Three polarization reflectometry methods for determination of optical anisotropy

Three polarization reflectometrymethods for determination of optical anisotropy

Gregory I. Surdutovich, Ritta Z. Vitlina, Aleksander V. Ghiner, Steven F. Durrant,and Vitor Baranauskas

Three novel methods for the determination of optical anisotropy are proposed and tested. The first, thespecial points method, may be applied to any uniaxially anisotropic medium and is based on themeasurement of s- and p-polarized light reflectances under near-normal or grazing angles ~or both! andof the Brewster angle. The second method is based on the use of the Azzam universal relationshipbetween the Fresnel s- and p-reflection coefficients. For a flat surface and an isotropic medium, theAzzam combination of coefficients becomes zero and thus is independent of the incidence angle, whereasfor a uniaxial or biaxial anisotropic sample it acquires a certain angular dependence, which may be usedto determine the anisotropy of the sample. Finally, for those cases in which the anisotropy of thematerial of a film deposited on an isotropic substrate is itself of interest, a third method, the interferencemethod, is suggested. This technique makes use of the different dependences of s- and p-polarized beamoptical path-length changes on the variation of the angle of incidence. © 1998 Optical Society ofAmerica

OCIS codes: 160.1190, 260.5430.

1. Introduction

The ellipsometric and reflectometric methods for thedetermination of the optical parameters of a sampleby measurement of its reflectances over all incidenceangles are, generally speaking, excessively informa-tive when the type of sample is known in advance, asin the case of a homogeneous isotropic or uniaxialsample, isotropic or anisotropic film on a substrate,etc. This consideration directly relates to the polar-ization reflectometry methods, when angular func-tions of the intensity reflectance coefficients Rp andRs for two polarizations represent an abundant set ofdata for determination of the anisotropy of a uniaxialsample with an optical axis normal to the surface ofthe film on the substrate. In this case it would be

G. I. Surdutovich, R. Z. Vitlina, and V. Baranauskas are withDepartamento de Semicondutores, Instrumentos e Fotonica, Fac-uldade de Engenharia Eletrica e de Computacao, Unicamp, CEP13083-970, Campinas Sao Paulo, Brazil. S. F. Durrant is withInstituto de Fisica Gleb Watagin, CEP 13083-970, Campinas, SaoPaulo, Brazil. A. V. Ghiner is with Departamento de Fisica, Uni-versidade Federal de Maranhao, CEP 65080-040, Sao Luis, MA,Brazil.

Received 11 March 1997; revised manuscript received 13 June1997.

0003-6935y98y010065-14$10.00y0© 1998 Optical Society of America

sufficient to perform a number of measurementsequal to the number of parameters needed. How-ever, it is important to choose the method and therange of the measurements correctly.

To find the needed optical parameters by the po-larized reflectometry technique we may use ~1! thevalues of the reflection coefficients only at definitepoints @the special points method ~SPM! approach#,~2! a definite invariant of the Fresnel reflection coef-ficients for a flat semi-infinite isotropic substrate @theAzzam universal relationship ~AUR! method#, and ~3!the relative shifts in the minima ~maxima! of s- andp-interference patterns @the interference method ~IM!technique#. All these methods may be applied todetermine the optical anisotropy of homogeneoussamples, the effective anisotropy of inhomogeneoussamples, and the anisotropy of a film deposited ontoan isotropic substrate.

An optical anisotropy is the inherent characteristicof many composite materials, ranging from three-dimensional composites to monolayers adsorbed onclean surfaces. When characteristic scales of a me-dium constituted from two isotropic components aremuch smaller, even if in only one dimension, than anoptical wavelength, then such a medium may be de-scribed by an effective susceptibility tensor ~or aneffective refractive index!. It may happen to be iso-tropic or anisotropic, depending on the form and the

1 January 1998 y Vol. 37, No. 1 y APPLIED OPTICS 65

orientations of inclusions and their relative positions,i.e., the geometry of the mesostructure. Thus, forexample, for randomly distributed spherical inclu-sions in a homogeneous host medium the MaxwellGarnett model1 predicts zero anisotropy, whereasthin-layered structures2–4 or porous Si ~PS! sampleswith a wire ~columnar! mesostructure demonstrateclear optical anisotropy.5–7 Evidently the anisot-ropy of a two-component medium with a given meso-structure depends on the concentrations of thecomponents and becomes zero for a small concentra-tion of either. An example of a strong dependence ofan anisotropy on the form of inclusions in the case ofa columnar mesostructure is given in Ref. 8. De-pending on the type of mesostructures ~cylindricalinclusions of bulk material in vacuum or of voids in ahost medium! the anisotropy of samples with thesame concentration of constituents for a mediumwith large dielectric permittivity ε may differ by anorder of magnitude.

In addition to the above-mentioned case of the bulkanisotropy of a composite medium, there are twoother well-known causes of the effective anisotropy ofa sample: surface roughness and profile inhomoge-neity ~Fig. 1!. When all characteristic sizes of thesurface relief are small compared with the wave-length of light ~a so-called low-relief surface! thensuch a surface layer may be considered as a specialcase of a two-component composite medium ~bulk ma-terial and vacuum!, and its anisotropy strongly de-pends on the shape of the irregularities. In somespecial cases of periodic elongated and random low-relief surfaces, analytical solutions for the effectivepermittivity tensor eef of a rough surface layer wereobtained.9,10 As a result, the problem is reduced toconsideration of the anisotropy of a standard struc-ture: an anisotropic film on an isotropic substrate ofknown refractive index.

The effective anisotropy due to the inhomogeneityof the refractive-index profile may be considered an-alytically for two simple models: a fine-layeredstructure and a thin isotropic film on an isotropicsubstrate. The first structure is always equivalentto a negative uniaxial crystal, whereas the latter maybe reduced to either an effective positive or negativeuniaxial crystal, depending on the sign and the valueof the difference between the film and substrate re-fractive indices.

In this paper we present a description of three

Fig. 1. Equivalence of a low-relief rough surface or an inhomoge-neous film to an anisotropic film.

66 APPLIED OPTICS y Vol. 37, No. 1 y 1 January 1998

simple optical reflectometry techniques for measure-ment of the total effective anisotropy of uniaxial sam-ples or the anisotropy of the material of a film ~orboth!. The paper is organized as follows. In Sec-tion 2 we discuss the SPM applied to a uniaxial me-dium and to a thin film on an isotropic substrate andanalyze the accuracy of the method and its depen-dence on the value of the refractive index of a sub-strate and on the surface roughness. Somenumerical estimates and a short analysis of the avail-able experimental data are presented. In Section 3we describe the AUR method and its application touniaxial and biaxial anisotropic media. The possi-bility of its employment in reflectometry is discussed.Two experimental measurements made with theAUR for a natural calcite crystal and a crystalline Si~c-Si! sample demonstrate the high sensitivity of thismethod. It may be particularly useful for the mea-surement of the effective anisotropy arising becauseof surface roughness. In Section 4 we extend ourconsideration to describe the IM and formulate thelimitations on film thickness for the reliable determi-nation of the anisotropy of a film by this method. Atest of the IM was performed with a specially pre-pared thin-layered structure whose anisotropy wascalculated analytically. Finally, in Section 5 we re-view the anisotropy properties of simple uniaxialcomposite structures and their dependence on theconcentration of their constituents. In the case of atwo-dimensional columnarlike structure, a strong de-pendence of the anisotropy on the form of the constit-uents is demonstrated. In Section 6 we compare theapplicability of the methods and their potentialities.

2. Special Points Method

A. Uniaxial Anisotropic Crystal

Any homogeneous uniaxial medium with an opticalaxis z normal to the surface may be characterized bytwo parameters: refractive indices nx 5 ny and nz@or, alternatively, nx and the anisotropy parameter b5 @~nzynx! 2 1#. Therefore it is sufficient, in princi-ple, to perform only two measurements to determinethese parameters. In reflectometry the natural can-didates for such measurements are the Brewster an-gle uB and the intensity reflection coefficients Rp andRs for p- and s-polarized light, respectively, undernormal- and grazing-incidence angles u. By mea-suring uB we obtain one constraint on the refractiveindices nx and nz:

tan uB 5 nzSnx2 2 1

nz2 2 1D

1y2

5 nx~b 1 1!F nx2 2 1

nx2~b 1 1!2 2 1G

1y2

. (1)

If these refractive indices satisfy the inequality

nx2 1

1nz

2 . 2, (2)

then from Eq. ~1! one can see that the value of theBrewster angle for any positive or negative anisot-ropy b is always more than py4, just as for isotropicmedia. However, for a refractive index nx , =2combined with a large nz, i.e., when b is large andpositive, inequality ~2! is violated. In this case theangle uB becomes smaller than py4 and, in the limitof nx3 1, uB tends to zero ~see Fig. 2!. This result isunderstandable because in the anisotropic case therefraction angle may become greater than the angleof incidence. The characteristic dependences of uBon the refractive index nx in the region of a positiveand negative anisotropy are shown in Fig. 2.

To find a second constraint on the refractive indicesnx and nz we may ~1! measure the reflectivity of asample at normal incidence when Rs~0! 5 Rp~0! 5R~0! and

R~0! 5 Snx 2 1nx 1 1D

2

, (3)

or ~2! measure the ratio a of the derivatives dRpyduand dRsydu at u 3 py2:

a 5

dRp

du

dRs

du*

u3 ~py2!

5 nxnzSnx2 2 1

nz2 2 1D

1y2

5 nx tan uB. (4)

In the isotropic case, a 5 ai 5 tan2uB. From Eqs. ~1!and ~3! we have

nx 51 1 @R~0!#1y2

1 2 @R~0!#1y2 ,

nz 5tan uB$1 2 @R~0!#1y2%

{tan2 uB@1 2 @R~0!#1y2#2 2 4@R~0!#1y2}1y2 , (5)

whereas from Eqs. ~1! and ~4! we obtain

nx 5 a cot uB, nz 5tan2 uB

~tan4 uB 1 tan2 uB 2 a2!1y2 . (6)

Equations ~5! and ~6! relate the refractive indices~and anisotropy! of a medium to the experimentallymeasured values of R~0!, uB, or a, uB. The regions ofnegative and positive anisotropy in the planes @R~0!,tan uB# and ~a, tan uB! are shown in Figs. 3~a! and3~b!, respectively. One may compare the relative ac-curacy of the anisotropy determination by use of Eqs.~5! and ~6!.

B. Accuracy of the Method

The absolute error in the determination of the anisot-ropy b is given by the relations

b 5 2nx

4 2 1nx

duB 1~nx

2 2 1!2

4nx

dR~0!

R~0!, (7)

b 5 22~nx

4 2 1!

nxduB 1 ~nx

2 2 1!da

a; (8)

~for this estimation we assume that a 5 nx2!. Un-

certainty in the determination of b due to the inac-curacy of measurement uB is approximately the samein both cases. However, with the same relative er-rors ~da!ya and @dR~0!#y@R~0!#, the R~0! coefficientmeasurements give better accuracy for the determi-nation of the anisotropy for not so great values of nxand approximately the same or slightly worse accu-racy than the a parameter measurements for nx .. 1.

From Figs. 3~a! and 3~b! one can see that the pos-itive anisotropy region is much narrower than thenegative one and so with the same accuracy of themeasured parameters a, R~0!, and uB, a negative an-isotropy may be determined more precisely than apositive one. One may, of course, use both R~0! anda measurements for the anisotropy determinationand compare the results.

C. Thin Isotropic Film on an Isotropic Substrate

The SPM may be applied to the determination of athickness d and the refractive index n1 5 =ε1 of athin isotropic film on an isotropic substrate with aknown refractive index n 5 =ε. The reflection co-efficients Rs and Rp of an ambient—film–substratesystem have the form11

Rs,p 5 U r01s,p 1 r12s,p exp~2id!

1 1 r01s,pr12s,p exp~2id!U2

, (9)

where r01s,p and r12s,p are Fresnel reflection coeffi-cients at ambient–film and film–substrate interfaces,d 5 @~4pd!yl#~ε1 2 sin2 u!1y2, and l is the free-spacewavelength. The presence of a thin film leads to theappearance of a minimum angle um ~the so-called

Fig. 2. Dependence of the Brewster angle uB on the refractiveindex nx for different positive and negative values of the anisotropyparameter b. The regions below and above the dashed curvecorrespond to greater and lesser solutions nx of Eq. ~1! for a givenuB and b , 0. Point Ca1 indicates the experimental data ~uB 560.5°, b 5 29.6%! obtained by the AUR method @see Fig. 10~a! inSection 3 below# for the natural calcite sample, and point Ca2 ~uB

5 60.78°, nx 5 1.658, b 5 210.3%! corresponds to the literaturedata. Points Si~ubu , 0.5%! and Sirough ~uB 5 74.38°, b 5 247.5%!correspond to the AUR measurements @Fig. 10~b!# for highly pol-ished and rough surfaces of the c-Si sample, respectively.


Fig. 3. Regions of negative ~vertical lines! and positive ~horizontallines! anisotropy are shown in the planes ~a! @R~0!, tan uB#, ~b! ~a, tanuB!. ~a! The boundaries of the positive anisotropy are outlined bycurves B,

R~0! 5 Stan uB 2 1tan uB 1 1D

2

, ~b 5 0!,

and C,

R~0! 5 F~1 1 tan2 uB!1y2 2 1~1 1 tan2 uB!1y2 1 1G

2

, ~b3 `!.

Points Ca1@R~0! 5 0.064, uB 5 60.5°, b 5 28%#, Sirough@R~0! 50.0443, uB 5 74.38°, b 5 231%#, and Si~ubu , 0.5%! represent theresults of R~0!, tan uB measurements of a natural calcite sample andthe c-Si sample with rough and highly polished surfaces, respec-tively. Point Ca2 represents literature data for the calcite sample.~b! Line A ~a 5 tan uB! and curve C @a 5 tan2 uB~1 1 cot2 uB!1y2#correspond to the boundaries of the physical domain. Curve B ~a 5


pseudo-Brewster angle! instead of the Brewster an-gle of a substrate uB.

For small film thicknesses, @~4pd!yl#2ε1 , 1, it ispossible to expand Rp in the parameter d and derivethe following relation for the Brewster angle shift Du5 um 2 uB:

Du 5 2k2 εÎε8ε1

2

~ε 2 ε1!~ε1 2 1!

~ε2 2 1!2~ε 1 1!@~ε2 1 1!~ε1 2 1!~ε 2 ε1!

1 2~ε12ε2 1 ε1

2 2 2ε2!#, (10)

and the reflection coefficient R~0! for u 5 0:

R~0! 5 qF1 2 k2Îε~ε1 2 1!~ε 2 ε1!

~ε 2 1!2 G ,

q 5 SÎε 2 1

Îε 1 1D2

, (11)

where k 5 @~4pd!yl#. Then, by eliminating the pa-rameter k2 from Eqs. ~10! and ~11!, we obtain a qua-dratic equation for the determination of the dielectricpermittivity ε1 of the film:

ε12@ε~ε2 1 1! 2 8qG~ε 1 1!3# 1 ε1ε~ε 1 1!~ε2 1 1!

2 ε2~ε2 1 4ε 1 1! 5 0. (12)

The virtue of Eq. ~12! is that it depends on only thesole parameter G:

G 5Du

R~0! 2 q, (13)

which does not depend on the film thickness, as boththe Brewster angle shift and the change in the nor-mal reflectivity, R~0! 2 q, are proportional to d2.When Eqs. ~10!–~12! are analyzed, it becomes clearthat, with the exception of a region of no interest, 1 ,ε1 , 1.25, the parameter G is always positive and hastwo critical values, G1 and G2. The first, G1, corre-sponds to the condition that the coefficient of thequadratic term in Eq. ~12! tends to zero:

G1 5ε~ε2 1 1!

8q~ε 1 1!3 . (14)

For 0 , G , G1, Eq. ~12! has only one physical solu-tion, ε1 . 1. The second critical value, G2, corre-sponds to the vanishing of the determinant of Eq.~12!:

G2 5~ε2 1 1!@~ε2 1 1!~ε 1 1!2 1 4ε~ε2 1 4ε 1 1!#

32q~ε 1 1!3~ε2 1 4ε 1 1!, (15)

tan2 uB! corresponds to an isotropic medium. Points Ca1 ~a 52.95, uB 5 60.5°, b 5 28.4%! and Ca2 correspond to the a, tan uB

measurements and the literature data for a natural calcite crystal.Point Sirough represents the experimental result for a c-Si samplewith a rough surface ~a 5 10.5, uB 5 74.38°, b 5 246.5%!.

which means the disappearance of the physical solu-tions of Eq. ~12! for G . G2.

The problem of choosing between two possible so-lutions in the region G1 , G , G2 may usually besolved on a physical basis when the difference be-tween the solutions of Eq. ~12! is large. However, ina rather narrow region in the vicinity of the curve ε15 ε1

crit ~Fig. 4!,

ε1crit 5

2ε~ε2 1 4ε 1 1!

~ε2 1 1!~ε 1 1!, (16)

which corresponds to the vanishing of the determi-nant of Eq. ~12!, the difference between solutions be-comes small and the choice grows difficult. All thisbecomes clear from inspection of Fig. 5, in which thedependence of the film’s dielectric permittivity ε1 onthe parameter G is shown for three different sub-strates: glass ~ε 5 2.75!, diamond ~ε 5 5.7!, and Si ~ε5 14.9!. In Fig. 5 points of the curve given by Eq.~16! correspond to the condition @~]G!y~]ε1!#ε 5 0.

Fig. 4. Critical curve ε1 5 ε1crit obtained from Eq. ~16!.

Fig. 5. Dependence of the dielectric permittivity ε of a film on theparameter G for three standard substrates: ε 5 2.72 ~glass!, 5.7~diamond!, and 14.9 ~Si!. The best accuracy in the determinationof the film permittivity ε on the basis of the experimentally mea-sured parameter G may be achieved outside the vicinity of thecurve given by Eq. ~16!.

After determination of the parameter ε1, we findthe thickness of the film from Eq. ~11!:

k2 5q 2 R~0!

qÎε~ε 2 1!2

~ε1 2 1!~ε 2 ε1!. (17)

The histograms, k 5 const, of this equation in theplane @ε1, q 2 R~0!# are given in Fig. 6 for the samesubstrates as in Fig. 5. One can see that the mostfavorable combination of values of the dielectric per-mittivities for the exact determination of a film thick-ness is the region ε1 ' @~ε 1 1!y2#, when even a smallchange in thickness leads to a large change in thenormal reflectance q 2 R~0!. In other words, for thevalues ε in this region, the destructive influence of afilm of a given thickness is a maximum. This resultfor a thin film is somewhat distinct from the well-known condition ε1 5 =ε for the maximum destruc-tive interference of a ly4 film. Such a distinctionbecomes significant for large values of ε. In two lim-iting cases of a small difference, either of ε 2 ε1 or ε12 1, the accuracy of the determination of film thick-ness by this method becomes poor, as it is evidentfrom Fig. 6.

1. Homogeneity Control of a Film In SituBesides the problem of the determination of both ofthe parameters ε1 and d of a film, there sometimesarises a more limited problem of monitoring the ho-mogeneity ~constancy of the dielectric permittivity! ofa film in situ during the growth process. In this casewe may take advantage of the fact that G does notdepend on the film thickness, as both the Brewsterangle shift Du and the change in the normal reflec-tivity R~0! 2 q are proportional to the parameter k2.The constancy of G in the course of growth guaranteesthe optical homogeneity of a film. Just as in theabove-mentioned case, the vicinity of the curve givenby Eq. ~16! is unfavorable for the application of thistechnique because, for such a specific relationshipbetween the dielectric permittivities of a substrateand a film, the parameter G is weakly sensitive tovariations in ε1.

Fig. 6. Contours k 5 const for the substrates with ε 5 2.72~dashed–dotted curves!, ε 5 5.7 ~dashed curves!, and ε 5 14.9 ~solidcurves!.


2. Integral Approach to the Thin-Film plusSubstrate SampleIn the case of a sufficiently thin film it is possible, andmay be useful, to consider a film on a substrate as ahomogeneous sample. One may introduce the con-cept of an effective uniaxial anisotropic crystal withrefractive indices nx

ef and nzef ~or effective anisotropy

bef! by postulating the equality of the reflectances ofthe real and the fictitious samples at a normal-incidence angle and equating the Brewster angle ofthis fictitious sample uB

ef to the real pseudo-Brewsterangle um. Then, by use of Eq. ~11!, we obtain

R~0! 5 Snxef 2 1

nxef 1 1D

2

,

i.e., nxef 5 ÎεF12

k2

4~ε 2 ε1!~ε1 2 1!

~ε 2 1! G . (18)

On the other hand, according to Eq. ~1!, the shift Du5 uB

ef 2 uB of the Brewster angle of the fictitiousanisotropic sample is given by

Du 5 S ]u

]nxD

b50

dnx 1 S]u

]bDnx5Îε

db 5Îε

ε2 2 1 Fε 2 1

Îεdnx 2 dbG. (19)

In our case, starting from a clean substrate, i.e., byputting nx 5 nx

ef 2 =ε and b 5 bef ~as the substrateis assumed to be isotropic! and by using Eqs. ~10! and~11!, we find

bef 5 218

k2 ~ε1 2 1!~ε 2 ε1!

~ε 1 1!2~ε 2 1!ε12 $ε~ε2 1 1!~ε1 2 1!~ε1 2 ε!

1 2@2ε~ε2 2 ε12! 1 ε1

2~ε2 2 1!#%. (20)

In Fig. 7 are given two examples of the Rs and theRp coefficient dependences, calculated by Eq. ~9! forreal inhomogeneous thin-film on a substrate sampleswith given values ε, ε1, and k and the same coeffi-cients for the equivalent fictitious homogeneous uni-axial crystals with parameters nx

ef and bef calculatedfrom Eqs. ~18! and ~20!. The chosen thicknesses ofthe films are already on the boundary of the applica-bility of the effective crystal approach, as the param-eter k=ε1 is equal to 0.6 and 0.49 ~for curves 1 and 2,respectively!. Nevertheless, as is visible from Fig. 7,the coefficients Rs of the real and effective crystalsamples are indistinguishable, that is, a vanishinglysmall difference in the Rp coefficients can be notedonly in the vicinity of the Brewster ~pseudo-Brewster!angles. The Brewster angle shifts ~Du 5 20.572°and Du 5 10.455° for curves 1 and 2, respectively!calculated by Eq. ~19! and the shifts obtained fromEq. ~9! by the numerical simulation ~Du 5 20.562°and Du 5 10.445°! are in good agreement. In thesenumerical examples the samples were chosen to havea rather small effective anisotropy because a largeranisotropy may be modeled with only even thickerfilms, i.e., beyond the limits of our approach.


In Fig. 8 the regions of positive and negative effec-tive anisotropy are shown together with the isolinesbef 5 befyk2 in the ε, ε1 plane. From Eq. ~19! itfollows that @~]uB!y~]b!#nx5=ε 5 2@=εy~ε2 2 1!# , 0,i.e., that for a fixed refractive index nx the shift of theBrewster angle and the sample’s anisotropy alwayshave opposite signs. The presence of the film, how-ever, simultaneously changes both nx and b. On in-

Fig. 7. Angular dependences of the Rs and Rp coefficients for twoinhomogeneous thin film on the substrate samples: 1, ε 5 15, ε1

5 9, k 5 0.2, uB 5 75.522°, um 5 74.96°, nxef 5 3.740, bef 5 10.0974,

uBef 5 74.95°; 2, ε 5 4, ε1 5 6, k 5 0.2, uB 5 63.435°, um 5 63.98°,

nxef 5 2.067, bef 5 10.0974, uB

ef 5 63.99°. The solid curves rep-resent the numerical simulations according to the exact formula ofEq. ~9! and the dashed curves correspond to the anisotropic uni-axial fictitious crystals with the parameters given by Eqs. ~18! and~20!. In the presented examples the solid and the dashed curvesconverge for all angles, except for those in the immediate vicinityof uB.

Fig. 8. Contours of the effective anisotropy b 5 befyk2 in the planeε, ε1. In the vicinity of the bisectrix ε 5 ε1, the sign of the effectiveanisotropy coincides with the sign of the difference ε1 2 ε. Belowthe bisectrix, for ε . 5.8, there is another region of positive anisot-ropy with asymptotes ε1 5 1 and ε1 5 ε 2 2 for ε 3 `. A verynarrow region of negative anisotropy exists where ε , 1.5, but inpractice this region has no physical importance.

spection of Eqs. ~10!, ~18!, and ~20!, it is evident thatin the vicinity of a bisectrix ε1 5 ε both

Du 5k2

4ε3y2

~ε 1 1!2 ~ε1 2 ε!,

bef 5k2

4ε 2 1ε 1 1

~ε1 2 ε!,

have the same signs. This means that in this regionthe shift in the Brewster angle Du due to the changeof the refractive index dnx

ef dominates, as is evidentfrom Eq. ~19!, over its shift in the opposite directionbecause of the anisotropy bef. However, for ε 2 ε1 .2 and ε $ 5.8 there is a region @its boundaries corre-spond to the vanishing of the braced expression in Eq.~20!# of positive anisotropy where both of these effectsshift the Brewster angle negatively.

The approach developed here gives a better insightinto the physical sense of the shift of the Brewsterangle induced by a rough low-relief surface that wasdiscovered12 in the course of the numerical simula-tion. The explanation of this effect was given interms of the average reflected fields.13,14 The provedcorrespondence9 of the elongated two-dimensional~one-dimensional in the notation of Refs. 12–14! low-relief periodic surface layer to an effective anisotropicfilm reduces the problem of the Brewster angle shiftto the one considered above. The anisotropy of thiseffective film depends on the shape of the relief andchanges from positive ~for steep! to negative ~forgently sloping! irregularities. Because a rough sur-face layer is a composite of the substrate medium andvacuum, then all components of the layer’s effectivepermittivity tensor e will be smaller than ε. In thespecific case of an isotropic film this means that thesusceptibility ε1 , ε and so, according to Eq. ~10!, inthe physical region the Brewster angle shift is alwaysnegative. A positive effective anisotropy, as observ-able from Eq. ~19!, only enhances the negativity of theBrewster angle shift. And, finally, we demonstratethat any possible effective negative anisotropy of asurface layer cannot compensate for the first term inEq. ~19! and give rise to a total positive shift of theBrewster angle. For this we must generalize ourapproach for the case of the anisotropic film on anisotropic substrate. It will be made elsewhere.The final result is that in Eq. ~10! the factor ε1 2 ε isreplaced by the difference ~tan2 uB!film 2 tan2 uB,which is maximum for a film with the greatest neg-ative anisotropy. The limiting case of such negativeanisotropy is realized in the fictitious model of a flat-layered surface roughness when screening of the elec-tric field along the x direction is absent $the tensorcomponent εxx 5 @~ε 1 1!y2# is maximal% and thescreening in the z direction is maximal $εzz 5 @~2ε!y~ε1 1!# is minimal% and the effective anisotropy of theequivalent surface film acquires the minimal nega-tive value

bfilmef 5 2

~Îε 2 1!2

ε 1 1

at the 50% concentration of constituents ~see Section5!. Now the substitution nx 5 =εxx and bfilm

ef intoEq. ~1! immediately gives ~tan uB!film 5 =ε, i.e., Du 50. Hence, even in this limiting unrealistic case offlat-layered surface roughness, the Brewster angleshift cannot become positive, exactly as was first ob-served in Refs. 12–14.

The result of a zero shift of the Brewster angle forgently sloping roughness matches with the results ofRefs. 12–14 in this limiting case as well as with theconclusion given in Ref. 15, i.e., that such irregular-ities with accuracy up to second-order terms do notcontribute to the Brewster angle shift. Finally, thesecond conclusion of Refs. 12–14, i.e., that because ofa low-relief roughness the reflectivity minimum isnever a true zero, becomes obvious as this is indeed sofor any surface covered by a film.

D. Influence of Scattering: Experimental Data

Let us discuss the effect of surface scattering on theaccuracy of the SPM measurements. Just as in anyconventional reflectometry method, the SPM mea-surements give integrated information about the op-tical anisotropy of a sample induced by all factors:bulk material anisotropy, sample inhomogeneity, andsurface scattering. What are the limitations of ap-plying the SPM to the determination of the bulk ma-terial anisotropy? First, as was shown above,surface roughness imitates a negative effective an-isotropy, so that the measured positive anisotropy ofa sample will be, in fact, a lower limit of the positivebulk material anisotropy. Such a situation takesplace with PS films.6 As is clear from Fig. 3, theregions of positive anisotropy are narrower and hencetheir detection is, in general, a more difficult task. Agently sloping surface roughness influences mainlythe value of the coefficient R~0!. The anisotropy ismuch more sensitive to a reduction in this coefficient~under a fixed value of the Brewster angle! for largevalues of ε ~tan uB .. 1! than for moderate values ofthe permittivity ~tan uB ' 2! typical for many con-ventional materials ~see Fig. 3!.

To determine the experimental scope of the SPMwe performed measurements of the natural anisot-ropy of a negative anisotropic uniaxial calcite crystalas well as of highly polished and unpolished opticallyisotropic c-Si samples. For the calcite crystal in bothR~0!, tan uB and a, tan uB types of measurement, theresults b 5 28% and b 5 28.4% are shown in Figs.3~a! and 3~b!, respectively. They match each other,although they differ from data in the literature. Wecompare these results with the data obtained by theAUR method in Section 3.

To elucidate the effect of surface scattering on theoptical anisotropy of a sample, we performed mea-surements on two sides, highly polished and unpol-ished, of several c-Si samples. The typical Rs and Rpdependences are shown in Fig. 9. For a highly pol-ished side we obtained zero anisotropy with an accu-racy of better than 60.3% in both variants of the SPMmeasurements. On the other hand, for the unpol-ished surfaces the shift of the pseudo-Brewster angle


Du was equal to 21.12° and the R~0! coefficients werean order of magnitude smaller than in the case of thepolished surfaces. For a ninefold reduction ~see Fig.9! of the value of R~0!, because of the scattering, theconventional Kirchhoff approximation in a one-dimensional model of a rough surface with a Gauss-ian distribution of heights h of the roughness leads tothe estimate h ' 0.12l, so that the thickness param-eter of the surface layer k 5 @~4ph!yl# ' 1. There-fore, strictly speaking, in this case we cannot use thelow-relief approximation and a thin-film plus sub-strate effective sample model or expect that Eqs. ~5!and ~6! will give identical results. If nevertheless weneglect this argument temporarily and substitute theparameters R~0! and a of the curves of Fig. 9 into Eqs.~5! and ~6!, we obtain significantly different results~nx 5 1.53, b 5 231%, and nx 5 2.9, b 5 246.5%significantly!, as is shown in Fig. 3. From Fig. 9 it isclear that surface scattering manifests itself muchmore strongly at normal incidence than at grazingangles because the roughness phase difference,@~4ph!yl# cos u, associated with the Rayleigh crite-

Fig. 9. ~a! Angular dependence of the coefficients Rs and Rp ~ar-bitrary units, logarithmic scale! for the highly polished ~solidcurve! and unpolished ~dashed curve! c-Si surfaces, ~b! the regionof grazing angles is shown on a linear scale.


rion, tends to zero under u 3 ~py2!. So it seemsprobable that the results of the a variant approach@Eq. ~6!, Fig. 3~b!# are closer to reality. We confirmthis assumption in Section 3 by analysis of these dataand data for a calcite crystal by using a differentmethod.

Thus the SPM approach is self-consistent in twoR~0! and a variants under conditions of not verystrong surface scattering and provides some qualita-tive estimates of the effective anisotropy even for thesamples with a very rough surface. From Fig. 3 it isclear that the appearance of scattering, i.e., the dim-inution of R~0! and a, may convert a sample of a bulkisotropic medium into an apparently negative aniso-tropic crystal, as was noted in experiments in whichPS samples with rough surfaces were used.6 Nowwe turn our attention to another method of determi-nation of the effective anisotropy, which exploits theangular dependences of the reflection coefficients.

3. Azzam’s Universal Relationship Method

The measurement of two complex Fresnel reflectioncoefficients, rp and rs, of monochromatic light at theplanar interface between vacuum and a linear homo-geneous isotropic medium implies the determinationof four parameters ~two amplitudes and two phases!,whereas these data are used in fact for the determi-nation of only two parameters: the complex dielec-tric constant ~or refractive index! and the extinctioncoefficient of the medium. Therefore, between thesecoefficients there must be two ~or one complex! uni-versal relationships that are independent of the angleof incidence and refractive index of the medium.Such a universal complex relationship was first foundby Azzam16 ~and later, independently, in Ref. 17! inthe form that a certain combination of the rp and rscoefficients becomes zero for any flat, isotropic, homo-geneous half-space. For an anisotropic medium, theAUR no longer holds.

It was found, however, for a uniaxial anisotropiccrystal with the optical axis oriented perpendicularlyto the surface, that the AUR acquires a certain an-gular dependence with the parameters nx and b,17

which we propose to use for measurement of the ef-fective anisotropy due to surface roughness.Clearly, to use the AUR it is necessary to have atone’s disposal the amplitudes and phases ds, dp, of thers, rp coefficients for a large range of incidence angles.Therefore the intensity self-reflectance measure-ments are not self-sufficient. Even additional ellip-sometric data, i.e., the knowledge of the phasedifference ds 2 dp, do not solve the problem, as oneparameter still remains indeterminate. In practice,however, for a weakly absorptive media it is oftensufficient to use a simple jump-phase model,11 whichcorresponds to an ideal isotropic homogeneous dielec-tric half-space when the phase ds equals p for allincidence angles, whereas the phase dp experiences ajump from 2p to p in the vicinity of the Brewsterangle. The applicability of this model for a dielectricwith rough surfaces is based on the fact that thephases ds, dp are second-order terms in the Rayleigh

criterion parameter. Although for anisotropic di-electric media, reflectometric and ellipsometric datagiven over all angles ~including the Brewster angle!

and, from Eqs. ~1! and ~27!, we may express the an-isotropy b through the experimentally measured val-ues of tan uB and K~py2!:

b 5tan uB

Htan uB

26 FStan uB

2 D2

1 KSp

2DG1y2JHtan2 uB

21 1 2 KSp

2D 7 tan uBFStan uB

2 D2

1 KSp

2DG1y2J1y2 , (28)

are equivalent to each other, for any given angle onlyreflectometry presents exhaustive information aboutthe substrate, whereas ellipsometry gives significantinformation about only the ratio ~RpyRs!

1y2. In thissense, in the discussion below of the anisotropic di-electrics, reflectometry has an advantage comparedwith ellipsometry.

Now we introduce the AUR in the form of the func-tion17

K~u! 5 Us2 cos2 u2

Us

Up1 sin2 u, (21)

where Us,p 5 @~1 2 rs,p!y~1 1 rs,p!#. For a crystalwith the optical axis normal to the surface we have17

K~u! 5 nx2F12

nz

nxSnx

2 2 sin2 u

nz2 2 sin2 uD

1y2G . (22)

In the limit of a weak anisotropy, b ,, 1, we obtain

K~u! < bnx

2 sin2 u

nx2 2 sin2 u

, (23)

whereas for small angles, sin2 u ,, nx2,

K~u! > b sin2 u (24)

depends on only the anisotropy parameter b.There is some ambiguity in determining both of the

parameters b and nx from the experimentally mea-sured function K~u!. It is possible, in principle, tofirst determine b from measurements at small an-gles, sin2 u ,, nx

2, when Eq. ~22! becomes

K~u! 5b 1 ~b2y2!

~b 1 1!2 sin2 u, (25)

@for small b Eq. ~25! turns into relation ~24!# and thendetermine nx from the relation

nx 5 sin uF K~u!

K~u! 2 b sin2 uG1y2

(26)

for not so small angles. We, however, proceed in adifferent way. If the angular measurements aredone up to the grazing angles u 3 ~py2! and theposition of the Brewster angle is known, then Eq. ~22!takes the form

K~py2! 5 nx~nx 2 tan uB!, (27)

where the upper signs apply for

K~py2! . Kcrit~py2! 5 2S1 1 tan2 uB

2 D1y2

3 Ftan uB 2 S1 1 tan2 uB

2 D1y2G ,

and the lower signs must be taken for K~py2! ,Kcrit~py2!. The critical value of K~py2! 5 Kcrit~py2!corresponds to the minimum value of the negativeanisotropy

b 5 bcrit 5 2~1 2 tan uB!2

1 1 tan2 uB,

which may exist for a given value of tan uB @for b ,bcrit and fixed uB, Eq. ~1! has no real solutions; com-pare with Fig. 2#. On the other hand, for all negativeb . bcrit and fixed tan uB there are two real solutions:

~nx2!6 5

1 1 tan2 uB

26 FS1 1 tan2 uB

2 D2

2tan2 uB

~1 1 b!2G1y2

.

The upper and the lower signs in Eq. ~28! correspondto the larger and the smaller values of the refractiveindex nx, respectively. In almost all cases, only thelarger root has a practical sense. For small anisot-ropy, when K~py2! ,, tan uB, Eq. ~28! simplifies to

b 5 KSp

2D tan2 uB 1 tan uB 2 22 tan uB

, (289)

i.e., for a standard situation, when tan uB . 1, thesign of K~py2! determines the sign of the anisotropy.

Up to now we have assumed a uniaxial medium,i.e., anisotropy in the surface plane: nx 5 ny. How-ever, this model may easily be generalized to the caseof a biaxial crystal when two of the three principalaxes ~x and y! are assumed to be parallel to the sur-face, while the z axis is perpendicular to it. In thiscase, the coefficient rp does not change, but in theexpression for the coefficient rs the refractive index nxmust be replaced by ny. Then Eq. ~22! becomes

Kx~u! 5 ny2F12

nxnz

ny2 Sny

2 2 sin2 u

nz2 2 sin2 uD

1y2G , (29)

where the index x denotes the ~xz! incident plane ofthe light. Now the function Kx~u! under normal in-


cidence is not equal to zero: Kx~0! 5 ny~ny 2 nx!.For the incident plane ~yz! after the evident substi-tution nxN ny, we obtain Ky~0! 5 nx~nx 2 ny!. If onedefines the optical anisotropy of the interface as byx 5@~nyynx! 2 1#, we come to the following simple expres-sion:

ny

nx5 byx 1 1 5 2

Kx~0!

Ky~0!. (30)

When the directions of the principal axes are un-known, it is necessary under azimuthal rotation ofthe incident plane to determine the maximum, de-noted as Ky~0!, and minimum, denoted as Kx~0!, val-ues of the function K~0! and substitute them into Eq.~30!. Being nonzero, they will always have differentsigns.

To verify the applicability and the precision of theAUR method, we analyzed reflectometry data ~theangular dependences of Rs and Rp! of the same calcitecrystal and c-Si samples that were used for the SPMmeasurements. The functions K~u! obtained fromfits to the experimental data are given in Fig. 10.For a calcite crystal @Fig. 10~a!# the measured value ofthe optical anisotropy, b 5 29.6%, is in better agree-ment with the literature data than the SPM results.For a highly polished side of the c-Si sample thefunction K~u! equals zero for all the angles with anaccuracy 60.003, i.e., the anisotropy ubu is less than0.5%. In Fig. 10~b! K~u! is shown for the unpolished~rough! side of this sample. The roughness imitatesa very strong effective negative anisotropy. The de-termined value of b 5 247.5% differs from the valueof the anisotropy obtained by the R~0! variant of theSPM and matches fairly well with the value b 5246.5% obtained with the a variant of the SPM.The reason why this variant with the measurementunder grazing angles is more suitable is the follow-ing. Under grazing angles the Rayleigh parameterdecreases, and this means an effective reduction inscattering. On the other hand, under normal inci-dence the influence of roughness is maximum, so thatin the given example only 10% of the light is reflectedcoherently. Naturally it is difficult to expect highprecision in this limiting case of so great an effectiveanisotropy. We gave this example to demonstratethe possibilities of the AUR method even under theseunfavorable conditions.

In the end, the AUR method is, in principle, moreprecise than the SPM, as it takes into account all thedata of all angular measurements. It would be in-teresting to test its applicability by use of the ellip-sometric technique and samples with differentdegrees of anisotropy.

4. Interference Method for Determination of FilmAnisotropy

The IM is of special interest in the cases in whichthere is no need to know the total effective anisotropyof a sample, but rather the anisotropic properties ofthe film material. The method is based on the dif-ference of the optical path lengths of s- and p-


polarized light in an anisotropic film. For s-polarized light the optical path length and theinterference pattern under reflection from the bound-aries of a layer depend on the horizontal refractiveindex nx only, whereas for p-polarized light this pat-tern depends on the vertical refractive index as well.In the case of an anisotropic film, light of each polar-ization experiences a different optical path, andtherefore each interference pattern observed as afunction of the incidence angle will have a differentnumber of interference peaks. When the angle ofincidence changes the interference patterns associ-

Fig. 10. Angular dependences of the function K~u! drawn on thebasis of the reflectometry data and the jump-phase model and theBrewster angle measurements @F and E represent the values ofK~u! calculated with Eq. ~21! and experimentally measured reflec-tion coefficients Rp and Rs. The function of Eq. ~22! ~solid curve!is fitted with the value K~py2! and the measured value of tan uB#for ~a! a natural calcite crystal sample: K~py2! 5 20.196, uB 560.5°. The use of Eqs. ~27! and ~28! gives b 5 29.6% and nx 51.648 ~literature data b 5 210.3%, nx 5 1.658!. ~b! An unpolishedrough surface c-Si sample ~see Fig. 9! with K~py2! 5 22.1, uB 574.38°. The use of Eqs. ~27! and ~28! leads to b 5 247.5% and nx

5 2.84.

ated with the s- and the p-polarized light, a differentnumber of oscillations are manifested, and thereforethe patterns will be shifted with respect to each other.If the angle of incidence is varied from zero to acertain angle u, the differences in phase angles,

ap~u 5 up! 5 2pdl

nx

nz~nz

2 2 sin2 up!1y2,

as~u 5 us! 5 2pdl

~nx2 2 sin2 us!,

are given by

Dp 5 ap~0! 2 ap~up! 52pd

l Fnx2nx

nz~nz

2 2 sin2 up!1y2G(31)

for the p component and by

Ds 5 as~0! 2 as~us! 52pd

l@nx 2 ~nx

2 2 sin2 us!1y2# (32)

for the s component.The phases Dp and Ds determine the number of

reflected light intensity oscillations, and their differ-ence18

D~up, us! 5 Dp 2 Ds

52pd

l F~nx2 2 sin2 us!

1y22nx

nz~nz

2 2 sin2 up!1y2G

(33)

gives the difference of these numbers, i.e., the totalshift in the interference pattern of maxima and min-ima of s- and p-reflected beams. If we choose anequal number m of oscillations in both of the pat-terns, i.e., Dp,s 5 pm, then D 5 0 and the relative shiftup 2 us of minima ~maxima! of the interference pat-terns will depend on only the anisotropy b. FromEq. ~33! we find

b 5sin up

sin us2 1. (34)

This expression for the anisotropy through the anglesof the extremum points is more visual and suitablethan the analogous Eq. ~6! of Ref. 18, which expressesb through us and the difference of the phase-angleshifts D. For small anisotropy, b ,, 1, Eq. ~34! be-comes

b 5 ~up 2 us!cot us. (349)

The sign of the anisotropy always corresponds to thesign of the difference up 2 us.

Figure 11 shows the calculated reflection s- andp-interference patterns for films with positive, zero,and negative anisotropy deposited on an isotropicsubstrate. With this value of the anisotropy, therelative shifts of the first minima and maxima are

2°–3°. This is a strong effect that may be easilymeasured even by coarse techniques.

Equation ~34! is derived with the assumption thatthe interference oscillations take place on the con-stant background, i.e., as if the derivatives Rs,p9~u! ofthe substrate were assumed to be zero which is trueonly for u 5 0. For not very large angles ~u , BB!they may be estimated as Rs,p9~u! > 6R~0!~4uynx!,which leads to positive ~negative! shifts @~umaxl!y~4pd!# of the maxima of the s ~p! patterns. Evi-dently, for minima the signs of all shifts are opposite.Therefore, even for an optically isotropic film, one willsee small negative shifts of p maxima and positiveshifts of p minima. A weak manifestation of thiseffect may be noted in Fig. 11~b!. To cancel thisfictitious anisotropy effect, one may apply Eq. ~34! toneighboring pairs of maxima and minima and thentake an average value. In general, the effect of thenonzero derivatives becomes noticeable only for thinfilms, when the interference extrema are situated atlarge angles and the proximity of the Brewster angledistorts the entire pattern.

We applied this method to the measurement of theanisotropy of a layered structure, which always has anegative effective anisotropy, and received satisfac-tory agreement between the measured value of theanisotropy18 b 5 24.1% with the theoretically calcu-lated value b 5 24.5%. We emphasize that becausesurface light scattering does not influence themaximum–minimum positions of the interferencepatterns and the number of oscillations in each in-terference pattern, then the IM is insensitive to sur-face roughness. This insensitivity of the IM is avaluable advantage compared with the advantages ofall the other methods.

5. Anisotropy of Some Simple Structures

Determination of the anisotropy of a sample has, initself, only limited significance. What other charac-teristics of the material may be obtained from anisot-ropy measurements? When the mesostructure of acomposite material is known, the value of the opticalanisotropy is directly related to the concentration of

Fig. 11. Calculated angular dependences of the Rs and Rp coeffi-cients for the films ~nx 5 2.664! with ~a! positive ~b 5 110%!, ~b!zero ~b 5 0!, ~c! negative ~b 5 210%! anisotropy. Refractive indexof the substrate n2 5 1.35, dyl 5 10. Hardly noticeable in theisotropic case, b 5 0, the negative and the positive shifts up 2 us

~equal to 20.4° and 10.6° for the first minima and maxima, re-spectively! are connected with the derivatives Rs,p9~u! as describedin the text.


the inclusions and their form. Therefore, in the caseof a PS sample, for example, there is a direct connec-tion between the anisotropy and the porosity, definedas the fraction of the voids in the sample. In itsturn, the porosity defines the photoluminescence ofthe PS material and therefore merits determination.We present examples of simple mesoscopic geome-tries when the connection between the concentra-tions of the constituents and the dielectricpermittivity may be calculated analytically.

A. Thin-Layered Geometry

The thin-layered structure is the limiting case of thegeometry with the maximum screening effect of theelectric field in one direction ~grazing-incidence an-gles! and the absence of any screening ~for normalincidence of the light!. In this case a composite op-tical medium consists of alternating layers of twoconstituents with the refractive indices n1 5 =ε1 andn2 5 =ε2. Each layer is much thinner than an op-tical wavelength. The volume fractions of these con-stituents are 1 2 c and c, respectively. For lightpolarized perpendicularly to the planes of the layers~along the z axis!, the effective refractive index ni 5~εii

ef!1y2 of the structure is given by

nz2 5 εzz

ef 5 S1 2 cε1

1cε2D21

,

whereas for light polarized parallel to the planes ofthe layers, nx

2 5 εxxef 5 ~1 2 c!ε1 1 cε2. An optical

Fig. 12. Effective anisotropy bef of a two-component ~with refrac-tive indices n1 and n2! medium as a function of the concentrationof the second constituent for the layered ~curves 1 and 2! andcolumnar ~curves 3 and 4! mesostructures. The different signs ofbef for curves 1–4 demonstrate the importance of the geometry ofthe mesostructure. The distinction between curves 3 and 4 dem-onstrates the importance of the form of the inclusions ~columnarstructure of c-Si in vacuum or vacuum holes in a bulk Si material!.Point A corresponds to the calculated anisotropy b 5 24.5% ofa-SiC:H ~n1 5 2.4! and a-Si:H ~n2 5 3.! layered structure with c 50.2 used in the IM measurements.18


anisotropy bef of such a structure depends on only therelative value ε 5 ~ε2yε1! 5 ~n2

2yn12! and has the form

bef 5 F εε 1 c~1 2 c!~ε 2 1!2G1y2

2 1. (35)

It is a symmetric function about the point c 5 1⁄2,where bef has a minimum bef~c 5 1⁄2! 5 2@~=ε 21!2y~ε 1 1!#. In Fig. 12 the function bef~c! is shownfor a multilayer structure with different ratios of therefractive indices.

B. Columnar Geometry

For a composite medium with inclusions of cylindri-cal form it is possible, by use of the method of integralequations, to obtain an analytical solution.8 If weassume that, on the one hand, each embedded cylin-der is sufficiently large to form the macroscopic di-electric permittivity but, on the other hand, is rathersmall compared with the wavelength, then it may beconsidered as a mesoscopic elementary radiator.Using the known solution for the cylinder in a uni-form field,19 we find the mesoscopic polarizability ten-sor of the radiator and, after its substitution into thegeneralized two-dimensional Lorentz–Lorenz rela-tion, obtain the dielectric permittivity tensor of a me-dium with a columnar ~oriented along the z axis!mesostructure:

εxxef 5 εyy

ef 5ε 1 1 1 c~ε 2 1!

ε 1 1 2 c~ε 2 1!, εzz

ef 5 1 1 c~ε 2 1!,

bef 5 Fε 1 1 1 cε~ε 2 1! 2 c2~ε 2 1!2

ε 1 1 1 c~ε 2 1! G1y2

2 1. (36)

We may now, in addition, calculate the dielectric per-mittivity tensor of the corresponding phase-transformed medium: a set of vacuum columns inthe bulk material. For this it is sufficient to substi-tute ε3 1yε and c3 1 2 c into Eqs. ~36!. Then weobtain

εxxef 5 εyy

ef 5 ε2 1 c~ε 2 1!

2ε 2 c~ε 2 1!, εzz

ef 5 1 1 c~ε 2 1!,

bef 5 F2ε 1 c~2ε 2 1!~ε 2 1! 2 c2~ε 2 1!2

2ε 1 cε~ε 2 1! G1y2

2 1. (37)

Curves 3 and 4 in Fig. 12 show the dependences givenby Eqs. ~36! and ~37! for the ratio n2yn1 characteristicof PS. They strongly differ at concentrations $50%.The mesostructure of Si columns in vacuum gives ananisotropy ;6 times that calculated for columnarvoids in Si bulk material.

C. Low-Relief Surface Layer

A low elongated periodic relief on the interface be-tween two dielectrics is shown to be equivalent, interms of electromagnetic wave reflection, to a homo-geneous but anisotropic thin film.9 The term lowmeans that all the sizes of the relief are small com-pared with the wavelength. The components of the

dielectric permittivity tensor can be calculated ana-lytically in the cases of gently sloping and steep ir-regularities, whereas they can be calculatednumerically for a relief with plane faces and arbitraryslopes. In the most relevant case of steep triangularirregularities, which corresponds to the concentra-tion c 5 0.5, the components of the tensor of effectivepermittivity of the film have the forms

εxxef 5

εε 2 1

ln ε, εzzef 5

ε 2 1ln ε

, bef 5ε 2 1

Îε ln ε2 1.

(38)

For such a geometry, the anisotropy of a film is al-ways positive.

6. Summary

We considered three simple polarization reflectom-etry techniques for determining the optical anisot-ropy of homogeneous uniaxial samples or films on anisotropic substrate. In the case of an isotropic filmon an isotropic substrate, the equivalence of such aninhomogeneous system to an effective uniaxial homo-geneous crystal is demonstrated for a film of rathersmall, dyl ,, 1, thickness. The procedure for thedetermination of the refractive index and the thick-ness of the film, as well as control of its homogeneityduring the growth in situ by the SPM, is proposed.With increasing film thickness, the interference pat-tern appears, and none of these methods may beused. However, for sufficiently thick films, dyl . 1,and fairly distinct interference patterns, the IM maybe applied to determine the anisotropy of the mate-rial of the film. Note that a large film thickness doesnot, generally speaking, prevent use of the IM. Oneneeds only to take the difference up 2 us, not of thefirst pair of the maxima ~minima!, but of one of thefollowing pairs, when this difference is already large,yet the relation up, us , uB still holds.

Because scattering always imitates negative an-isotropy, its neglect means that using the SPM for thedetermination of a positive bulk sample’s anisotropygives only the lower limit of this anisotropy. Al-though both the SPM and the AUR methods are ap-plicable for determining the effective anisotropy, theAUR approach is, in general, more precise even in itscoarse interference reflectometry variant. The ad-vantage of the IM consists in its insensitivity to sur-face scattering, up to the blurring of the interferencepattern. It may be especially useful for determiningthe optical anisotropy of high-quality free-standing orsubstrate-disposed films, now available with a poros-ity of .80%.20

The self-consistency of these methods was tested ina number of cases for natural crystals, PS samples,and a specially prepared thin-layered film structure.Because the AUR method, which assumes a knowl-edge of the amplitudes and the phases of the Fresnelcoefficients, gave fairly satisfactory results evenwhen only reflectometry data were used, one may

expect better accuracy when using experimental el-lipsometric data.

The polarization reflectometry methods may beused for the qualitative and, in many cases, quanti-tative determination of the optical anisotropy. Acomparison of the results obtained by use of thesesimple techniques with experimental data obtainedby more elaborate methods of reflectance and inter-ference spectroscopy21,22 or spectroscopic ellipsom-etry23 should allow us to define the limits of theirapplicability.

We are grateful to the referee for valuable com-ments. This work is supported by the Conselho Na-tional de Desenvolvimento Cientifico e Tecnologico-CNPq and Fundacao de Amparo a Pesquisa do Estadode S.Paulo-FAPESP-,Brazil.

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Three polarization reflectometry methods for determination of optical anisotropy

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