ANALYSIS OF PHOTOTHERMAL RESPONSE OF THIN SOLID FILMS BY ANALOGY WITH PASSIVE LINEAR ELECTRIC NETWORKS

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1 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp ANALYSIS OF PHOTOTHERMAL RESPONSE OF THIN SOLID FILMS BY ANALOGY WITH PASSIVE LINEAR ELECTRIC NETWORKS by Sloboanka P. GALOVI], Zlatan N. [O[KI], an Marica N. POPOVI] Orig i nal sci en tific pa per UDC: 60.: /.43 DOI: 0.98/TSCI09049G The pa per pres ents con i tions that shoul be met in or er to make the photothermal in uce tem per a ture vari a tions of a soli sam ple anal o gous to the volt age vari a - tions of the elec tric net ork ith pas sive lin ear el e ments. Fur ther anal y sis shos that such anal ogy en hances ex per i men tal e ter mi na tion of the ther mal prop er ties of thin soli lay ers by photothermal fre quency metho. Key ors: photothermal response, thermal ave, transmission-line theory Introuction Photothermal (PT) mea sure ment tech niques are be ing in ten sively e vel ope an ap - plie ith in crease suc cess to the mea sure ment of ther mal, op ti cal, an other re late phys i cal prop er ties, as ell as for the in ves ti ga tion of subsurface struc ture an mac ro scopic e fects of var i ous sam ples [-6]. PT meth os are base on i rect or in i rect re cor ing of sur face tem per a ture vari a tions that arise from the gen er a tion an trans fer of heat pro uce as a con se quence of the ab sorp tion of la ser ra i a tion of mo u late in ten sity by a sam ple. The re sult ing sig nal e pens on the amount of gen er ate heat (e pen ing on the co ef fi cients of op ti cal ab sorp tion an the ef fi ciency of the heat-to-light con ver sion of the sam ple) an on the heat trans fer pro cess (hence on the sam - ple s ther mal con uc tiv ity, co ef fi cient of ther mal iffusivity an other thermophysical prop er - ties). There fore, PT meth os have broa ca pa bil i ties as tool for non-e struc tive char ac ter iza tion of var i ous ma te ri als. In or er to e ter mine the phys i cal prop er ties of the in ves ti gate struc ture, it is nec es - sary, as the first step, to e velop a math e mat i cal moel that suf fi ciently ell e scribes phys i cal pro cesses lea ing from the op ti cal ex ci ta tion to the ther mal re sponse (i rect prob lem), an then to solve the in verse prob lem of e ter min ing the phys i cal prop er ties of the sys tem once the op ti - cal ex ci ta tion, ther mal re sponse an moel are knon. The in verse prob lem is usu ally solve by ap pli ca tion of curve-fit ting. Ho ever, non-lin ear fit ting be ing a rather com plex pro ce ure, quite of ten gives am big u ous re sults [7, 8]. There fore it is or thy of ef fort to in ves ti gate ap prox i mate so lu tions of the prob lem that en able e ter min ing the phys i cal prop er ties of the sys tem ith out non-lin ear curve-fit ting, by an a lyz ing only cer tain ranges of PT sig nal in func tion of the mo u - la tion fre quency.

2 30 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... This pa per eals ith one such so lu tion of the in verse prob lem in anal y sis of the PT re - sponse, the case hen a math e mat i cal moel of the PT in uce sur face tem per a ture vari a tions is anal o gous to the moel of the flo of elec tric cur rent through an elec tric net ork con sist ing of pas sive lin ear el e ments (re sis tors, ca pac i tors, an coils). In that case, al reay e vel ope meth - os of solv ing the in verse prob lem in pas sive lin ear elec tric net orks may also be ap plie to solve the in verse prob lem of char ac ter iza tion of ma te ri als by PT meth os. The first part of the pa per pres ents an anal y sis of a moel of the PT heat prop a ga tion in or er to e ter mine con i tions un er hich the in ves ti gate anal ogy hols. Af ter that, the an a - lytic ex pres sion for the PT re sponse un er those con i tions is e ter mine, an fi nally, a metho of solv ing the in verse prob lem in the case un er con si er ation is pre sente, an its po ten tials for ap pli ca tion for char ac ter iza tion of ma te rial prop er ties are is cusse. Moel Prop a ga tion of the PT in uce heat through the sam ple The anal y sis pre sente in the pa per con si ers a typ i cal PT con fig u ra tion sche mat i cally pre - sente in fig.. A soli sam ple of length s, mounte on back ing of length b, is ex pose to ra i - a tion of op ti cal beam pass ing through air layer ith length a. Air an back ing pres ent the en vi - ron ment of the sam ple sur roune by am bi ent ith tem per a ture T amb. As a con se quence of the ab sorp tion of light by the sam ple, the sam ple is heate, an ue to heat trans fer, tem per a tures of the sam ple (T s ), air layer (T a ), an back ing (T b ) are change. Tem per a ture vari a tions ue to time-e pen ent PT heat ing, e fine as J = T - Tc T amb, are pro por tional to the PT Figure. Typical PT experiment setup re sponse [5]. T c is the steay-state tem per a ture MOB stans for moulate optical beam vari a tion. As the heat ing of the sam ple s sur face can be con si ere uni form over the en tire heate sur face, it can be as sume that tem per a ture is uni form over all cross-sec tions an that it var ies only along the i rec tion of the in ci ent light beam (x-axis in fig..). Hence, the heat trans - fer may be con si ere one-i men sional, an the va li ity of this as sump tion is al reay rec og - nize [5]. There fore, the PT re sponse shoul be rep re sente as a func tion of only to vari ables, J = Jx, t. Tem per a ture vari a tions arise ue to the trans fer of heat gen er ate by the ab sorbe ra i - a tion, an they are, in one-i men sional case, e scribe by en ergy bal ance equa tion: C n T ( x, t ) q( x, t) S ( x, t) () t x here C n stans for the vol u met ric spe cific ther mal ca pac ity of the me ium, S(x, t) for the vol - u met ric heat gen er a tion rate, an q(x, t) for the heat flux. Tak ing into ac count ther mal mem ory an heat con uc tion ef fects, the heat flux e pens on the tem per a ture gra i ent, an this fact is ex presse by the fol lo ing equa tion [5, 9-]: q( x, t) J( x, t) q( x, t) t k t x ()

3 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp here k is the heat con uc tiv ity, an t the re lax ation time of ther mal pro cesses in the me ium. In the case of a ho mo ge neous an opaque sam ple, it can be as sume that the in ci ent light is ab sorbe by a very small re gion of the sam ple, so that the heat gen er a tion rate can be ex - presse as S(x, t) = S(t) (x), an thus ther mal vari a tions an heat flux may be e scribe by a set of hy per bolic ho mo ge neous if fer en tial equa tions of the sec on or er: J i ( x, t) Ji ( x, t) i Ji ( x, t) t (3) x D t D t i i qi ( x, t) qi ( x, t) i qi ( x, t) t (4) x D t D t i here the in ex i e notes part of the con si ere sys tem (i = a, s, an b stans for air, sam ple, an back ing, re spec tively) an D stans for ther mal iffusivity of the me ium (D = k/c). To fi nal ize the moel for the e scrip tion of the PT re sponse, eqs. (3) an (4) are to be com plete ith ho mo ge neous ini tial con i tions: ho mo ge neous boun ary con i tions: J i J a (x, t = 0) = 0 an q a (x, t = 0) = 0, (5) ( x, t) 0 an J ( x, t) 0, (6) a a b b an con i tions of con ti nu ity of tem per a ture an heat flux on in ter fa cial sur faces: J ( x 0, t) J ( x 0, t) an q ( x 0, t) q ( x 0, t) S ( t) a s a s s J b s b J ( x, t) ( x, t) an q ( x, t) q ( x, t) (7) Anal ogy ith flo of elec tric cur rent through lines Anal o gies be teen heat con uc tion an elec tric con uc tion pro cesses ere al reay in ves ti gate an ap plie to solv ing ther mal prob lems [6, -6], but pre sent ing mo els nei ther in clues pas sive lin ear elec tri cal net ork anal ogy nor con i tions that shoul be met in or er to make this anal ogy. In the con si ere case of the moel of the PT re sponse pre sente in the pre vi - ous chap ter, an anal ogy ith a moel of elec tric cur rent flo through ho mo ge neous lines may eas ily be e rive. The volt age be teen the lines, u(x, t), an the elec tric cur rent pass ing through them, i(x, t), sat isfy the fol lo ing equa tions: c u ( x, t ) i( x, t) (8) t x u( x, t) i ri l x t here r stans for is trib ute re sis tance (r = R/x), c for is trib ute ca pac i tance (c = C/x), an l for is trib ute in uc tance (l = L/x) of the line. From eqs. (8) an (9), hy per bolic ho mo ge neous if fer en tial equa tions of the sec on or er e scrib ing volt age an elec tric cur rent, the so-calle te leg ra phy equa tions, may eas ily be e rive: u( x, t) rc u ( x, t ) u( x, t) lc (0) x t t (9)

4 3 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... i( x, t) rc i ( x, t ) i( x, t) lc () x t t An anal ogy be teen the if fer en tial equa tions e scrib ing the PT re sponse eqs. (3) an (4), an the flo of elec tric cur rent through lines eqs. (8) an (9), is es tab lishe by in tro - uc ing the fol lo ing re la tion ships: t u J, i q, r, l, c Cv () k k Ho ever, by in tro uc ing anal ogy (), it is state only that heat con uc tion through one ho mo ge neous layer (air, sam ple or back ing) can be e scribe by an elec tric cur rent flo through ho mo ge neous lines [6, -6]. Since in PT fre quency meth os the ex ci ta tion of the sys tem (light beam) is mo u late by am pli tue, the gen er ate heat can be e scribe as S(t) = S 0 cos (t), an it is suit able to em - ploy the Fou rier trans form to an a lyze the prob lem. Sym bol sig ni fies mo u la tion fre quency. Tem per a ture vari a tion an heat flux can be rep re sente by their com plex rep re sen ta tives Ji ( x ) an qi (x): J ( x, t) J ( x)cos[ t + q ( x)] Re{ J ( x )} (3) i i i i q ( x, t) q ( x)cos[ t + y ( x)] Re{ q ( x )} (4) i i i i The anal o gous prob lem is solve by ap pli ca tion of com plex rep re sen ta tives for volt - age an cur rent, U an I, respectively: u( x, t) U ( x)cos[ t q ( x)] Re{ U ( x )} (5) i( x, t) I( x)cos( t y ( x)) Re{ I ( x )} (6) The sym bols q an y sig nify the phase lag of y namic tem per a ture J(x,t), or volt age u (x, t) an heat flux q(x, t) or elec tric cur rent i(x,t), from y namic source. Then, the tele graph equa tions be come or i nary lin ear if fer en tial equa tions of the sec - on or er in the com plex o main (j is imag i nary unit): U ( x ) ( jrc lc) U ( x) (7) x I ( x ) ( jrc lc) I ( x) (8) x It is con ve nient to rite lin ear if fer en tial equa tions of the sec on or er in the fol lo - ing form: ( ) U x I ( x ) s U ( x) an s I ( x) (9) x x an it fol los that s ( x ) is given by: s i z y, z r jl, y jc (0) s ( x ) is calle com plex trans mis sion co ef fi cient, hile z an y are the is trib ute im pe ance an a mit tance of the line. The gen eral so lu tion of eqs. (7) an (8), tak ing into ac count the re la tion be teen volt age an elec tric cur rent, is given by ex pres sions: U ( x) A exp ( s x) A exp ( s x) () i i i

5 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp A I ( x) exp ( A ) s x exp( i s ix ) () Zc Zc here Z c stans for the char ac ter is tic im pe ance of the line: z Zc (3) y hile A an A are con stants hich are to be e ter mine from the boun ary con i tions. Moel of the en vi ron ment an sur face ther mal sources In or er to e velop an elec tric net ork that ill be the an a log of the PT in uce sys - tem un er con si er ation, the boun ary an in ter faces con i tions eqs. (6) an (7), shoul be em ploye. The boun ary con i tions, eq. (6), es sen tially mean that the lay ers of air an back ing are much lon ger than the layer of the sam ple, sug gest ing that the en vi ron ment of the sam ple shoul be mo ele as very long (semi-in fi nite) lines. In the case of very long (semi-in fi nite) lines, the con stant A has to be zero, to pro vie fi nite val ues for volt age an cur rent hen x. There fore, for very long lines there hols the equa tion (in e pen ent of x) U ( x ) I ( x Zc const. (4) ) sho ing that a very long line can be mo ele as elec tric el e ment ith im pe ance equal to the char ac ter is tic im pe ance of the line. Be sies, the fact that A = 0 also im plies that near the en of very long lines volt age an elec tric cur rent ten to zero. The e rive con clu sion may be use to e velop an anal o gous elec tric moel for anal y - sis of PT phe nom ena in opaque sam ples, hich is pre sente in fig.. Lay ers of air an back ing, from the point of vie of their in ter ac tions ith the sam ple, may be mo ele ith im pe ances an that have val ues: jta jt b Zca, Zcb (5) jk C jk C a a The sam ple is mo ele by elec tric lines of length an by the is trib ute im pe ance an a mit tance: z ( jt ), s y j Cs (6) k The gen er ate heat at x = 0 is mo ele by ieal cur rent source giv ing elec tric cur rent S 0, thus sat is - fy ing boun ary con i tions eq. (7). The val ues of volt ages an elec tric cur rents in this cir cuit are equal to the val ues of tem per a ture vari a tions an heat flux at the re spec tive points of the sam ple. Moel of the sam ple s PT ap pli ca tions to char ac ter iza tion an im ag ing of ma te ri als im ply the i rect or in i - rect mea sure ment of tem per a ture vari a tions of one sur face of the sam ple. There fore, from the point of vie of the anal y sis of the PT re sponse, a rel e vant moel of the sam ple shoul be able to e scribe tem per a ture vari a tions on the sam ple s sur faces, hile tem per a ture vari a tions ithin b b Figure. Analogous electric circuit here sample is moele by electric lines

6 34 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... the sam ple are not of par tic u lar in ter est. There fore, the sim plest rel e vant moel of the sam ple shoul e scribe only the re la tions be teen sur face tem per a ture vari a tions an heat fluxes, i. e. the sim plest rel e vant anal o gous elec tric cir cuit can be e scribe only by volt ages an elec tric cur rents at the ens of the line. Putt ing x = 0 an x = into eqs. () an (), it can be eas ily shon that the volt ages an the elec tric cur rents at the ens of the line of fi nite length sat isfy the fol lo ing ma trix equa tion: ( ch ) Z sh( ) U ( 0 ) s s I ( 0 ) sh( ) ch( c s s U ( ) ) (7) I ( ) Zc It is usual in the the ory of elec tric cir cuits to for mally in tro uce to tal line im pe ance Z z s an to tal line a mit tance Y y s, hich can be use to re rite the pre vi ous ma trix equa tion in the form here only those pa ram e ters are use: Z ( ) ( ) ( ch ZY ) U Y sh ZY 0 I ( 0 U ( ) a a ) Y I U ( ) ( ) a a (8) sh( ZY ) ch( ZY ) I ( ) Z Re la tion (8) is knon in the the ory of elec tric cir cuits as the rep re sen ta tion of the elec tric net ork by a-pa ram e ters. It is also es tab lishe that, for any given ma trix of a-pa ram e - ters, it is pos si ble to com pose a four-ter mi nal elec tric net ork con sist ing of el e ments ith con - cen trate pa ram e ters, hich is e scribe by the given ma trix. There fore, it is pos si ble to com - pose an elec tric net ork con sist ing of the el e ments ith con cen trate (not any more is trib ute) pa ram e ters, hich has the same re la tions be teen the volt ages an elec tric cur rents at the ens, e scribe by eq. (8), as the con si ere line. The scheme of the net ork is pre sente in fig. 3, hile the val ues of the im pe ance an a mit tance in the net ork are: th Z Z eq ZY ZY an sh ZY Yeq Y ZY (9) Fig ure 3. Elec tric net ork mo el ing the PT inuce sample As the con si ere line of fi nite length rep re - sents an elec tric moel of the PT in uce sam ple, the elec tric net ork from fig. 3 is an elec tric moel of tem per a ture vari a tions an heat fluxes at the sur faces of the sam ple. The im pe ance an the a mit tance rep re sent - ing the sam ple in fig. 3 shoul have the val ues: Z eq ( jt) k th Dt Dt ( t) ( t) jt, jt Y eq jc sh Dt Dt ( t) ( t) jt jt (30)

7 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp The spe cial case con si ers multi-layer struc - tures, here each layer has if fer ent ther mal char ac ter is tics. Ho ever, since the boun ary con i tions for the heat trans fer be teen the lay - ers con sist of con ti nu ity of heat flux an tem - per a ture on in ter fa cial sur faces, hile the boun ary con i tions be teen the elec tric lines con sist of con ti nu ity of volt age an elec tric cur - rent, multi-layer struc tures may be mo ele by com plex elec tri cal net ork con sist ing of sev eral stages, each stage rep re sent ing one layer of multi-layer struc ture by elec tri cal net ork in fig. 3. Such elec tric moel for anal y sis of the PT re sponse of a struc ture ith n lay ers is pre sente in fig. 4. Each layer L m is rep re sente by im pe ances an a mit tance that have val ues as e ter - mine by eq. (30). While the cir cuit in fig. 3 has a rather sim ple struc ture, the im pe ance an the a mit - tance can not be rep re sente by real elec tric el e ments (re sis tors, ca pac i tors, an coils); so, the ob - taine anal o gous cir cuit is not suit able for ap pli ca tion to stan ar meth os of anal y sis of elec tric cir cuits. Anal ogy ith pas sive lin ear net orks Ex pres sions for the equiv a lent im pe ance an a mit tance, eq. (9), of the PT in uce sam ple may be ex presse us ing the fol lo ing se ries ex pan sions: sh( ZY ) ZY Z Y an (3) ZY 6 0 th There fore, hen the con i tion: is sat is fie, it hols that: ZY ZY Z Y 4 90 ZY ZY Z Y ZY 6 In that case, the anal o gous elec tric net ork from fig. 3 may be pre sente by us ing only pas sive lin ear el e ments (re sis tors, ca pac i tors, an coils), as it is shon in fig. 5. The el e ments in a pas sive lin ear net ork (in fur - ther text also e note by the ab bre vi a tion PLN) that is the moel of a sur face tem per a ture vari a tions of a PT in uce sam ple has the fol lo ing val ues: (3) (33) Z Z ( r j), Y Y j C (34) eq Fig ure 4. Anal o gous elec tri cal net ork for the PT in uce sam ple ith n lay ers eq Figure 5. Analogous passive linear netork moel of the PT in uce sam ple

8 36 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... R t, L, C C v k k eq eq eq When the PT re sponse can be mo ele by PLN, the ma trix equa tion (8) re uces to: ZY ZY Z U ( 0 ) 4 I ( 0 U ( ) ) ZY Y I ( ) The spe cial case con si ers multi-layer struc tures, here each layer sat is fies the con i - tion eq. (33). These struc tures may be mo ele by PLNs con sist ing of sev eral stages, each stage rep re sent ing one layer of multi-layer struc ture by PLN in fig. 5. Dis cus sion Con i tions for mo el ing PT re sponse by pas sive lin ear net orks It is al reay es tab lishe that the con i tion to moel anal o gous elec tric cir cuit by pas - sive lin ear el e ments is ex presse by eq. (33). Re rit ten by elec tric line pa ram e ters, the re la tions take the form: ( lc + jrc) 6 (37) Elec tric line pa ram e ters may be use to ex press con i tion for the mo el ing in terms of the prop er ties of the PT in uce sam ple: t s j 6D 6D (38) s s hich, be ing ex presse by com plex num bers, may be turne into con junc tion of to con i tions ex presse by real num bers: an (39) here the fol lo ing sym bols are in tro uce: r D r D an D 6 D t The con i tions given by eq. (39) can also be ex presse as: In tro uc ing crit i cal thick ness t, e fine so that: s D s (35) (36) (40) D an (4) t t t t t 6D s t s (4) D t t (43)

9 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp the con i tion for mo el ing the PT re sponse by PLN eq. (39), re uces to: n t, t t, here t, t 6D s t s, n (44) ts, t The e rive con i tion (44) shos that the PT re sponse of any sam ple can be mo ele by PLN for suf fi ciently lo fre quen cies. For a sam ple ith thick ness t, the lim it ing fre quency is t ; for sam ples thin ner than t, the lim it ing fre quency in creases in versely pro por tion ally to the thick ness of the sam ple; for sam ples thicker than t the lim it ing fre quency e creases in - versely pro por tion ally to the square of the thick ness of the sam ple. At the same time, it shoul be note that hen the mo el ing of the sam ple by PLN is pos si ble, the sam ples thicker than t can be mo ele by net orks con sist ing only of re sis tor an ca pac i tor (RC net orks), be cause ther - mal mem ory ef fects may be ne glecte at fre quen cies much loer than t. The con i tion (44) is graph i cally pre sente in fig. 6: in or er to ap ply the mo el ing by PLN, the mo u la tion fre quency an epth of the PT in uce sam ple shoul e scribe a point eep ithin the shaoe area. Ta ble pres ents a list of typ i cal rep re sen ta tives of var i ous classes of ma te ri als, to - gether ith re spec tive crit i cal thick nesses t an fre quen cies t, cal cu late on the ba sis of bulk ma te rial prop er ties. Figure 6. Graphical presentation of conitions for moeling PT inuce sample by linear passive electric netork The ta ble sug gests that thin lay ers (up to the or er of mi crom e ter) of crys tal line ma te ri - als may be mo ele by PLN for all ex per i men tally achiev able mo u la tion fre quen cies (of the or - er of 0 Hz to 00 MHz), hile the thick nesses of the sam ples of non-crys tal line ma te ri als, hich may be mo ele by PLN, are of the orer of millimeter. Characterization of PT sample properties by analogy ith passive linear netorks Ta ble. A list of the crit i cal thick ness an the lim it ing frequencies Ma te rial (rep re sen ta tive) t [m] t [s ] Metal (alu mi num) Semiconuctor (silicon) Polymer (PVC) Po rous (ry san) Organic (oo) This chap ter is cusses the pos si bil i ties of ap pli ca tion of the e vel ope anal ogy in the e ter mi na tion of the prop er ties of PT in uce sam ples. The anal ogy ill be ap plie by us ing meth os for fre quency anal y sis of PLN to anal y sis of fre quency e pen ence of the PT re sponse. In or er to char ac ter ize the ma te rial prop er ties of thin lay ers, fre quency re sponse of if fer en tial PT re sponse of the so-calle free stan ing sam ple is mea sure. The if fer en tial PT

10 38 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... re sponse is the ra tio of tem per a ture vari a tions on the sam ple s sur faces, J()/J(0), an free-stan ing sam ple is a sam ple that is sur roune by air or some other gas (ie ally vac uum). Heat con uc tiv i ties of gases are very lo com pare ith sol is; so, as sum ing that k air 0, it can be con si ere that the char ac ter is tic im pe ances of air an back ing in anal o gous elec tric moel, Z ca an Z cb, are in fi nite, so that elec tric cur rents pass ing through them are equal to zero. The the ory of PLN shos that fre quency e pen ence of any volt age may be ex presse in the com plex o main by ra tio nal func tions of fre quency, an an a lyze by ap ply ing anal y sis of ze roes an poles of the func tions, as it ill be pre sente later in the text. In the case of the free-stan ing sam ple, I ( ) in the anal o gous elec tric cir cuit from fig. 5 equals to zero, an thus eq. (36) gives re sult: ZY ( U ( 0) U ) (45) There fore ap ply ing the anal ogy be teen the elec tric cir cuit an the PT in uce sam ple, an the al reay es tab lishe re la tions (), (33), (35), (40), an (4), it can be con clue that: J ( ) Dt G( j) J ( ) 0 3 j 3 t 3s 3 s t j t D t t here s = j. The func tion G(s) is, in fact, the trans fer func tion of the PT in uce sam ple if the tem per a ture vari a tion at the il lu mi nate sur face J(0) is con si ere to be the sys tem in put, an the tem per a ture vari a tion on the op po site sie of the PT sam ple J() the out put of the in ves ti - gate sys tem. G(s) is a trans fer func tion of the sec on or er, an its be hav ior e pens on the roots of its char ac ter is tic poly no mial hich is the poly no mial in e nom i na tor of G(s) [7]. The phys i cal in ter pre ta tion of G(j) is that its moulus G(j) shos fre quency e - pen ence of the ra tio of am pli tues of tem per a ture vari a tions J()/J(0), an its ar gu ment arg[g(j)] shos fre quency e pen ence of the phase if fer ence be teen the tem per a ture vari a - tions J() an J(0). The ex pres sions for G(j), hich is the am pli tue-fre quency char ac ter is - tic of the sys tem, an arg[g(j)], hich is the phase-fre quency char ac ter is tic of the sys tem, are: J( ) G( j) G( ) J( 0) 3 t t 4 9 t t t (46) (47) q( ) q( 0) arg G( j) arctg (48) t t 3 t The usual metho of an a lyz ing PLN is ap pli ca tion of Boe plots, as ymp totic log-log plots of the am pli tue-fre quency char ac ter is tic. If the thick ness of the PT sam ple sat is fies the con i tion: t (49) 3 then the roots of the char ac ter is tic poly no mial are real, an the am pli tue-fre quency char ac ter is - tic of the trans fer func tion can be ap prox i mately pre sente by the Boe plot in fig. 7.

11 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp There fore, the log-log graph of am pli tue of the ex - per i men tally mea sure if fer en tial PT spec tra has to is tinct fea tures, fre quen cies an, here the graph changes the slope. The eas i est ay to e ter mine those fre quen cies from the re core spec tra coul be to ra the lo-fre quency as ymp tote of the graph (hich is par - al lel to ab scissa), the high-fre quency as ymp tote of the graphic (hav ing a slope of 40 B) an the me ium range as ymp tote (ith a slope of 0 B); then, the in ter - sec tion of the lo-fre quency as ymp tote an the me ium range as ymp tote has the ab scissa, an the in ter sec tion of the high-fre quency as ymp tote an the me ium range as ymp tote has ab scissa. Fre quen cies an can eas ily be cal cu late to be: t t t t , sho ing that the fre quency e creases from t / to zero, hile the fre quency in creases from t / to t ith the in crease of sam ple thick ness. Tak ing into con si er ation that the PLN moel of the PT sam ple can be ap plie only to fre quen cies sat is fy ing the con i tion e scribe by eq. (44), the con clu sion about the val ues of an means that the Boe plot anal y sis may be ap plie only in the e ter mi na tion of, since the PLN moel is not vali for the fre quency range to hich be longs. Nev er the less, by e ter min ing from the ex per i men tal ata for suf fi ciently thick PT sam ples, it is pos si ble to es tab lish one an a lyt i cal re la tion be teen the ma te rial pa ram e ters t an t (an con se quently t an D). It means that the fit ting pro ce ure for the e ter mi na tion of the ma te rial pa ram e ters may be sig nif i cantly sim pli fie by ap pli ca tion of PLN moel, thus im - prov ing re li abil ity of the ata ob taine by fit ting. If the thick ness of the PT sam ple oes not com ply ith the con i tion (49), the char ac - ter is tic poly no mial has ou ble real, or com plex con ju gate, roots. The Boe plot in that case has one is tinc tive fea ture, fre quency 0 here the Boe plot rap ily changes its slope from 0 B to 40 B. This fre quency coul eas ily be e ter mine from log-log graphics of the ex per i men tal if fer en tial PT spec tra as the in ter sec tion of their lo-fre quency an high-fre quency as ymp - totes. Ho ever, by an a lyz ing eq. (46), it can eas ily be shon that the fol lo ing equa tion hols: 0 t t (5) 3 here from it can be con clue that in the case hen the con i tion ex presse by eq. (49) is not sat is fie, the high-fre quency as ymp tote be longs to a fre quency range here mo el ing by PLN is not ap pli ca ble, an con se quently, Boe plot anal y sis is not use ful for thin sam ples. Ho ever, it is ell knon in the the ory of elec tric cir cuits that a trans fer func tions of the form (46), hich can be re rit ten in the form: Dt 3 G( s), T, x 3s 3 s T s xts 3 t 3 D (5) t D t D t Fig ure 7. Boe plot (thick line) an log-log graph (thin line) of am pli tue of if fer en tial PT re sponse for PT in uce sam ple that sat is fies eq. (49) here an are roots of the char - ac ter is tic poly no mial of G(s) t (50)

12 40 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... has the res o nant max i mum [5]: G max. x x at frequency x (53) hen the con i tion x < 0.5 is statisfie, as it is shon in fig. 8 Con si er ing eq. (5), it means that PT spec tra of thin sam ples, sat is fy ing con i tion / t < /3 /, ex hibit a peak at a fre quency max., hich can still be ap prox i mately con si ere to be long to the fre quency range here PLN moel is ap pli ca ble, be cause the va li ity of the moel for thin sam ples is e ter mine by con i tion (44). There fore, hile the Boe plot anal y sis can not be ap plie for thin sam ples, the PLN moel pro vies the pos si bil ity of e ter min ing 0 an xof thin sam ples from the ex per i men tal ata ith out ap pli ca tion of the fit ting pro ce ure. Equa tions (44), (5), an (53) en able fur ther e ter mi na tion of the val ues of the sam ple pa ram e ters D s an t s, thus pro vi ing the pos si bil ity of per form ing char ac ter iza tion of a sam ple ith out fit ting. Be sies, it shoul be note that the ex pres sion given by eq. (46), e rive by ap pli ca tion of the PLN mo el - ing, is suit able for non-lin ear curve fit ting. Thus, even Figure 8. Spectra of ifferential PT response (eq. 53) for three ifferent values of x max. 0 in the cases hen con ven tional meth os of anal y sis of fre quency anal y sis of elec tric cir cuits are not ap pli ca - ble, the mo el ing of the PT re sponse by PLN sim pli fies the char ac ter iza tion of PT in uce sam ples. The spe cial case that ill be is cusse is the case of the free-stan ing multi-layer PT sam ple. In or er to moel the PT re sponse of such struc ture by lin ear pas sive elec tric net ork pre sente in fig. 4, it is nec es sary that the con i tion (44) be sat is - fie for each of the lay ers. If that is the case, a PT in uce sam ple ith n lay ers may be mo ele by lin ear pas sive elec tric net ork that, base on eq. (36), can be rep re sente by a-pa ram e ters as fol los: U ( 0 ) ( ) ( ) T T... T U T U n a I ( 0 ) I ( (54) ) I ( ) here T m (m =,,,n) stans for the ma trix of a-pa ram e ters of m-th layer. In the case of the free-stan ing multi-layer PT in uce sam ple, sim i lar to the case of the free-stan ing sin gle layer sam ple, the if fer en tial PT re sponse J()/J(0) can be cal cu late by anal y sis of the anal o gous net ork in the case hen I ( ) = 0. From eq. (54) then fol los that: U( ) ( T ) U( ) (55) 0 Con si er ing the form of ma trix T m, it can be con clue that the el e ments of the ma trix T a are poly no mial ex pres sions of the gen eral form: n u n ( T ) Z u vy v sk b a (56) u0v 0 uv here a k an b uv are real con stants. Hence, the if fer en tial PT re sponse of the multi-layer sam - ple can be ex presse in the fol lo ing form: J ( ) G( s) (57) J ( ) n 0 a sk k 0 k k 0 k 0

13 THERMAL SCIENCE: Vol. 3 (009), No. 4, pp There fore, the rec om mene non-lin ear fit ting pro ce ure for e ter min ing ther mal prop er ties of multi-layer sam ples is to ap ply mo el ing by lin ear pas sive elec tric net ork in or - er to fin the e pen ence of co ef fi cients a k on the ther mal prop er ties of the ma te ri als of lay ers, an then to per form non-lin ear fit ting of square of the in verse of the if fer en tial PT re sponse spec tra to poly no mial of or er 4n, as: J( 0) n J ( ) ( ) ( ) k 0 G j k a k k n ( ) k a k k (58) k An ex am ple of mo el ing of a three-layer struc ture (metal-poly mer-metal) by elec tric net ork of the sec on or er is pre sente in pa per [6]. It is also of in ter est to point out that the ex pres sions (5) an (53) pres ent the o ret i cal ba sis for Pae s ap prox i mate pro ce ures for the e ter mi na tion of ther mal prop er ties of bi po lar tran sis tors pre sente in [8]. Con clu sions This pa per pres ents an anal y sis of the pos si bil ity an po ten tials of mo el ing of the PT re sponse of a thin sam ple by lin ear pas sive elec tric net ork. It is es tab lishe that such a pos si bil - ity e pens on the mo u la tion fre quency of PT ex ci ta tion an the thick ness of the sam ple. If this con i tion is sat is fie, the en vi ron ment of the sam ple may be mo ele by im pe ances an the sam ple it self by lin ear pas sive net ork pre sente in fig. 5. If the free-stan ing PT in uce sam ple can be mo ele by pas sive lin ear net ork, its if fer en tial PT re sponse is e ter mine an it is shon that ther mal prop er ties of the ma te rial of the sam ple can be e ter mine by the anal y sis of the peak in PT spec tra (for the sam ples thin ner than t /3), the com bi na tion of the Boe plot anal y sis an non-lin ear fit ting (for the sam ples thicker than t ) or by lin ear fit ting of poly no mial ex pres sion (for the thin sam ples of ar bi trary thick ness). More over, by anal y sis of the case of the multi-layer thin film sam ple, it has been shon that the mo el ing of the PT re sponse by elec tric net ork may be use ful even hen the ob taine net ork is not lin ear or not sim ple, since the ex pres sions for the PT re sponse may be more eas ily e rive an an a lyze, of fer ing eas ier in sight into the phe nom e non com pare ith other mo els. It can be con clue that mo el ing of the PT re sponse by lin ear pas sive net ork is an easy-to-use an use ful tool for the anal y sis of PT sig nal in func tion of mo u la tion fre quen cies, since the e rive mo els can sig nif i cantly sim plify the anal y sis of the PT re sponse. Be sies, ob - taine re sults coul be em ploye in ex trac tion of ther mal prop er ties by other lock-in thermography meth os. Acknolegments This ork as sup porte by Min is try of Sci ence an Tech nol ogy Development of the Re pub lic of Ser bia (pro ject no. 403). References [] Manelis, A., Diffusion-Wave Laser Raiometric Diagnostic Quality-Control Technologies for Materials, NDT&E International, 34 (00), 4, pp [] Vargas, H., Mirana, L. C. M., Photoacoustic an Re late Photothermal Tech niques, Phys. Rep., 6 (988),, pp. 45-0

14 4 Galovi}, S., [o{ki}, Z., Popovi}, M.: Analysis of Photothermal Response of Thin Soli... [3] Rosencaig, A., Gersho, A., The ory of the Photoacoustic Ef fect ith Sol is, J. Appl. Phys., 47 (976),, pp [4] Toorovi}, D. M., et al., Thermoelastic an Elec tronic Strain Con tri bu tions to the Fre quency Trans mis - sion Photoacoustic Ef fect in Semi con uctors, Phys. Rev., B 55 (997), 3, pp [5] Galovic, S., Kostoski, D.,Photothermal Wave Prop a ga tion in Me ia ith Ther mal Mem ory, J. Appl. Phys., 93 (003), 3, pp [6] Chen, G., Hui, P., Pulse Photothermal Mo el ing of Com pos ite Sam ples Base on Trans mis sion-line The ory of Heat Con uc tion, Thin Soli Films, 339 (999), pp [7] Hansen, P. C., Nu mer i cal Tools for Anal y sis an So lu tion of Freholm In te gral Equa tions of the First Kin, Inverse Problems, 8 (99), pp [8] Nagasaka, Y., Thermophysical Prop er ties Mea sure ment of Liq uis an Sol is by Photoacoustic an Photothermal Tech niques, Pro cee ings (Es. F. Scuieri, M. Bertolotti), 0 th International Conference of Photoacoustic an Photothermal Phe nom ena, Rome, It aly, 998, AIP Press, Woobury, Ne York, USA, 998, Vol. 463, pp [9] Tzou, D. Y., An En gi neer ing As sess ment to the Re lax ation Time in Ther mal Wave Prop a ga tion, Int. J. Heat Mass Trans fer, 36 (993), 7, pp [0] Novikov, I. A., Har monic Ther mal Waves in Ma te ri als ith Ther mal Mem ory, J. Appl. Phys., 8 (997), 3, pp [] Ulbrich, C. W., Ex act Elec tric Anal ogy to the Vernotte Hypothesis, Phys. Rev., 3 (96) 6, pp [] Suszynski, Z., Ther mal Moel Base on the Elec tri cal Anal ogy of the Ther mal Pro cesses, Proceeings (Es. F. Scuieri, M. Bertolotti), 0 th International Conference of Photoacoustic an Photothermal Phe - nom ena, Rome, It aly, 998, AIP Press, Woobury, Ne York, USA, 998, Vol 463, pp [3] Hui, P., Tan, H. S., A Trans mis sion-line The ory for Heat Con uc tion in Multilayer Thin Films, IEEE Trans. Comp. Pack. Manufact. Tech.-Part B, 7 (994), 3, pp [4] Hui, P., Tan, H. S., Mo el ling of Ther mal Diffusivity Mea sure ment of Di a mon Thin Films Us ing a Pulse La ser Tech nique, Surf. Coat. Tech nol ogy, 6 (993), pp [5] Kaminski, W., Hy per bolic Heat Con uc tion Equa tion for Ma te ri als ith a Nonhomogeneous In ner Struc - ture, ASME J. Heat Trans fer, (990), pp [6] Galovi}, S., Šoški}, Z., Photothermal Char ac ter iza tion of Poly mer-base Three-Lay ere Nanoelectronic Struc tures, Pro cee ings, ISSE, Bu a pest, 008, pp [7] Stoji}, M., Con tin u ous Sys tem of Au to matic Con trol (in Ser bian), Nauna knjiga, Bel grae, 990 [8] Nenaovic, N., et al., Ex trac tion of Ther mal Net ork Func tion for Bi po lar RF De vices an ICs, Pro cee - ings, XLVII ETRAN Con fer ence, Herceg-Novi, Ser bia an Montenegro, 003, Vol. 4, pp. 5-8 Author's affiliations: S. P. Galovi} (corresponing author) Laboratory for Raiation Chemistry an Physics, Vina Institute of Nuclear Sciences P. O. Box 5, 00 Belgrae, Serbia bobagal@vinca.rs Z. N. [o{ki} Faculty of Mechanical Engineering Kraljevo, University of Kragujevac, Serbia M. N. Popovi} Laboratory for Thermal Engineering an Energy, Vina Institute of Nuclear Sciences Belgrae, Serbia Paper submitte: July 4, 009 Paper revise: November 6, 009 Paper accepte: November 7, 009