Analyss f ellpsmetrc data btaned frm curved surfaces J. Křepelka Jnt Labratry f Optcs f Palacky Unversty and Insttute f Physcs f Academy f Scences f the Czech Republc, 7. lstpadu 5a, 77 7 Olmuc, Czech Republc Abstract: Ths artcle deals wth quanttatve errr analyss resultng frm ellpsmetrc data btaned frm measurement n curved surfaces ncludng the nfluence f nn-cllmated beams. Numercal mdel based n the cmbnatn f gemetrcal and wave ptcs s restrcted t the example f sngle delectrc layer depsted n the substrate wth cmplex ndex f refractn. Three methds fr averagng measurable ellpsmetrc data are cmpared. Keywrds: ellpsmetry, curved surface, nn-cllmated beam. Nrmal t the curved surface Let z = z( x, y) s a functn descrbng the curved surface n a Cartesan c-rdnate system, then a vectr perpendcular t the tangental plane n the pnt r ( x, y) = ( x, y, z( x, y)) f the ( x, y) surface s u ( x, y) = grad( z( x, y) z). Hence the unt nrmal vectr s n ( x, y) = u, u( x, y) where u( x, y) = u ( x, y) s a sze (nrm) f the vectr u ( x, y). Especally fr the upper half- sphere z( x, y) = R x y, x + y R, where R s a sphere radus (and e f the sphere cncdes wth the rgn f c-rdnate system), we have a unt vectr nrmal t the surface at the pnt r( x, y) = ( x, y, z( x, y)) ( x, y, R x y ) r( x, y) r( x, y) n ( x, y) = =. (a) R r( x, y) R Ths surface appears t be cnvex frm the pnt f vew f ncdent beam. Smlarly, fr the bttm half-sphere z( x, y) R x y = the unt vectr nrmal t the cncave surface s ( x, y, R x y ) ( x, y) ( x, y) r r n ( x, y) = =. (b) R r( x, y) R Prbably the frmula x + y R R x y = R R (c) x + y + R can be useful fr numercal purpses when x + y R.. Lght beam ncdentng the curved surface Usually fr the beam ncdentng the curved surface we can defne the al (mean r chef) ray that s fr nstance pssble t dentfy wth the symmetry axs f a wde beam f crcular crss-sectn. Fr the cnverged beams wth angle f cnvergence β > r dvergent beams wth angle f dvergence β < r cllmated beams ( β = ) we can defne the prpagatn
drectn f the al ray by a unt vectr n, = ( n,, x, n,, y, n,, z ). Usng standard sphercal c-rdnates ϕ,π, θ π/,π ( csθ, ) ths vectr can be expressed as n = (csϕ sn θ,snϕ sn θ, cs θ ). () Let r = (,, R) s a pnt where the al beam ncdents the sphere surface, then a crcle wth e r and dameter d n the plane perpendcular t n, s a set f pnts C r: r r d / n ( r r ) =. (3) { }, We may cnstruct C (that can be generalsed fr arbtrary beam crss sectns) usng tw unt vectrs a, a cmpsng the rthnrmal base tgether wth n, n the fllwng way: π (a) Whenθ = (r n z =,.e. tangental ncdence), then a = ( n, n,) (sn ϕ, cs ϕ,). (4),, y,, x π (b) When θ >, (.e. n,, z = csθ < ), then fr nstance ( ) a = ( n,, n ) / n ( cs θ,,csϕ sn θ ) / snϕ snθ,, z,, x,, y. (5) Of curse n these bth cases the scalar prduct a n, =. Especally ϕ = π/ gves n = (,sn θ, cs θ ) and a = (,, ), snce, θ θ =. The ther requred cs / sn vectr a can be btaned as a crss prduct: a = a n,. (6) Especally we have π (a) fr θ =, a = (,,), π (b) fr θ >, a = ( n,, xn,, y, n,, y, n,, yn,, z ) / n,, y and descrptve example ϕ = π/ prvdes a = (, cs θ,sn θ). We can express an arbtrary element f C usng lcal Cartesan c-rdnates ( x, y ) n the frm f a sum f tw vectrs r + a ( x ', y '), where a( x, y ) = a x + a y (7) s a lnear cmbnatn f the base vectrs a, a meetng the cndtn x + y d / 4. Anther pssblty fr cnstructn f C emplys plar c-rdnates ( γ, q) defned by denttes x = γ cs( q), y = γ sn( q), where γ d / and q < π. The unt vectr charactersng the drectn f the sngle ray prpagatn, hwever changng lnearly (as pstulated) wth the dstance a = a, s then d β n, a tan n =, (8) d β n, a tan ther varances f ths smple premse are als acceptable. Fr the denmnatr n (Eq. 8) we can fnd
β d β n a = + a. (9) d, tan tan t + representng sngle ray f the ncdent A vectr lne equatn wth parameter ( ), beam s therefre: r ( t) = r + a + tn () and a frst ntersectn f ths ray wth the reflectng surface derved frm the quadratc equatn r ( t) = R s n detal r = r + a + tn, (a) ( ) t = Rn + a n ( Rn + a n ) ( a + Ra ), (b) z z z where ndex z means z -cmpnent f the vectr. Generally (except cllmated beams) vectrs a, n are nt mutually perpendcular. The dscrmnant (.e. an expressn under the rt sgn n Eq. b) f the slved quadratc equatn has t be pstve fr exstng ray ntersectn wth the sphere and s just equal t zer fr tangental ncdence. 3. Plarsatn f ncdent rays The nrmal t the surface s n = r / r ( r s defned n Eqs. a, b) and therefre tw lcal unt vectrs, prjectng tw rthgnal cmpnents and allwng t defne n ths way the plarsatn state f the electrc felds, are n n s = n n, p = s n. (a) T be nted that the plarsatn state s nt defned n the case f nrmal ncdence when n = n. Vectrs p, s, n create rght-handed rthnrmal system, s s plarsatn cmpnent f electrc feld s ts prjectn t the vectr s, smlarly fr p plarsatn cmpnent. In addtn t the lcal quanttes (Eq. a), varyng wth partcular cnvergent r dvergent rays, we can defne smlar useful quanttes related t the al ray by frmulae: n, n s =, p = s n,, where n = (,,). (b) n n, The plarsatn state f the ncdent lght, charactersed by the cmplex electrc feld vectr E = ( E, E, E ), prvded that t s the same fr all beam rays (what can be certanly, x y z generalsed t arbtrary beam plarsatn prfles), can be determned just usng vectrs specfed n (Eq. b). The cmplex varables E,,s = E, s, E,,p = E, p (3) are s and p cmpnents f electrc feld f the al beam and presumably als f all rays f the ncdent beam, hence the electrc feld vectr f the arbtrary ray s E = E s + E p. (4),,s,,p Actually, fr each ray the s and p cmpnents f the feld are the same as fr the al ray n cnsstency wth pstulatn E,s E s = E,,s, E,p E p = E,,p. (5) The plarsatn state f the ncdent beam can be als defned applyng ellpsmetrc parameters ψ, accrdng t the frmula E / E E / E = tanψ exp( ), (6a) frm whch we have,p,s,p,,s,
E,p = E tanψ exp( ), E,s = E, (6b) where E s nnzer cmplex cnstant wth abslute value prprtnal t the square rt f ncdent lght ntensty. It s bvus that nstead f the vectr base a, a we can chse vectrs s, p satsfyng the transfrmatn denttes a = cs( q) s + sn( q) p, a = sn( q ) s + cs( q) p, (7a) where cs( q ) = a s = a p, sn( q ) = a p = a s. (7b) 4. Plarsatn f the reflected beam Unt vectrs defnng the drectns f the lcally reflected rays can be btaned frm the equatn n = n ( n n) n (7) r and lcal angles f ncdence α (dentcal t the angles f reflectn α r ) are α αr = arccs( n nr ) = arccs( n n ). (8) Desgnatng the lcal ampltude reflectvtes f the curved surface fr s and p electrmagnetc waves by r s, r p (they are cmplex numbers dependng n the angle f ncdence, wavelength and ptcal prpertes f reflectng materal) we can calculate the ampltudes f reflected electrc feld fr each plarsatn state f the ray Er,s = rs E,s, Er,p = rp E,p, (9) s that the electrc feld vectr f the reflected ray s E = E s + E p () r r,s r,s takng nt accunt that unt vectr s (nrmal t the plane f ncdence) and unt vectr p (lyng n the plane f ncdence) are bth dentcal fr the ray ncdentng the surface and ray reflected frm the surface. The crrect ellpsmetrc parameters ψ crr, crr f the reflected wave are then defned frm the relatn tanψ exp( ) = r / r, () hence tanψ exp( ) crr crr crr p s E E E. () tan exp( ) r,p,s r,p crr = = Er,s E,p Er,s ψ Usng (Eq. ) t s pssble t slve the nverse ellpsmetrc prblem, fr nstance fr sngle delectrc layer depsted n the substrate f knwn refractn ndex, fr all rays ncdentng the curved surface under dfferent lcal angles f ncdence, and then t btan the same ndex f refractn and thckness f the layer, althugh evdently the nput ellpsmetrc parameters lcally dffer. Otherwse, n the case when a lght detectr (fr nstance deal rtatng analysatr capable t dstngush and analyse sngle beam rays wth suffcent space reslutn) s calbrated fr the al ray (see defntn f vectrs s and p by Eq. b), we btan (ntentnally) naccurate ellpsmetrc parameters ( Er p ) tanψ exp( ) =. (3) ( Er s ) tanψ exp( ) It s clear that quanttes ψ a frm (Eq. 3) hld true precsely nly fr the al ray, hwever fr all ther rays f the lght beam allw t determne quanttatvely the devatns
frm the crrect values and therefre t estmate the errr emerged frm measurement ellpsmetrc parameters f curved surfaces n cmparsn wth a flat surface and als t analyse the effect f beam cnvergence r dvergence n cmparsn wth cllmated beam. 5. Inverse ellpsmetrc prblem fr a sngle thn delectrc layer The am f ths sectn s t fnd the thckness and ndex f refractn f sngle delectrc layer frm knwn ellpsmetrc parameters ψ and. Fr ths purpse we defne quanttes n ndex f refractn f external envrnment (superstrate, ar), n (unknwn) real ndex f refractn f the layer wth (unknwn) thckness h, n g ndex f refractn f the substrate (generally cmplex number), α angle f ncdence measured n the superstrate and λ wavelength f lght n vacuum. We als use the admttance f vacuum y = ε / µ ( µ, ε permeablty and permttvty f vacuum) fr defntn f relatve admttances fr bth lght plarsatns f all meda cnsderng the law f refractn n the frm n snα = n snα = n snα, see fr nstance [] g g (p) Y = n ( n sn α ) = n csα, Y = n / n ( n sn α ) = n / csα, (s) (s) ( p) Y = n ( n sn α ) = n csα, Y = n / n ( n sn α ) = n / csα, (4) (s) g g g g ( p) Y = n ( n sn α ) = n csα, Y = n / n ( n sn α ) = n / csα. g g g g g The phase shft f plane wave prpagatng nce thrugh the layer ϕ s ndependent n the plarsatn and s equal t π ϕ = h n ( n sn α). (5) λ The slutn f Maxwell equatns fr plane mnchrmatc waves n strpc medum can (s,p) be expressed by matrx S physcally representng transfrm peratrs fr tangental cmpnents f electrc feld vectrs, hwever separated nt cunter-prpagatng waves (s,p) (s,p) Y csϕ snϕ (s,p) S = Y (s,p) (s,p) (6) Y (s,p) g Yg (s,p) Y sn ϕ cs ϕ Y (s,p) fr bth plarsatn states. Frm matrx elements f S we can cmpute eght ampltude ceffcents, namely ampltude reflectvtes fr waves prpagatng n the drectn frm superstrate t the substrate n the frm (s,p) S rs,p =. (7) (s,p) S Emplyng the defntn f ellpsmetrc parameters rp tanψ exp( ) =, (8) rs cnsdered n ths mment knwn (fr nstance as a result f measurement) we can derve frm (Eqs. 7, 8) the quadratc equatn - a cndtn fr the wanted layer parameters [] a exp( 4 ϕ) + b exp( ϕ) + c =. (9) Here (s) (p) (s) (p) a = b d tanψ exp( ) d b,
b = a d + b c d a + c b, (3) (s) (p) (s) (p) (s) (p) (s) (p) ( ) tanψ exp( ) ( ) (s) (p) (s) (p) = ψ, c a c tan exp( ) c a where quanttes wth s r p ndces (nt all are ndependent) (s,p) (s,p) (s,p) (s,p) (s,p) (s,p) (s,p) (s,p) (s,p) (s,p) a = ( Y / Y )( + Y / Y ), b = ( + Y / Y )( Y / Y ), (3) g c = ( + Y / Y )( + Y / Y ), (s,p) (s,p) (s,p) (s,p) (s,p) g g d = ( Y / Y )( Y / Y ). (s,p) (s,p) (s,p) (s,p) (s,p) g The slutn f (Eq. 9) has t be n the frm f cmplex unts (due t the fact that ϕ s real), therefre the cmplex cnjugated equatn c exp( 4 ϕ) + b exp( ϕ) + a =, (3) where astersk means cmplex cnjugated quanttes, have t be fulflled t. Therefre we can reduce the quadratc equatn fr exp( ϕ) t the lnear equatn and a cndtn fr ndex f refractn Admttable layer thcknesses are then σ hk = k h π where b c ba exp( σ ) = a c per f ( n ) b c b a a c =. (33) per, k =,,..., (34) λ <, hper = n ( n sn α ), σ π, (35) h s a perd f the layer thckness. Ths methd s mre effcent than least square methd appled fr tw unknwn varables at the same tme. Frm lcal ellpsmetrc varables ψ crr, crr (Eq. ) and usng the abve relatns we can cmpute a lcal layer parameters dentcal wth the gven layer ndex f refractn and ts thckness because these quanttes have t be ndependent n the ray selectn. Certanly, we have dfferent values when cmputng the ellpsmetrc parameters btaned frm the detectr calbrated fr the al beam (Eq. 3), whch are ntentnally affected by the systematc errr due t the measurng methd. 6. Averagng f ellpsmetrc parameters T average values f quanttes ver beam crss-sectn we can use an equdstant grd s that the electrc feld f each ray s cnsdered wth the same weght what allws t apprxmate ntegrals ver surface suffcently. Fr ths purpse the fllwng equdstant square grd spread ver C may be sutable d k x k = + N, k =,..., N, (36a) d l y l = + N, l =,..., N. (36b) But nly the subset M f the rdered pars f natural numbers refers actually t the beam, namely n the case f the crcular beam crss-sectn t s a set d M = ( k, l) <, N > <, N > : x k + y l. (37)
The pnts x k + y l k, l M (ther number s m( M ) ) may be used fr calculatn f averaged varables expectng the mre precse result fr larger N. There are at least three ways f averagng: (a) Averagng lcal ndex f refractn n ( k, l ) and thckness h (, ) k l fllwng the relatns r + a a, ( ) n = n ( k, l), h (a),aver m( M ) ( k, l ) M (a),aver m( M ) ( k, l ) M = h ( k, l). (38) (b) Averagng lcal ellpsmetrc parameters btaned va (Eq. 3) (b) (b) ψ aver = ( k, l) m( M ) ( k ψ, aver = (, ), l ) M ( ) ( k k l (39) m M, l ) M and then applyng the algrthm explaned n the sectn 5 fr calculatn averaged quanttes (b) n and h.,aver (b),aver (c) Averagng the electrc feld vectr f the reflected beam (Eq. ) E = r,aver E r ( k, l), (4) m( M ) ( k, l ) M then usng (Eq. 3) n the frm ( Er,aver p ) tanψ exp( ) = (4) ( E s ) tanψ exp( ) and fnally calculatng 7. Numercal smulatn (c) n,aver and r,aver (c) h,aver frm such btaned ellpsmetrc parameters. The fllwng example demnstrates the effect f the curved surface and cnvergence r dvergence f the ncdent beam n the evaluatn f ellpsmetrc data: the sphercally shaped slcn substrate wth radus R = m and a cmplex ndex f refractn n g = 3.855.4 s cvered by a thn layer wth ndex f refractn n =.46 (slcn xde) and thckness h = nm, the ndex f refractn f superstrate s n = (ar). The crcular beam wth dameter d (varable quantty) and wavelength λ = 63.8 nm ncdents the sphere frm the drectn ϕ = 9, θ =, t means that the angle f ncdence measured n the superstrate s α α = and a al beam s parallel t the plane x =. Cnsdered ellpsmetrc 7 parameters f the lnearly plarsed ncdent beam are ψ = 45, = and a number f grd dvsn N = 5. Fgures llustratng maps f lcal quanttes are depcted fr the dameter f the cllmated beam d = R( sn α).6 mm, t s an extreme case when just ne bundary ray ncdents the sphere tangentally. It s clear that the prblem exhbts the dmensnless characterstc number d / R. The change f the lcal angle f ncdence f the cllmated beam ( β = ) culd be seen n Fg., the values vary between 6,57 and 9.
Fg. Lcal angle f ncdence f the cllmated beam wth the dameter.6 mm mpngng the sphere wth radus m; angle f ncdence f the al beam s 7. Fg. Lcal ellpsmetrc parameters ψ (left) (rght) fr same cndtns as n Fg.. Fg. 3 Lcal ndex f refractn (left) and thckness (rght) f the layer wth (ntentnal) errr fr same cndtns as n Fg.. Fg. demnstrates lcal changes f ellpsmetrc parameters. Usng lcal angles f ncdence t s pssble t calculate the crrect layer ndex f refractn and ts thckness. The Fg. 3 shws lcal errneus values f layer ndex f refractn and ts thckness btaned frm ellpsmetrc parameters accrdng t (Eq. 3) when the lcal plarsatn state f the each beam s replaced by the plarsatn state f the al beam.
Fg. 4 Averaged values f the layer ndex f refractn (left) and ts thckness (rght) btaned by three methds n dependence n cllmated beam dameter d, the radus f the sphere s R = m. Frm Fg. 4 we can deduce the errr rate fr determnatn f thn layer parameters wth ncreasng dameter f the cllmated beam. The mre precse seems t be (a) methd n cmparsn wth (b) methd, the result s nearly cnstant fr d / R rughly less than 5 %. The mre adverse methd s (c) usng fr detectn the resultng electrc feld f the reflected beam, but t s just a methd suppsed t apprxmate the physcal measurement cndtns mre realstcally. Hwever, als fr relatvely wde beam wth dameter 5 mm we btan the devatn nly. frm the crrect ndex f refractn.46 and fr layer thckness the value nm whle the crrect value s nm. The effect f beam cnvergence r dvergence n the measurement precsn s demnstrated n Fgs. 5 8. Fg. 5 Averaged ndex f refractn f thn layer btaned by three averagng methds (a, b, c frm left t rght) frm measurement f ellpsmetrc parameters fr cnvergent beam wth angle f cnvergence β =. Fg. 6 Averaged thckness f thn layer btaned by three averagng methds (a, b, c frm left t rght) frm measurement f ellpsmetrc parameters fr cnvergent beam wth angle f cnvergence β =.
Fg. 7 Averaged ndex f refractn f thn layer btaned by three averagng methds (a, b, c frm left t rght) frm measurement f ellpsmetrc parameters fr dvergent beam wth angle f dvergence β =. Fg. 8 Averaged thckness f thn layer btaned by three averagng methds (a, b, c frm left t rght) frm measurement f ellpsmetrc parameters fr dvergent beam wth angle f dvergence β =. These fgures allw t estmate quanttatvely the effect f beam cnvergence r dvergence n the naccuracy f results btaned frm ellpsmetrc measurements. Surprsngly we can expect better results fr cnvergent beams n cmparsn wth cllmated beams but nly fr sutable dameter f the beam. In the smulated case the smallest devatns f the ndex f refractn and thckness frm the crrect values ccur when the cnvergent angle s abut fve degrees. Dvergent beams unambguusly exhbt wrse results fr all angles f beam dvergence wth ncreasng beam dameter and angle f dvergence. The averagng methd usng the lcal ndex f refractn and thckness btaned frm lcally measured ellpsmetrc parameters (.e. methd (a)) s mnmally senstve t the beam dameter and ts degree f cnvergence r dvergence. 8. Cnclusn Ths paper usng the cmbnatn f gemetrcal and wave ptcs quanttatvely examnes the effect f curved surface and als beam cnvergence r dvergence n the accuracy f the results btaned frm ellpsmetrc measurements. Fr ths purpse three averagng methds fr estmatn the measurable quanttes were prpsed and results numercally cmpared applyng the example f ne thn delectrc layer depsted n the sphercally shaped substrate. Acknwledgement Ths wrk was supprted by the prject N. M6 f the Mnstry f Educatn, Yuth and Sprts f the Czech Republc. References [] Křepelka J.: Optka tenkých vrstev, Unverzta Palackéh v Olmuc, 993