Kinoform using an electrically controlled
`birefringent liquid-crystal spatial light modulator
`
`Jun Amako and Tomb Sonehara
`
`A programmable kinoform using an electrically controlled birefringent liquid-crystal spatial modulator
`(ECB-LCSLM) is discussed. The LCSLM is capable of continuous phase modulation from 0 to 2'r. For the
`kinoform generation, the phase distribution is calculated by iterative methods and recorded on the
`LCSLM with 16 quantizing levels. We discuss the characteristics and the structure of the LCSLM for the
`implementation of the programmable kinoform while comparing the computed results and optical
`reconstructions.
`Key words: Kinoform, spatial light modulator, liquid crystal, thin-film transistor active matrix,
`electrically controlled birefringent mode, phase shift.
`
`Introduction
`A programmable hologram is an attractive and useful
`device in many applications including laser beam
`steering, optical computing, and three-dimensional
`motion pictures. One of the ways of making a holo-
`gram is to record it on a spatial light modulator by a
`computer-synthesized method. Specifically, the liquid-
`crystal spatial light modulator (LCSLM) is suitable
`because of operation at low voltage, low power con-
`sumption, matrix-addressing capability, and flexi-
`bility of design. Recently there have been reports on
`the programmable computer-generated hologram
`(CGH)'' when commercially available twisted-nem-
`atic, liquid-crystal (TN-LC) display devices were used.
`Since both amplitude and phase modulation occur
`simultaneously in the TN mode, the TN-LC devices
`are not applicable for a phase-only modulator. To
`obtain phase-only modulation, the applied voltage
`must be kept in the range of constant amplitude
`modulation, which is limited below the threshold
`voltage of the amplitude modulation.4 Because the
`phase modulation depth is limited to under 2'r, the
`conventional phase-type CGH requires binary quanti-
`zation of 0 to Tr stable states with equal magnitude of
`amplitude."2 The CGH has high quantization noise.
`On the other hand, when recording an amplitude-
`type CGH, low light efficiency remains a problem,
`
`The authors are with the R&D Division, Seiko Epson Corpora-
`tion, Owa 3-3-5, Suwa-shi, Nagano 392, Japan.
`Received 12 October 1990.
`0003-6935/91/324622-07$05.00/0.
`© 1991 Optical Society of America.
`
`4622
`
`APPLIED OPTICS / Vol. 30, No. 32 / 10 November 1991
`
`although quantization noise can be reduced by using
`to the applied
`a good linearity of transmittance
`voltage. 3
`The objective of this research is to examine the
`feasibility of a programmable kinoform5 by using a
`LCSLM. To record the kinoform, we designed the
`ECB (electrically controlled birefringent)-LCSLM,
`which is characterized by a homogeneous LC layer in
`which the molecular alignment on the input and
`output face is parallel and different from the 900
`alignment of the TN mode. This alignment enabled
`us to utilize the maximum change of birefringence
`when the polarizaiton of the incident light is parallel
`to the LC directors. Therefore, our LCSLM has phase
`modulation from 0 to 27r, keeping the magnitude of
`amplitude constant.
`To drive the LCSLM, we employed a thin-film
`transistor (TFT) active matrix method that has the
`advantages of low cross talk and linear phase modula-
`tion to the applied voltage. By using iterative meth-
`ods,6' 7 we calculated the phase distribution of the pixel
`in the TFT-LCSLM.
`For high-quality reconstruction, we discuss the
`following points: how to calculate and record the
`kinoform, how to optimize the phase modulation
`characteristics of the LCSLM, and how to modify the
`structure of the LCSLM.
`
`Liquid-Crystal Spatial Light Modulator
`Recording the kinoform requires a continuous phase
`modulation between 0 and 2iT and a constant ampli-
`tude. We employed an ECB-LCSLM.
`When fabricating the ECB-LCSLM, the material,
`molecular alignment, and thickness of the LC layer
`
`FNC 1034
`
`

`
`are important factors. These factors should be chosen
`by taking the following points into consideration:
`
`(1) Phase modulation over 2 and linearity of
`phase shift to applied voltage. Nonlinearity increases
`the quantization noise in the reconstructed image.
`(2) No amplitude modulation duringthe full range
`of the phase modulation. Amplitude modulation
`causes degradation in the quality of the reconstructed
`image.
`(3) Thin LC layer. The response of the LCSLM
`becomes slow in proportion to the square of the LC
`layer thickness. Slow response limits the versatility of
`the LCSLM as a programmable optical device.
`
`The LCSLM we fabricated has a homogeneous LC
`layer in which the LC molecules at the off state were
`aligned parallel with the panel faces. The retardation
`is set to 1.25 jim at 632.8-nm wavelength, where the
`birefringence at the off state was 0.209 and the
`thickness of the LC layer was 6.0 m. In a display
`area of 25.6 x 19.8 mm, there are 320 x 220 pixels,
`and the pixel size is 80 x 90 im. Our LCSLM is an
`active matrix type and has a polycrystalline Si TFT
`circuit in each pixel. Our polycrystalline Si TFT
`driving method has been developed for LC television'
`and projection displays.9 An active matrix LCSLM
`displays lower cross talk and better
`linearity of
`electro-optic response to the applied voltage than a
`simple multiplexed-type LC display device. The over-
`view of the LCSLM and its pixel structure is shown in
`Figs. 1(a) and 1(b), respectively. The operating princi-
`ple of the LCSLM has been described elsewhere8 and
`is not covered here.
`
`Light Modulation Characteristics
`First, we measured the light modulation characteris-
`tics of the LCSLM. The experimental results are
`shown in Fig. 2, where the phase shift and intensity
`transmittance are plotted as a function of applied
`voltage. In all the experiments, a linearly polarized
`He-Ne laser (632.8 nm) was used as the light source.
`The collimated laser beam enters the device that is
`located between a pair of parallel polarizers. The
`transmission axes of these polarizers are set in paral-
`lel with the LC directors. Higher diffraction orders
`due to the regular pixel structure of the device were
`filtered out by using a long-focal-length lens and a
`slit. A Mach-Zehnder interferometer was used to
`measure the dependence of the phase modulation on
`applied voltage. The applied voltage in the following
`figures was measured at the external terminal on the
`peripheral circuit of the LCSLM.
`As shown in Fig. 2, we can obtain more than 2 r
`continuous phase modulation and good linearity be-
`tween 2.0 and 3.5 V. However, an unexpected 10%
`modulation of the transmittance is observed. This is
`caused by imperfect alignment of the LC molecules on
`the TFT substrate, on which the transistors appear
`as projections on the surface of the substrate.
`While the + 10% modulation of the transmittance
`still remains, our ECB-LCSLM can produce high-
`performance kinoforms as described in the next sec-
`tion.
`Next, we record a phase grating on the LCSLM as a
`parameter of quantization
`to confirm the blazed
`effect featured by kinforms. The distribution of the
`phase for a grating period and the intensity profile of
`a Fraunhofer diffraction pattern with quantizing
`levels of 2, 4, 8, and 16 are shown in Figs. 3(a)-3(d),
`respectively. One grating period consists of 16 data
`lines of the LCSLM. The intensity profile was de-
`
`1.0
`
`~4
`
`0.5 a)
`0
`0
`
`Cd
`
`i E
`
`s
`
`0
`
`0 .Ad
`* 0
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0
`
`0 0 S SO
`
`k 2.0
`It0
`
`Zua
`
`U2 1.0
`
`H
`
`C 1
`
`1
`
`0
`
`1
`
`5
`
`0
`3
`4
`2
`Applied Voltage (v)
`Fig. 2. Phase shift and transmittance versus applied voltage: 0
`phase shift; *, transmittance. The LCSLM was in the ECB mode
`with the off-state molecular orientation aligned parallel to the
`panel faces.
`
`10 November 1991 / Vol. 30, No. 32 / APPLIED OPTICS
`
`4623
`
`10
`I
`
`20
`I
`
`40
`I
`
`3 30
`(a)
`!
`(a)
`
`50
`
`60
`
`(b)
`Fig. 1. (a) Overview of the LCSLM, (b) its pixel structure.
`
`

`
`a) 27r-
`U]
`Cd4 7 -
`
`P4
`
`0 L
`
`a) 27r
`Co
`4 7r
`
`P4
`
`°
`
`position
`
`d
`
`position
`
`2:
`
`a)
`cn
`
`P4
`
`2:
`
`a)
`Eacd
`
`4
`
`(a)
`
`I
`(b)
`
`I
`
`(c)
`
`+1 0 -1
`
`+1
`
`+1
`
`1P
`
`position
`
`(d)
`Fig. 3. Phase distribution and Fraunhofer diffraction pattern of
`the grating recorded on the LCSLM. The quantizing levels are (a)
`2, (b) 4, (c) 8, and (d) 16. One grating period is marked with the
`letter d. Dead regions between the pixels are omitted in the
`drawings of the phase functions.
`
`tected by a video camera. Two peaks of ± 1 diffraction
`appear with the same strength in Fig. 3(a). This is
`always observed for a binary phase grating. On the
`other hand, the peak of +1 order in Figs. 3(b)-3(d)
`becomes stronger with an increase of quantizing
`levels, while other peaks are well suppressed. The
`residual energy of the zero order, which should
`theoretically be zero, is probably due to a combination
`of some of the following effects: nonlinearity at
`relatively high voltage regions as seen in Fig. 2,
`nonuniformity of phase modulation among pixels,
`and insufficient resolution of 12 lines/mm.
`To support the experiments, some theoretical dis-
`cussion is required. For the multilevel phase grating,
`mth-order diffraction efficiency -rim can be expressed
`by
`
`sinc('rrm/Q)
`
`Q
`I
`k-1 i
`
`exp[i(- - 2'rm)k/Q]/Q
`
`2
`
`(1)
`
`levels and 4, is
`where Q is the number of quantizing
`the phase difference between the top and the bottom
`
`4624
`
`APPLIED OPTICS / Vol. 30, No. 32 / 10 November 1991
`
`of the grating. Equation (1) is derived from the
`Fourier transform of the phase distribution of the
`grating. When the phase-matching condition is met,
`that is, + = 2'T, the efficiencies 'r+, of the + 1 order for
`Q = 2, 4, 8, and 16 are obtained as 0.41, 0.81, 0.95,
`and 0.99, respectively, while the efficiency 'rn of 0
`order is 0 for any values of Q, and the efficiency -, of
`the -1 order is 0.41 for Q = 2 or 0 for others. In
`general, however, some effects as described earlier
`(cid:144) 2wr). Therefore
`introduce phase mismatching (
`+, is decreased while both 'q0 and
`-i are increased,
`which means residual energy exists for 0 and + 1
`orders.
`From these results and analyses, we confirm that
`our system can properly supply the signal to the
`LCSLM, and as a result of this, the LCSLM can
`function as a high-efficiency phase grating.
`
`Writing and Reading a Kinoform
`We describe the method of recording the kinoform on
`the LCSLM and the results of its reconstruction.
`We calculate the phase distribution of the kinoform
`by using iterative methods.6'7 These methods enable
`us to reduce quantization errors by taking into
`account the display performance of a device (the
`LCSLM in our case) and iteratively optimizing the
`kinoform. The principal steps of the calculation pro-
`ceed as follows.
`The kth trial solution for the input image gk(x) is
`Fourier transformed, yielding
`Gk(u) = I Gk(u) I exp1i4(u)],
`
`(2)
`
`where the vector x may be a spatial coordinate, and
`the vector u may be a spatial frequency coordinate.
`Gk(u) is then made to fulfill the constraints in the
`Fourier domain. These constraints are that the Fou-
`rier transform has the constant amplitude of Ak and
`the multilevel quantized phase of k(U). We note that
`Ak should be the average of I Gk(u) and the number of
`phase levels should be determined considering the
`signal resolution supported by the LCSLM and its
`driving unit. In this way the modified transform
`Gk'(u) is given by
`
`Gk'(u) = Ak exp[i4(u)].
`
`(3)
`
`G,'(u) is inverse Fourier transformed yielding the
`complex amplitude of the reconstructed image g,'(X).
`Then the kth iteration is completed by forming a new
`trial gk+1(x) for the next iteration by making gk'(x)
`fulfill the constraints in the object domain. The first
`iteration is started by multiplying an input image
`f (x) with random phase. The iterations continue
`until the error in the reconstructed image is reduced
`and approaches a limit.
`By using computer simulations, we tested the
`ability of two approaches: the error-reduction ap-
`proach6 and the input-output approach.7 We used the
`constraints in the object domain that are defined in
`
`

`
`quantization error. For 16 levels, the error can be
`substantially reduced after a few iterations and shows
`little change after 10-20 iterations. According to
`these results, the phase distribution after 50 itera-
`tions can be recorded on the LCSLM by using a 128 X
`128 pixel field in each case. The optical reconstruc-
`tions of these kinoforms are shown in Fig. 5. When
`the kinoform is quantized into two levels [Fig. 5(a)],
`double images and large nonuniformity of the inten-
`sity are clearly seen as in Fig. 5(b). This is why a
`binary kinoform is not used in practical applications.
`When the kinoform is quantized into 16 levels [Fig.
`5(c)], improvement is noticed. As a consequence, the
`energy dissipation to the conjugate image is well
`suppressed and only the desired image can be brightly
`reconstructed as shown in Fig. 5(d). The phase distri-
`butions shown in the figures are displayed with 16
`gray levels.
`The second example is the binary image of an array
`of spots. The image was sampled on a 128 X 128 grid.
`Figure 6 shows the range of the computed image
`intensity as a function of the number of iterations.
`The image was a line of 1 x 13 spots. The optimized
`phase distribution after 50 iterations was recorded on
`the LCSLM when a 128 x 128 pixel field was used.
`The 1 x 13 spots were optically reconstructed and
`their intensity profiles are shown in Figs. 7(a) and
`7(b), respectively. Figure 8 is the range of computed
`image intensities for an array of 13 x 13 spots. The
`kinoform was recorded on the LCSLM in a way that is
`similar to that in Fig. 7. The 13 x 13 spots were
`optically reconstructed and their intensity profiles
`are shown in Figs. 9(a) and 9(b), respectively, where
`
`! fX
`
`(a)(b
`
`the error-reduction approach by
`
`gk+l
`
`= I f (X) I g'(X)/ gk(X)I,
`
`and in the input-output approach by
`
`gk+i = g9(x) + I[ f(x) I gk '(X)/ gk'(X)I - gk(X)]
`+ En If (x)Igk'(x)/ Igk'(x)| -
`f(x)|gk(x)/Igk(x)].
`
`(4)
`
`(5)
`
`In Eq. (5), f3 is the dumping factor. We set f3 to 1.0 in
`our simulations described below.
`In a series of calculations, parameters are the
`number of quantizing levels, the number of sampling
`points, and the design of input binary images. We
`verified that the input-output approach results in
`faster uniformity with less quantization error than
`the error-reduction approach as pointed out by
`Fienup.7 For this reason we decided to employ the
`input-output approach in our experiments.
`After the phase distribution of the kinoform is
`computed, the corresponding video signal is provided
`by a personal computer. The signal was supplied to
`the LCSLM through buffer memories. Quantization
`of the signal is limited to up to 16 by the analog-to-
`digital conversion board (4-bit resolution) used in our
`experimental system. Fast rewriting of the kinoform
`becomes possible by reading preloaded pattern infor-
`mation from the video memories of the personal
`computer.
`By using the experimental system, reconstruction
`of the kinoform is implemented. The first example is
`for a binary (= 0 or 1) image (a bird). The image was
`sampled on a 128 x 128 grid to compute the kino-
`form. The image intensity is represented by one (one
`bit) for the white part of the sampled area or by zero
`(zero bit) for the black part. Figure 4 shows the range
`of computed image intensities as a function of the
`number of iterations. Here, the results for different
`quantizing levels are compared. For the quantization
`of 2 levels, the iterative process could barely reduce
`
`r. Pone
`
`bit
`
`4.,
`
`1.0
`
`4.,
`P4
`
`ao-
`
`°
`
`0
`
`#
`
`/
`
`one bit;l
`( 2 levels)
`
`1
`
`10
`5
`Number of Iterations
`Fig. 4. Range of the computed image intensity versus the number
`of iterations. A binary image (a bird) was sampled on a 128 x 128
`grid: dashed curves, 2 quantizing
`levels; solid curves, 16 quantizing
`levels.
`
`50
`
`100
`
`(C)
`(d)
`(a), (c) Kinoform of the binary image (a bird), (b), (d) its
`Fig 5
`optical reconstruction. A 128 x 128 pixel field was used to record
`the kinoform. on the LCSLM. The quantizing
`levels were (a), (b) 2
`and (c), (d) 16.
`
`10 November 1991 / Vol. 30, No. 32 / APPLIED OPTICS
`
`4625
`
`

`
`oW§.00X
`
`4.,
`
`04bi
`
`o
`
`50
`
`100
`
`0
`
`1
`
`,zerobi
`1^.-
`10
`5
`Number of Iterations
`Fig. 6. Range of the computed image intensity versus the number
`of iterations. A binary
`image of 1 x 13 spots was sampled on a
`128 x 128 grid. The number of quantizing
`levels was 16.
`
`4.,
`
`4
`
`4.,
`
`0
`
`1
`
`0
`
`o /
`
`z~~~oe bit
`
`06
`5
`zero bit
`
`0
`
`1
`
`10
`5
`Number of Iterations
`Fig. 8. Range of the computed image intensity versus the number
`of iterations. A binary image of 13 x 13 spots was used. The other
`conditions are the same as those in Fig. 6.
`
`50
`
`100
`
`the peak of the zero order was deliberately suppressed
`to observe details of the spots. As seen in the figures,
`the energy of the incident light is well focused on the
`desired points of the output plane. Considering the
`good agreement between the deviation of the com-
`puted intensity and that of the optical reconstruction,
`the nonuniformity of the spots can be mainly due to
`the lack of quantizing levels rather than the nonlinear-
`ity and the nonuniformity of the phase modulation
`characteristics.
`The generation of an array of spots as described
`above can be used for an array illumination in optical
`data processing. An array illuminator efficiently con-
`verts a wide beam into an array of bright spots. These
`
`spots are to be used to illuminate the array of
`microdevices in an optical computer.
`
`Discussion
`As mentioned in the previous sections, the kinoform
`recorded on our LCSLM exhibits successful recon-
`structions. However, a few problems remain to be
`solved. These problems could be solved as follows:
`The first solution is to optimize the LC molecular
`alignment and the retardation of the LC layer. This
`would improve the linearity of phase modulation to
`applied voltage. Another solution would be to compen-
`sate for nonuniformity of the phase modulation char-
`acteristics among pixels. This can be achieved by
`
`(a)
`
`(a)
`
`(b)
`image of 1 x 13 spots, (b) the intensity
`(a) Reconstructed
`Fig. 7.
`profile in the output plane. A 128 x 128 pixel field was used to
`record the kinoform.
`
`(b)
`image of 13 x 13 spots, (b) the intensity
`(a) Reconstructed
`Fig. 9.
`profile in the output plane (including the spot of the zero order).
`The other conditions are the same as in Fig. 7.
`
`4626
`
`APPLIED OPTICS / Vol. 30, No. 32 / 10 November 1991
`
`

`
`controlling the applied voltage to each pixel with
`reference to a lookup table. The nonlinearity and the
`nonuniformity superimpose the undesirable random
`phase error on the optimized phase distribution of the
`kinoform, and the phase error causes quantization
`noise in the reconstructed image. To prove this, we
`performed the computer simulations shown in Fig.
`10. After a phase error is superimposed onto the
`computed kinoforms by using a random number
`generator, the reconstructed images of the letter P
`are displayed on a cathode ray tube. The reconstruc-
`tion without phase error is shown in Fig. 10(a).
`Similarly, reconstructions with phase errors of 2r/
`16, 2/8, and 2wr/4 are shown in Figs. 10(b)-10(d),
`respectively. These computer simulations indicate
`that, as a phase error becomes large, the quantization
`noise in reconstruction increases as well.
`Next we discuss the experimental system. It is
`necessary to use a sophisticated system that can deal
`with quantizing levels of more than 16. To know the
`effect of quantization on the reconstruction of a
`kinoform, we estimated uniformity in the range of the
`image intensity as a function of quantizing levels by
`using computer simulation. The results indicate that
`at least 64 quantizing levels are necessary to obtain
`good uniformity for the binary input image. A typical
`example of a calculation is shown in Fig. 11. The
`array of 13 x 13 spots in Fig. 8 was used and the
`number of iterations
`is 50.
`The final topic is the efficiency of the LCSLM. A
`reflection-type LCSLM has the possibility of higher
`efficiency than a transmission-type LCSLM. In our
`experiments we observed some energy loss due to the
`dead region between pixels and some energy dissipa-
`
`(a)
`
`(b)
`
`(d)
`(C)
`Fig. 10. Reconstructed image from the kinoform with a random
`phase error (computer simulation). The amplitudes of the phase
`error are (a) 0, (b) 2ar/16, (c) 2'rr/8, and (d) 2r/4.
`
`,2.0-
`
`4.,a)
`4.,
`
`a)
`cd 10-
`
`04 ° | X
`
`0
`
`one bit
`
`/ ~~~zero
`bitl
`
`1
`
`10
`5
`100
`50
`Number of Quantizing Levels
`Fig. 11. Range of the computed image intensity versus the
`number of quantizing levels. The binary image of 13 x 13 spots in
`Fig. 8 was used and the number of iterations is 50.
`
`500
`
`tion to higher-order carriers, which produce unde-
`sired images around the reconstructed image. As the
`pixel density increases, undesired images disperse
`from the reconstructed image. This phenomenon is a
`kind of diffraction. The 12-lines/mm pixel density of
`our LCSLM is insufficient to avoid this phenomenon.
`A reflection-type LCSLM enables us to place all the
`elements including TFT's under the pixel electrodes,
`so that the aperture of each electrode and the spatial
`resolution are improved. In addition, a reflection-type
`LCSLM is used to reduce the size of the optics and
`enlarge the output area.
`
`Conclusions
`The ECB-LCSLM displays a wide linear range with a
`phase modulation of 2r suppressing amplitude modu-
`lation without degradation of the reconstruction.
`Using the LCSLM, we recorded the kinoform and
`demonstrated successful reconstructions. Good agree-
`ment was obtained between computed intensity and
`optical reconstruction that helps to clarify that lack of
`quantizing levels, nonlinearity, and nonuniformity of
`the LCSLM dominate the degradation of the recon-
`struction.
`Next we will try to improve the LCSLM and extend
`the study to applications of the kinoform.
`We thank Hirotsuna Miura and Eiji Chino of the
`R&D Division, Seiko Epson Corporation, for their
`technical assistance and Toshio Honda of the Tokyo
`Institute of Technology for his helpful comments.
`
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`
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