Complex amplitude reflectance of the liquid crystal light valve Kanghua Lu and Bahaa E. A. Saleh The complex amplitude reflectance of the liquid crystal light valve (LCLV) is determined as a function of the writing intensity and applied voltage using an approximate model. The input and output polarizers are assumed to have arbitrary directions. The theoretical results based on this model match our experimental measurements. This theory allows us to optimize the operation of the LCLV as an intensity or phase-only spatial light modulator. When the polarizers are orthogonal and the input polarizer is at -34° with the front liquid crystal director, the intensity reflectance reaches 100% (compared to 81% for the conventional configuration). Phase-only modulation is realizable by use of appropriate applied voltage bias and configuration of polarizers. Key words: Spatial light modulator, liquid crystal displays, optical processing. 1. Introduction The liquid crystal light valve (LCLV) was developed in the 19 7 0s14 and has been extensively used as a spatial light modulator (SLM). 5,6 However, no explicit analytical expressions are available for its reflectance as a function of writing intensity and applied voltage. This paper aims to fill in this gap. Using a model for light propagation through the inhomogeneous medium of the twisted nematic liquid crystal, we use Jones calculus to derive the Jones matrix of the device. Under some approximation we determine an expression for the dependence of the complex amplitude reflectance on the writing light and the applied voltage. Input and output polarizers are assumed to have arbitrary directions. This theory is verified by experimental measurements of intensity reflectance and phase shift in different conditions. Using our theory we undertake a systematic study of the polarization configurations and the device parameters that optimize the operation of the LCLV as an intensity or phase-only spatial modulator. We find that, when the polarizers are orthogonal and the input polarizer is oriented at -34° with the front liquid crystal director, the efficiency of intensity modulation The authors are with University of Wisconsin, Department of Electrical & Computer Engineering, Madison, Wisconsin 537061691. Received 20 October 1989. 0003-6935/91/172354-09$05.00/0. ©1991 Optical Society of America. 2354 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991 reaches 100% for monochromatic reading light (compared to 81% for the conventional configuration). Phase-only modulation is also found to be realizable by use of appropriate applied voltage bias and a configuration in which both polarizers are parallel to the front liquid crystal director. The intensity distortion can be reduced below 10%. II. Complex Amplitude Modulation of the LCLV A liquid crystal material is an anisotropic medium which can be treated locally as a uniaxial crystal whose optical axis is parallel to the director of the molecules. In a twisted nematic liquid crystal cell (LC),7 - 21 the molecules rotate gradually in a helical fashion, and the material is therefore inhomogeneous. It is locally a uniaxial crystal whose optical axis rotates helically in the direction of twist. To examine the propagation of polarized light along the twist axis of an LC, the Jones calculus may be used.9"12"19 For an LC of thickness d, when no electric field is applied, there is no molecular tilt, i.e., the tilt angle = 0. Twist angle ois linearly distributed along the twist axis z with a maximum twist angle a at z = d, so that o = azid. The material can be divided into incremental slices orthogonal to the z-axis. Each slice acts as a homogeneous uniaxial crystal8"19 whose Jones matrix can be written as a function of its ordinary and extraordinary indices of refraction, n and ne, and its direction. The overall Jones matrix is the product of the individual matrices:' 9 JT= f N j=1 J, Neinatic Liquid Crystal where Jj is the incremental Jones matrix of a layer between zj and zj+l and N is the number of layers. In the limit N JT = Diclectric MilTor X,19,21 - exp(-i)R(a) )[ - cos(y) - i sin(y) -sin(-y) -sin(y) cosl(y) + isin(y) CdS-CdTe Heterojunction , 4 (1) .4- Writing Light _ where R(a) is the rotation matrix: R(a) = [cos(a) [-sin(a) sin(a)] (2) cos(a)j' ^ = V.E2+ gl2 1= A In, - (3) no] =rd = n,+nj Function Generator Configuration of the LCLV modulator with normal incident light and arbitrary input and output polarizations. Fig. 1. When an electric field is applied in the direction of the twist axis, the equilibrium tilt and twist angles are altered in accordance with the Oseen-Frank equation;13 in general, the tilt and twist angles vary nonlinearly along the z-direction, i.e., 0 = f(z,V), and so = g(z,V), where V is the applied voltage. Each incremental Jones matrix Jj is a function of the applied voltage. Because of the nonlinear distributions of the tilt and twist angles, it has not been possible to determine an explicit expression for JT similar to Eq. (1). The problem can only be addressed numerically. To circumvent this difficulty we consider the following approximation: (1) Since the tilt angle 0 = f(z, V) is a smoothly varying function with even symmetry around the midpoint (z = d/2),113 , 7 it is approximated by its mean value, so that it is treated approximately as a homogeneous tile. The dependence of this constant value of 0 on V has been expressed in the form15 21 = A no (5) = d [ne(O) + n0 ] = 0° + /3, (6) [ne(O) where ne(O) is the equivalent extraordinary index of refraction given by cos 2(O)+ sin 2 (0) 1 n2 n2 n,1(O) (7) ' and 4O is a constant phase shift, which is ignored thereafter. For a given LC, 3 is defined in a region (,flnmaxl, where = /max 7rd (n - n.); represents the maximum value of ,-which occurs in the absence of the applied voltage. Equations (1)(7) are applicable to a transmissive twisted nematic liquid crystal cell. Our experimental measurements have demonstrated that the accuracy of this simplified model is quite satisfactory. 2 ' Furthermore, the predictions of this model are consistent with experimental results obtained by others.2 2 - 2 4 An LCLV is a reflective twisted nematic liquid crystal cell (see Fig. 1) that can be equivalently unfolded in the form of a cascade of two identical transmissive cells with mirror-symmetric structures., By use of Eq. (1), the Jones matrix of the LCLV is l#max 0, a= 2 f[e(- V-Vc)] V< Vc where V is the rms voltage applied on the LC, V, is the threshold voltage, and V0 is the excess voltage. (2) Since the twist angle so = g(z,V) is a smoothly varying function with odd symmetry around the midpoint (z = d/2)," ' 3 17 it is approximated by a linear function of z, so = azid. When these two assumptions are used, an explicit expression for JT can be obtained. Using the approximate model, Eqs. (1)-(3) continue to apply but with - (J = exp(-i 2 ) cos(2y) - i - sin(2y) + i-; [1 - cos(2y)] i [1 - cos(2y)] 1 // 3 2 cos(2'y) + i -/3sin(2Y)J (a2+ 10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS I ' (8) 2355 90 LQ Lu, z GLu 0 N~ 0 -90 a 90 180 270 360 BETA (b) (a) Fig. 2. (a) Dependence of intensity reflectance R on : and input polarization angle 6. The input and output polarizers are orthogonal. (b) A topographic view of the relation in (a); R is a periodic function of with a period of 900. The first optical threshold occurs when A = 174.3°. where a, Al,and y are as before. Considering the effects of a polarizer and an analyzer at arbitrary angles 61 and 02 as shown in Fig. 1, the complex amplitude reflectance of the entire device is R = Vit exp(-ib), where + (i) R = f(-) +{ cos(2y)] cos(V t2)} sin(2,y) coS(it 1 + 02)} (9) is the intensity reflectance, and - is~ phas ~ shift. ~ ~~~[(O) a [1 -cos(2y)] 2 + nr, n cos(2p + sin(2y) sin(2l)2 142. In this case, the dependence of R on and A when a = 450, is shown in Fig. 2. We find that when 2 2 = n - (n7r) -a , n = 1,2,3,.... R = 0 regardless of A/; therefore, O3n acts as the nth optical threshold. Furthermore, R is a periodic function of ipwith a period of 90°. The phase has a linear dependence on with jumps of iroccurring whenR = 0. 2356 1 th osP -I is the phase shift. If a polarizing beam splitter is used instead of the common beam splitter with a pair of polarizers (Fig. 1), the condition A -2 = 90° must be satisfied. When ^61 = , Eqs. (9) and (10) become R =-l where S(I) is a monotonically increasing function of I, and s() = S(I)/S(o) is called the switching ratio, which may be determined from an electrical model of the circuit.' However, it is convenient to measure the switching ratio experimentally, as explained in the [1 - cos(2y)] sin(\61 + 1k2) - sin(2y) cos(P + 'I2) (a) + () cos(2)] cos(tj(10) tan' (11) V = S() Vext, - [1 - cos(2y)] sin(ik1 + Vt2 ) - =2 Equations (9) and (10), together with Eqs. (3)-(7), describe the reading side of the LCLV modulator. At the writing side, for the LCLV using a CdS-CdTe heterojunction photosensor, voltage V across the LC layer is related to the intensity of writing light I and the rms value of the externally applied voltage Vext by APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991 next section. Therefore, the normalized voltage in Eq. (4) can be written in the form v ~S(O) V,, V~ V S)Vet s~) Vet V. C V. S(O) In summary, the LCLV is characterized by the internal parameters a, /3ma., VIS(), VIS(0), and s(I). The operating point is characterized by the external variables Vext, I, and the setting of polarizers 4(1 and 02. The dependence of the complex amplitude reflectance A? = R exp(-ib) on these variables is established as follows: (1)R and 5are expressed as functions of the auxiliary variable is and parameters ihl, 2, and a, given by Eqs. (9), (10), and (3). For example, for an LCLV with a = I :I Zr LCLV Fig. 3. Dependence of phase shift on j3and input polarization angle 46.The input and output polarizers are parallel. 1.0 <0.8 LI o X 0.6 LL\ N 0.4 0 0.2 0.0 0 1 2 3 4 NORMALIZED VOLTAGE Fig. 4. Dependence of the normalized parameter /#max on the normalized voltage (V-V,)/V6, where n0 = 1.5 and ne = 1.7. When ne is changed by +0.1, this curve is changed only very slightly. 450 when 11 - 12 = 900, the R-3 dependence for different q1 is plotted in Fig. 2. When 11 = 412, the b-j dependence for different 461 is plotted in Fig. 3. (2) The ratio W//max is itself related to the normalized voltage (V - V,)/V,, by Eqs. (4), (5), and (7), as illustrated in Fig. 4 for several values of ne. This relationship is monotonically decreasing and is approximately linear when f/lmax is within the region from 0.3 to 0.9, with an average slope of -0.75. The ne has a small effect on this relationship. (3) Finally, the normalized voltage (V - V,)IV0 is determined by the product s(I) Vext. The features of s(I) are discussed in the next section. Ill. Experimental Verification A. Verification of Theory To verify the theory and to estimate the parameters of the device, we conducted some experiments in which CRT Fig. 5. Experimental system to measure the dependence of the intensity reflectance R on external voltage Vext and writing intensity I. the dependences of R and on I and Vext were measured for several polarization angles i61 and P2 We use RP1,42 and 414'2 to denote the setting of the polarizers. The LCLV used in our experiments is model H-4060 made by Hughes Aircraft Co. The total twist angle a = 450 is given by the manufacturer. The system used to measure the dependence of the intensity reflectance R on Vext and I is shown in Fig. 5. A uniform pattern displayed on the screen of a CRT monitor is imaged on the writing side of the LCLV. The writing intensity I is controlled by varying the Brightness. An external voltage Vext is applied using a function generator. A polarizing beam splitter is used and the LCLV is rotated to change ^i6. This limits us to the case of V/1 - '12 = 900. However, i1p= 02 can be inferred by using the relation Rp = 1 -Rpo+0. We have measured Rp,9,o+ vs Vext with fixed I for several values of A. These results are compared with the theoretical relationship between Rp,9 o+, and : displayed in Fig. 2. For example, in the case of / = -34°, the experimental relationship of R-3 4,5 6 to Vext is shown in Fig. 6(a). The corresponding theoretical relationship between R- 34 56 and A is copied from Fig. 2(a) into Fig. 6(b) for convenience. We find that these two functions have similar structures with the same number of lobes and the same values of maxima and However, the minima within the region A e (#max]. order in which these maxima and minima appear is inverted. This is an indication of a decreasing 3 as a function of Vext, as is also expected from the theory (see Fig. 4). Similar comparisons are held for other values of A,as shown in Fig. 6(e) where the T-axis is reversed for convenience. In all cases, the number and values of the maxima and minima were identical to those in the theoretical plots. Our experiment has demonstrated that the same relationship between Rp,9 o+p and f is repeated for each 90° period of , as predicted by theory. Another theoretical prediction, the optical 10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS 2357 1.0 LUi z 0.8 LU -i 0.6 LILUi 0.4 zLU 1- 0.2 0.0 Vext ca) (b) 1.0 Ld LU ()0.8 so Illlilllll LU LI * 0.6 1111111111111111111111111111 a m 11X111 slzllllllilllllllllllllllll . X H t 111111llllT tE t : ::: :: : i eLlllllllll II II I . .,. , LL r . I- 0.4 W m h a ._.__...___.._..... .J i 1 1111 I I g g , . .... .. ....... En l l llllllll II : _ s _ _ : : . _ _ _ . . _ : n : : . . _ 1111111111 H * ._ ..v .: .___ ; _ 4 4 I I I I 1 1TTTT = II : _ : i S . _ _ . : _ : 7 . . . . n I 11111111 X n 1n| N Xr s H = * ._. * .. .x_ _ 4 = : _: = : : __ ' ; _: = n 1 1 1 1 1 1 1 F I , z- 0.2 _ 4 _r I z _ = _ = = 7 . . . n 0.0 , .,:,, I,,,, = = : : : : . r : -r n . . . -r _, I I I 1_1_'_'_'_' l l ,, . _. l l : .. X J; J _ 7 WRITING INTENSITY I (d) (c) =O 0 / tY vex = -22.50 0.0. 0A 27U Vex A 180 90 0 P I =-450 vex: C0.5 270 18U 90 0 1 XA A vexe 2358 = -67.5' 04 :5 0 270 Ce) A 180 90 0 13 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991 Fig. 6. (a) Measured intensity reflectance R-34,56 VS Vext at a fixed writing intensity I = 500 W cm-2 . At V8,t = 0, corresponds to 3max = 2480 when X = 633 nm. (b) Theoretical dependence of R-34,56 on 13. The peak value of the second lobe from the left is 100%. (c) Measured intensity reflectance R-34,56 vs I when the external voltage bias is V.xt = 12.2 V. (d) Theoreticalresultcorrespondingto (c). (e) Experimental dependence (left) of R on Vext is compared with the theoretical dependence (right) of R on 13for different input polarization angles ip. Since 13is a monotonically decreasing function of Vext, it is plotted in the reverse direction so that experiment and theory can be conveniently compared. Film 1.4 Lj 1.2 1.0 0 r)0.8/ N/ 0.6 y 0.4- 0 0.2 0.0 0 4 . 8 .,. . 12 . 16 EXPERIMENTAL VOLTAGE Fig. 7. Correspondence between the experimental Vext and the theoretical normalized voltage (V - V,)/V 0 when s(I) = 2.1. threshold, was also observed. When setting the value of Vext so that fi = fli, R = 0 for all iA. Thus, rotating the LCLV does not alter the intensity reflectance, as is evident in Fig. 6(e). We, therefore, conclude that the theoretical model, although approximate, leads to device characteristics that are generally consistent with experimental observations. By comparing the ordinates in Figs. 6(a) and (b), a one-to-one correspondence between points of the abscissas can be established. Ambiguities due to the nonmonotonic nature of the functions are resolved by using the order in which the maxima and minima appear. The device parameter flmax = 2480 can be determined by the experimental R when Vext = 0. Once the f3-Vext relation is known, the theoretical relationship between 3 and the normalized voltage (V -V,)IV (shown in Fig. 4) can be used to determine the correspondence between the theoretical normalized voltage (V - V,)/V 0 and the experimental voltage Vext. This relationship is plotted in Fig. 7. Its linearity is reasonably good. A best-fit straight line gives an estimation of the parameters V0 IS(O) = 28 V and VcIS(O) = 2.9 V. The experimental setup used to observe the phase modulation is shown in Fig. 8(a). The LCLV replaces the mirror in one arm of a Michelson interferometer. A pattern with a grey value varying in one direction as a sawtooth function is addressed on the CRT. The angle it' = t'2 can be changed by rotating the LCLV. The displacements of the interference fringes are observed. Although no attempts were made to measure a accurately, the qualitative results are consistent with the theoretical plot in Fig. 3. For example, 50, has the maximum modulation sensitivity and 690,9o (which is identical to -9-o,-9o) undergoes little change when varying I or Vext Similar experimental results have been reported in Ref. 24. B. Measurement of Switching Ratio We now turn to the estimation of switching ratio S(1). Here we rely on the experimental measurements of function R vs Vext for fixed I [see Fig. 6(a), for example] and function R vs I for fixed Vext [see Fig. 6(c), for example]. Since R is determined completely (a) (b) Fig. 8. (a) Michelson interferometer with an LCLV phase-only modulator replacing the mirror in one arm. A sawtooth grey level pattern is addressed on the CRT. (b) Interference pattern demonstrating phase modulation. by the products(I) Vext. We identify matching pairs of points (1, Vextj) and (I2, Vext2) on the two functions that have the same R; ambiguities due to the nonmonotonic nature of the functions are resolved by using the order in which the maxima and minima appear. Therefore, we equate S(Ii)Vextl = S2)Ye.12, where Vext2 and I, are of fixed values. When I2 = 0, the tj. When I2 is equal to an corresponding Vext = V%_ arbitrary nonzero value I, the corresponding Vexti = V tj, then the switching ratio can be determined by ex SU) = VIL V=exti This method was applied to the data in Figs. 6(a) and (c). The switching ratio s(I) obtained is shown in Fig. 10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS 2359 2.2 2.0 0 <1.8 Z 1.6 I-1.4 1.2 1.0 260 400 60 WRITING INTENSITY Fig. 9. Dependence of switching ratio s(I) on writing intensity I. Experimental data are based on Figs. 6(a) and (c). Its best match with Eq. (12) gives the device parameters Sat = 2.4 and k = 1.94% cm2 .'W1 9. Since this relationship exhibits initial linearity for small I and saturation for large I, we fit it to a twoparameter function: = kI 1 + (12) / Ssat which has an initial value s(0) = 1, initial slope k, and a saturation value Ssat for large IL The best fit is accomplished for sat = 2.4 and k = 1.94% cm 2 _ jLW-1 The quality of fit is excellent as is evident from Fig. 9. Equation (12) is consistent with an equivalent electrical circuit in which the main contribution of the writing light is to change the capacitance of the CdS-CdTe heterojunction. Now that we have estimated all the parameters of the device, we provide one more verification of the theory by comparing experimental results for R- 34 .56 as a function of I at a fixed V,a = 12.2 V, with the corresponding theoretical relationship. These results are consistent as Figs. 6(c) and (d) show. IV. Operation of the LCLV as a Spatial Intensity or Phase-Only Modulator As shown earlier, the dependence of the complex amplitude reflectance of the LCLV on writing intensity I can be changed by selection of polarizer angles Al and V'2 (or only one angle p when a polarizing beam splitter is used) and voltage bias Vt. This offers great flexibility in the operation of the device for different applications. A. LCLV as a Spatial Intensity Modulator If an LCLV is used as an optical intensity modulator, high efficiency, high modulating sensitivity, and good linearity are usually expected. Because of the high efficiency of the polarizing beam splitter in comparison with other beam splitters, we limit our discussion to this case. Figure 2(b) shows the dependence of the intensity reflectance R on V/and Al,where the control variables Vext and I are embedded in . The design involves 2360 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991 selection of the angle and the interval over which ,B varies so that R is changed from 0 to 1. Within the range of a single period, A'e [-90°,0] and 3 (,flma], where &m. = 2480 in our LCLV, there are three points (A,B,C) at whichR = 1. There are also lines at whichR = 0, one at /3I = 174.30 and an oblique line extending from the points ( = 0,# = 0) and ( = -90', = 173.40). There are several possible choices for the operating lines, each corresponds to a horizontal line passing through points A,B, or C and extending to the right or left to the first zero. Lines extending to the right correspond to positive modulations, those extending to the left correspond to negative modulations. The lines associated with point A are not suitable since small values of require high voltages and involve nonlinearity. Point C involves only negative modulation. Operation at point B appears to be the most suitable one. This choice requires / = - 3 4 ° and a voltage bias Vext = 12.2 V, corresponding to iB1 for positive modulation. The experimental and theoretical R-I dependences of this choice are shown in Figs. 6(c) and (d). We now compare the option of s =-34° to the conventional option of A/= 00. In the former case, the maximum value of R is 1; whereas in the latter it is only 0.81. Furthermore, when ,6=-34°, the slope of R vs f at R = 0.5 is -5.5hr. For , = 0, the slope at R = 0.41 is -2.9/7r. One may be interested in the phase distortion caused by the intensity modulator. As discussed in Sec. II, when using the polarizing beam splitter, the phase distortion is at a rate of 2f regardless of Af. The maximum phase distortion for the operating value of i is 90° in the VI = -34° case, and 180° in the t = 0 case. Thus, the proposed operating angle A = -34° offers a higher sensitivity, greater efficiency and dynamic range, and smaller phase distortion. These conclusions, however, are applicable only for monochromatic reading light because ,3depends on wavelength X. An investigation of the nonmonochromatic case has been presented elsewhere.2 5 B. LCLV as a Spatial Phase-Only Modulator When the LCLV is used as a spatial phase modulator, it is expected to have a dynamic range of 2r, good linearity, and high sensitivity. For phase-only operation, the intensity reflectance should be kept constant and large. As is clear from Fig. 2, which is applicable to the polarizing beam splitter, there is no extended region for which R is flat and high. We must therefore examine other polarizer configurations not allowed by the polarizing beam splitter. We found that the configuration A1 = V'2 = 0 satisfies the above requirements most closely. This configuration is realized by use of a single polarizer. oo and Ro,0 are plotted in Figs. 10(a) and (b). When A is large, the -: dependence is sufficiently linear with a slope of 4, and R varies between 0.9 and 1 for > 127°. The requirement of a 27r dynamic range necessitates that the operating range of extends between two values 03, and A1 (as the writing intensity varies be- 720 I U- / Co 360 0~ 0 180 90 BETA (a) 270 (DEG.) 1.0 U0.8-\t LU () zJ 10.6 LU U- 0.4- z 0.4 Z \\ 0.0- 0 90 180 270 BETA (DEG.) (b) Fig. 10. (a) Theoretical dependence of bo,o on /. When: > 90°, the relation is approximately linear with a slope of 4. (b) Theoretical dependence of R0 ,0 on /3. The efficiency is high and undergoes relatively small variation when :3is large. find the corresponding Imax. These points are represented by the solid curve in Fig. 11. Thus, for each Vext, the operating line is a vertical line extending from I = 0 to I = Imax on the solid curve. The dotted curve represents values of I and Vext for which ,3= 13c. Clearly, we must operate below this line. Since the solid curve crosses the dotted line at Vext = 9.2 V, the operating range must be limited to the shaded region. Within this region any value of Vext satisfies all -requirements. However, to minimize Imax we must operate near the critical line Vext = 9.2 V. Although the relationship between phase shift 6 and f3 in Fig. 10 is reasonably linear for large A3, the relationship between fl and I is not linear, mainly because of the nonlinearity of the switching ratio (see Fig. 9). This nonlinear effect may be alleviated by precompensation of the CRT used to produce the writing light. The previous analysis was limited to an LCLV with twist angle a = 450 (the only commercially available choice). This is not the optimal choice for phase-only operation. In fact, a device with zero twist angle a = 0 and Alj = VP2 = 0, yields R = 1 and = 4, as may be readily seen from Eqs. (9) and (10), with the curves in Figs. 4 and 9 remaining unchanged. In this case, phase modulation is achieved by tilting the liquid crystal molecules and thus changing ne(O). The device then operates as in the well-known tunable birefringence effect. 2 6 4 In this case, a liquid crystal material of large birefringence should be selected so as to obtain a large value of /3max, and therefore to increase the dynamic range and reduce the effect of nonlinearity of the switching ratio. References V/) z z z 81 12 14 16 APPLIED VOLTAGE (VOLTS) Fig. 11. Operation lines within the shaded area have a dynamic range for phase modulation >3600 and associated intensity distortion <10%. tween 0 and Imax), such that ,B-,/I = 90°. The requirement of high reflectance R is satisfied if / > ti, where &3is some critical value [see Fig. 10(b)]. For example, if R must be greater than 0.9, /3 = 1270. The distortion of R is then <10%. For each value of Vxt we determine the value of /3 for I = 0 and determine /3I = AU- 900, from which we 1. J. Grinberg, A. Jacobson, W. P. Bleha, L. Miller, L. Fraas, D. Boswell, and G. Myer, "New Real Time Noncoherent to Coherent Light Image Converter: Hybrid Field Effect Liquid Crystal Light Valve," Opt. Eng. 14, 217-225 (1975). 2. W. P. Bleha, "Progress in Liquid Crystal Light Valves," Laser Focus/Electro-optics, 111-120 (Oct. 1983). 3. U. Efron, P. 0. Braatz, M. J. Little, R. N. Schwartz, and J. Grinberg, "Silicon Liquid Crystal Light Valves: Status and Issues," Opt. Eng. 22, 682-686 (1983). 4. U. Efron, S. T. Wu, J. Grinberg, and L. D. Hess, "Liquid-CrystalBased Visible-to-Infrared Dynamic Image Converter," Opt. Eng. 24,111-118 (1985). 5. W. P. Bleha, L. T. Lipton, E. Wiener-Avnear, J. Grinberg, P. G. Reif, D. P. Casasent, H. B. Brown, and B. V. Markevitch, "Application of Liquid Crystal Light Valve to Real Time Optical Data Processing," Opt. Eng. 17, 371-384 (1978). 6. A. Fisher and L. Lee, "The Current Status of Two-Dimensional Spatial Light Modulator Technology," Proc. Soc. Photo-Opt. Instrum. Eng. 634, 352-371 (1986). 7. M. Schadt and W. Helfrich, "Voltage-Dependent Optical Activity of a Twisted Nematic Liquid Crystal," Appl. Phys. Lett. 15, 127-128 (1971). 8. D. W. Berreman, "Optics in Stratified and Anisotropic Media: 4 X 4-Matrix Formulation," J. Opt. Soc. Am. 62,502-510 (1972). 9. R. M. A. Azzam and N. M. Bashara, "Simplified Approach to the Propagation of Polarized Light in Anisotropic Media-Application to Liquid Crystals," J. Opt. Soc. Am. 62,1252-1257 (1972). 10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS 2361 10. D. W. Berreman, "Optics in Smoothly Varying Anisotropic Planar Structures: Application to Liquid-Crystal Twist Cells," J. Opt. Soc. Am. 63, 1374-1379 (1973). 11. L. A. Goodman, "Liquid Crystal Displays," J. Vac. Sci. Technol. 10, 804-823 (1973). 12. C. H. Gooch and H. A. Tarry, "Optical Characteristics of Twisted Nematic Liquid-Crystal Film," Electron. Lett. 10,2-4 (1974). 13. D. W. Berreman, "Dynamics of Liquid-Crystal Twist Cells," Appl. Phys. Lett. 25, 12-15 (1974). 14. C. H. Gooch and H. A. Tarry, "The Optical Properties of Twisted Nematic Liquid Crystal Structures with Twisted Angle <90O," Appl. Phys. D 8, 1575-1584 (1975). 15. P. G. de Gennes, The Physics of Liquid Crystals (Clarendon, Oxford, 1975), Chap. 3. 16. J. Grinberg and A. D. Jacobson, "Transmission Characteristics of a Twisted Nematic Liquid-Crystal Layer," J. Opt. Soc. Am. 66, 1003-1009 (1976). 17. G. Buar, "Optical Characteristics of Liquid Crystal Displays," in Physics and Chemistry of Liquid CrystalDevices, G. J. Sprokel, Ed. (Plenum, New York, 1980), pp. 61-78. 18. R. J. Gagnon, "Liquid-Crystal Twist-Cell Optics," J. Opt. Soc. Am. 71, 348-353 (1981). 19. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, New York, 1984), Chap. 5. 2362 APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991 20. F. Leenhots and M. Schadt, "Optics of Twisted Nematic and Supertwisted Nematic Liquid-Crystal Displays," J. Appl. Phys. 60, 3275-3281 (1986). 21. K. Lu and B. E. A. Saleh, "Theory and Design of the Liquid Crystal TV as an Optical Phase Modulator," Opt. Eng. 29, 240246 (1990). 22. D. A. Yocky, T. H. Barns, K. Matsumoto, N. Ooyama, and K. Matusda, "Simple Measurement of the Phase Modulation Capability of Liquid Crystal Phase-Only Light Modulators," Optik 84, 140-144 (1990). 23. U. Efron, S. T. Wu, and T. D. Bates, "Nematic Liquid Crystals for Spatial Light Modulators: Recent Studies," J. Opt. Soc. Am. B 3, 247-252 (1986). 24. N. Konforti, E. Marom, and S.-T. Wu, "Phase-Only Modulation with Twisted Nematic Liquid-Crystal Spatial Light Modulators," Opt. Lett. 13, 251-253 (1988). 25. K. Lu and B. E. A. Saleh, "Optimal Twist and Polarization Angles for the Reflective Liquid-Crystal Light Modulators," OSA Annual Meeting Technical Digest Series, Vol. 15 (Optical Society of America, Washington, DC, 1990), p. 255. 26. F. J. Kahn, "Electric-Field-Induced Orientational Deformation of Nematic Liquid Crystals: Tunable Birefringence," Appl. Phys. Lett. 20, 199-201 (1971).
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