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Modeling of scattering and depolarizingelectro-optic devices. I. Characterizationof lanthanum-modified lead zirconate titanatePaul E. Shames, Pang Chen Sun, and Yeshaiahu FainmanWe describe a new method of modeling electro-optic EO devices, such as lanthanum-modified leadzirconate titanate polarization modulators, that resolves two deficiencies of current methods: i theinclusion of depolarization effects resulting from scattering and ii saturation of the EO response atstrong electric-field strengths. Our approach to modeling depolarization is based on describing thetransmitted optical field by superposition of a deterministic polarized wave and a scattered, randomlypolarized, stochastic wave. Corresponding Jones matrices are used to derive a Mueller matrix todescribe the wave propagation in scattering and depolarizing EO media accurately. A few simple opticalmeasurements can be used to find the nonlinear behavior of the EO phase function, which is shown todescribe accurately the material’s EO behavior for weak and strong applied electric fields. © 1998Optical Society of AmericaOCIS codes: 160.2100, 190.5890, 080.2730, 190.3270, 260.2130.1. IntroductionPolycrystalline lanthanum-modified lead zirconatetitanate PLZT is an electro-optic EO material thathas many advantages over single-crystal EO materials,including ease of manufacturing, low operatingvoltages, and reduced long-range strain effects. 1PLZT-based devices with compositions of 8.8–9.56535 are used in a variety of optoelectronic applicationsincluding high-speed scanning, 2 dynamiclenses, 3 optical switches, 4 and shutters and eyeprotectiondevices. 5Bulk PLZT can be highly scattering, which hasconstrained the geometry of many devices to the useof thin wafers with interdigital surface-electrode designs.These design geometries can require the applicationof very strong electric fields. For example,consider a programmable phased-array device thatuses PLZT 9.06535 with surface electrodes separatedby 40 m. To achieve an average phase shiftof for light passing through such a deviceThe authors are with the Department of Electrical and ComputerEngineering, University of California, San Diego, 9500 GilmanDrive, La Jolla, California 92093-0407.Received 13 October 1997; revised manuscript received 2 March1998.0003-693598173717-09$15.000© 1998 Optical Society of Americarequires applying field strengths greater than 1.5 10 6 Vm throughout the gap region. At these fieldstrengths, we have found that even thin wafers ofPLZT become highly scattering. 6 This scatteringcan significantly reduce the optical signal strength,create off-axis noise, and depolarize the transmittedlight. 7 Therefore design geometries that reduce theoperating electric-field strength and the associatedscattering and depolarizing effects are critical forPLZT-based devices.Computer-aided design and modeling, or CAD–CAM, can be a very useful tool in the development ofEO device geometries for improved performance.Traditionally, 8–13 these EO devices are modeled byuse of a simple Jones calculus 14 model that cannotaccount for the effects of depolarization. Additionally,PLZT 8.8–9.56535 device phase modulation isusually modeled by a classical Kerr quadratic EOeffect 1,3 or a combination of linear and quadratic effects5,15,16 i.e., second-order effects. These modelsfail to describe accurately both the weak and thestrong electric-field-induced EO response.To achieve accurate computer simulation, particularlyfor devices that experience strong electric fields,it is necessary to take into account scattering and theresultant depolarization effects, and a higher thansecond-order EO phase function must be used. Toderive a model that can account for depolarization,10 June 1998 Vol. 37, No. 17 APPLIED OPTICS 3717

we describe the optical field transmitted through ascattering EO medium by superposition of a deterministicpolarized wave and a scattered, randomlypolarized, stochastic wave. The vector properties ofsuch optical fields are then described by the correspondingsuperposition of a deterministic and a stochasticJones matrix. The resulting Jones matricesare used to derive a Mueller matrix to describe accuratelythe wave propagation in a scattering EO medium.A few simple optical measurements can beused to characterize the material for the Mueller matrixcoefficients. In addition, using the same set ofmeasurements we solve for an EO function that accuratelydescribes the change in relative phase betweentwo orthogonal polarization componentsparallel and perpendicular to an applied electric field.Applying this method to PLZT 8.8–9.56535, wefind that, in contrast to the traditional quadratic EOmodel, the derived EO phase response is best fittedwith a fifth-order polynomial function. This higherorderphase function accounts for the experimentallyobserved EO response that is due to weak as well asstrong applied electric fields.In this manuscript part 1 of two papers we deriveour simplified Mueller matrix model and discuss itsapplication to characterizing PLZT; the application ofthis technique to device modeling is reported in thesecond paper, in this issue see pp. 3726–3734. Weshow how a few simple optical transmission measurementsaccount fully for the electric-field-induced responseof the material while allowing for the solutionof an accurate EO function. In Section 2 we beginour discussion with a review of the various electricfield-inducedeffects that are included in our model.In Section 3 we use Stokes vectors and Mueller matricesto describe optical beam transmission throughdepolarizing systems. We then present in Section 4our experimental characterization of a sample ofPLZT material and compare the results to our newmodel. Finally, in Section 5 we give a summary andconclusions.Fig. 1. a PLZT sample with transverse electrodes oriented togenerate an electric field along the y axis. A crossed polarizer–analyzer pair is oriented at 45° and 45°, respectively, to the yaxis. Linear polarized light is rotated 90° if the induced birefringenceresults in a phase difference of . b Experimentalmeasurement results filled circles of the transmission intensity oflight passing through the system as a function of the electric-fieldstrength for a PLZT sample of L 388 m. The solid curveshows the simulated performance by use of Eq. 1 and a best-fitquadratic EO model. To account for the weak EO response at lowelectric fields, a constant field of E 0 2 10 5 Vm, opposite to theapplied field direction, is included in the model.2. Electric-Field-Induced EffectsPLZT is a ferroelectric ceramic material composed ofcrystallites, varying 1–10 m in grain size, 17 whosematerial phase and electrical characteristics aresensitive to changes in material composition, temperature,and applied electric field. In the absence of anapplied electric field the PLZT is optically isotropic,whereas an applied electric field induces refractiveindexanisotropy. This leads to an optical phase difference for two orthogonally polarized lightcomponents passing through the material e.g.,parallel to and perpendicular to the direction of theapplied field. Figure 1a shows a typical experimentalsetup for the measurement of transmittedlight intensity through a PLZT sample by use ofcrossed polarizers set at 45° relative to the appliedfielddirection. Traditionally, the relative phasechange i.e., the EO response of the material is assumedto be quadratic, that is, E 2 , where E is themagnitude of the applied electric field. The transmissionintensity I of light through such a system isusually modeled by use of Jones calculus by the relationI 1 2 sin2 2 , (1)where the intensity reaches maxima and minima seeFig. 1b when n where n is an integer.However, a typical experimental result of intensitymeasurements as shown in Fig. 1b illustrates how areal device’s EO response departs from this simpleanalytic model. At low field strength there is almostno EO response, and the quadratic phase modelneeds to be shifted i.e., E E 0 2 to take thisweak response into account. At high field strengthwe can see that the EO effect diminishes i.e., theperiod increases compared to the quadratic approx-3718 APPLIED OPTICS Vol. 37, No. 17 10 June 1998

imation. We can conclude that the quadratic EOmodel is accurate for only a limited range of the appliedelectric-field strength. The technique to generatea more accurate EO phase function, which canaccount for the weak and the strong electric-field responseof this material, is discussed in Section 4.In Fig. 1b we can also see that the transmittedoptical intensity, as well as the contrast ratio, decreaseas functions of the applied electric-fieldstrength. The decrease in the measured intensity isdue to scattering, which in PLZT has been found to bea function of material composition, 1 grain size, 17 andstrength of the applied electric field. 7 The fieldinducedscattering can be related to index changes atgrain and domain boundaries and with small strainrelievingdomain formations. 18 For the majority ofspatial light modulator devices, scattering introducesunwanted signal attenuation. Furthermore, thescattered light is depolarized, 7 and the portion thatremains within the solid angle of the system’s exitpupil creates noise in the detected signal. As theapplied electric field is increased, the intensity of thedetected depolarized light also increases, which canbe seen as increases in the minima intensities for thetransmission curve shown in Fig. 1b. At high fieldstrength the light is completely unpolarized, and wecan no longer detect any change in relative phase.3. Mueller Matrix Model for PLZT-Based DevicesJones 1 2 vectors and 2 2 matrices are convenienttools for describing polarized light propagatingthrough nondepolarizing optical devices and systems.However, to model depolarizing optical devices suchas PLZT-based modulators, one must use a higherordermatrix representation. 19 In our case we useStokes vectors 20 and Mueller matrices 21,22 1 4 and4 4, respectively. Consider a normally incidentquasi-monochromatic optical beam with a mean frequency. Using a right-handed Cartesian coordinatesystem, as shown in Fig. 1a, with the directionof propagation along the z axis, we obtain the twoorthogonal complex electric-field components byE x t E 1 texpi 1 t 2t,E y t E 2 texpi 2 t 2t, (2)where the absolute phase 1 t, 2 t and amplitudesE 1 t, E 2 t of each component are real functionsof time. The Stokes parameters, which areused to describe the polarization state of the light, aregiven in vector form by0SS S 13E 2 1 E 2 2 E 2 1 E 2 2 (3)S 2 2E 1 E 2 cos S 2E 1 E 2 sin t,where t 1 t 2 t is the relative phase differencebetween the two orthogonal components.Transformation of the Stokes vector by a nonimaging21 device, such as a polarizer or a phase retarder,is described by a 4 4 Mueller matrix M. An opticalsystem comprising a cascade of N such devices isrepresented by the product of their respective Muellermatrices:M system M N M N1 ...M n ...M 1 , (4)where M system is the Mueller matrix representation ofthe entire optical system and M n is the matrix of thenth device of the system. The transformation of theStokes vector by an optical system is given byS out M system S in . (5)Given a known Jones matrix J see Appendix A,Subsection 1, describing a nondepolarizing opticalelement, the corresponding Mueller matrix is given 21by the transformationM TJ J*T 1 , (6)where the symbol V denotes the Kronecker product 23see Appendix A, Subsection 2, the asterisk denotesthe complex conjugate, and T is the 4 4 lineartransformation between the Stokes parameters andthe coherency matrix parameters 24 see Appendix A,Subsection 3. As an example, for an ideal nonscatteringand nondepolarizing EO device the Jones matrixis the same as a linear phase retarder and isgiven 10 byJ expi 2 00 expi 2, (7)where is the relative phase difference for the x- andy-polarization components of the light. SubstitutingEqs. 7, A5, and A9 into Eq. 6, we find the correspondingMueller matrix 25 :0 0 00 1 0 0M PLZTideal 1 . (8)0 0 cos sin0 0 sin cosHowever, for a real PLZT device that has randomscattering and depolarization one must use an ensembleaverage of Jones matrices. 24 To create suchan ensemble, we distinguish in our model betweenthe spatial frequency of the light incident upon andthe spatial frequency of the light exiting the PLZTdevice, i.e., because of scattering we expect that thelight exiting the device will have additional spatialfrequencycomponents. For example, in our experimentwe use a plane wave, i.e., with a single spatialfrequency, which we define as the dc spatialfrequencyoptical field parallel to the optical axis,incident upon our PLZT device, whereas the exitinglight is scattered and consists of a wide band ofspatial-frequency components. Under the assumptionthat the scattering field from every point can bemodeled as a zero-mean Gaussian random process, 26we can separate the optical field transmitted throughthe PLZT device into two components: a nonscat-10 June 1998 Vol. 37, No. 17 APPLIED OPTICS 3719

Fig. 2. Experimental setup for the characterization and modelingof PLZT samples. The PLZT sample is fully illuminated to avoidlocalized photorefractive effects. The low-pass filter was onlyused for experimental measurements for which the scattered fieldsneeded to be filtered out.tered dc field and the scattered field. By using alow-pass filter see Fig. 2 in the spatial Fouriertransform plane of the output lens, we are able tofilter the scattered light out from the output signal.Experimental measurements of such a system showsee Fig. 3 that the dc field is polarized and has anegligibly small scattered or depolarized component,which justifies the use of our superposition model.Therefore we conclude that only the scattering fieldcontributes to the depolarization, which can be describedby an ensemble of randomly polarized fields. 24We describe a typical PLZT-device Jones matrix bya superposition of a deterministic matrix, which accountsfor the DC field, and an ensemble of nondeterministicmatrices, which represents the scatteredunpolarized field, indicated by the superscript e,given byJ e a expi 2 00 b expi 2 u 1a expi 2 u 1eu 4eu 3eu 4u 3eu 2b expi 2 u 2e, (9)Fig. 3. Using a low-pass filter see Fig. 2, we remove the scatteredlight from the detected signal. The remaining dc componentcontains no optical bias, allowing us to conclude that the dc light ispolarized. In this measurement we used parallel polarizers,which resulted in a maximum intensity at zero voltage. ThePLZT sample used was 1 mm thick, which accounts for the increasein phase variation compared with Fig. 1b.where a and b are the real transmittance amplitudesof the nonscattered light, is the relative phase differencefor the two orthogonal polarization components,and u i are the complex transmittanceeamplitudes that account for the randomly polarizedlight. The coefficients u 1 and u 2 represent the amplitudesof the depolarized x and y light components,respectively, and u 3 and u 4 are the coupled or crossdepolarizationamplitudes.We assume that u e i are independent random variableswith a zero mean, u i e 0, and under theensemble over a large detection area e.g., a powermeter much larger than the coherence area of thescattered light the unpolarized components can beconsidered uncorrelated with each other as well aswith the deterministic polarized components, i.e.,u i u k * e 0 if i kU i if i k , (10)u i expj 2e 0, (11)where . .. e is shorthand notation for the averageover the ensemble and U i u i 2 . The transformationfor the ensemble average of Jones matrices intoa Mueller matrix then takes the form 24 ofM TJ e J e* e T 1 . (12)Using Eqs. 10 and 11 allows the resultant Muellermatrix to be written as2M PLZT 1A B U 11 U 2 U 3 U 4 2 A B U 1 U 2 U 3 U 4 0 012 A B U 11 U 2 U 3 U 4 2 A B U 1 U 2 U 3 U 4 0 0 , (13)0 0 AB cos AB sin0 0 AB sin AB cos3720 APPLIED OPTICS Vol. 37, No. 17 10 June 1998

where we have used the definitions A a 2 and B b 2 , which represent the intensity of the nonscatteredlight for the two orthogonally polarized light components,and U i are the intensities of the depolarizationcomponents as defined in Eq. 10. As discussed inSection 2, the nonscattered and depolarized intensitiesas well as the change in phase are all functions ofapplied electric field. However, for simplicity we usethe abbreviated formsAE A, BE B,U i E U i , E , (14)where E is the magnitude of the applied electric field.A similar matrix 27 has been obtained for a reflectorwith incoherent scattering i.e., depolarization.However, in that case the cross-scattering is assumedto be negligible i.e., U 3 U 4 0. Applying such amatrix in our system, we find that the resultantStokes parameters do not coincide with our experimentaldata. Therefore we need to include thecross-depolarization terms in our derivation.Having derived an appropriate Mueller matrix fora PLZT device, we need to calculate the transformationof our complete experimental optical system anddetermine what set of intensity measurements willallow us to solve for the unknown coefficients. Ourmeasurement apparatus uses normally incident coherentmonochromatic light, polarized at 45° relativeto the x axis see Fig. 1a. For our experiments theincident Stokes vector is determined by use of Eq. 3with the maximum intensity normalized to 1 andunder the assumption that E 1 t E 2 t and t isconstant, which result inS in 1010. (15)The light passes through a cascade of optical componentsmade up of a polarizer, a PLZT device, and asecond polarizer. The Mueller matrix for an idealpolarizer 25 isM Pol 1 21 cos2 sin2 0cos2 cos 2 2 cos2sin2 0sin2 cos2sin2 sin 2 2 00 0 0 0,(16)where is the angle between the polarizer axis andthe x axis. The output Stokes vector is obtainedwith the transformation given by Eq. 5 applied toour system, yieldingS out M Pol a M PLZT M Pol p S in , (17)where p is the angle of the first polarizer and a is theangle of the second polarizer usually referred to asthe polarizer and the analyzer, respectively, withM PLZT from Eq. 13.From the Mueller matrix Eq. 13 for the PLZTdevice we have seven unknown coefficients: A, B,U 1 , U 2 , U 3 , U 4 , and , which represent the intensitiesof the nonscattered polarized light, the depolarizedlight, and the change in the relative phase betweenthe orthogonally polarized components. Our objectiveis to find equations of measured intensity thatallow us to solve for the seven unknowns that willcharacterize the PLZT sample as functions of E, theapplied electric-field strength. The first componentof the output Stokes vector represents the measuredoptical intensity. It can be described as a function ofthe two polarizer angles: S 0out I a , p. Using fiveconvenient pairs of polarizer–analyzer angles in Eq.17, we solve for S 0outand findI 45°,45° 1 4 A B U 1 U 2 U 3 U 4 2AB cos, (18)I 0°,0° 1 2 A U 1, (19)I 90°,90° 1 2 B U 2, (20)I 0°,90° 1 2 U 3, (21)I 90°,0° 1 2 U 4, (22)where we have normalized the incident light intensity.Equations 19 and 20 each contain one polarizedcomponent and one depolarized component.By subtracting the depolarized component by use ofthe low-pass filtering technique described above, wecan reduce these equations toI 0°,0°Filter 1 A, (23)2I 90°,90°Filter 1 B. (24)2Here we normalize the measurements with respect tothe intensity at zero applied voltage after the lowpassfilter. Including Eqs. 23 and 24, we havederived seven equations Eqs. 18–24, representingseven experimental measurements required to characterizethe PLZT sample fully, i.e., the seven coefficientsthat represent the intensity of the polarizedcomponents A and B, the depolarized componentsU 1 , U 2 , U 3 , and U 4 , and the change in relative phase.If we wish to reduce the number of required measurements,we can assume that the scattering isisotropic. This results in the amplitudes and intensitiesof the polarized components being identi-10 June 1998 Vol. 37, No. 17 APPLIED OPTICS 3721

This decomposition shows us how our ensembleaveragederivation and isotropic-scattering assumptionsresult in two known Mueller matrices: Thefirst is a combination of a linear attenuator and retardermatrix, 28 and the second is a depolarizer. 29Fig. 4. Seven optical intensity measurements normalized to illustratethe envelope functionality of the vertical and horizontalcomponents to the I 45°,45° data.cal, i.e., A B, and the various depolarization intensitycoefficients U i would also be equivalent, i.e.,Equation 13 reduces toU U i , i 1, 2, 3, 4. (25) 2U 0 0 00 A 0 0M PLZT A . (26)0 0 A cos A sin0 0 A sin A cosWe now require only three equations to solve for thecoefficients, for example:I 45°,45° 1 A 2U A cos, (27)2I 0°,0° I 90°,90° 1 A U, (28)2I 0°,90° I 90°,0° 1 U. (29)2In Section 4 we describe experimental measurementdata Fig. 4 showing that, for the PLZT samplesused, the isotropic approximation is acceptable.Finally, the Mueller matrix of Eq. 26 can be decomposedinto deterministic and nondeterministicparts:0 0 00 A 0 0M PLZT A 0 0 A cos A sin0 0 A sin A cos0 0 00 0 0 02U(30)0 0 0 00 0 0 0.4. Experimental MeasurementsWe have developed experimental procedures that allowus to measure the effects of the electric field inPLZT on such characteristics as scattering, depolarization,and the EO response. In these experimentswe constructed a setup that produces a homogeneouselectric field within our sample that is achieved byuse of a parallel-plate capacitor configuration seeFig. 1a. By sandwiching our PLZT sample betweentwo parallel gold-plated copper plates, whichare much larger than the sample, we generate a constantelectric field inside and outside the ceramic andavoid fringing-field inhomogeneities. The distancebetween plates d and the applied voltage V give usthe electric-field strength E Vd, and the thicknessL of the PLZT is the optical interaction length.In this experiment we use a He–Ne laser beampolarized at 45° relative to the direction of the appliedelectric field. The coherent light is transmittedthrough a spatial filter and recollimated to define thedc spatial frequency of the input optical field illuminatingthe entire PLZT sample see Fig. 2. We usedichroic polarizers with an extinction ratio betterthan 1 10 3 . The transmitted light is captured bya lens and directed onto a Newport Si detector, Model818-SL, and the total intensity is measured with aNewport Model 835 optical power meter.Using the experimental setup outlined above, wemeasure the transmitted optical intensity versus anapplied electric field see Fig. 4 by using the fivepolarizer–analyzer orientations given in Eqs. 18–22. For comparison purposes we show twice theintensity values of the vertical and horizontal measurements,as we discuss below. We can see thatI 0°,0° I 90°,90° , I 0°,90° I 90°,0° . (31)Therefore in this case we may assume that theseterms are equivalent and use the isotropic scatteringapproximation made in deriving the simplified Muellermatrix Eq. 26. We can also see from the plotthat all the intensity curves converge to the samevalue above 1.75 10 6 Vm. This convergence indicatesthat the light becomes completely unpolarizedunder the influence of a very strong electric field.In Fig. 4 the maxima of the I 45°,45° data are approximatelyequal to the 2I 0°,0° and the 2I 90°,90° data sets,which is consistent with our equations, since themaxima values, i.e., cos 1, of Eq. 27 yieldI 45°,45°max AE UE 2I 0°,0° 2I 90°,90° . (32)Additionally, the minima areI 45°,45°min UE 2I 0°,90° 2I 90°,0° , (33)3722 APPLIED OPTICS Vol. 37, No. 17 10 June 1998

Fig. 5. Change in the relative phase calculated from measureddata for PLZT 8.96535 circles. For comparison we also showthe calculated change in phase obtained by use of a shifted E 0 1.5 10 5 Vm electric field in the standard quadratic model solidcurve with the quadratic EO coefficient R 17.0 10 16 mV 2 .The change in the index of refraction is calculated by use of thelinear relation L 2n.which again coincides with the measurements shownin Fig. 4. In Eqs. 32 and 33 we use a more completenotation showing the electric-field dependenceof the coefficients.Substituting Eqs. 28 and 29 into Eq. 27 wesolve, using the simplified Mueller matrix, for thechange in relative 1 phase: cos I 45°,45° I 0°,0° I 0°,90°I 0°,0° I 0°,90° . (34)The calculated phase function is shown in Fig. 5.The smoothness of this curve represents the accuracyof our isotropic approximation. That is, intensitymeasurements with strong anisotropic scattering willresult in a phase function with clear discontinuities.Therefore for some materials a fine resolution of thephase function could require including the variationsof transmitted intensity owing to anisotropic scattering,i.e., making all seven of the required measurementsEqs. 18–24. As we noted above see Fig.1b, upto2 10 5 Vm electric-field strength thePLZT has relatively no EO response and the phasefunction remains flat. From 2 10 5 Vmto1 10 6Vm the EO phase response is roughly quadratic, andat higher electric-field strengths the response diminishessignificantly, i.e., saturates, as discussed in Section2.Previous research has produced similar phasefunctions 1,10 that show the saturation of the EO effectat high field, with a maximum change in index ofapproximately 1 10 2 . However, the weak EOresponse of PLZT at low electric-field strength is oftennot present in published experimental data. Webelieve that this is due to the use of a surfaceelectrodeconfiguration to characterize the materials’EO response. 10 In this configuration the electricfield is not homogeneous, and it can be very difficultto calculate accurately the phase versus the electricfieldfunction.For simplification these PLZT compositions areusually assumed to have a quadratic EO response ora combination of a linear and a quadratic response.However, as can be seen from Fig. 5, the quadraticapproximation does not account for the saturation athigh electric fields, and neither model can account forthe weak response at low field strength. We findthat the use of a fifth-order polynomial achieves anexcellent fit to the change in phase–index versus theelectric-field data available. Although this material’sEO response is dominated by the quadratic Kerreffect, the linear Pockels response might also bepresent. The higher-order terms in the fifth-orderpolynomial fit can be associated with material constraintsthat freeze the polarizability under weakfields and limit molecular deformation under highfields.5. ConclusionIn this paper we have presented a new method forcharacterizing the polarization properties of scatteringand depolarizing EO materials by using as anexample PLZT 8.8–9.56535 compositions. Weshowed that the optical field passing through bulkPLZT can be described by superposition of a nonscatteredpolarized component and a scattered depolarizedcomponent. Using the corresponding deterministicand stochastic Jones matrices, we havederived a simplified Mueller matrix representation inwhich only seven coefficients are required to describethe scattering and depolarizing EO device. UsingStokes vectors, we were then able to calculate thetransmitted optical intensity for light passingthrough our device in terms of the seven correspondingcoefficients.By placing a PLZT device that has a parallel-plateelectrode geometry between a pair of polarizers, weperform relatively simple compared with Muellermatrix ellipsometry 30 experimental measurementsto solve for the unknown coefficients. As we varythe electric potential across the PLZT device, we generatea data set describing the coefficients of scattering,depolarization, and EO phase response asfunctions of the applied electric field. Experimentalresults show practically no EO response for PLZTunder a weak electric field. This effect might not beobserved with a traditional surface-electrode geometry,since the electric field can be very high near theelectrode edges, even under small applied voltages.Our characterization shows distinctly different EOresponses for weak, medium, and strong electric-fieldstrengths. Therefore, for accurate simulation ofchanges in phase or index of refraction resultingfrom large electric-field gradients e.g., surfaceelectrodedevices, we must use a higher than secondorderEO model.Although PLZT has been shown to have anisotropicscattering, to reduce the number of unknowns in ourMueller matrix, we have assumed the simpler modelof isotropic scattering. This can be justified on the10 June 1998 Vol. 37, No. 17 APPLIED OPTICS 3723

asis of the similarity between the measured intensitiesfor the various polarized and depolarized components.Using the experimental data shown in Fig.4, we have compared the intensities of the two orthogonalpolarized components A, B and have foundthat the average difference is less than 2%, with amaximum of 5%. Comparing the intensities of thefour depolarization-component intensities U 1 , U 2 ,U 3 , U 4 gives an average difference of less than 4%when dividing by the maximum depolarized intensityand a maximum difference of 13%. Thereforewithin a less than 5% average error we can use theisotropic-scattering approximation to simplify ourMueller matrix to have only three unknown coefficientsA, U, and and reduce the correspondingnumber of required experimental measurements.The application of our characterization study, asdiscussed in this paper, to computer-aided design anddevice simulation is discussed in a second paper, alsoin this issue. We also continue to characterize bulkPLZT in terms of temperature and ac electric-fielddependencies, as well as to characterize alternativeEO materials.Appendix A1. Jones Matrix FormulationFor a uniform monochromatic TE plane wave incidenton a nondepolarizing optical system, the outgoingplane wave can be described byE x J 11 E x J 12 E y ,E y J 21 E x J 22 E y ,which can be combined in matrix form as E xE y J 11 J 12J 21 J 22 E xE y ,(A1)(A2)(A3)or by the simplified notation E JE, where thetransformation by the optical element or system,is called the Jones matrix. 20J J 11 J 12J 21 J 22 , (A4)2. Kronecker ProductThe Kronecker 23 or direct product for 2 2 matricesis given by11 J 11 J 11 J 12 J 12 J 11 J 12 J 12 JJ J J 11 J 21 J 11 J 22 J 12 J 21 J 12 J 22 (A5)J 21 J 11 J 21 J 12 J 22 J 11 J 22 J 12 J 21 J 21 J 21 J 22 J 22 J 21 J 22 J 22 .3. Stokes to Coherency-Parameter TransformationThe coherency matrix, like the Stokes vector, can alsobe used to describe the polarization state of light andis defined byC E xE x * E x E y , (A6)E y E x * E y E y *where E x and E y are defined in Eq. 2. Although thestandard notation for the coherency matrix is J, weuse the notation C to avoid confusion with our use ofJ for the Jones matrix. By use of Eq. 3 the relationbetween the Stokes and the coherency parameterscan be written asS 0 C xx C yy ,S 1 C xx C yy ,S 2 C xy C yx ,S 3 jC xy C yx .(A7)Presenting the coherency matrix as a 4 1 vectoryieldsxxCC C xyC yxC yy;(A8)then the transformation from the Stokes parametersto the coherency parameters can be given by the relationS TC, where0 0 11 0 0 1T 1 . (A9)0 1 1 00 j j 0The authors thank James Thomas for his generousassistance. This study was funded in part fromgrants from the National Science Foundation, the AirForce Office for Scientific Research, and Rome Labs.References1. G. Haertling, “Electro-optic ceramics and devices,” ElectronicCeramics: Properties, Devices, and Applications, L. Levinson,ed. Marcel Dekker, New York, 1988, pp. 371–492.2. J. Thomas and Y. Fainman, “Programmable diffractive opticalelements using a multichannel lanthanum-modified lead zirconatetitanate phase modulator,” Opt. Lett. 20, 1510–15121995.3. Q. W. Song, X. M. Wang, and R. Bussjager, “Lanthanummodifiedlead zirconate titanate ceramic wafer-based electroopticdynamic diverging lens,” Opt. Lett. 21, 242–244 1996.4. T. Utsunomiya, “Optical switch using PLZT ceramics,” Ferroelectrics109, 235–240 1990.5. J. T. Cutchen, J. O. Harris, and G. R. Laguna, “PLZT electroopticshutters: applications,” Appl. Opt. 14, 1866–18731975.6. P. Shames, P. C. Sun, and Y. Fainman, “Modeling and optimizationof electro-optic phase modulator,” in Physics and Simulationof Optoelectronic Devices IV, W. W. Chow and M.Osinski, eds., Proc. SPIE 2693, 787–796 1996.3724 APPLIED OPTICS Vol. 37, No. 17 10 June 1998

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