methd f beam shapng mpses thee man cnstants hen desgnng the efactng ptcal system. These ae: () Optcal path length: all the ays that ente the fst lens and leave the secnd lens must have the same ptcal path length. () Ray Paallelsm: the utput ays that exst fm the system must be paallel t the nput ays. () Enegy cnsevatn: the at f the nput beam pe t the utput beam pe must be a cnstant. Fg. llustates the gemetcal cnfguatn f the tlens efactng system f Gaussan t annula-unfm beam tansfmatn. The same set-up can als be used f the beam shapes by makng sme apppate mdfcatns. The t lenses ae made fm glass (ndex f efactn n = 7) and ae sepaated (by a hee ndex f efactn n = ) by a dstance D. A ay entes nput lens th an angle θ and then s efacted by an angle θ. Ths ay tavels n a and eaches the secnd lens th an angle θ and then t s efacted by an angle θ and exts paallel t the fst ncdent ay. In the fgue, the nput and utput lenses ae epesented by the mathematcal sufaces functns y ( ) and y ( ), espectvely. Fm the gemety f the set-up n Fg., ne can deduce that: tan( ) tan( ) () D y y N, the abve fst cnstant s tanslated nt the fllng equatn: ny ( ) ( Dy y ) n( dy ) f () hee f s a cnstant. Addng (-nd) tem t bth sdes f Eq. (), yelds: ndny ny ( ) ( Dy y ) f ndnd Let f = f-nd-ndand use Eq. (), e btan: ' n sec( ) f () tan( ) tan( ) Next, the ay s paallelsm cnstant dctates that bth the nput and the utput sufaces slpes must be equal t each the, namely: dy dy tan tan () d Futhe, by applyng Snell s la at the nput lens (nsnθ = snθ ) and usng tgnmetc denttes, ne can btan fm Eq. () and Eq. () the fllng mptant elatnshp: dy dy () f' n Futheme, the thd cnstant mples the fllng equatn: Enegy k cnstant () Enegy Eq. () and Eq. () ll be usedt btan the mathematcal expessns y ( ) and y ( ) f the nput and the utput Page

2 sufaces f the lenses. The desgn pcedue stats th the selectn f the system length and the pe at k fm hch ne btans the elatnshp beteen the adal dstances and. Next, the sufaces slpes dy / and dy / n Eq. () ae numecally btaned fm hch e get the numecal values f the sufaces y ( ) and y ( ) by numecal ntegatn. A plynmal cuve-fttng utne s used t btan the mathematcal expessns f these sufaces. Nte that e need t apply apppate ntal values n the sufaces t btan the cnstants f ntegatns. It s th mentnng that f pactcal ssues, bth the veall system length and the statng suface slpe ae pefeed t be as small as pssble. Futhe, a lage adus f cuvatue ρ() f the sufaces s pefeable t help manage the fabcatn pcess. ρ() f any suface s gven by: / [( dy ) ] ( ) (7) d y Eventually, the secnd devatve f the statng suface slpe d y needs t be small. Glass A Glass R y y D d Ie k () I( R ) Whee s the adus f the nput Gaussan beam; R and ae the nne and the ute ad f the utput annula beam. Eq. () ll be slved t btan: (e ) k R (9a) ln( k ( R ) ) (9b) These equatns alng th Eq. () ae used t desgn the t-lens efactng system. T demnstate the desgn pcedue, let us cnsde the tansfmatn f a Gaussan beam ( = cm) t an annulaunfm beam (R = cm) hee the pe at f the t beams s k =. An apppate system length D needs t be chsen. In ths egad, e have pvded a plt (Fg. a) that llustates the system length, the statng suface slpe, and the stang adus f cuvatue vesus the statng dffactn angle θ. Fm the fgue, bth the system length and the adus f cuvatue decease hen the efactng angle θ nceases. Cnsequently, e have selected a mdeate value f θ =.77 hch cespnds t D =.9 cm and f =. Usng these values n equatns (), (), and (), e can btan the numecal values f the suface slpes dy and d dy. The mathematcal expessns f these slpes ae acheved usng a plynmal cuve-fttng utne t yeld: dy (.) (.) (.) (a) (.) dy (9) (.77) (.99) (.7). Integatng the abve expessns t btan appxmate equatns f lenses sufaces as: - - y ( ) (.x ) (.) (.7x ) (a) (.) ().7 (b) Fg. : The gemetc set-up f the t-element efactn system beam tansfmatn.. LASER BEAM TRANSFORMATIONS In ths sectn, the mathematcal expessns y ( ) and y ( ) f the nput and the utput sufaces f the lenses ae deved f thee dffeent types f beam tansfmatns.. GAUSSIAN TO ANNULAR-UNIFORM BEAM TRANSFORMATION As shn n Fg., a Gaussan nput beam pfle ll be edstbuted by the t-lens set-up t fm an annulaunfm utput beam pfle. The cnsevatn f enegy cnstant mples that ntenstes must be cnseved ve the css-sectnal aeas f the t beams: y ( ) (.7) (9) (.99) (b) (9) -(.).99 Equatn () s pltted n Fg. b. The ad f cuvatue f the sufaces f ths desgn ae shn n Fgs. c and d. The adus f cuvatue f the nput lens stats th a value f. cm t each a maxmum value f 7 cm and then deceases t a value f 7.9 cm; hle f the utput suface, t stats th a value f.9 cm t each a maxmum value f cm and then deceases t a value f. cm. In geneal f annula beam, f a fxed pe at k, the statng efactng angle θ = sn - (nsnθ ) fm the nne adus s ctcal n decdng the length f the t-element lens desgn, n addtn t the facts such as the ntal suface slpe and the ntal adus f cuvatue f the suface. Fg. shs the lenses sufaces f anthe desgn hee the nne adus f the annula beam s set t R = cm nstead f cm. Page

3 . f' = ; R = ; k = f' = ; R = ; k = D: System length (cm) dy/: Intal suface slpe Intal adus f cuvatue.. X:.77 Y:.9 Radal dstance (cm) (a) - efactng angle n degee (b) f' = ; R = ; k = 7 x f' = ; R = ; k = suface adus f cuvatue (cm) X:. Y:.7e+ suface adus f cuvatue (cm) X:. Y:.e+ Radal dstance Radal dstance (c) (d) Fg. : Gaussan t annula-unfm desgn: (a) effect f the efactng angle n the system length; (b) the nput and the utput sufacesy ( ) and y ( );(c) and (d) the ad f cuvatue. Fm Fg. a, ne ecgnzes that f a efactng angle θ =.77, the system length s almst dubled (D = 7.7 cm). T btan a system length f D =.9 cm, θ must ncease t 7. as ndcated by the a shn n Fg. a.. ANNULAR-GAUSSIAN TO ANNULAR-UNIFORM BEAM TRANSFORMATION The efactng t-lens beam tansfmatn desgn f annula-gaussan t annula-unfm pfle s demnstated n ths subsectn. Statng th the enegy balance equatn beteen the t dffeent beams: I [ Re ] e R g k () I( Ra) Whee R a and ae the nne and the ute ad f the utput annula-unfm beam and I s the peak nput beam ntensty. F the annula-gaussan beam, R g and ae the nne and the ute ad; s the annula-gaussan beam adus; ( M ) s the beam spt sze n the lage Fesnel numbe lmt; M s the magnfcatn; R s the eflectvty f the cental m. Page 7

4 R = ; k = D: System length (cm) dy/: Intal suface slpe Intal adus f cuvatue f' = ; R = ; k = X:.77 Y: Refactn angle (degee) Radal dstance (a) (b) Fg. : Gaussan t annula-unfm desgn: (a) effect f nne adus R f the annula beam n the efactng angle and n the system length; (b) the nput and the utput sufaces f R = cm. Ra = ; Rg = ; k = D: System length (cm) dy/: Intal suface slpe Intal adus f cuvatue. X:.77 Y:.9 Radus f cuvatue. - Refactng angle (degee) Radal dstance Radal dstance (a) (b) (c) Fg. : Annula-Gaussan t annula-unfm desgn: (a) chsng the system length f a specfc efactng angle; (b) the nput and the utput ad f cuvatue; (c) the nput and the utput sufaces. The ntegal n Eq. () can be slved t btan the utput adus: R/ ( [ ) R ( g ) ] e e ( ) {( ) } R a k R g () ( e e )/ Nte that hen M.Cnsequently, Eq. () s smplfed t: ( Rg )/ {( ) R e e R } a k R g () ( e e )/ The desgn shn n Fg. s f tansfmng beams th R a =, R g =, =, k =, R =.9, hee e have selected a mdeate value f θ =.77 hch cespnds t D =.9 cm and f =. Usng these values n equatns (), (), and (), e can btan the numecal values f the suface slpes dy and d dy. The mathematcal expessns f these Page

5 a = ; g = ; Rg = ; k = D: System length dy/: Intal suface slpe Intal adus f cuvatue Radus f cuvatue. x a = ; g = ; Rg = ; k =.... a = ; g = ; Rg =; k = - Refactng angle (degee) Radal dstance Radal dstance (a) (b) (c) Fg. : Annula-Gaussan t Gaussan desgn: (a) chsng the system length f a specfc efactng angle; (b) the nput and the utput ad f cuvatue; (c) the nput and the utput sufaces. slpes ae acheved usng a plynmal cuve-fttng utne t yeld: dy (.) (.9) (.979) (a) (.) dy (.) (.9) (.77) (.99).7 N, ntegate the abve equatns t btan the appxmate expessns f lens sufaces as: - (b) y ( ) (7.7x ) (.7 (9) (a) (.) (). y ( ) (.) (.) () (b) (9) -(.7) 9 Nte that the adus f cuvatue f nput suface changes fm a mnmum value f. cm t maxmum value f 9.7 cm. And the adus f cuvatue f utput suface changes fm a mnmum value f. cm and eaches a maxmum value f 9 cm.. ANNULAR-GAUSSIAN TO GAUSSIAN BEAM TRANSFORMATION The pe at f the annula-gaussan beam t the Gaussan beam s: k I [ R e R g a ] e (7) g I ( e ) g Whee g s the adus f the utput Gaussan beam; a s the annula Gaussan beam adus; and R g ae the ute and the nne ad f the nput annula-gaussan beam; I s the ( M ) peak nput beam ntensty; s the beam spt sze n the lage Fesnel numbe lmt; M s the magnfcatn; R s the eflectvty f the cental m. Nte that a hen M. The ntegal n Eq. (7) s slved t btan the utput adus f the Gaussan beam: a a R ( e e a) k g () g { ln } R R a a ( ) e e a k g In ths subsectn, e pvde a desgn f tansfmng beams th R g =, a =, g =, k =, R =.9. In de t btan a system length smla t the pevus t desgns,.e. D =.9 cm and f =, e have selected θ = 7. (see Fg. ). Agan, usng these values n equatns (),(), and (), e can btan the numecal values f the sufaces slpes dy dy and, fm hch the mathematcal expessns f d these slpes ae: dy (.) (.) (.7) (9a) (9) 9 dy (.7) (.7). () (7) (9b) Accdngly, the appxmate equatns f the lens sufaces ae: y( ) (.x (.9) ) (9) - (.).9 (.7) (a) y ( ) (.9x ) (.) (.7) (b) (.) -(.) Page 9

6 The adus f cuvatue f the nput suface changes fm a value f 9. cm t a value f.x cm; hle the adus f cuvatue f utput suface changes fm a mnmum f 9.7 cm eachng.x cm.. CONCLUSION A desgn pcedue f tansfmng lase beam pfles s dscussed n ths k usng t-element ptcal efactng system. The pcedue s appled t thee types f beam tansfmatns, hch ee nt adessed pevusly: () Gaussan t annula-unfm; () annula-gaussan t annulaunfm; and () annula-gaussan t Gaussan. The mathematcal expessns f the t plan-asphec sufaces ae deved f the thee types f lase beams. In addtn, fe mptant paametes such as the length f the set-up, the ad f cuvatue f the sufaces, and the ntal efactng angle f annula beam ee dscussed. REFERENCES [] J. Wlsn, and J. F. B. Hakes, Lase Pncples and Applcatns, Pentce Hall Intenatnal, Hetfdshe, UK, 97. [] J. C. In, Lase Pcessng f Engneeng Mateals: Pncples, Pcedue, and Industal Applcatn, Elseve Butteth-Henemann, Bulngtn, MA,. [] Naena B. Dahte, Sandp Hamka, Lase Fabcatn and Machnng f Mateals, Spnge, Ne Yk, NY,. [] C. Bschff, E. Jäge, and U. Umhfe, Beam Shapng Optcs f Pcess Acceleatn: Inceasng the pductvty f lase mcmachnng, Lase Technk Junal, vl., n., pp. -7,. [] M. Ducastella, and C. B. Anld, Bessel and annula beams f mateals pcessng, Lase Phtncs Rev., vl., n., pp. 7,. [] M. Allmen, and A. Blatte, Lase-Beam Inteactns th Mateals, Spnge-Velag, Beln, pp., 99. [7] F. M. Dckey, Lase Beam Shapng: They and Technques, Secnd Edtn, CRC Pess Publshng, 7. [] F. M. Dckey, and T. E. Lztte, Lase Beam Shapng Applcatns, Secnd Edtn, CRC Pess, publshng, 7. [9] D. L. Shealy, and S. Cha, Gemetc Optcs-based desgn f lase beam shapes, Opt. Eng., vl., n., pp. -,. [] A. J. Cx, and D. C. Dbble, Hlgaphc epductn f a dffactn-fee beam, Appl. Opt., vl., n., pp. -, 99. [] M. L, Y. Meuet, M. Vevaeke, H. Thenpnt, and F. Due, Desgn f fcal beam shapng system thugh adance and phase cntl, SPIE Pc. In Optcal Mdellng and Desgn IV (Vl. 99),. [] F. Due, and H. Thenpnt, Refactve lase beam shapng by means f a functnal dffeental equatn based desgn appach, Opt. Exp., vl., n. 7, pp. -,. [] G. Ede, G. Szavas, E. Lncz, and S. Vakny, Sngle-element efactve ptcal devce f lase beam pflng, Pc. SPIE Senss, Sens System, and Sens Data Pcessng, vl., pp., 997. [] J. A. Hffnagle, and C. M. Jeffesn, Beam shapng th a plan-asphec lens pa, Opt. Eng., vl., n., pp. 9-99,. [] M. Af, N. M. Hssan, A. A. S. Aal, and M. N. Islam, T-element efactng system f annula Gaussan-t-Bessel beam tansfmatn, Appl. Opt., vl. 7, n. 9, pp. -9, 99. [] H. K. Iftekhauddn, A. A. S.Aal, and M. A. Kam, Gaussan-t-Bessel beam tansfmatn usng a splt efactng system, Appl. Opt., vl., n., pp. -, 99. [7] M. A. Kam, A. K. Che, A. A. S.Aal, A. Bast, Refactng system f annula lase beam tansfmatn, Appl. Opt., vl., n., pp. - 9, 97. [] S. R. Jahan, and M. A. Kam, Refactng systems f Gaussan-t-unfm beam tansfmatns, Opt. Lase Technl., vl., n., pp. 7-, 99. [9] K. Thees, M. A. Kam, and A. A. S. Aal, Dffactn-fee beam geneatn usng efactng systems, Opt. Lase Technl., vl., n., pp., 99. [] M. Af, M. N. Hssan, A. A. S. Aal, and M. N. Islam, Refactng system f annula Gaussan-t- Bessel beam tansfmatn, Appl. Opt., vl. 7, n., pp. 9-, 99. Page