Full-range, continuous, complex modulation by the

Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions Luiz Gonc¸alves Neto, Danny Roberge, and Yunlong Sheng

Switchable, continuous, complex-amplitude modulation is demonstrated with two cascaded, twisted nematic liquid-crystal televisions ~LCTV’s!, both operating in phase- and amplitude-coupled modulation modes. The condition for full-range complex modulation is that one of the LCTV’s must provide a 2p-range phase modulation. A look-up table encoding method is proposed that permits the compensation of phase–amplitude coupling and nonlinearity in the two individual LCTV modulations. Experimental techniques for determining the LCTV-device parameters, for maximizing the phase-mostly modulation range and the amplitude-mostly modulation contrast, and for testing the complex-amplitude modulation are developed. Optical complex-amplitude Fresnel holograms are shown. © 1996 Optical Society of America Key words: Complex-amplitude modulation, twisted nematic liquid-crystal devices.

1. Introduction

An ideal support for holograms and holographic filters is a spatial light modulator ~SLM! that provides complex-amplitude modulation. Such a SLM is a real-time wave-front manipulator. The SLM is addressed by a single electrical or optical signal. The phase and amplitude at each pixel of the SLM cannot be controlled independently and simultaneously. Thus, continuous complex modulation can be obtained only with a combination of at least two SLM’s,1– 4 or of two subpixels in a micropixel, which is optically underresolved.5 A nematic liquid-crystal device ~LCD! can provide continuous phase modulation.6 When the liquidcrystal ~LC! molecules are tilted by the applied voltage, the effective birefringence of the LCD decreases with the increase in the LC molecules tilt.7 The use When this work was performed, the authors were with the Centre d’Optique, Photonique et Laser, De´partement de Physique, Universite´ Laval, Que´bec, Canada G1K 7P4. L. G. Neto is now with the Laboratory of Integrated Systems, Departimento d’Engenharia, Electronica, Sa˜o Paulo University, C.P. 8174, 01065-970 Sa˜o Paulo, Brazil. D. Roberge is now with MYTEC Technology, 10 Gateway Boulevard, Suite 430, Don Mills, Ontario M3C 3A1, Canada. Received 3 April 1995; revised manuscript received 11 December 1995. 0003-6935y96y04567-10$10.00y0 © 1996 Optical Society of America

of two LCD’s, one for phase and another for amplitude modulation, have been proposed1 and demonstrated.2 To implement this approach Amako et al.2 fabricated a homogeneously aligned nematic LCD for phase-only modulation with the molecular orientations aligned parallel to the panel faces and to each other without twist. A twisted nematic LCD was used for amplitude modulation. This approach requires special fabrication of the phase LCD, which still has a coupled amplitude modulation of more than 10%; the amplitude-twisted nematic LCD has a typical coupled phase modulation.2 In the twisted nematic LCD, the phase and amplitude modulations are inherently associated. Based on the Berreman theory of the dynamics of the twisted nematic LC cell, Konforti et al.7 showed that phase-only modulation can be obtained when the applied voltage is below an optical threshold and the incident light is polarized parallel to the front molecular director of the cell. Similar conditions may be obtained for the phase-mostly modulation by use of the Jones-matrix theory for twisted nematic LC cells, proposed by Lu and Saleh.8,9 Pezzaniti et al.10 determine the eigenpolarization states experimentally with Mueller-matrix measurements with the imaging polarimeter. The elliptical eigenpolarizationstate input can be modulated for phase only modulation. This regime has no limitation on the bias voltage.10 However, the theoretical explanation to this regime has yet, to our knowledge, to be 10 August 1996 y Vol. 35, No. 23 y APPLIED OPTICS

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developed. Recently, coupled-mode SLM’s found application in optimized correlation filters,11 phasemostly holograms12 and filters,13 ternary-phase and ternary-amplitude filters,14 and ternary wavelet filters.15 Complex-amplitude modulation with two coupledmode SLM’s was discussed by Juday and Florence.4 They proposed the combination of two SLM operating curves by addition. In this paper we demonstrate complex-amplitude modulation with two cascaded, commercially available, coupled-mode liquid-crystal televisions ~LCTV’s!. We develop experimental techniques to manipulate the modulation characteristics of the LCTV to maximize the phase-mostly modulation range and the amplitude-modulation contrast. We use a look-up table encoding method to combine two coupled-mode modulations into the required complex modulation. The cascade architecture performs the multiplication of two modulations. This architecture is easy to align and permits the removal of the moire´ pattern with a spatial filter. Full-range complex modulation is obtained when one of the LCTV’s provides a 2p-range phase shift. We test the complex-amplitude modulation by a two-dimensional ~2-D! complex-valued grating. A number of switchable, optical, complex-valued Fresnel- and Fourier-diffraction holograms are implemented. Our experimental techniques are based on the Jones-matrix theory of the twisted nematic LCTV8,9 developed by Lu and Saleh. This simple theory is verified by experiments,8,9,16 including the new experiments reported in this paper. We determine the twisted angle, the LC molecular direction, and the birefringence parameter by using specially designed experiments and the theory of Lu and Saleh. Those device parameters are predetermined by the manufacturer of the commercial LCTV. 2. Electro-optic Property of the Twisted Nematic LCTV

Characteristics of the twisted nematic LC cells have been studied extensively. Gooch and Tarry17 applied the Jones-calculus approach to determining the optical characteristics of the twisted nematic LC cells.17 Lu and Saleh8,9 introduced a simplified theory with an assumption that the tilt angle of LC molecules is homogeneous inside the LC cell. In this section we briefly review this theory and apply the theory to determine experimentally the device parameters of the LCTV. A.

Jones-Matrix Calculus

A twisted nematic LC cell can be considered as a stack of N birefringent, uniaxial crystal slices, whose extraordinary ~e! axes are parallel to the slice and are rotated helically from the front to the back of the LC cell through the twist angle a. The light propagates along the z axis normal to the LC cell. Assume that the twist angle a is linearly distributed along the z axis. Each slice is rotated by ayN with respect to the previous slice. When there is no applied voltage, no tilt occurs among the LC molecules, and the number 4568

APPLIED OPTICS y Vol. 35, No. 23 y 10 August 1996

Fig. 1. Schematic diagram of the coordinate systems for the polarizer–LCTV–analyzer sandwich.

of slices N shows the trend N 3 `; the overall Jones matrix of the LC cell is expressed in the coordinate system along the extraordinary and ordinary ~o! axes of the LC slice at the front of the LC cell as18 JT 5 ejf

F

3

cos a sin a

2sin a cos a

G

3

4

sin g sin g a g g , sin g sin g 2a cos g 2 jb g g

cos g 1 jb

(1)

with the birefringence parameters b5

pd ~ne 2 no!, l

g 5 Œa 2 1 b 2, f5

pd ~ne 1 no! 5 f0 1 b l

(2)

where d is the LC cell thickness, l is the wavelength, no and ne are the ordinary and extraordinary indices of refraction, respectively, b is the birefringence parameter, and f0 is a constant phase shift, which is ignored hereafter. When an electrical voltage is applied across the LC cell along the z axis, the LC molecules are tilted toward the electric field. With approximations, introduced by Lu and Saleh, that the tilt is independent of the position z and the twist is still linearly distributed along the z axis, the Jones matrix of Eq. ~1! continues to apply. Only ne is replaced by the effective extraordinary index of refraction ne~u!, as a function of the tilt angle u. The value of b depends on the applied voltage. When the applied voltage increases, b decreases with the increase of u. The function ne~u! and the relation between the tilt angle u and the applied voltage Vrms are nonlinear. However, Lu and Saleh8 found a monotonic relation between b and Vrms, which is approximately linear for a large range of b. The LCTV operates between a polarizer and an analyzer. Let c1, c2, and cD be the orientations of

the polarizer, the analyzer, and the molecular director at the front of the LCTV, respectively. All three angles are defined with respect to the vertical axis in a clockwise direction, as shown in Fig. 1. In the coordinate system along the extraordinary and ordinary axes at the front of the LCTV, the incident beam after passage through the polarizer is

FG F G

Ee cos j1 5 , Eo sin j1

(3)

where j1 5 c1 2 cD is the angle between the polarizer and the LC molecular director at the front of the LCTV. The output beam after the analyzer is Eout 5

F G F

Ee9 cos2 j2 5 Eo9 sin j2 cos j2

GF G

sin j2 cos j2 cos j1 JT , 2 sin j2 sin j1 (4)

same LCTV video projector. We need to know cD and a to determine the orientations of the polarizer and analyzer and bmax to determine the range of b. Recently, Soutar and Lu16 proposed a method that determines simultaneously the values of a, bmax, and cD and also an additional scale factor by using three nonlinear curve-fitting procedures. We evaluate a, cD, and bmax directly and individually. The linearregression technique used in our method is simpler than the nonlinear curve-fitting method. First we look for a regime where g 5 np, with n 5 1, 2, . . . . From Eq. ~2! we can see that g depends on b and a. For a given LCTV, b decreases approximately linearly with an increase of the applied voltage, as mentioned in Section 2.A. When we set the analyzer and polarizer orientations parallel, i.e., c1 5 c2 5 c, and rotate them simultaneously, Eq. ~5! becomes

F

with j2 5 c2 2 cD. The intensity transmittance T of the polarizer–LCTV–analyzer sandwich can be obtained from Eq. ~4! as

T 5 cos g cos a 1 a

T 5 uEe9u2 1 uEo9u2

H

F

1 b

a 5 sin g sin@a 1 ~c1 2 c2!# 1 cos g cos@a 1 ~c1 g

J H

J

2

2

b 2 c2!# 1 sin g cos@a 2 ~c1 1 c2! 1 2cD# . g

(5)

The phase shift is

d 5 tan21

G

sin g sin a g

2

G

2

sin g cos~a 2 2c 1 2cD! , g

(7)

where the first term on the right-hand side is independent of the orientation c of the polarizer and analyzer. The second term on that side varies with c. With the rotation of c, maximum transmittance occurs when

Im~Ee9! Im~Eo9! 5 tan21 Re~Ee9! Re~Eo9!

5 b 1 tan21

b sin g cos@a 2 ~c1 1 c2! 1 2cD# g

(6)

a sin g sin@a 1 ~c1 2 c2!# 1 cos g cos@a 1 ~c1 2 c2!# g

Equations ~5! and ~6! show the inherently coupled phase and amplitude modulations of a twist nematic LCD as functions of the orientations of the polarizer and analyzer c1 and c2, respectively, and the birefringence parameter b. The maximum value of b, bmax, occurs when the gray level of the signal is zero and the brightness-bias voltage is minimum. The value of b , bmax decreases with the applied voltage. When b tends toward zero and the LC molecular tilt tends toward 90°, the above theory no longer holds. The twist angle a, the orientation of the front LC molecule direction cD in Eqs. ~5! and ~6!, and the value of bmax are unknown parameters of the device. B. Determination of a, cD, and bmax

The values of the unknown device parameters a, cD, and bmax change from one LCTV to another and are different for the red, blue, and green panels in the

cmax 5

a np 1 cD 6 , 2 2

F

(8)

G F G 2

Tmax 5 cos g cos a 1 a

2

sin g sin g . sin a 1 b g g

(9)

Minimum transmittance occurs when cmin 5

a ~2n 1 1!p 1 cD 6 , 2 4

F

(10)

G

2

sin g sin a . Tmin 5 cos g cos a 1 a g

(11)

According to Eqs. ~8! and ~10!, cmax and cmin are independent of b. To verify this relation we rotated c in a step of 10° and measured T with gray levels of 0, 25, 50, . . . , 150. The results are shown in Fig. 2. 10 August 1996 y Vol. 35, No. 23 y APPLIED OPTICS

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Fig. 2. Intensity transmittance T as a function of the orientation c of the parallel polarizer and the analyzer at five different graylevel values: 0, 50, 75, 100, and 150.

Fig. 4. Linear relation between cos21~=T! and c2, when g 5 np and c1 5 0.

T given in Eq. ~5! becomes The values of cmax 5 38° and cmin 5 82° remain constant for such a large variation in the gray level and the value of ucmin 2 cmaxu 5 44° is also in agreement with the theoretical value ucmax 2 cminu 5 45°, as predicted by Eqs. ~8! and ~10!. This good agreement shows the validity of the above-described theory. According to Eq. ~7!, when g 5 np or b 5 0, the transmittance value of T 5 cos2 a is constant with the simultaneous rotation of c and Tmax 5 Tmin. Figure 3 shows the ratio TminyTmax as a function of the gray level. There is a distinct narrow peak of the ratio TminyTmax at gray level 68, which corresponds to g 5 np. In the high gray-level range TminyTmax increases again, which corresponds to b 5 0, when the above-mentioned theory no longer holds. Now we relax the constraint of c1 5 c2. In the regime in which g 5 np the intensity transmittance

T 5 cos2@a 1 ~c1 2 c2!#.

(12)

For evaluating the twist angle a it suffices to set the analyzer orientation value at c1 5 0 and measure the output intensity as a function of the polarizer orientation c2. The experimental data shown in Fig. 4 show good linearity between cos21=T and c2. The values of a are determined by a linear regression. After a is determined, the orientation cD of the front LC molecules can be obtained from Eqs. ~8! or ~10!. We use the mean value of cmax or cmin determined in the above experiments while simultaneously rotating the polarizer and analyzer at different gray levels. We obtained, for a green LCTV panel, values of cmax 5 38°, cmin 5 82°, a 5 282°, and cD 5 79°. On the basis of Eq. ~5! T will not be changed when cD is changed by py2. The extraordinary and ordinary axes cannot be distinguished by means of the intensity transmittance measurement. An additional measurement of the phase shift d is needed, as shown in Section 3.A, below. Finally, we turned the brightness to minimum and the gray level to zero. Under these conditions we have the value b 5 bmax. We rotated the polarizer and analyzer independently to find the null intensity. The value of bmax can be obtained by the solution of Eq. ~5! numerically, with T 5 0 and the use of the values of a, c1, c2, and cD determined in the abovedescribed experiment. 3. Phase-Mostly and Amplitude-Mostly Regimes A.

Theoretical Regimes

In theoretical equations ~5! and ~6! two special regimes are of interest: Fig. 3. Ratio of the maximum and minimum intensity transmittances, Tmax and Tmin, respectively, for the simultaneously rotated parallel polarizer and analyzer c as a function of gray-level values. 4570

APPLIED OPTICS y Vol. 35, No. 23 y 10 August 1996

1. Phase-mostly modulation: When the polarizer is parallel to the front LC molecular director and the analyzer is parallel to the rear LC molecular

Fig. 5. Intensity transmittance T and phase shift d of the twisted nematic LCD as functions of b for four configurations of the polarizer c1 and the analyzer c2 described in Eqs. ~5! and ~6!: a is the twist angle, and cD is the orientation of the front LC molecules.

director, we have the values c1 5 cD and c2 5 cD 1 a, and Eq. ~5! becomes T512

SD

a a2 2 2 sin g 5 1 2 g p

2

sinc2

g . p

(13)

The intensity transmittance has a set of maxima at g 5 np, where n 5 1, 2, . . . . When g . p or b . ~p2 2 a2!1y2, the variation of T is small. This is a phasemostly regime. Equation ~6! becomes d 5 b 1 tan21

S D

b tang . g

S

D

b tan g . g

T5

(15)

The value of the phase shift becomes very small, as shown in Fig. 5. Thus, the values of cD and cD 1 py2 can be distinguished by the measurement of the phase shifts in two orthogonal orientations of the polarizer and analyzer. 2. Amplitude-mostly modulation: When the polarizer is parallel to the front LC molecular director and the analyzer is perpendicular to the rear LC molecular director, we have the values c1 5 cD and

SD SD a p

2

sinc2

g . p

(16)

High-contrast amplitude modulation can be obtained when the values are g # p or b # ~p2 2 a2!1y2. The coupled phase modulation described in Eq. ~6! becomes

(14)

The phase shift is approximately linear to b when b . ~p2 2 a2!1y2 as shown in Fig. 5. To avoid the region where b , ~p2 2 a2!1y2, where a strong drop in the amplitude occurs, the bias voltage should be turned to minimum such that b0 5 bmax. Thus, in the phase-mostly regime the polarizer should be parallel to the front LC molecular direction and the bias should be minimum, which is similar to the conditions for phase-only modulation in the Berreman model.7 If we rotate both the polarizer and analyzer by an amount py2, then we have the values c1 5 cD 1 py2 and c2 5 cD 1 a 1 py2, Eq. ~6! becomes d 5 b 2 tan21

c2 5 cD 1 a 6 py2, the intensity transmittance T is the complement of the value of T in the phase-mostly mode:

d 5 b 6 mp,

m 5 0, 1, . . . .

(17)

which is equal to b and has a jump of the amount p each time that g passes through the value of p when b increases. B.

Experimental Techniques

From the experiments we need to determine the range of b. Let b0 and b1 be the value of b corresponding to zero and to the maximum gray level, respectively, b0 $ b $ b1. The minimum bias voltage corresponds to b0 5 bmax. Adding the bias makes b0 , bmax. We have another degree of freedom, which comprises the orientations of the polarizer and the analyzer. Changing orientations c1 and c2 can change the curves T 2 b and d 2 b. 1. Phase-mostly regime: We want to maximize the phase-modulation range and avoid the regime in which b , ~p2 2 a2!1y2. Thus, the bias is turned to the minimum such that b0 5 bmax. The maximum phase shift that is due to the maximum gray level, corresponding to the range b1 2 b0, is determined by the manufacturer. The Epson Crystal Image LCTV can produce a 2p phase shift. We minimize the coupled amplitude modulation experimentally in the following way: We use the wedged shear-plate interferometer to measure the 10 August 1996 y Vol. 35, No. 23 y APPLIED OPTICS

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phase shift. This interferometer is very simple and easy to align, permitting the in situ characterization of the SLM.12 We display two vertical bars of grayscale levels of 0 and 255 side by side on the LCTV. Starting from the values c1 5 cD and c2 5 cD 1 a, we rotate the polarizer and analyzer independently until the two gray-level beams have equal intensities after passing through the polarizer–LCTV–analyzer device. During the adjustment we survey and keep the 2p phase shift unchanged. We found that when c1 5 cD 1 16° and c2 5 cD 1 a 2 14°, the amplitude transmittances with gray levels 0 and 255 were equal, the operating curve was a closed circle, and the kinoforms encoded on the LCTV had no zero-order spot in the reconstruction.12 The theoretical explanation is that the rotation of the polarizer and the analyzer can shift the curve T 2 b along the b axis with respect to the fixed point of the value b0 5 bmax, such that T~b0! 5 T~b1!, as shown in Fig. 5. 2. Amplitude-mostly regime: We look for the high-contrast amplitude modulation, which can be obtained by setting a value of b0 5 ~p2 2 a2!1y2. First, we set the gray level to zero, turn the bias to minimum, and look for the null intensity by independently rotating the polarizer and the analyzer. Then, we increase the bias voltage so that we have b0 , bmax, and observe the intensity transmittance T increasing and decreasing again to a new minimum. We continued to increase the bias voltage, but no new null intensity was found. We chose the bias voltage of the last null intensity as the starting point of the amplitude modulation. This value for b0 is at the first minimum of sinc ~gyp!, when T~b0! 5 0. The variable bmax is a device parameter of the LCTV determined by the manufacturer. A value of T~bmax! 5 0 does not necessarily occur when the values c1 5 cD and c2 5 cD 1 a 6 py2 occur, as described by Eq. ~16!. We found orientation values of c1 5 cD 2 1° and c2 5 cD 1 a 2 88° for the green panel and c1 5 cD 1 8° and c2 5 cD 1 a 2 100° for the red panel. In both cases the null-intensity value T~b0! 5 0 guarantees the high contrast in amplitude modulation. 4. Complex-Amplitude Modulator

Figure 6 shows a diagrammed cascade of two LCTV’s that performs the full complex-amplitude modulation for a parallel beam. The two LCTV’s are combined with an afocal imaging system and are aligned for pixel-to-pixel superimposition under the control of a microscope. After careful alignment, a spatial filter is placed in the focal plane of the afocal system to eliminate the grid structure in the image of the first LCTV and to minimize the moire´-pattern effect of the residual alignment error. Each LCTV is placed between a polarizer and an analyzer that are adjusted with the methods mentioned in Section 3.B. The polarizer is replaced by a ly2 plate to rotate the input linear polarization without a loss of energy. LCTV1 is in the phase-mostly regime; LCTV2 is in the amplitude-mostly regime. Both LCTV’s are the green panels of two Epson Crys4572

APPLIED OPTICS y Vol. 35, No. 23 y 10 August 1996

Fig. 6. Schematic diagram of an experimental cascade of two LCTV’s for complex-amplitude modulations: Phase mod., phase modulator; Amp. mod., amplitude modulator.

tal Image™ video projectors, connected to the computer through a frame grabber. Our computer codes permit addressing two monochromatic LCTV’s simultaneously and independently, with a single-color video signal. The phase and amplitude modulations of both LCTV’s were measured with the wedged shearplate interferometer and the photodetector. LCTV1 provides a 2p range of phase modulation. Its coupled-amplitude modulation is less than 6% over the gray-level range of 0 –255. LCTV2 provides high-contrast amplitude modulation coupled with a linear phase modulation from 0 to 0.76p over the gray-level range of 0 –255, as shown in Fig. 7. A.

Full-Range Complex Modulation

The first question we must investigate is whether an arbitrary complex transmittance can be realized by cascading two LCTV’s in the coupled modes. In the cascade architecture the complex transmittances of the two LCTV’s are multiplied. Therefore, the realizable complex-amplitude transmittance is A exp~ jd! 5 A1A2 exp@ j~d1 1 d2!#,

(18)

where amplitude A1 is coupled with phase d1, and amplitude A2 is coupled with d2. The complex transmittances of each LCTV are constrained to be on an operating curve in the complex plane. The multiplication corresponds to rotation by the phase-shift d1 in the complex plane of the operation curve of LCTV2, plus a small radial fluctuation caused by the variation of A1. Obviously, if LCTV1 provides a 2p-range phase modulation, the realizable complex-transmittance values can cover a complete circle whose radius is equal to ~ A2!max~ A1!min, which can be normalized to unity. The effect of the coupled-amplitude modulation in LCTV1 is to reduce the maximum amplitude transmittance. If the maximum phase shift of LCTV1 is less than 2p, the realizable values can no longer cover a complete circle, and the full-range com-

Fig. 7. Coupled phase d and amplitude A modulations: ~a! d from LCTV1, ~b! d from LCTV2, ~c! A from LCTV1, and ~d! A from LCTV2.

plex-amplitude modulation cannot be achieved, as shown in Fig. 8. B. Encoding

The second question we need to investigate is, for a given complex transmittance, how to determine the gray levels, gl1 and gl2, of the two coupled-mode LCTV’s, LCTV1 and LCTV2, respectively. This can be done with a look-up table. First, for all combinations of uniformly sampled gl1’s and gl2’s with 0 # gl1 and gl2 # 255, we map the realizable complex transmittances into a phase–amplitude plane, d 2 A, as shown in Fig. 9, according to the measurement data shown in Fig. 7 and the relation given by Eq. ~18!.

Because d1 is approximately linear to gl1, the distribution of the sampling points on the map along the d axis is uniform. The concentration of the sampling points along the A axis in the region of large value of A is due to nonlinearity in the relation A2 2 gl2. The coupled-phase shift d2 of LCTV2 causes the map to shift up on one side along the d axis because, for the value gl1 5 255, we have values of d1 5 0 and A1 ; 1. Then, when gl2 5 0, we have A 5 A1A2 5 0 and d 5 d2 5 0.76p and, when gl2 5 255, we have A 5 A1A2 5 1 and d 5 0. We then make an inverse map to build the look-up table that yields the values of gl1 and gl2 as functions of the required amplitude A and phase d, as shown in Figs. 10. Both LCTV1 and LCTV2 contribute to the complex transmittance. When a pure amplitude modulation of A is required, gl2 varies in the region from 0 to 255; gl1 also varies to compensate the coupled-phase shift in LCTV2. When a pure phase modulation of d is required, gl1 varies in the region from 0 to 255, and gl2 also varies to compensate the coupled-amplitude modulation of the LCTV1. In the look-up table, gl1 2 ~ A, d!, some complex transmittances in the region d , 0.76p correspond to gl1 5 0. Those transmittances may be realized solely by the amplitude and the coupled-phase modulations of LCTV2. The look-up table encoding method combines the two phase–amplitude coupled modulations to the required complex-amplitude modulation. As the look-up table is based on the experimental data, the effects of the coupled modulations and nonlinearities of the phase and amplitude modulations of both individual LCTV’s are corrected. Figure 11 shows the interferograms of the wedged shear-plate interferometer. First, both LCTV’s display a uniform gray level of 255 @Fig. 11~a!#. Then, LCTV2 displays two vertical bars of 255 in the lefthand side and 125 on the right-hand side @Fig. 11~b!#. The two sides of the interferogram then have different gray levels. In the middle, the shift of the fringes indicates the phase shift associated with the coupled phase modulation in the LCTV2. Finally, in Fig. 11~c!, we use the look-up table to encode a complex amplitude transmittance whose phase is uniform and whose amplitude is equal to 255 and 125 in the left- and right-hand sides, respectively. The interferogram shows that the coupled phase shift in the LCTV2 was corrected by the encoding. C.

Testing

To test the complex modulation realized by cascading two LCTV’s we encode a 2-D complex-amplitude grating by

S D S D

H~u, v! 5 sin 2p

u v exp j2p , a b

(19)

which is a sinusoidal grating in the horizontal u direction and a prism in the vertical v direction. Its Fraunhofer diffraction has two shifted peaks at values of x 5 61ya and y 5 1yb. The 2-D grating 10 August 1996 y Vol. 35, No. 23 y APPLIED OPTICS

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Fig. 8. Example of operating curves for ~a! LCTV1, ~b! LCTV2, and ~c! the realizable complex amplitude in the complex plane. When LCTV1 provides phase modulation that is lower than 2p, the realizable complex amplitude does not cover a complete circle.

function was input to the look-up table, which generated the gray-level functions gl1 and gl2 for LCTV1 and LCTV2, respectively. Figures 12 show the onedimensional plots of the optical diffraction patterns on the horizontal line y 5 1yb and the vertical line x 5 1ya, respectively. These are on-axis diffraction patterns of the complex-amplitude SLM. The encoding errors can cause noise in the position of the zero and the high orders of diffraction: x 5 y 5 0, where x 5 6nya, y 5 6myb. Figure 12~a! shows that noise caused by the encoding error is small. Figure 12~b! is the diffraction of the same grating encoded without the use of the look-up table, but with the assumptions that LCTV1 is phase only, LCTV2 is amplitude only, and both the phase and the amplitude are linear to the gray level. The noise caused by the encoding errors is higher than that shown in Fig. 12~a!. D. Complex-Amplitude Holograms

For computing the Fresnel holograms, the free-space propagation from the Fresnel hologram f ~x, y, 0! to the reconstructed image f ~x, y, z! is considered to be

linear spatial filtering19:

F

F~u, v, z! 5 F~u, v, 0!exp j2p

G

z Œ1 2 ~lu!2 2 ~ln!2 l

(20) where F~u, v, 0! and F~u, v, z! are the Fourier transform of f ~x, y, 0! and f ~x, y, z! with respect to x and y, respectively. For a given image f ~ x, y, z! the complex-valued Fresnel hologram f ~ x, y, 0! is calculated from Eq. ~20! through their Fourier transforms and then encoded with the look-up table on the cascade of the two LCTV’s. The LCTV’s are 2.42 cm 3 2.42 cm in size with 220 3 320 pixels illuminated with a parallel He–Ne laser beam. The two lenses in the afocal imaging system have a focal length of f 5 300 mm. The reconstructed images were observed at a diffraction distance of z 5 3 m. The optical reconstructed images are shown in Figs. 13. We note absence of speckle noise that was observed in the reconstruction of the phase-mostly holograms12 designed with the Gerchbourg–Saxton iterative method, which introduced random phase in the image plane. In the encoding of the complex-amplitude holograms no such random phase was used. The noise in the reconstructed images is caused by the grid structure of the LCTV’s. 6. Conclusion

Fig. 9. Mapped plot of all realizable complex amplitudes A exp~ jd! as functions of gray levels ~a! gl1 and ~b! gl2. 4574

APPLIED OPTICS y Vol. 35, No. 23 y 10 August 1996

We have demonstrated complex-amplitude modulation with two cascaded, coupled-mode LCTV’s. The condition for full-range complex-amplitude modulation is that one of the LCTV’s provides a 2p-range phase modulation. We have developed experimental techniques for determining the parameters of the LCTV and maximizing the phase-mostly modulation range and the amplitude-mostly modulation contrast, minimizing the coupled amplitude modulation in the phase-mostly regime. The look-up table encoding method can correct the phase–amplitude coupling and nonlinearity in the two individual LCTV’s.

Fig. 12. Fraunhofer diffraction patterns of a 2-D grating: ~a! patterns encoded with the look-up table, and ~b! the encoded phase in LCTV1 and amplitude in LCTV2.

(a)

Fig. 10. Look-up tables used to generate gray levels for any required complex-transmittance value: ~a! gray level gl1, and ~b! gray level gl2.

(a)

(b)

Fig. 13. Reconstructed images of switchable complex-valued Fresnel holograms.

(b)

(c)

Fig. 11. Interferograms of the wedged shear-plate interferometer: ~a! the gray levels of both LCTV1 and LCTV2 are uniform, ~b! LCTV2 has two vertical bars of gray levels 255 and 125 and the coupled phase shift can be seen in the middle region, and ~c! the complex amplitude encoded with the look-up table. 10 August 1996 y Vol. 35, No. 23 y APPLIED OPTICS

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The results of experimental testing and the Fresnel hologram show that this approach provides fullrange, continuous, complex-amplitude modulation. This programmable complex-amplitude modulator can have applications in real-time holography and optical pattern recognition. The authors acknowledge Philippe Gagne´ for his help with the computer codes for video signals. This work is supported by the Fonds pour la Formation de Chercheurs et l’Aide a` la Recherche of Que´bec and the Natural Sciences and Engineering Council ~Canada!. L. G. Neto also received support from the Conselho Nacional de Disenvolvimento Cientifico e Technologicio @formerly the Conselho de Pasquisas ~name changed in October 1994!#, Brasilia, Brazil. The e-mail address for L. G. Neto is lgneto@lsi. usp.br. References 1. D. A. Gregory, J. C. Kirsch, and E. C. Tam, “Full complex modulation using liquid-crystal televisions,” Appl. Opt. 31, 163–165 ~1992!. 2. J. Amako, H. Miura, and T. Sonehara, “Wave-front control using liquid-crystal devices,” Appl. Opt. 32, 4323– 4329 ~1993!. 3. H. Bartelt, “Computer-generated holographic component with optimum light efficiency,” Appl. Opt. 23, 1499 –1502 ~1984!. 4. R. D. Juday and J. M. Florence, “Full complex modulation with two one-parameter SLM’s,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 499 –504 ~1991!. 5. J. M. Florence and R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE 1558, 505–516 ~1991!. 6. 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!.

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7. N. Konforti, E. Marom, and S. T. Wu, “Phase-only modulation with twisted liquid-crystal spatial light modulators,” Opt. Lett. 13, 251–253 ~1988!. 8. K. Lu and B. E. A. Saleh, “Theory and design of the liquid crystal TV as an optical spatial light modulator,” Opt. Eng. 29, 240 –245 ~1990!. 9. K. Lu and B. E. A. Saleh, “Complex amplitude reflectance of the liquid crystal light valve,” Appl. Opt. 30, 2354 –2362 ~1991!. 10. J. L. Pezzaniti and R. A. Chipman, “Phase-only modulation of a twisted nematic liquid-crystal TV by use of the eigenpolarization states,” Opt. Lett. 18, 1567–1569 ~1993!. 11. R. D. Juday, “Optimal realizable filters and the minimum Euclidean distance principle,” Appl. Opt. 32, 5100 –5111 ~1993!. 12. L. Gonc¸alves, D. Roberge, and Y. Sheng, “Programmable optical phase-mostly holograms with coupled-mode modulation liquid-crystal television,” Appl. Opt. 34, 1944 –1950 ~1995!. 13. Y. Sheng and G. Paul-Hus, “Optical on-axis imperfect phaseonly correlator using liquid-crystal television,” Appl. Opt. 32, 5782–5785 ~1993!. 14. A. Au, C. S. Wu, S. T. Wu, and U. Efron, “Ternary phase and amplitude modulations using a twisted nematic liquid-crystal spatial light modulator,” Appl. Opt. 34, 281–284 ~1995!. 15. T. H. Chao, A. Yacoubian, B. Lau, and W. J. Miceli, “Optical wavelet processor for target detection,” in Optical Computing, Vol. 10 of OSA Technical Digest Series ~Optical Society of America, Washington, D.C., 1995!, pp. 242–245. 16. C. Soutar and K. Lu, “Determination of the physical properties of an arbitrary twisted-nematic liquid-crystal cell,” Opt. Eng. 33, 2704 –2712 ~1994!. 17. C. H. Gooch and H. A. Tarry, “The optical properties of twisted nematic liquid-crystal structures with twist angles #90°,” J. Phys. D. 8, 1575–1584 ~1975!. 18. A. Yariv and P. Yeh, Optical Waves in Crystals ~Wiley, New York, 1984!, Chap. 5. 19. J. W. Goodman, Introduction to Fourier Optics, ~McGraw-Hill, New York, 1968!, Chap. 3.