Schubert et al. Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 875

Extension of rotating-analyzer ellipsometryto generalized ellipsometry:

determination of the dielectric functiontensor from uniaxial TiO2

Mathias Schubert and Bernd Rheinlander

Faculty of Physics and Geoscience, Department of Semiconductor Physics,University of Leipzig, Linnestrasse 5, Leipzig 04103, Germany

John A. Woollam

Department of Electrical Engineering and Center for Microelectronicand Optical Materials Research, University of Nebraska, Lincoln, Nebraska 68588

Blaine Johs and Craig M. Herzinger

J. A. Woollam Company, 650 J Street, Suite 39, Lincoln, Nebraska 68508

Received June 29, 1995; revised manuscript received October 19, 1995; accepted October 24, 1995

For what is the first time, to our knowledge, we report on the extension of spectroscopic rotating-analyzerellipsometry to generalized ellipsometry to define and to determine three essentially normalized elementsof the optical Jones matrix J [R. M. A. Azzam and N. M. Bashara, J. Opt. Soc. Am. 62, 1521 (1972)]. Theseelements are measured in reflection over the spectral range of 3.5–4.5 eV on different surface orientations ofuniaxial TiO2 cut from the same bulk crystal. With a wavelength-by-wavelength regression and a 4 3 4 gen-eralized matrix algebra, both refractive and absorption indices for the ordinary and the extraordinary waves,no, ko, ne, and ke, are determined. The inclinations and the azimuths of the optic axes with respect to thesample normal and plane of incidence were determined as well. The latter are confirmed by x-ray diffractionand polarization microscopy. Hence the spectrally dependent dielectric function tensor in laboratory coor-dinates is obtained. Very good agreement between measured and calculated data for the normalized Joneselements for the respective sample orientations and positions are presented. This technique may become animportant tool for investigating layered systems with nonscalar dielectric susceptibilities. 1996 OpticalSociety of America

1. INTRODUCTION

The extension of standard ellipsometry to systems withnondiagonal Jones matrices has historically been termedgeneralized ellipsometry (GE).1 The Jones matrices oflinear nondepolarizing systems connect the two modes ofthe incident and the emerging electromagnetic planewaves with respect to a Cartesian laboratory coordi-nate system2 (Fig. 1). Depending on the emerging planewaves used for the optical investigation in layered media,the Jones matrix is equal to the reflection matrix r or thetransmission matrix t. Their elements provide the com-plex reflectance (transmittance) coefficients for incidentlight that is s or p polarized into reflected (transmitted)s- or p-polarized light. Standard ellipsometry is relatedto structures that reflect or transmit p- and s-polarizedlight into p- and s-polarized light, respectively. Herethe Jones matrices are definitely diagonal. If the Jonesmatrices have nonvanishing off-diagonal elements, whichis the most general case, new ellipsometric parametersmust be defined. Also, to let us acquire and analyze themore complex sample information, new techniques mustbe established and discussed. The basis of GE is simplyto define and to determine three normalized Jones ma-

0740-3232/96/040875-09$06.00

trix elements for a given sample setup. The standardarrangement for rotating-analyzer ellipsometry (RAE)can be extended to measure these complex reflectance ortransmittance ratios in terms of normalized Jones matrixelements. However, the signs of the phases of these ra-tios cannot be observed by this method. Here we reporton for what is the first time to our knowledge the exten-sion of RAE to GE by use of a conventional IR-visible-UVspectroscopic ellipsometer. This technique is applied tomeasure the anisotropic reflectivity from samples of dif-ferent surface orientations of uniaxial TiO2 cut from thesame bulk crystal. The 4 3 4 matrix algebra introducedby Berreman3 and extended by further developments fromseveral authors is then used for a regression analysis.4 – 6

This algebra allows for the calculation of the optical Jonesmatrices as a function of the optical properties of layeredmedia and of the sample setup such as the orientationof the crystal coordinate system, the angle of incidenceFa, and the angular frequency v of the incident planewave. We obtain the complete dielectric function tensorof each sample. Therefore nonscalar dielectric suscep-tibilities that may be parts of the optical properties oflayered systems are measurable by combining the resultsfrom GE with an anisotropic regression algorithm. We

1996 Optical Society of America

876 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 Schubert et al.

Fig. 1. Incident and emerging electromagnetic plane waves ina nondepolarizing optical system. The Jones matrix connectsboth plane waves with respect to a chosen Cartesian coordinatesystem, such as the p and s planes in Fig. 2.

believe that this technique opens a new and wide fieldfor ellipsometric investigations. In Section 2 we givea short review and definition of GE. The extension ofthe standard rotating analyzer (or polarizer) techniqueis then explained in Section 3. Here the basics of RAEare briefly mentioned, and some necessary formulas arederived again to make this extension more transparent.We also rule out existing experimental limits for the de-termination of the off-diagonal Jones matrix elements.The experimental setup and the results of the TiO2

samples are described in Section 4. There is also a briefannotation of the regression analysis used for the deter-mination of the spectrally and directionally dependentdielectric function.

2. GENERALIZED ELLIPSOMETRYThe plane of incidence is defined through the wave vectorsof the incident and the emerging wave (Fig. 2), in whichthe p (electric field vector E parallel to the plane of in-cidence) and the s (electric field vector E perpendicularto the plane of incidence) modes of the plane waves be-fore and after the optical system are related by the Jonesmatrix J:"

Bp

Bs

#­ J

"Ap

As

#­

"jpp jsp

jps jss

#"Ap

As

#, (1)

where Ap, As, Bp, and Bs denote the p and the s modesof the incident and the emerging waves, respectively2

(Fig. 1). Let us define the laboratory coordinate system,using the sample surface as the (x–y) plane, where thex axis is parallel to the plane of incidence, and the direc-tion of the z axis is parallel to the surface normal andtoward the half-infinite substrate material. The originis set at the sample surface. Standard ellipsometry isconventionally used to analyze surfaces that do not causeconversions from incident p (or s) into emerging s (or p)polarized light. The Cartesian components of the inci-dent and the emerging electric field components along twomutually perpendicular directions in the wave fronts aresimply related through a diagonal Jones matrix.

Azzam and Bashara1 have adopted the term general-ized ellipsometry (GE) to denote the application of el-lipsometry to nondepolarizing systems with nondiagonalJones matrices. They considered Eq. (1) as a bilineartransformation between the complex ratios of the incidentand the emerging plane-wave amplitudes.2 The linear,

nondepolarizing system can therefore be categorized byuse of the coefficients of this transformation. In fact,these coefficients are equal to Jones matrix elementsnormalized to a common factor.7 Depending on thestructure of the optical system and the orientations of theoptical axes with respect to the laboratory coordinate sys-tem, the off-diagonal elements of the Jones matrix J maybe symmetrical or antisymmetrical, Hermitean or anti-Hermitean, completely different or simply zero (see, e.g.,Ref. 4 and references therein).

We restrict ourselves to the case of reflection ellip-sometry for light reflected from layered systems. Thenthe Jones matrix is equal to the reflection matrix r. Notethat all formulas and considerations made for the reflec-tion arrangement in this paper are valid for the trans-mission case as well. For the transmission case one hassimply to replace the Jones reflection matrix elementswith their respective transmission matrix elements.These matrix elements provide the complex reflectance(transmittance) coefficients for incident light that is ei-ther s or p polarized into reflected (transmitted) s- orp-polarized light. We refer the reader to Ref. 8 for fur-ther definitions of these elements.

The complex reflectance ratio has been traditionallydefined as4

r ; tan C expsiDd ; zyx, x ­ ApyAs, z ­ BpyBs ,(2)

which depends for nonvanishing off-diagonal reflectioncoefficients on the ratio of the incident wave amplitudesx through the elements of the Jones reflection matrix,5

r ­z

x­

rpp 1 rspx21

rss 1 rpsx, (3)

or in a slightly different form

r ­srppyrssd 1 srspyrssdx21

1 1 srppyrssdsrpsyrppdx. (4)

Fig. 2. Definition of the plane of incidence ( p plane) and theincidence angle Fa through the wave vectors of the incident andthe emerging (reflected here) plane waves. Ap, As, Bp, andBs denote the complex amplitudes of the p and the s modes(parallel to the p and the s plane) before and after the sample,respectively. P and A are the azimuth angles of the linearpolarizers used in the standard arrangement of RAE. P or Ais equal to zero if its selected direction is parallel to the p plane.P or A move counterclockwise with respect to the light propaga-tion; see also comments in Section 3.

Schubert et al. Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 877

As is seen in Eq. (4), the complex reflectance ratio r

is then a combination of three ratios formed by the ele-ments of the Jones reflection matrix. The basis of GE isto define and to determine three linear independent nor-malized reflection matrix elements for one combinationof the sample azimuth f, the angle of incidence Fa, andthe angular frequency v. (The sample azimuth f is anarbitrarily defined angle between the x axis of the labora-tory system and a direction parallel to the sample surface.This arbitrary direction may be, for convenience, a crys-tallographic orientation parallel to the x–y plane.) Wedefine a set of normalized reflection matrix elements fordetermination by GE. For convenience we suggest thefollowing choice, which deviates slightly but without lossof generality from those introduced elsewhere1:

rpp

rss; Rpp ­ tan Cpp expsiDppd ,

rps

rpp; Rps ­ tan Cps expsiDpsd ,

rsp

rss; Rsp ­ tan Csp expsiDspd . (5)

This results in Eq. (4):

r ­Rpp 1 Rspx21

1 1 RppRpsx. (6)

Note that we used diagonal elements to normalize thereflection coefficients to themselves. This definition isnot an end in itself and may be changed by other users.However, there exist three basic independent normalizedcoefficients, and all other definitions can be understoodto be a linear combination in terms of this basic set.In our notation the complex ratios Rps and Rsp behavesymmetrically if the polarization transfer function of thesample for s and p polarized light is unique (e.g., anoptically active medium in transmission arrangement ora uniaxial medium with its optic axis perpendicular tothe sample normal and tilted to the x axis by 45±). Theratio Rpp is similar to the complex ratio determined bystandard ellipsometry. When Rps and Rsp vanish, Rpp

gives exactly the standard ellipsometric parameters C

and D as defined in Eq. (2).

3. EXTENSION OF ROTATING-ANALYZER ELLIPSOMETRY TOGENERALIZED ELLIPSOMETRYThe fundamentals of RAE have often been described anddiscussed. Nevertheless to give a short review of thebasic formulas so that the reader can follow the exten-sion to GE made by the authors. For a more detailedintroduction to this technique we recommend Ref. 5 andreferences therein.

A completely nonpolarized monochromatic light beambecomes linearly polarized on passing through a linear po-larizer P (Fig. 2). Because of the reflection at the samplesurface the light beam generally changes its polarizationstate. A second linear polarizer acting as an analyzeris then used to determine this polarization state. Theintensity of the light beam after passing through the ana-

lyzer can be expressed as a function of the sample proper-ties and the polarizer and analyzer azimuth angles P andA. (P or A is equal to zero if the chosen direction of thelinear polarizer is parallel to the plane of incidence; P or Amoves counterclockwise, locking onto the incident or thereflected beam, respectively.) If Ei and Edet denote theelectric field components of the polarizer incident beamand the detected beam, the polarization state transfer,beginning with the nonpolarized light Ei and ending withthe field components at the detector Edet, can be describedas a successive matrix multiplication:

Edet ­ RsAd ? P ? Rs2Ad ? r ? RsP d ? P ? Rs2P d ? Ei. (7)

Here P is the projection matrix of a linear polarizer andRsAd and RsP d are the rotation matrices with respect tothe laboratory coordinate system:

P ­

"1 00 0

#, Rsad ;

"cos a 2 sin a

sin a cos a

#. (8)

The matrix r is the Jones reflection matrix and charac-terizes the optical properties of a given sample:

r ­

"rpp rsp

rps rss

#. (9)

The analyzer rotates with a constant angular frequencyA ­ Vt. Hence Eq. (7) becomes√

Ep

Es

!det

­ Epi

√R1 cos Vt 1 R2 sin Vt

0

!, (10)

where we have for brevity

Edet ; Rs2Ad ? Edet, Ei ; Rs2P d ? Ei,

R1 ­ rpp cos P 1 rsp sin P ,

R2 ­ rps cos P 1 rss sin P . (11)

If the time-dependent intensity jEdetj2 is the subject ofa Fourier analysis with respect to the analyzer angularfrequency V, the only nonvanishing Fourier coefficientsare defined as a and b as follows:

I sVtd ­ I0h1 1 a cos 2Vt 1 b sin 2Vtj , (12)

a ;jR1j2 2 jR2j2

jR1j2 1 jR2j2, b ;

sR1R2 1 R2R1djR1j2 1 jR2j2

, (13)

where the overbar denotes the complex conjugate. Thestandard arrangement for RAE is usually applied for mea-suring systems with diagonal Jones matrices. In thiscase Eqs. (7)–(13) simplify and Eqs. (13) can be used im-mediately to calculate the complex reflectance ratio r asdefined in Eq. (2) in terms of the ellipsometric parame-ters C and D:

tan C ­

√1 1 a

1 2 a

!1/2

tan P , cos D ­b

s1 2 a2d1/2. (14)

Note also that in this case r is independent of the polar-izer azimuth P because of the isotropic behavior of thesample. Note further that this technique does not allow

878 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 Schubert et al.

Fig. 3. Differences between simulated Fourier coefficientsasP d and bsP d, where 2py2 , P , py2, with and without Rpsand Rsp as examples. safRps, Rsp fi 0g 2 afRps, Rsp ­ 0g:short-dashed curve; bfRps, Rsp fi 0g 2 bfRps , Rsp ­ 0g:long-dashed curve); RpphC ­ 19.14±; jDj ­ 50.13±j,RpshC ­ 1.33±; jDj ­ 125.4±j, RsphC ­ 0.82±; jDj ­ 153.4±j.

us to determine the handedness of the polarization stateof the light beam after the sample. The phase of theemerging plane wave D appears only as the argument ofthe even cosine function.

In general, and as mentioned above, the Jones matrixof a sample may be nondiagonal. Then the Fourier coef-ficients a and b become more complex [see Eqs. (5) for thedefinitions of the normalized reflection matrix elements]:

a ­jRpp 1 Rsp tan P j2 2 jRppRps 1 tan P j2

jRpp 1 Rsp tan P j2 1 jRppRps 1 tan P j2,

b ­2 RefsRpp 1 Rsp tan P d p sRppRsp 1 tan P dg

jRpp 1 Rsp tan P j2 1 jRppRps 1 tan P j2, (15)

where Re[ ] denotes the real part of a complex number.Our extension of RAE to GE consists of the measure-ment of the Fourier coefficients at many polarizer azimuthangles P and of the determination of the three unknowncomplex parameters Rpp, Rsp, and Rps with a regressionanalysis performed by use of Eqs. (15). Similar to thefact that RAE is generally unable to detect the sign ofthe phase of the emerging plane wave, the signs of allphases of the normalized reflection coefficients defined inEq. (5) cannot be determined. In fact, Rpp, Rsp, and Rps

are determined up to their complex conjugates. This factcan be seen by evaluating a and b by use of an arbitraryset of Rpp, Rsp, and Rps and, once again, their complexconjugates. In both cases a and b will be the same.

In principle only three azimuth settings P are neededto determine Rpp, Rsp, and Rps from a given sample.However, we have overdetermined the number of mea-

surements in relation to the number of parameters to bedetermined. When the p–s mode conversion of thesample is small compared with the mode-preserving re-flectivities rpp and rss, the changes in the Fourier coeffi-cients caused by nondiagonal normalized elements mayvanish below the noise in asP d and bsP d. Here the num-ber of azimuthal angle settings must be increased. Tomake such a critical situation more transparent, we showa simulation of the differences in asP d and bsP d with andwithout an assumed p –s mode conversion of a sample.These values correspond to the typical lower limits for thedetection of Rsp and Rps during our experiments, as canbe seen in Section 4. Figure 3 presents the differences a,bsP ; Rpp, Rps, Rspd 2 a, bsP ; Rpp, 0, 0d, where 2py2 ,

P , py2. We used Rpp ­ tans19.14±dexpsij50.13±jd,Rps ­ tans1.33±dexpsij125.4±jd, and Rsp ­ tans0.82±dexpsij153.4±jd. The short-dashed line shows the differ-ences between the Fourier coefficient a, whereas thelong-dashed line represents the same for b. These dif-ferences are very small. Also, the Fourier coefficientsexhibit sensitivity to values Rps and Rsp at different po-larizer azimuths. Hence many polarizer settings insidethe entire azimuth span 2py2 , P , py2 are necessarytoo yield a definite functional relation among asP d, bsP d,and Rpp and especially between Rps and Rsp. It is ourexperience that this technique is no longer sensitive top –s mode conversion coefficients rps and rsp that are lessthan 1% of the mode-preserving coefficients rpp and rss.

Finally, we note that the mathematical developmentin this section for an RAE arrangement can be easilytransferred to the rotating polarizer ellipsometry (RPE)configuration: Changes must be made through Eq. (11)by swapping A for P when the polarizer azimuth angle Protates as Vt. The azimuth setting of the fixed analyzerA appears instead of P . All the subsequent formulas anddiscussions are then still valid for RPE.

4. APPLICATION TO UNIAXIAL TiO2

Three different samples of uniaxial TiO2 (rutile) were cutfrom a bulk crystal with nominal [100] (sample 380), [110](sample 375), and [111] (sample 588) surface orientations.The lattice constants for all samples were determined tobe a ­ b ­ 4.492 A and c ­ 2.893 A by x-ray investiga-tions. The crystal orientations were confirmed by Lauebackscattering and polarization microscopy. The inclina-tion angles of the optic axes were determined to be par-allel to the sample surface (Q ­ 90± 6 2±, 375 and 380)or bent from the sample normal by Q ­ 42± 6 3± (588).

Table 1. Measurement Setup and Results for the Euler Angles

Sample

380 375 588Surface Orientation [100] TiO2 [110] TiO2 [111] TiO2

Fa 70± 70± 60±, 70±

Dfi 0±, 15±, 90±, 105± 0±, 90±, 180±, 270± 0±, 90±, 180±, 270±

Euler anglesf, GE x-ray diffraction 35± 6 3± 15± 6 3± 50± 6 3±

33.1± 6 0.1± 14.5± 6 0.1± 52.2± 6 0.1±

Q, GE x-ray diffraction 90± 6 2± 92± 6 2± 42± 6 3±

90±a 92±a 40.5± 6 0.5±

aQ fixed during the final regression. Here the optic axes were assumed to lie parallel to the sample surfaces.

Schubert et al. Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 879

Fig. 4. Simulated three-dimensional plot of Rpp versus wave-length and orientation angle f for the extracted ordinary andextraordinary optical constants of TiO2 from sample 380 ([100]surface, Fa ­ 60±). The anisotropic reflectivity expressed interms of normalized Jones matrix elements is equal for 180±-opposite sample orientations f, as is seen from Figs. 4–6. Notethat Rps and Rsp vanish when the optic axis is oriented parallelto the x or the y axis of the laboratory coordinate system.

We define the sample orientation f by choosing the anglebetween the x axis of our laboratory coordinate systemand the projection of the optic axis through to the sam-ple surface. The Euler angles f, c, and u are associatedwith the respective crystal coordinate system and providethe orthogonal rotation matrices that diagonalize the di-electric function tensors e, which is given in laboratorycoordinates.6 Here the Euler angles f and Q are the azi-muth and the inclination angles of the optic axis with re-spect to the x and the z axes, respectively, whereas c isequal to zero. Note that the Euler angle f is the sameas the sample azimuth defined in this work. All sampleswere mounted on a rotation stage in order to rotate thesamples around the z axis by discrete angles Df. Oncethe samples were adjusted, the optic axis azimuth anglesf were defined but still unknown with respect to thelaboratory coordinate system. The measurements weretaken in the spectral range 3.5–4.5 eV at multiple anglesof incidence Fa,i and multiple sample orientations fi ­f 1 Dfi. The Fourier coefficients a and b were deter-mined at one set of wavelength, sample azimuth fi, andincidence angle Fa,i for many polarizer settings. A re-gression algorithm was then used to fit for Rpp, Rps, andRsp as described in Section 3. We obtained these ratiosby minimizing the mean-square error function and usingEqs. (15) and the experimentally determined standard de-

Fig. 5. Same as Fig. 4 for Rps .

Fig. 6. Same as Fig. 4 for Rsp .

880 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 Schubert et al.

viations da and db:

MSE ­1N

NXi­1

("ai

m 2 aicalcsRpp, Rps, Rsp; P d

daim

#2

1

"bi

m 2 bicalcsRpp, Rps, Rsp; P d

dbim

#2)

. (16)

A recently discussed 4 3 4 matrix algebra6 was then usedto fit for the optical constants of the TiO2 substrate ma-terial and the Euler angles of their respective crystal co-ordinate system, again minimizing a mean-square errorfunction:

MSE ­1N

NXi­1

"√Rpp 2 Rpp

calc

dRpp

!2

i

1

√Rps 2 Rps

calc

dRps

!2

i

1

√Rsp 2 Rsp

calc

dRsp

!2

i

#. (17)

Fig. 7. Experimental and generated data obtained from best fit for No, Ne, optic axis inclination u, and azimuth f by use of experi-mental data from many sample azimuth angles and one incidence angle (see Table 1). The values represented by the symbols indicatethe azimuths f; [100] TiO2 sample, Fa ­ 70±. Here the optic axes are parallel to the sample surface and hence the inclinations u

were fixed at 0± during the regression.

This matrix algebra allows for the calculation of the Jonesmatrix elements as a function of the dielectric propertiesof layered media. In particular, for the analysis of theanisotropic TiO2 substrates two matrices need to be calcu-lated and multiplied to yield the so-called general transfermatrix T . The Jones reflection and transmission coeffi-cients are then derivable from T . The first matrix con-tains the incidence conditions and the optical propertiesof the surrounding medium, e.g., air. The second ma-trix contains the optical eigenmodes of the half-infiniteanisotropic substrate material. After these eigenmodesare determined from an eigenproblem algorithm, both ma-trices can be multiplied. The calculated Rpp, Rps, andRsp ratios can then be compared with the measured ratios.The angle differences Dfi were used as known values forthe regression just as were the angles of incidence Fa,i.In addition to the optical constants Ne ; ne 1 ike, andNo ; no 1 iko (refractive index 1 i 3 absorption index)

Schubert et al. Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 881

Fig. 8. Same as Fig. 7 for [110] TiO2 sample, Fa ­ 70±. The differences Df between adjacent azimuths fi for sample 375 ([110]TiO2) were intentionally 90±. Hence the results in each Rpp are twofold degenerate. The Rsp values demonstrate the experimentallimits for the determination of vanishing off-diagonal elements as discussed in Section 4.

at every wavelength, the initial sample position of the op-tic axis azimuth angle f and the inclination u (only forsample 588) were found from the fit combining the Rpp,Rsp, and Rps data acquired at different sample azimuthsfi and incidence angles Fa,i. Owing to the use of mul-tiple sample orientations, the number of parameters tobe determined at each wavelength is much less than thenumber of data points acquired at each wavelength. Inaddition, the multiple sample orientations also provided adirect observation of the variable anisotropic reflectivityof the surface caused by the different inclinations of the di-electric function tensor axes with respect to the laboratorycoordinate system. Table 1 shows the measurement set-ups for all three samples. For sample 588 ([111] surface)we used data acquired with multiple incidence angles Fa,i

to extract both Euler angles for the optic axis.To make the anisotropic reflectivity of a uniaxial sub-

strate (e.g., optic axis parallel to the sample surface) obvi-ous, in Figs. 4–6 we show simulations of the normalized

Jones matrix elements defined in this work. We used ourextracted ordinary and extraordinary optical constants ofTiO2 inside the spectral range from 3.5 to 4.5 eV. Thethree-dimensional plots of tansCppd, cossDppd; Cps, Dps;and Csp, Dsp versus wavelength and orientation angle f

show significant sensitivity to the sample orientationangle f. Hence, with combined data gained at differ-ent sample azimuth angles f, the orientation of the crys-tal axes can be determined precisely. In particular theknowledge of the phases Dps and Dsp causes Rps and Rsp

to appear twofold degenerate instead of fourfold degener-ate as they would appear from the knowledge of Cps andCsp alone. These three-dimensional surfaces may act asmaps wherein the reader can find our original data andverify conclusions during the following discussion.

Figures 7 and 8 show the measured and the fitted datafrom samples 380 and 375. Two angles of incidence wereincluded in the regression analysis for sample 588 ([111]TiO2, Fa1 ­ 60±, Fa2 ­ 70±) to yield enough information

882 J. Opt. Soc. Am. A/Vol. 13, No. 4 /April 1996 Schubert et al.

for both Euler angles of the optic axis position. The mea-sured Jones matrix elements were qualitatively equal tothose of samples 375 and 380. With the best fit for No,Ne, u, and f, very good agreement between the measuredand the calculated normalized Jones reflection matrixelements was achieved (see, e.g., Fig. 7).

The azimuth angle step Dfi for sample 375 was inten-tionally exactly 90± (Fig. 8). Hence the coefficients Rpp

and Rps, as well as Rsp, obtained at 180±-opposite sam-ple positions, should be twofold degenerate as seen inFigs. 5 and 6. We measured at 180± opposite azimuthsthe same values for Cpp and Dpp. Therefore only twographs are visible, and exact agreement is seen in thelower diagrams for f ­ 14±, 194± and f ­ 104±, 284±, re-spectively (Fig. 8). However, influences from the samplesurface obviously cause deviations from the theoretical(fitted) values in the results for Rsp. Depending on thesurface quadrant in which the measurements were taken,surface depolarizations decrease or increase Csp and Cps

as well. The Csp values gained at azimuths f ­ 104±,284± (filled triangles, filled squares) lie above the best fit,whereas the Csp values gained at f ­ 14±, 194± (opentriangles, open squares) are below the theoretical val-ues (solid lines). A small p–s mode conversion of thesample caused by a nonideal surface increases or de-creases rps and rsp. A similar behavior is well knownfrom applications of GE to the determination of the prefer-ence direction of diffraction gratings.1 Furthermore, andas mentioned above, if the sample p–s mode conversionis small, the phases and the absolute values of the re-flection coefficients Rsp and Rps become nondetectable forour technique: below 4.0 eV (upper-left diagram, Fig. 8),the absolute value of Rsp, given by Csp, is less then 1±

for sample azimuths of f ­ 14±, 194± (open triangles,open squares) indicating a p–s mode conversion of lessthen 1% of the mode preserving coefficients rpp and rss.Here the determination of the corresponding phase Dsp

is extremely difficult (upper-right diagram, Fig. 8). Themeasured phases have a large uncertainty and deviatestrongly from the best fit (solid lines, upper half of theupper-right diagram). This behavior is due to the un-favorable sample azimuths. During the measurementson sample 375 the optic axis was unintentionally close tothe x or y axis, and therefore the off-diagonal elementsrsp and rps were very small compared with their diago-nal elements rpp and rss. (That Cps and Csp vanish ateach 90±-type position of the optic axis can be seen fromour surface plot of Rps and Rsp in Figs. 5 and 6.) Weconclude from these results that Rsp was somewhat de-tectable when the intrinsic p–s mode conversion of thesample had the same orientation as the nonideal-surface-induced anisotropic reflectivity. We have included thatresult to demonstrate our current experimental limits.

The best way to get the maximum information aboutthe extraordinary and the ordinary optical constants aswell as their three-dimensional orientations is to mea-sure at sample azimuths of f ­ 45±, 135±, 225±, and 315±.Here the p–s mode conversions (rsp and rps) are at theirmaxima. Note again that measuring the sample at thesefour positions (optic axis perpendicular to the sample nor-mal) shows that Dps and Dsp are each twofold degenerate(Cps, Dps, Fig. 5; Csp, Dsp, Fig. 6). Note in addition thatthe p–s mode conversion vanishes if the optic axis be-

comes parallel to the sample normal. We were not ableto differentiate between ordinary and extraordinary opti-cal constants using data from a [001] TiO2 sample (opticaxis perpendicular to the sample surface). The Eulerangles shown in Table 1 are determined in the same wayas the optical properties from the same wavelength-bywavelength regression. Therefore, a spectral depen-dence of the optic axis orientation would also be ob-servable by this method. We obtained straight lines forthe Euler angles Q and f as functions of the wavelength.This is in fact the expected behavior of the uniaxialTiO2 in the tetragonal rutile configuration. The result-ing Euler angles shown in Table 1 are averaged over allwavelengths.

Figure 9 shows the extracted ordinary and extraor-

Fig. 9. Ordinary and extraordinary optical constants from TiO2extracted for our three samples with different surface orienta-tions. Note that no overlayer correction was performed duringthe model fit. The relative error for each value is less than 1%(380, 588) or 3% (375).

Schubert et al. Vol. 13, No. 4/April 1996 /J. Opt. Soc. Am. A 883

Table 2. Ordinary and Extraordinary OpticalConstants at Selected Spectral Positions in

Comparison with Results Reviewed in Ref. 9

v (eV) ne no ke ko

4.31 3.56a 2.83a 3.55a 1.82a

2.76b 2.44b 2.55b 1.78b

4.17 3.00c 1.68c

3.16b 2.64b 2.52b 1.91b

4.05 3.84c 1.95c

3.59b 3.02b 2.33b 1.96b

3.90 5.38a 4.00a 2.18a 1.79a

3.96b 3.54b 1.89b 1.64b

3.79 3.87c 0.81c

4.07b 3.69b 1.55b 1.21b

3.70 4.22c 0.79c

4.10b 3.65b 1.25b 0.89b

aRef. 10.bPresent work, obtained from sample 380.cRef. 11.

dinary optical constants from all three samples. Therelative errors at every wavelength, estimated with theexperimental uncertainties and a 90% confidence limitduring the regression analysis for every value, are notgreater than 1% (380, 588) and 3% (375). The extractedoptical constants should be essentially the same if allsamples contain the same material but with different ori-entations. For TiO2 we obtained fairly good agreementamong the optical constants determined from our threesamples. Note that we have not made any overlayerreduction to account for any thin surface films. It iswell known from ellipsometric investigations of semicon-ductor materials that these ultrathin films renormalizethe extracted optical properties of a substrate material.4

In Table 2 we have summarized the optical constantsfrom TiO2 reviewed in Ref. 9 together with our results atparticular wavelengths. Unfortunately, the data fromdifferent authors reported in Ref. 9 are significantly dif-ferent from each other inside the spectral region usedin our work. The results reported there were obtainedfrom power reflectivity measurements in combinationwith a Kramers–Kronig analysis. This includes the un-certainty that is due to the truncation of the reflectivityto a finite frequency range. Our values are somewhatin the midrange of those reported in Ref. 9. Becauseour optical constants are obtained from a phase sensitivemeasurement technique we believe that our results pre-sented here for TiO2 are appropriate (i) for demonstratingthe validity for the technique introduced here for investi-gations of anisotropic materials and (ii) for giving reliabledata for the optical constants of TiO2.

5. SUMMARYA new technique to investigate anisotropic layered me-dia is introduced. We report what is, to our knowledge,the first extension of rotating analyzer ellipsometry togeneralized ellipsometry. The basis of generalized ellip-sometry is to define and to determine three normalizedJones matrix elements. A set of three independent ratios

of Jones matrix elements are defined in this work. Thistechnique is applied to samples of uniaxial TiO2 with arutile structure and different surface orientations. Theanisotropic reflectivity could also be obtained at differentangles of incidence, surface orientations, and azimuths.With a 4 3 4 matrix algebra6 a regression analysis is per-formed to extract the complete dielectric function tensor ofeach sample inside the energy range measured. We ob-tained the crystal angles and the optical properties fromthe measurement at samples with different surface ori-entations. This technique was performed on a commer-cially standard IR-visible-UV spectroscopic ellipsometerand may become an important tool for investigations ofoptically anisotropic materials.

ACKNOWLEDGMENTSThe authors thank Dan Thompson, University of Ne-braska at Lincoln, for helpful comments and techni-cal support of the work. The cooperation of VolkerGottschalch, University Leipzig, Germany, in the polar-ization microscopic aspects of this work is very much ap-preciated. We thank also Roger French at DuPont forhis kindness in providing samples and performing x-rayinvestigations.

REFERENCES AND NOTES1. R. M. A. Azzam and N. M. Bashara, “Generalized ellipsome-

try for surfaces with directional preference: application todiffraction gratings,” J. Opt. Soc. Am. 62, 1521–1523 (1972).

2. See, e.g.: R. M. A. Azzam and N. M. Bashara, “Polarizationtransfer function of a biaxial system as a bilinear transfor-mation,” J. Opt. Soc. Am. 62, 222–229 (1972).

3. D. W. Berreman, “Optics in stratified and anisotropic media:4 3 4-matrix formulation,” J. Opt. Soc. Am. 62, 502–510(1972).

4. Depending on what causes the optical anisotropy; see, e.g.,H. Wohler, M. Fritsch, G. Haas, and D. A. Mlynski, “Charac-teristic matrix method for stratified anisotropic media: op-tical properties of special configurations,” J. Opt. Soc. Am. A8, 536–540 (1991); K. Eidner, “Light propagation in strati-fied anisotropic media: orthogonality and symmetry prop-erties of the 4 3 4 matrix formalism,” J. Opt. Soc. Am. A 6,1657–1660 (1989); Ref. 5.

5. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polar-ized Light (North-Holland, Amsterdam, 1977).

6. M. Schubert, “Polarization dependent parameters of arbitrar-ily anisotropic homogeneous epitaxial systems,” Phys. Rev. B53(7) (1996).

7. P. S. Hauge, “Generalized rotating-compensator ellipsome-try,” Surf. Sci. 56, 148–160 (1976); see also D. J. De Smet,“Generalized ellipsometry and the 4 3 4 matrix formalism,”Surf. Sci. 56, 293–306 (1976); M. Elshazly-Zaghloul, R. M. A.Azzam, and N. M. Bashara, “Explicit solutions for the opticalproperties of a uniaxial crystal in generalized ellipsometry,”Surf. Sci. 56, 281–292 (1976).

8. P. Yeh, “Optics of anisotropic layered media: a new 4 3 4matrix algebra,” Surf. Sci. 96, 41–53 (1980).

9. E. D. Palik, Handbook of Optical Constants of Solids (Aca-demic, New York, 1985), pp. 798–804.

10. K. Vos and H. J. Krusemayer, “Reflectance and electro-reflectance of TiO2 single crystals: I. Optical spectra,”Phys. C 10, 3893–3915 (1977).

11. M. Cardona and G. Harbeke, “Optical properties and bandstructure of wurtzite-type crystals and rutile,” Phys. Rev. A137, 1467–1476 (1965).