inverse relationships for reflection diagnostics of uniaxially anisotropic nanoscale films on...

Inverse relationships for reflection diagnosticsof uniaxially anisotropic nanoscale films

on isotropic materials

Peep AdamsonInstitute of Physics, University of Tartu, Riia 142, Tartu 51014, Estonia

([email protected])

Received 6 December 2010; revised 2 February 2011; accepted 3 February 2011;posted 10 February 2011 (Doc. ID 139243); published 9 June 2011

The possibilities of determining the parameters of uniaxially anisotropic ultrathin nonabsorbing dielec-tric films on absorbing or transparent isotropic substrates by surface differential reflectance measure-ments are analyzed. The analysis is based on analytical reflection formulas obtained in the framework ofa long-wavelength approximation. It is shown that, in the case of transparent substrates, it is alwayspossible to determine the thickness of a uniaxially ultrathin film and its four parameters of anisotropy(optical constants no and ne and angles θ and φ) simultaneously. However, for such films on absorbingsubstrates, it is possible to decouple the thickness and optical constants by differential reflectance mea-surements only if θ ≠ 0. The accuracy of the obtained analytic formulas for determining the parameters ofultrathin films is estimated by computer simulations where the reflection problem was solved numeri-cally on the basis of the rigorous electromagnetic theory for anisotropic layered systems. © 2011 OpticalSociety of AmericaOCIS codes: 260.2110, 240.0310, 240.6645, 310.6860, 260.2130, 120.4530.

1. Introduction

The surface differential reflectance (SDR) method,which is founded on the direct measurement of thecontribution of an ultrathin layer to the reflectance,has gained widespread acceptance in optical diagnos-tics of ultrathin isotropic films [1–6]. This is becauseoptical methods are nondestructive, noninvasive,and fast (can be performed in real time and, there-fore, are ideal for many in situ applications). Physi-cally in the category of ultrathin films, which ratherarbitrarily can be defined as films less than 5 [7] or10nm [8] thick, fall such layers as native oxides, ad-sorbed monolayers, Langmuir–Blodgett films, andgate oxides (nanometer-sized insulating films) on mi-croelectronic devices. In fact, at the moment, also ofinterest are the possibilities of optical diagnostics forthe analysis of utrathin films less than 1nm thick [9]because, in particular, this technique is of primary

importance in real-time control of deposition and ap-plications involving preparation of next-generation(opto)electronic devices.

On the other hand, it is well known that isotropicbehavior of optical properties in the near-surface re-gion is rare in occurrence. In contrast, these proper-ties often show a quite anisotropic nature: one simplereason that causes anisotropy is that the propertiesof the ultrathin layer in the direction perpendicularto this layer differ from those in the direction parallelto the layer [10,11]. For example, thin overlayers inisland form and surface roughness will often have anuniaxial character with the optic axis normal to thesurface. At the same time, the classical reflectionmethods [12–17] for determining the optical con-stants of anisotropic ultrathin films and surfacelayers suffer many disadvantages. Therefore, the ela-boration of novel optical diagnostics techniques withgreater capabilities for the analysis of anisotropic ul-trathin films is a subject of much current interest[18–24].

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10 June 2011 / Vol. 50, No. 17 / APPLIED OPTICS 2773

On principle, simply measurable SDR is generallyrelated to the film parameters by rather complicatedmathematical expressions, especially in the case ofanisotropic films. But for ultrathin films, it is possi-ble to simplify the theoretical treatment significantlyby applying a long-wavelength approximation (thick-ness d is much less than optical wavelength λ). Thespecial convenience of long-wavelength approxima-tion is that it enables obtaining final analyticalexpressions for reflection coefficients. These rela-tively simple analytical relationships not only givephysical insight into the reflection problem, but alsoare especially advantageous for tackling the inverseproblem, i.e., the determination of parameters of anultrathin film on the basis of SDR measurements.This approximation is of critical importance ex-pressly for anisotropic ultrathin films because the ex-act solution of even the direct reflection problem;moreover, the creation of a solution for the inversionproblem of an anisotropic system is rather compli-cated and the solution of such problems can generallybe found by the use of numerical methods.

The possibilities of SDR measurements for an ul-trathin film with biaxial anisotropy on absorbingsubstrate were recently considered in [23]. Noticethat, compared to transparent substrates, in the caseof an absorbing substrate, the analysis is more sim-ple because the contribution of an ultrathin film tothe reflectance already differs from zero in the firstorder with respect to the small parameter d=λ. How-ever, on the basis of these first-order expressions forSDR, it is not possible to simultaneously determinethe thickness and all six parameters of biaxial aniso-tropy for an ultrathin dielectric film [23]. For trans-parent substrates, the situation is more complicatedbecause SDR differs from zero only in the secondorder in d=λ. The possibilities of determining theparameters of an ultrathin anisotropic film on atransparent substrate are analyzed only when thefilm thickness is predetermined [24].

In practice, however, primarily uniaxially aniso-tropic materials are of considerable use. In theearly literature about uniaxially anisotropic ultra-thin films, the consideration is restricted to thederivation of general formulas for the reflection coef-ficients [10]. An elaborated analysis of the potentialfor determining the thickness and optical constantsof uniaxially anisotropic ultrathin films on the basisof reflectance measurements is missing. A purpose ofthis paper is to analyze theoretically the possibilitiesfor determining the parameters of uniaxially aniso-tropic nonmagnetic ultrathin films on the basis of dif-ferential reflectance that is given in the long-wavelimit.

The paper is organized as follows. The Section 2 isconcerned with the analytical solution of the inverseproblem for uniaxially anisotropic ultrathin films onabsorbing substrates. In Section 3, the possibilities ofSDR diagnostics for such films on transparent sub-strates are considered.

2. Absorbing Substrate

Consider an ultrathin (d ≪ λ) uniaxially anisotropicnonabsorbing dielectric film that is located upon asemi-infinite isotropic, homogeneous, and absorbingsubstrate with complex dielectric constant ε̂s ¼ εsRþiεsI ≡ n̂2

s ¼ ðnsR þ insIÞ2. The dielectric tensor foruniaxially anisotropic material in the xyz coordinatesystem is given by

24 ε11 ε12 ε13ε21 ε22 ε23ε31 ε32 ε33

35 ¼ A

24 εo 0 00 εo 00 0 εe

35A−1; ð1Þ

where A is the coordinate rotation matrix [25].Therefore,

ε11 ¼ εo cos2 φþ ðεo cos2 θ þ εe sin2 θÞ sin2 φ; ð2Þ

ε12 ¼ ε21 ¼ ðεo − εeÞsin2θ sinφ cosφ; ð3Þ

ε22 ¼ εo sin2 φþ ðεo cos2 θ þ εe sin2 θÞ cos2 φ; ð4Þ

ε13 ¼ ε31 ¼ ðεe − εoÞ sin θ cos θ sinφ; ð5Þ

ε23 ¼ ε32 ¼ ðεo − εeÞ sin θ cos θ cosφ; ð6Þ

ε33 ¼ εo sin2 θ þ εe cos2 θ; ð7Þwhere εo ≡ n2

o and εe ≡ n2e are the principal dielectric-

tensor components in the crystal-coordinate system,and θ and φ are the Euler angles with respect to afixed xyz coordinate system (the Cartesian labora-tory coordinate system). The laboratory x, y, and zaxes are defined as follows. The reflecting surfaceis the x–y plane, and the plane of incidence is thez–x plane, with the z axis normal to the surface ofthe layeredmedium and directed into it. The incidentlinearly polarized time harmonic (the complex repre-sentation is taken in the form expð−iωtÞ, where ω ¼2πc=λ and λ is a vacuumwavelength) electromagneticplane wave in a transparent ambient medium withisotropic and homogeneous dielectric constant εa ≡n2a makes an angle ϕa with the z axis. Assume that

all the media are nonmagnetic.Dealing directly with first-order Maxwell equa-

tions, one can calculate the reflection characteristicsof an anisotropic layered system from a wave trans-fer matrix of rank 4 [26,27] (note that the other wayis to work with corresponding second-order waveequations [28,29]). As shown in [30] for SDR of s-and p-polarized light, ΔRss=Rs ¼ ðRss − RsÞ=Rs andΔRpp=Rp ¼ ðRpp − RpÞ=Rp, where Rss;pp and Rs;p arereflectances from the substrate covered with an ul-trathin film and from the bare substrate, respec-tively, one can obtain the following expressionsaccurate to the first order in small parameter d=λ:

2774 APPLIED OPTICS / Vol. 50, No. 17 / 10 June 2011

ΔRss=Rs ≈ 8πK0εsIna cosϕaðεa þ ε223ε−133 − ε22Þðd=λÞ;ð8Þ

ΔRpp=Rp ≈ 8πKεsIna cosϕa½εað1 − εaε−133 sin2 ϕaÞ− ðε11 − ε213ε−133Þ cos2 ϕa�ðd=λÞ; ð9Þ

where

Rs ¼ jðna cosϕa − n̂s cosϕsÞ=ðna cosϕa þ n̂s cosϕsÞj2;ð10Þ

Rp ¼ jðna cosϕs − n̂s cosϕaÞ=ðna cosϕs þ n̂s cosϕaÞj2;ð11Þ

K ¼ ½1 − 2εaαsin2ϕa�=½fεað1 − εaαsin2ϕaÞ − εsRcos2ϕag2þ ε2sIðcos2ϕa − ε2ajε̂sj−2sin2ϕaÞ2�;

cosϕs ¼ ð1 − εaε̂−1s sin2ϕaÞ1=2; α ¼ εsR=jε̂sj2;jε̂sj2 ¼ ε2sR þ ε2sI; K0 ¼ Kðϕa ¼ 0Þ: ð12Þ

The reflectancesRsp andRps (in this paper, the firstsubscript always indicates the incident light) areequal to zero in the first order in d=λ. The second-order formulas take the form:

Rσ ≈ 16π2εacos2ϕajðna cosϕs þ n̂s cosϕaÞj−2× jðna cosϕa þ n̂s cosϕsÞj−2jðε12 − ε13ε23ε−133Þ cosϕs

þ Pσnan̂sε23ε−133 sinϕaj2ðd=λÞ2; ð13Þ

where σ ¼ sp or ps, and Psp ¼ þ1, Pps ¼ −1.In the case of an uniaxially anisotropic layer, gen-

erally, we have five unknown parameters: εo, εe, θ, φ,and d. As follows from Eq. (8), on the basis of the dif-ferential reflectance ΔRss=Rs, one can simply obtainthe quantity ðε22 − ε223ε−133 − εaÞd (it is suggested, cer-tainly, that the optical constants of a substrate andan ambient medium are known). From the measure-ments of ΔRpp=Rp at two different incident angles,one can obtain ðε11 − ε213ε−133 − εaÞd and ðε−133 − ε−1a Þd.This is apparent from Eq. (9) by replacing therethe unity term inside brackets by sin2 ϕa þ cos2 ϕa.Finally, from the measurements of Rps and Rsp bothat the same incident angle or from one reflectance(Rps or Rsp) at two different incident angles, onecan obtain ðε12 − ε13ε23ε−133Þd and ðε23ε−133Þd.

Therefore, the major practical issue is how to de-termine the five unknown (desired) parameters εo,εe, θ, φ, and d on the basis of measurable quantitiesðε22 − ε223ε−133 − εaÞd, ðε11 − ε213ε−133 − εaÞd, ðε−133 − ε−1a Þd,ðε12 − ε13ε23ε−133Þd, and ðε23ε−133Þd. We consider, first,the general case, where θ ≠ 0 and φ ≠ 0, and then weanalyze more closely the two special cases: (i) θ ≠ 0,φ ¼ 0, and (ii) θ ¼ 0.

In the general case, (θ ≠ 0 and φ ≠ 0) fromΔRpp=Rp, ΔRss=Rs, and Rσ, one can determine thefollowing five quantities:

ðε11 − ε213ε−133 − εaÞd ¼ p1; ð14Þ

ðε−133 − ε−1a Þd ¼ p2; ð15Þ

ðε22 − ε223ε−133 − εaÞd ¼ p3; ð16Þ

ðε23ε−133Þd ¼ p4; ð17Þ

ðε12 − ε13ε23ε−133Þd ¼ p5: ð18Þ

Here,

p1 ¼ P2 sin2 ϕð1Þap − P1 sin2 ϕð2Þ

ap

sin2 ϕð2Þap cos2 ϕð1Þ

ap − sin2 ϕð1Þap cos2 ϕð2Þ

ap

; ð19Þ

p2 ¼�

P1 cos2 ϕð2Þap − P2 cos2 ϕð1Þ

ap

sin2 ϕð2Þap cos2 ϕð1Þ

ap − sin2 ϕð1Þap cos2 ϕð2Þ

ap

�1

ε2a; ð20Þ

Pi ¼λ

8πnaK cosϕðiÞapεsI

ΔRppðϕðiÞapÞ

RpðϕðiÞapÞ

; ð21Þ

where ϕð1Þap and ϕð2Þ

ap are two different incident anglesfor p-polarized light, i ¼ 1; 2,

p3 ¼ −λ

8πnaK0 cosϕasεsIΔRssðϕasÞRsðϕasÞ

; ð22Þ

where ϕas is the incident angle of s-polarized light,

p4 ¼ ��ηps þ ηsp

4β

�

×�1�

�1 −

4αβγ2

�ηps − ηspηps þ ηsp

�2�

1=2��

1=2; ð23Þ

p5 ¼ ηsp − ηps2γp4

; ð24Þ

ησ¼Rσjn̂scosϕaσþnacosϕsσj2jnacosϕaσþ n̂scosϕsσj2

16π2εacos2ϕaσλ2;

ð25Þwhere ϕaσ is the incident angle where Rσ is mea-sured, cosϕsσ ¼ ð1 − εaε̂−1s sin2 ϕaσÞ1=2, α ¼ j cosϕsj2,β ¼ jn̂sj2εa sin2 ϕa, γ ¼ 2na sinϕaRe½n̂�

s cosϕs�, and theasterisk (�) denotes the complex conjugate [the


reflectances Rsp and Rps are both measured at thesame incident angle (ϕasp ¼ ϕaps)].

On the basis of Eqs. (14)–(18) and (2)–(7), one canobtain the following set of equations for the four un-known quantities εo, εe, θ ≠ 0, and φ ≠ 0:

ðεo sin2 θ þ εe cos2 θÞ½2ðεo − εaÞ þ ðεe − εoÞ sin2 θ�− sin2 θ cos2 θðεe − εoÞ2

¼ ðp1 þ p3Þp−14 cosφ sin θ cos θðεo − εeÞ; ð26Þ

1 − ε−1a ðεo sin2 θ þ εe cos2 θÞ¼ p2p−1

4 cosφ sin θ cos θðεo − εeÞ; ð27Þ

εo ¼ ðp1 − p3Þp−14 cosφðcos 2φ tan θÞ−1

¼ p5p−14 ðsinφ tan θÞ−1: ð28Þ

Equation (28) gives

tan 2φ ¼ 2p5ðp1 − p3Þ−1; ð29Þ

tan θ ¼ p5ðp4εo sinφÞ−1; ð30Þ

and Eqs. (27) and (30) yield

εe ¼p25 þ p2

4ε2o sin2 φ − p25ε−1a εo − p2p5ε2o sinφ cosφ

p24ε−1a ε2o sin2 φ − p2p5εo sinφ cosφ :

ð31Þ

The final equation for the desired quantity εo followsfrom Eq. (26) through the use of Eqs. (29)–(31). Thisequation can be solved numerically with a computer.If the material parameters are known, then thethickness d can simply be determined fromEqs. (14)–(18), for example,

d ¼ p2=ðε−133 − ε−1a Þ: ð32Þ

If φ ¼ 0 and θ ≠ 0, we have only four unknownparameters: εo, εe, θ, and d. From Eqs. (2)–(7), itfollows that

ε11 ¼ εo; ð33Þ

ε22 ¼ εo þ ðεe − εoÞ sin2 θ; ð34Þ

ε33 ¼ εe − ðεe − εoÞ sin2 θ; ð35Þ

ε23 ¼ ðεo − εeÞ sin θ cos θ; ð36Þ

ε12 ¼ ε13 ¼ 0; ð37Þ

and, therefore, on the basis of Eqs. (14)–(17), weobtain the following set of equations for desiredparameters εo, εe, θ, and d:

ðεo − εaÞd ¼ p1; ð38Þ

�1

εe − ðεe − εoÞ sin2 θ −1εa

�d ¼ p2; ð39Þ

�εo þðεe − εoÞsin2 θ − sin2 θ cos2 θðεe − εoÞ2

εe − ðεe − εoÞsin2 θ − εa�d¼ p3;

ð40Þ

�sin θ cos θðεo − εeÞεe − ðεe − εoÞ sin2 θ

�d ¼ p4; ð41Þ

where

p4 ¼ λjna cosϕsσ þ n̂s cosϕaσ jjna cosϕaσ þ n̂s cosϕsσj2πεajn̂sj sin 2ϕaσ

×ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRσðϕaσÞ

p; ð42Þ

as follows from Eqs. (23) and (25). Solving the systemin Eqs. (38)–(41) gives

εo ¼1εa

�p1ðp1 − p3Þ

p24 þ p2ðp3 − p1Þ

�; ð43Þ

εe ¼1εo

�p3ðεo − εaÞ þ p1εap2ðεo − εaÞ þ p1ε−1a

�; ð44Þ

sin2 θ ¼ εeðεe − εoÞ

�1 −

εoεa þ ðεo − εaÞp3p−1

1

�: ð45Þ

If the quantities εo, εe, and θ are known, thenthe thickness d can simply be determined fromEqs. (38)–(41), for example,

d ¼ p1=ðεo − εaÞ: ð46ÞIf θ ¼ 0, then ε11 ¼ ε22 ¼ εo, ε33 ¼ εe, and εij ¼ 0 if

i ≠ j. Hence, Rsp ¼ Rps ≡ 0 and, in the case beingconsidered, we can determine only two quantities:ðεo − εaÞd and ðε−1e − ε−1a Þd, for example, fromequations

ðεo − εaÞd ¼ p1; ð47Þ

ðε−1e − ε−1a Þd ¼ p2: ð48Þ


Of course, the quantities ðεo − εaÞd and ðε−1e − ε−1a Þdcan also be determined by using both p- ands-polarized light, i.e., on the basis of Eqs. (15) and(16), which, in this special case, take the form

ðεo − εaÞd ¼ −λ

8πnaK0 cosϕasεsIΔRssðϕasÞRsðϕasÞ

; ð49Þ

ðε−1e − ε−1a Þd ¼ λ8πnaε2asin2ϕapεsI

�cos2ϕap

K0 cosϕas

ΔRssðϕasÞRsðϕasÞ

−1

K cosϕap

ΔRppðϕapÞRpðϕapÞ

�: ð50Þ

However, the quantities ðεo − εaÞd and ðε−1e − ε−1a Þdcontain three unknown parameters. Therefore, inthis simple case of anisotropy, the correlation-freeSDR measurements are impossible for an ultrathinfilm, i.e., we cannot determine the thickness andoptical constants simultaneously (the material con-stants εo and εe can be determined for such ultrathinfilms only if their thicknesses are known in advance).This result has come as a little surprise because, for amore simple case, i.e., for an isotropic nonabsorbingultrathin film on an absorbing isotropic substrate, wecan simply determine the thickness and optical con-stant simultaneously in the framework of the long-wave limit [31].

On the other hand, the physical reason of the oc-currence of such correlation is easily understood.Namely, in this simplest case of anisotropy, Rsp ¼Rps ≡ 0, therefore, we have only two independentmeasurable quantities: ΔRpp=Rp and ΔRss=Rs. It isevident that we can determine as many unknownparameters as there are independent equations or,in other words, the number of equations must beequal to the number of unknown parameters. On thebasis of ΔRpp=Rp and ΔRss=Rs, we can create onlytwo independent equations for three desired para-meters εo, εe, and d. Because of this, it is impossibleto simultaneously determine the thickness and opti-cal constants of such anisotropic films. Naturally, it ispossible for an isotropic film because it is character-ized by only one optical constant.

For reference, we have included a computer simu-lation for the possible errors of the approximateformulas in Eqs. (29), (31), and (32) (Fig. 1) andEqs. (47)–(50) (Fig. 2). Computer simulations offera clearer view of how the approximate formulas workbecause such an approach makes it possible to ana-lyze more complicated situations than we can createin real experiments. It can be said with confidencethat such an application is a perfect example of thebenefit to the harnessing of computer simulations.

In order to calculate the error of approximateequations, we give certain exact values for all un-known parameters εo, εe, θ, and φ, and then calculateby the exact electromagnetic theory the values ofΔRpp;ss=Rp;s and Rσ (the relevant computational

technique is outlined, for example, in [30]). Next, weuse these quantities in the form of ΔRpp;ss=Rp;sð1 −

vpp;ssÞ and Rσð1 − vσÞ (where vpp;ss and vσ representthe relative errors of ΔRpp;ss=Rp;s and Rσ, respec-tively) in Eqs. (19), (20), and (22)–(24) for calculating,first, the quantities pi (i ¼ 1–5) and then for deter-mining εðcalcÞo , εðcalcÞe , θðcalcÞ, and φðcalcÞ on the basis ofcorresponding equations. The machine-performedcomputations of the relative errors ðεo − εðcalcÞo Þ=εo,ðεe − εðcalcÞe Þ=εe, ðθ − θðcalcÞÞ=θ, and ðφ − φðcalcÞÞ=φ asfunctions of λ or d=λ for different vpp, vss, vps, andvsp are plotted in Figs. 1 and 2. Note that, if vpp ¼vss ¼ vps ¼ vsp ¼ 0, then we obtain the pure mathe-matical error of the approximate formulas that hasnothing to do with the error of ΔRpp;ss=Rp;s and Rσthat occurs in the experimental measurements ofthese quantities. The results of this simulation showthat the wavelength plays a role in the error forma-tion only in the short-wavelength region where d=λ isnot a sufficiently small quantity. But, in the long-wavelength region, the error of approximate formu-las is of no concern: the error of desired anisotropicconstants is completely defined by instrumental er-ror. The computations also show that the uncertaintyof Rσ is not of first importance (the error of

Fig. 1. Relative errors of (a) εe and (b) φ (solid curves) and d(dashed curves) determined by Eqs. (31), (29), and (32), respec-tively, as functions of λ for an anisotropic ultrathin film withd ¼ 1nm, no ¼ 3:4, ne ¼ 3:0, θ ¼ φ ¼ 40° at n̂s ¼ 2:5þ i2, vpp ¼vss ¼ 0 (curves 1, 3) and 2% (curves 2), and vsp ¼ vps ¼ 0 (curves1, 2) and −5% (curves 3). Incident angles ϕð1Þ

ap ¼ 20°, ϕð2Þap ¼ 70°,

ϕas ¼ 20°, and ϕaps ¼ ϕasp ¼ 70°.


ΔRpp;ss=Rp;s has a dominant role). Notice that, ifεa ¼ 1, then Eq. (50) to evaluate the quantity εe isconsiderably inexact [Fig. 2(b)] in comparison withEq. (49) for determining εo [Fig. 2(a)]. This derivesfrom the fact that jε−1e − ε−1a j ≪ εo − εa, which is whythe terms on the right side of Eq. (50) for εa ¼ 1 con-sist of the difference of closely related approximatequantities with an uncertainty, which is greater thanthe uncertainty of the right side of Eq. (49). Analo-gous is the situation in the case of Eqs. (48) and (47)(if εa ¼ 1), where, by the difference of two approxi-mate quantities in one case [Eq. (48)], a significantlysmaller quantity is defined than in another case[Eq. (47)]. If εa > 1, then the accuracy of Eqs. (50)and (48) will be enhanced [becomes practically com-parable to the accuracy of Eqs. (49) and (47)].

Furthermore, we present the results of a computersimulation for a specific example. Namely, we consid-er an uniaxially anisotropic ultrathin TiO2 (Rutile)film with d ¼ 2nm on an isotropic Ge substrate forλ ¼ 500nm. Let us assume that n̂s ¼ 4:4þ i2:4 andthe exact values for the parameters of film anisotro-py are the following: no ¼ 2:5, ne ¼ 2:77, φ ¼ 0°, andθ ¼ 50°. For these prescribed values, by the rigorouselectromagnetic theory, we calculate the exact differ-ential reflectances ΔRpp=R

ð0Þp , ΔRss=R

ð0Þs , and Rsp

at incident angles ϕð1Þap ¼ ϕas ¼ 30° and ϕð2Þ

ap ¼ϕasp ¼ 70°. Thereafter, assuming that the relative er-rors vpp ¼ 2%, vss ¼ −2%, and vsp ¼ 5%, we calculatethe values of the quantities p1, p2, p3, and p4 byEqs. (19), (20), (22), and (42), respectively. Then, byEqs. (43)–(46), we can determine approximate valuesfor no, ne, θ, and d. The results are the following: no ¼2:499, ne ¼ 2:776, θ ¼ 60:4°, and d ¼ 2:18nm.

Finally, we consider some difficulties, which canemerge in the design of a spectroscopic experiment.Because the film thickness is very small, it is evidentthat a central problem is the quantifiability ofdifferential characteristics ΔRpp;ss=Rp;s and Rσ. Cal-culations [Fig. 3(a)] show that, for uniaxially aniso-tropic ultrathin films with nanometric thicknesseson absorbing substrates, SDR for the dominantquantities Rpp and Rss are of sufficiently high values.For the measurements of such quantities, excellentexperimental techniques exist [1–6,9].

However, the detection of Rσ is a serious challengefrom an experimental standpoint. For example, if thethickness of an anisotropic layer is several nano-meters, then the typical value of Rσ in this regionof wavelengths where approximate formulas workreasonably well (Fig. 1) is roughly in the range from10−8 to 10−6 [Fig. 3(b)]. For such measurements, themajor criterion is the minimum difference in signalthat can be detected or, in other words, the signal-to-noise ratio is bound to be quite large values. As is

Fig. 2. Relative error (a) of the quantity ðεo − εaÞd determined byEq. (49) (solid curves) or (47) (dashed curves) and (b) of the quan-tity ðε−1e − ε−1a Þd determined by Eq. (50) (solid curves) or Eq. (48)(dashed curves) as functions of d=λ for an anisotropic ultrathin filmwith no ¼ 2:4, ne ¼ 2:2, and θ ¼ φ ¼ 0° at n̂s ¼ 3:5þ i1:5, vpp ¼vss ¼ 0 (curves 1), and vpp ¼ vss ¼ −1% (curves 2). Incident anglesϕð1Þap ¼ 45°, ϕð2Þ

ap ¼ 70°, and ϕas ¼ 45°.

Fig. 3. (a) Differential reflectances ΔRpp=Rp (solid curves) andΔRss=Rs (dashed curves) and (b) reflectances Rps (solid curves)andRsp (dashed curves) as functions of λ for an absorbing substrateat two different incident angles ϕa ¼ 20° (curves 1) and 70° (curves2). The other parameters are the same as in Fig. 1.


shown in [32,33] by special effort, the measurementof reflectance of p-polarized light around the Brew-ster angle of the order of 10−7 is made possible. Itmust be emphasized that the physical situation fordetermining Rσ is quite analogous to the reflectivitymeasurement at the Brewster angle. Namely, thisfact that, for a Fresnel interface, the reflectance ofp-polarized light vanishes at the Brewster angle,leads to a high sensitivity to adsorbed layers. Inour case, the quantity Rσ is also equal to zero ifthe surface layer is missing (d ¼ 0), i.e., the reflectedsignal is free from the s-polarized wave if the incidentlight has pure p polarization, and, vice versa, the re-flected signal is free from the p-polarized wave if theincident light has pure s polarization. Besides, thiscircumstance offers good possibilities for determinga crucial apparatus parameter—the residual inten-sity for s-polarized (p-polarized) light if the incidentwave has p polarization (s-polarization). Thus, inconclusion, it may be said that the measurementsof the reflectances Rps and Rsp are in principle prac-ticable, although difficult to accomplish accurately.

3. Transparent Substrate

In the case of a transparent substrate (εsI ¼ nsI ¼ 0,εsR ≡ εs, nsR ≡ ns) the differential quantities ΔRpp=Rp and ΔRss=Rs are also equal to zero in the first or-der in d=λ. The second-order formulas for ΔRss=Rs,ΔRpp=Rp, and Rσ take the form [34]

ΔRss=Rs ≈16π2nans cosϕa cosϕs

ðεa − εsÞ

×�ðε22 − ε223ε−133 − εsÞðε22 − ε223ε−133 − εaÞ

ðεa − εsÞ−ðn3

ansε223ε−233 sin2ϕaþðε12 − ε13ε23ε−133Þ2 cosϕa cosϕsÞðnans cos2ϕsþ εs cosϕa cosϕsÞ

��dλ

�2; ð51Þ

ΔRpp=Rp ≈16π2nans cosϕa cosϕs

ðεa cos2 ϕs − εs cos2 ϕaÞ

×�ððε11 − ε213ε−133Þ cos2 ϕs − ð1 − εaε−133 sin2 ϕaÞεsÞððε11 − ε213ε−133Þ cos2 ϕa − ð1 − εaε−133 sin2 ϕaÞεaÞ

ðεa cos2 ϕs − εs cos2 ϕaÞ

þ ðεaεsε223ε−233 sin2 ϕa − ðε12 − ε13ε23ε−133Þ2 cos2 ϕsÞðnans cosϕa cosϕs þ εs cos2 ϕsÞ

��dλ

�2; ð52Þ

Rσ ≈ 16π2εa cos2 ϕa½ðε12 − ε13ε23ε−133Þ cosϕs − Pσnansε23ε−133 sinϕaÞ�2ðna cosϕa þ ns cosϕsÞ2ðna cosϕs þ ns cosϕaÞ2

�dλ

�2; ð53Þ

where cosϕs ¼ ð1 − εaε−1s sin2 ϕaÞ1=2. Note that thedifferential quantities ΔRss=Rs and ΔRpp=Rp aremeaningful if and only if εs ≠ εa and, additionally,in the case of p polarization, the relative differentialreflectance (ΔRpp=Rp) makes no sense for ϕa ¼ ϕB.

On the basis of the measurements of ΔRpp=Rp at

three different incident angles ϕa ¼ 0, ϕa ¼ ϕð1Þa ,

and ϕa ¼ ϕð2Þa , we can determine the quantities

ε11 − ε213=ε33 ≡ x, ε−133 ≡ y from the following systemof equations:

a11x2 þ a12y2 þ a13xyþ a14xþ a15yþ a16 ¼ 0;

a21x2 þ a22y2 þ a23xyþ a24xþ a25yþ a26 ¼ 0; ð54Þ

in which

ai1 ¼ cos2 ϕðiÞs cos2 ϕðiÞ

a − Pi; ð55Þ

ai2 ¼ ε3aεs sin4 ϕðiÞa ; ð56Þ

ai3 ¼ εa sin2 ϕðiÞa ½εa cos2 ϕðiÞ

s þ εs cos2 ϕðiÞa �; ð57Þ

ai4 ¼ ðεa þ εsÞPi − εa cos2 ϕðiÞs − εs cos2 ϕðiÞ

a ; ð58Þ

ai5 ¼ −2ε2aεs sin2 ϕðiÞa ; ð59Þ

ai6 ¼ εaεsð1 − PiÞ; ð60Þ

Pi ¼ðεacos2ϕðiÞ

s − εscos2ϕðiÞa Þ2

ðεa − εsÞ2 cosϕðiÞa cosϕðiÞ

s

�ΔRppðϕðiÞa Þ

RpðϕðiÞa Þ

þ Si

�

×�ΔRppðϕa ¼ 0Þ

Rpðϕa ¼ 0Þ þ S0

�−1; ð61Þ


Si ¼ �ðna cosϕðiÞa þ ns cosϕðiÞ

s Þðna cosϕðiÞs þ ns cosϕðiÞ

a Þnaðna cosϕðiÞ

s − ns cosϕðiÞa Þ cosϕðiÞ

a

× ½RpsðϕðiÞa ÞRspðϕðiÞ

a Þ�1=2; ð62Þ

where i ¼ 1; 2, and S0 ¼ SiðϕðiÞa ¼ 0Þ.

The system of two nonlinear equations in Eq. (54)can be solved with a computer. On the other hand,rather than solve the nonlinear system, the problemcan be reduced to a quartic equation for one un-known. This approach has an advantage over thefirst method because, for solving the quartic equa-tions, foolproof methods exist. For unknown y, for in-stance, one can obtain the following quartic equation:

Ay4 þ By3 þ Cy2 þDyþ F ¼ 0; ð63Þ

in which

A ¼ a11f 21 þ a12f 24 − a13f 1f 4; ð64Þ

B ¼ 2ða11f 1f 2 þ a12f 4f 5Þ − a13ðf 2f 4 þ f 1f 5Þ− a14f 1f 4 þ a15f 24; ð65Þ

C ¼ a11ðf 22 þ 2f 1f 3Þ þ a12f 25 − a13ðf 3f 4 þ f 2f 5Þ− a14ðf 2f 4 þ f 1f 5Þ þ 2a15f 4f 5 þ a16f 24; ð66Þ

D¼2a11f 2f 3−a13f 3f 5−a14ðf 3f 4þ f 2f 5Þþa15f 25

þ2a16f 4f 5; ð67Þ

F ¼ a11f 23 − a14f 3f 5 þ a16f 25; ð68Þ

where f 1 ¼ a12a21 − a11a22, f 2 ¼ a15a21 − a11a25,f 3 ¼ a16a21 − a11a26, f 4 ¼ a13a21 − a11a23, and f 5 ¼a14a21 − a11a24. If the quantities x and y are known,then the thickness d can simply be determined fromthe following expression:

ðd=λÞ2 ¼ ðεa − εsÞ2½16π2nansðx − εsÞðx − εaÞ�−1

×�ΔRppðϕa ¼ 0Þ

Rpðϕa ¼ 0Þ þ S0

�: ð69Þ

The quantities ε12 − ε13ε23=ε33 and ε23=ε33 can beobtained on the basis of Eq. (53):

ε12 −ε13ε23ε33

¼ Kps − Ksp

2 cosϕs; ð70Þ

ε23ε33

¼ Ksp þ Kps

2nans sinϕa; ð71Þ

where

Kσ ¼�R1=2σ

ðna cosϕaþns cosϕsÞðna cosϕsþnscosϕaÞ4πna cosϕa

λd;

ð72Þ

and ϕa ≠ 0 if φ ¼ 0.As is evident from the foregoing, the mea-

surements of differential reflectances make it possi-ble to determine the four quantities: ε33, ε23,ε11−ε213ε−133≡x, and ε12 − ε13ε23ε−133 ≡ t. Consequently,the desired parameters of uniaxial anisotropy εo,εe, θ;, and φ can be obtained from the following sys-tem of equations:

εo sin2 θ þ εe cos2 θ ¼ ε33; ð73Þ

ðεo − εeÞ sin θ cos θ cosφ ¼ ε23; ð74Þ

εo cos2 φþ ðεo þ εe − ε33Þ sin2 φ − ε223ε−133 tan2 φ ¼ x;

ð75Þ

ðεo − εeÞ sin2 θ sinφ cosφþ ε223ε−133 tanφ ¼ t: ð76Þ

Solving the system in Eqs. (73)–(76) gives

εo ¼12

�xþ t2ε33

ε223�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�xþ t2ε33

ε223

�2− 4

�tε33ε23

�2

s �;

ð77Þ

εe ¼xε33εo

−ε223

εo − ε33; ð78Þ

sin2 θ ¼ ε33 − εeεo − εe

; ð79Þ

cos2 φ ¼ ε223ðε33 − εeÞðεo − ε33Þ

: ð80Þ

In the special case φ ¼ 0 (θ ≠ 0), the parameter εo ¼ x(t ¼ 0) and, if θ ¼ 0, then εo ¼ x and εe ¼ ε33(ε23 ¼ t ¼ 0). The relative errors of the approximaterelations in Eqs. (77)–(80) versus λ are demonstratedin Figs. 4 and 5.

It is necessary to stress that a quartic equationgives generally four different solutions; then it maybe that Eq. (63) yields several real positive solutionsfor ε33. However, computer simulations show that,frequently, only one solution of Eq. (63) is physically


meaningful, i.e., is greater than unity. On the otherhand, if two (or even more) real solutions of Eq. (63)are >1, then, of course, one needs to find all final so-lutions for εo, εe, θ, φ, and d that follow immediatelyfrom different solutions of Eq. (63). It is apparentthat a mathematical model generates all possiblecombinations of parameters that possess identicalreflection characteristics (the greater the numberof parameters, the greater the probability that suchdifferent combinations exist). Hence, if two (or more)dissimilar combinations of optical constants andthickness that are all physically meaningful existamong the final solutions, then more information(or a combination of optical methods and alternatemetrology techniques) is needed for the separationof a true value of the set of optical parameters for theuniaxial ultrathin film to be investigated. Note that,the elucidation of the proper solution in computer si-mulations should present no problems because, withreduction in d=λ, one can always increase the accu-racy of such calculations in a way that one possiblesolution virtually agrees with given film parameters.

As an illustration for a transparent substrate, letus consider a uniaxially anisotropic ultrathin TiO2(Rutile) film with d ¼ 5nm on an isotropic Si (crys-talline silicon) substrate (ns ¼ 3:48) at λ ¼ 1500nm.The exact values for the parameters of film anisotro-

py are the following: no ¼ 2:48, ne ¼ 2:75, θ ¼ 30°,and φ ¼ 60°. For these prescribed values, by the rig-orous electromagnetic theory, we calculate the exactreflection characteristics ΔRpp=Rp at incident anglesϕa ¼ 0°, ϕa ¼ ϕð1Þ

a ¼ 20°, and ϕa ¼ ϕð2Þa ¼ 60°; and Rps

and Rsp both at the same incident angle ϕa ¼ ϕð2Þa ¼

60°. Thereafter, assuming that the relative errorsvpp ¼ 2% and vps ¼ vsp ¼ 4%, we calculate the valuesof the quantities no, ne, θ, φ, and d by approximateEqs. (77)–(80) and (69), respectively. The resultsare the following: no ¼ 2:481, ne ¼ 2:747, θ ¼ 29:88°,φ ¼ 59:68°, and d ¼ 4:949nm.

Next we point out some features of the sensibilityproblem for a transparent substrate. At first, the re-flectances Rsp and Rps are basically of the same orderof magnitude as for absorbing substrates [compareEqs. (13) and (53) and Figs. 3(b) and 6(b)]. Butthe values of differential reflectances ΔRss=Rs andΔRpp=Rp are generally significantly smaller thanin the case of absorbing substrates because, for trans-parent substrates, these quantities are proportionalto ðd=λÞ2 (for absorbing substrates, they are ex-pressed by first-order formulas). The sole exceptionin this situation is the reflection of p-polarized lightat the Brewster angle where the reflectance vanishesand, therefore, in the vicinity of this incident angle,ΔRpp=Rp is sufficiently large. This is also apparentfrom Fig. 6(a), where, for the substrate withns ¼ 1:46, the Brewster angle is equal to 55:6°. That

Fig. 4. Relative errors of (a) εo and (b) εe determined by Eqs. (77)and (78), respectively, as functions of λ for an anisotropic ultrathinfilm with d ¼ 2nm, no ¼ 2:49, ne ¼ 2:86, θ ¼ 40°, φ ¼ 30° atns¼1:46, vpp ¼ vps ¼ vsp ¼ 0 (solid curves), vpp¼1%, vps ¼ vsp ¼ 0(dotted curves), and vpp ¼ 0, vps ¼ vsp ¼ 4% (dashed curves). Inci-dent angles ϕð1Þ

a ¼ 0°, ϕð2Þa ¼ 30°, and ϕð3Þ

a ¼ 60°.

Fig. 5. Relative errors of (a) θ and (b) φ determined by Eqs. (79)and (80), respectively, as functions of λ for an anisotropic ultrathinfilm with the same parameters as in Fig. 4.


is the reason that SDR measurements at theBrewster angle are commonly used [32,33,35]. Atthe same time, the values of ΔRss=Rs may be ofthe order of 10−3 in the visible region [Fig. 6(a)].

That kind of outstanding distinction betweentransparent and absorbing substrates arises fromthe fact that the reflectance for the interface betweentwo different transparent materials is significantlysmaller than for the interface between dielectric andstrongly absorbing material, e.g., the dielectric–metal interface. Because of this, a change in the re-flectance of a transparent interface generated by aweak perturbance, e.g., by deposition of an ultrathinfilm, on principle, may be a very small quantity, i.e.,the differences ΔRss;pp −Rs;p must generally be ex-pressed in the second order in d=λ. However, thiscircumstance does not necessarily mean that thecoefficient before ðd=λÞ2 is always less than or equalto unity. On the contrary, in certain combinations ofthe parameters of ultrathin films and substrates,this coefficient may even be of the order of 102–103and, in this case,ΔRss=Rs for a transparent substratedoes not drastically differ from ΔRss=Rs for an ab-sorbing substrate [this is also evident from Fig. 6(a)].

Therefore, in a nutshell, it may be said that, fromthe measurement standpoint, a transparent sub-

strate does not substantially differ from an absorbingone: first of all, the serious problem is that the mea-surement of reflectances Rsp and Rps is difficult butnot impossible to realize. Note that, in the infraredregion of wavelengths, the exactness of approximateequations is significantly better. However, in this re-gion, the requirements for the sensitivity also grownoticeably because, first and foremost, the valuesof Rsp and Rps diminish progressively downstream.But, on the other hand, a very high exactness levelfor equations is actually meaningless inasmuch asinstrumental errors exist. Referring to Figs. 3 and 6,in the near-infrared region, the measurements are inprinciple possible even for ultrathin films with thick-nesses of a few nanometers. Of course, in the infraredregion of wavelengths, the method in hand isundoubtedly a long-term technique for diagnosticsof ultrathin films with thicknesses of several tensof nanometers.

In closing, we emphasize that the special conveni-ence of the analytical approach developed in this pa-per for the solution of the inverse problem lies in thefact that it enables us to find the situations where itis (im)possible to decouple the optical constants andthe thickness of a uniaxially anisotropic ultrathinfilm. For example, it is shown that, for a uniaxiallyultrathin film with θ ¼ 0 on an isotropic absorbingsubstrate, it is impossible to decouple the optical con-stants and the thickness, otherwise (θ ≠ 0) it is pos-sible. Thus, in the light of this work, a viewpoint thatthe correlation between parameters is an ineluctablephenomenon turns out to be correct in the case of an-isotropic ultrathin films, as well.

Notice that, in many respects, this effect is gov-erned by the nature of mathematical procedures.Therefore, a simple and “see-through” mathematicalapproach is of first importance. Our method gives aphysical insight into the reflection problem thatpermits clarifying how one can adopt reflection char-acteristics for correlation-free measurements. Incontrast to the generally accepted picture, our calcu-lations show that simple algebraic equations elimi-nate most, if not all, of the nonphysical roots inthe inversion of reflection data. In addition, thedeveloped analytic method is free from the necessityof reasonable initial values for the parameters to besolved.

4. Conclusions

The long-wavelength limit makes it possible to elabo-rate the relatively simple analytical methods for de-termining the optical constants and thickness ofultrathin uniaxially anisotropic films from SDRdata. It must be emphasized that a key capability ofsuch analytical approximations is to decouple theusual correlations in the optical constants and thethickness of ultrathin films. Namely, the latter fea-ture of obtained analytical expressions for reflectioncharacteristics is currently of first importance inoptics of nanoscale films, because the standard re-gression analysis for determining the parameters

Fig. 6. (a) Differential reflectances ΔRpp=Rp (solid and dashed–dotted curves) and ΔRss=Rs (dashed and dotted curves) and (b) re-flectances Rps (solid and dashed–dotted curves) and Rsp (dashedand dotted curves) as functions of λ for a transparent substrateat two different incident angles ϕa ¼ 30° (curves 1) and 60° (curves2). The other parameters are the same as in Fig. 4 (solid anddashed curves) and Fig. 1 [dashed–dotted and dotted curves(nsI ¼ 0)].


of such ultrathin films on the basis of reflection mea-surements are characterized by a strong correlationbetween film thickness and dielectric response. Con-currently performed numerical simulations showthat the accuracy of the long-wavelength approxima-tion is reasonable if d=λ ≤ a few hundredths.

References

1. P. Chiaradia, A. Cricenti, S. Selci, and G. Chiarotti, “Differen-tial reflectivity of Sið111Þ2 × 1 surface with polarized light: atest for surface structure,” Phys. Rev. Lett. 52, 1145–1147(1984).

2. J. F. McGilp, “Optical characterization of semiconductorsurfaces and interfaces,” Prog. Surf. Sci. 49, 1–106 (1995).

3. R. Lazzari, J. Jupille, and Y. Borensztein, “In situ study of athin metal film by optical means,” Appl. Surf. Sci. 142,451–454 (1999).

4. H. Proehl, R. Nitsche, T. Dienel, K. Leo, and T. Fritz, “In situdifferential reflectance spectroscopy of thin crystalline films ofPTCDA on different substrates,” Phys. Rev. B 71,165207 (2005).

5. S. Ohno, H. Kobayashi, F. Mitobe, T. Suzuki, K. Shudo, andM. Tanaka, “Monolayer oxidation on Sið001Þ-ð2 × 1Þ studiedby means of reflectance difference spectroscopy,” Phys. Rev.B 77, 085319 (2008).

6. L. Simonot, D. Babonneau, S. Camelio, D. Lantiat, P. Guérin,B. Lamongie, and V. Antad, “In situ optical spectroscopy dur-ing deposition of Ag:Si3N4 nanocomposite films by magnetronsputtering,” Thin Solid Films 518, 2637–2643 (2010).

7. H. G. Tompkins, A User’s Guide to Ellipsometry (Academic,1993).

8. H. G. Tompkins and W. A. McGahan, Spectroscopic Ellipsome-try and Reflectometry: A User’s Guide (Wiley, 1999).

9. I. K. Kim and D. E. Aspnes, “Analytic determination of n, κ,and d of an absorbing film from polarimetric data in thethin-film limit,” J. Appl. Phys. 101, 033109 (2007).

10. J. Lekner, Theory of Reflection of Electromagnetic and ParticleWaves (Martinus Nijhoff, 1987).

11. D. Bedeaux and J. Vlieger, Optical Properties of Surfaces(Imperial College Press, 2004).

12. I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, andJ. J. Wharton, “Measurement of the principal refractive in-dices of thin films deposited at oblique incidence,” J. Opt.Soc. Am. A 2, 1693–1697 (1985).

13. F. Flory, D. Endelema, E. Pelletier, and I. Hodgkinson, “Aniso-tropy in thin films: modeling and measurement of guided andnonguided optical properties: application to TiO2 films,” Appl.Opt. 32, 5649–5659 (1993).

14. H. Wang, “Propagation and reflection of plane waves in amedium with the 3-dimensional columnar structure inducedanisotropy,” Optik 106, 140–146 (1997).

15. I. Hodgkinson, Q. H. Wu, and J. Hazel, “Empirical equationsfor the principal refractive indices and column angle of obli-quely deposited films of tantalum oxide, titanium oxide,and zirconium oxide,” Appl. Opt. 37, 2653–2659 (1998).

16. G. I. Surdutovich, R. Z. Vitlina, A. V. Ghiner, S. F. Durrant, andV. Baranauskas, “Three polarization reflectometry methodsfor determination of optical anisotropy,” Appl. Opt. 37, 65–78(1998).

17. Y. J. Jen, C. Y. Peng, and H. H. Chang, “Optical constantdetermination of an anisotropic thin film via polarizationconversion,” Opt. Express 15, 4445–4451 (2007).

18. M. K. Kelly, S. Zollner, and M. Cardona, “Modeling the opticalresponse of surfaces measured by spectroscopic ellipsometry:application to Si and Ge,” Surf. Sci. 285, 282–294 (1993).

19. K. Hingerl, D. E. Aspnes, and I. Kamiya, “Comparison of re-flectance difference spectroscopy and surface photoabsorptionused for the investigation of anisotropic surfaces,” Surf. Sci.287/288, 686–692 (1993).

20. B. Lecourt, D. Blaudez, and J. M. Turlet, “Specific approach ofgeneralized ellipsometry for the determination of weak in-plane anisotropy: application to Langmuir–Blodgett ultrathinfilms,” J. Opt. Soc. Am. A 15, 2769–2782 (1998).

21. S. Visnovsky, Optics in Magnetic Multilayers and Nanostruc-tures (Taylor & Francis, 2006).

22. M. Gilliot, A. En Naciri, L. Johann, J. P. Stoquert, J. J. Grob,and D. Muller, “Optical anisotropy of shaped oriented cobaltnanoparticles by generalized spectroscopic ellipsometry,”Phys. Rev. B 76, 045424 (2007).

23. P. Adamson, “Reflection characterization of anisotropic ultra-thin dielectric films on absorbing isotropic substrates,” Surf.Sci. 603, 3227–3233 (2009).

24. P. Adamson, “Optical diagnostics of anisotropic nanoscalefilms on transparent isotropic materials by integrating reflec-tivity and ellipsometry,” Appl. Opt. 48, 5906–5916 (2009).

25. H. Goldstein, Classical Mechanics (Addison-Wesley,1965).

26. D. W. Berreman, “Optics in stratified and anisotropicmedia: 4 × 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510 (1972).

27. R. M. A. Azzam and N. M. Bashara, Ellipsometry andPolarized Light (North-Holland, 1977).

28. P. J. Lin-Chung and S. Teitler, “4 × 4-matrix formalisms foroptics in stratified anisotropic media,” J. Opt. Soc. Am. A 1,703–705 (1984).

29. P. Yeh, Optical Waves in Layered Media (Wiley, 2005).30. P. Adamson, “Reflection of electromagnetic plane waves in a

long-wavelength approximation from a multilayer system ofanisotropic transparent films on absorbing medium,” WavesRandom Complex Media 18, 651–668 (2008).

31. P. Adamson, “Differential reflection photometry of ultrathindielectric layers on strongly absorbing materials,” Opt.Spectrosc. 86, 408–414 (1999).

32. L. Heinrich, E. K. Mann, J. C. Voegel, G. J. M. Koper, and P.Schaaf, “Scanning angle reflectometry study of the structureof antigen-antibody layers adsorbed on silica surfaces,” Lang-muir 12, 4857–4865 (1996).

33. R. C. van Duijvenbode, G. J. M. Koper, and M. R. Böhmer,“Adsorption of poly(propylene imine) dendrimers on glass.An interplay between surface and particle properties,”Langmuir 16, 7713–7719 (2000).

34. P. Adamson, “Reflection of electromagnetic plane waves in along-wavelength approximation from a multilayer system ofanisotropic transparent films on non-absorbing isotropicmedium,” Waves Random Complex Media 20, 443–471(2010).

35. P. Landry, J. Gray, M. K. O’Toole, and X. D. Zhu, “Incidence-angle dependence of optical reflectivity difference from anultrathin film on solid surface,” Opt. Lett. 31 531–533(2006).


inverse relationships for reflection diagnostics of uniaxially anisotropic nanoscale films on...

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