Method for forming a computer generated hologram

New methods for producing computer generated holograms to be used for a liquid crystal spacial light modulator having a plurality of pixels, wherein the amplitude component of a coherent wave disturbance is corrected by taking into account the phase distortion due to the twist of liquid crystal molecules of the liquid crystal spacial light modulator.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to a method for forming a computer generated 
hologram to be used for optical information processing by displaying the 
computer generated hologram with a liquid crystal spatial light modulator. 
2. Description of Related Art 
Holography is a technique of three-dimensional optical image formation for 
recording, and later reconstructing, the amplitude and phase distributions 
of a coherent wave disturbance. A hologram is a photographic recording 
obtained by recording the interference fringes between the waves reflected 
from an object and the mutually coherent waves called the reference light 
from the same laser. 
A computer generated hologram is optical information in the form of the 
digital data of the above-mentioned amplitude and phase distributions of a 
coherent wave distributions at a position for recording, and it is 
obtained by computer simulation on the basis of wave optics. Such a 
computer generated hologram is used to display an optical image with use 
of a liquid crystal spatial light modulator. In other words, an electric 
voltage applied to each pixel of the liquid crystal spatial light 
modulator is controlled according to the data of computer generated 
hologram so as to modulate spatially the transmittance or the reflectance 
of pixels. 
In a layer of twisted nematic type liquid crystal of a spatial light 
modulator, the longer molecular axes of liquid crystal molecules are 
twisted by 90.degree. from the incident side to the outgoing side, and the 
polarization of the linearly polarized, incident light is rotated along 
the longer molecular axes of liquid crystal molecules. By applying an 
electric voltage to the liquid crystal molecule layer, the twist of the 
liquid crystal molecules decreases, and the twist of the polarizing 
direction decreases so that the transmittance of the liquid crystal 
molecule layer vary with the applied electric voltage. Thus, the 
transmittance is modulated spatially by controlling the applied electric 
voltage. 
However, when the amplitude component of the incident light is modulated by 
controlling the applied electric voltage, the length of optical path of 
the transmitting or reflecting light varies with the transmittance or the 
twist of liquid crystal molecules according to the applied electric 
voltage. Therefore, the phase distortion is caused by the optical path 
difference between pixels divided by the wavelength of the incident light, 
and the phase distortion varies with the applied electric voltage. If such 
a phase distortion arises in optical information processing in a coherent 
optical system wherein both amplitude and phase of light are processed, 
the modulation of the amplitude component of light accompanies inevitably 
an undesirable change in the phase component. Thus, required optical 
information processing cannot be carried out by using a liquid crystal 
spatial light modulator. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a method for producing 
a computer generated hologram for display with a liquid crystal spatial 
light modulator, which computer generated hologram not being affected by 
the phase distortion. 
A first method according to the present invention for producing a computer 
generated hologram to be used for a liquid crystal spatial light modulator 
having a plurality of pixels to be controlled open completely or to close 
completely, wherein a cell being composed of a plurality of pixels is a 
unit for displaying the amplitude component and the phase component of a 
coherent wave disturbance, and a cell may have an aperture composed of 
pixels adjacent to each other to express the amplitude and phase 
components, the amplitude component for a cell being expressed by the area 
of the aperture, the phase component being expressed by the distance of 
the aperture from the center of the cell, comprises the steps of: (a) 
calculating the amplitude component and the phase component for each cell; 
(b) adding the phase distortion component to the phase component by using 
experimental data of phase distortion of a pixel; and (c) determining the 
center of the aperture in each cell according to the result of the adding 
step and the area of the aperture according to the amplitude component. 
A second method according to the present invention for producing a computer 
generated hologram to be used for a liquid crystal spatial light modulator 
having a plurality of pixels to be controlled to change the transmittance 
of liquid crystal layer continuously, wherein a cell being composed of a 
linear array of pixels is a unit for displaying the amplitude component 
and the phase component of a coherent wave disturbance, the phase 
component being expressed as the position of a pixel assigned to the phase 
in the linear array of pixels, the amplitude component in correspondence 
with the phase component being expressed as the amount of transmitted 
light in said pixel assigned to the phase, comprises steps of: (a) 
calculating the amplitude component and the phase component in a cell; (b) 
estimating the phase distortion in each cell according to the amplitude 
component data obtained in the calculating step by using experimental data 
of phase distortion of a pixel; and (c) correcting the amplitude component 
and the phase component by adding the phase distortion to the phase 
component. 
It is an advantage of the present invention that a computer generated 
hologram for display with a liquid crystal spatial light modulator without 
the effect of the phase distortion can be produced.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
Embodiments of the present invention will be explained below with reference 
to the accompanying drawings. First, the phase distortion when a computer 
generated hologram is displayed with a liquid crystal spatial light 
modulator will be explained. 
FIG. 1 shows an example of a liquid crystal spatial light modulator and its 
driver. A liquid crystal spatial light modulator 1 is composed of a 
plurality of pixels 6 arranged as a two-dimensional matrix. A driver of 
the liquid crystal spatial light modulator 1 consists of a first part 2 
for applying signal electric voltages and a second part 3 for applying 
pixel selection electric voltages. The first part 2 and the second one 3 
are connected to the X electrode lines 4 and the Y electrode lines 5, 
respectively. 
FIG. 2 shows a section of a pixel 6 of the liquid crystal spatial light 
modulator 1. A first transparent electrode 9 and a second transparent 
electrode 10 are applied to a first glass substrate 7 and a second glass 
substrate 13, respectively, and twisted nematic type liquid crystal is 
filled between the two transparent electrodes 9, 10 to form a liquid 
crystal molecule layer 14. A switching element 11 is formed for each pixel 
6 on the second transparent electrode 10, and it is connected to an X 
electrode line 4 and a Y electrode line 5. An analyser 8 and a polarizer 
12 are arranged outside the substrates 7, 13 in the parallel Nicol state. 
Light is incident on the side of the polarizer 12 and is outgoing on the 
side of the analyzer 8. As shown in FIG. 2 schematically, liquid crystal 
molecules of elliptic shape are aligned so that their longer molecular 
axes are twisted by 90.degree. from the incident side to the outgoing 
side. 
The liquid crystal spatial light modulator 1 is driven as follows. The 
light incident on the liquid crystal light modulator 1 is converted to a 
linearly polarizing light in the X.sub.1 direction by the polarizer 12 to 
come into the liquid crystal molecule layer 14. The linearly polarized 
incident light has optical rotator power that the polarization is rotated 
along each longer molecule axis of liquid crystal molecules in the layer 
14. Because the liquid crystal molecules are twisted by 90.degree. in the 
layer 14, the linearly polarized light in the X.sub.1 direction is 
converted to the linearly polarized light in the X.sub.2 direction in the 
outgoing side if no driving voltage is applied to the pixel 6. On the 
other hand, because the analyser 8 and the polarizer 12 are arranged in 
the parallel Nicol state, the outgoing light from the pixel 6 to which no 
electric voltage is applied is absorbed by the analyser 8. 
Next, the driving method of the pixels 6 of the liquid crystal spatial 
light modulator 1 is explained below. The liquid crystal light modulator 1 
is composed of a plurality of pixels 6 arranged as a two-dimensional 
matrix. In order to drive a specified pixel among the pixels 6, the first 
part 2 and the second one 3 apply a pixel selection signal (not shown) and 
a signal voltage V.sub.s (not shown) to the switching element 11 of the 
specified pixel via the corresponding X and Y electrode 4, 5, 
respectively. 
If the signal voltage is zero, the polarizing direction of the incident 
light is perpendicular so that of the outgoing light because liquid 
crystal molecules are twisted by about 90.degree. in the layer 14, as 
shown in FIG. 3(a). If the signal voltage V.sub.s is increased, the twist 
of the polarizing direction decreases, as shown in FIG. 3(b). If the 
maximum electric voltage V.sub.max is applied, the polarizing direction of 
the outgoing light becomes parallel to that of the incident light, as 
shown in FIG. 3(c), so that the light incident to the analyzer 8 goes out, 
while not absorbed by the analyzer 8. 
FIG. 4 shows an example of a relation of the transmittance T of a liquid 
crystal spatial light modulator 1 with the signal voltage V.sub.s for 
driving (hereinafter referred to as V-T characteristic). The transmittance 
T can be modulated spatially for each unit by controlling the magnitude of 
the signal voltage to be applied to a pixel 6 of the liquid crystal 
special light modulator 1. 
However, the alignment of liquid crystal molecules changes by changing the 
applied electric voltage V.sub.s on modulation, so that the optical path 
of the transmitting or reflecting light in a pixel changes according to a 
change in the applied voltage V.sub.s or in the transmittance. Therefore, 
the phase distortion obtained as the difference in optical paths between 
pixels 6 varies with the transmittance as shown in FIG. 5. In optical 
information processing in a coherent optical system, both amplitude and 
phase have to be controlled. However, if a computer generated hologram is 
used to display an image, the modulation of the amplitude component 
accompanies an undesirable change in the phase component shown in FIG. 5. 
This undesirable phase component can be corrected by methods according to 
the present invention, as will be described below. 
EXAMPLE 1 
A first example of a method for forming a computer generated hologram is 
explained below with reference to FIG. 6, which shows the structure of a 
computer generated hologram called generally a Lohmann type. A cell 21 
consists of m n (8 8 in this example) of pixels 6 of a liquid crystal 
spatial light modulator 1. A reference numeral 22 represents an aperture 
located in a cell 21. In a cell 21, the transmittance of the pixels 6 in 
the aperture 22 is one, whereas that of the other pixels 6 displayed with 
crossed hatch lines is zero. The amplitude component for a cell 21 is 
expressed as the area of an aperture 22. Because the width W of an 
aperture 22 is taken as constant, the amplitude component is expressed by 
the height A of the aperture 22. On the other hand, the phase component 
.PSI. is expressed as a distance .DELTA.P of the center of the aperture 22 
from the center of a cell 21 in the horizontal direction. Thus, the phase 
component .PSI. is expressed by the following equation: 
EQU .PSI.=2.pi.(.DELTA.P/m) (1) 
wherein -.pi..ltoreq..PSI.&lt;.pi. and m designates the number of pixels 
corresponding to the distance between the centers of the aperture 22 and 
of the cell 21. In other words, the distance .DELTA.P' is expressed by the 
following equation: 
EQU .DELTA.P'=m.PSI./2.pi. (2) 
As explained above, in a Lohmann type computer generated hologram, a cell 
21 is composed of a plurality of pixels 6, and a pixel 6 in a liquid 
crystal spatial light modulator 1 is controlled to transmit or reflect 
light completely or not. The amplitude component is expressed as the area 
of an aperture 22 in a cell 21, while the phase component is expressed as 
the position of the cell in a direction; the pixels 6 belonging to an 
aperture 22 is controlled to transmit light completely. 
If a Lohmann type computer generated hologram is constructed by using .PSI. 
obtained only by the calculation of wave optics, and such a computer 
generated hologram is displayed in the liquid crystal spatial light 
modulator 1, the phase distortion explained above with reference to FIG. 5 
arises owing to the modulation of the transmittance of the pixels in an 
aperture 22 by applying a signal voltage to the pixels 6 so as to make the 
transmittance one. 
In this example, in order to remove such a phase distortion, another phase 
distortion .phi. which arises from in the pixels 6 owing to the change in 
the optical path of the transmitting or reflecting light is added to the 
phase component .PSI.. That is, the distance .DELTA.P for the phase 
component is calculated as follows: 
EQU .DELTA.P=m(.PSI.+.phi.)/2.pi. (3) 
The phase distortion .phi. is .pi./2 as shown in FIG. 5. Thus, the position 
of the pixels for displaying the phase component is changed according to 
the corrected distance .DELTA.P of Equation (3). 
By using such a computer generated hologram for a liquid crystal spatial 
light modulator 1, the deterioration of a reproduction image of the 
computer generated hologram caused by the phase distortion in the pixels 
can be prevented. Therefore, good coherent holographic optical information 
processing can be realized. 
EXAMPLE 2 
A method for forming a computer generated hologram called in general a Lee 
type will be explained below with reference to FIGS. 7-14. 
FIG. 8 shows the structure of a computer generated hologram of Lee type, 
wherein a cell 31 consists of arrays 31 in the horizontal direction. An 
array 31 has four pixels 31, 32, 33 and 34 in the horizontal direction and 
displays the amplitude and phase components of a Lee type computer 
generated hologram expressed in Equation (4). 
EQU g=(a.sub.1 +a.sub.2)+i (b.sub.1 +b.sub.2), (4) 
wherein g is a complex transmittance function of the Lee type computer 
generated hologram in a coherent optical system, a.sub.1 is the positive 
real part of g, a.sub.2 is the negative real part of g, b.sub.1 is the 
positive imaginary part of g, and b.sub.2 is the negative imaginary part 
of g. 
In other words, a.sub.1 is the amplitude component at phase .PSI.=0, 
a.sub.2 is the amplitude component at phase .PSI.=.pi., b.sub.1 is the 
amplitude component at phase .PSI.=.pi./2, and b.sub.2 is the amplitude 
component at phase .PSI.=3.pi./2. Then, the display of this Lee type 
computer generated hologram with the liquid crystal spatial light 
modulator 1 is carried out by modulating the amplitude transmittance in 
correspondence with the amplitude component for each pixel 32-35 assigned 
for the phase component .PSI.(=0, 2/.pi., .pi., 3.pi./2), as will be 
explained with FIG. 9. 
As explained above, in a Lee type computer generated hologram, a cell is 
composed of a linear array 31 of pixels. A pixel 6 in a liquid crystal 
spatial light modulator 1 is controlled to change the transmittance so as 
to express gradation. The positions of the pixels in an array 31 represent 
the phase component, while the amplitude component is expressed as the 
transmittance of the pixels, that is, as the amplitude transmittance 
Ta.sub.1, Ta.sub.2, Tb.sub.1 and Tb.sub.2 defined as the projections of 
the complex transmittance function to the a.sub.1, a.sub.2, b.sub.1 and 
b.sub.2 axes as shown later in FIG. 9. 
FIG. 9 shows a vector of complex transmittance function g in a complex 
plane. The complex transmittance function expressed as g(a,b) is 
decomposed into two axes, a.sub.1 -axis (.PSI.=0) and b.sub.1 -axis 
(.PSI.=.pi./2), and the components, Ta.sub.1 and Ta.sub.2, along the two 
axes are defined as the amplitude transmittance in correspondence with 
a.sub.1 and with a.sub.2 shown in FIG. 8, respectively. 
If a Lee type computer generated hologram is constructed by using the 
complex transmission function g obtained only by the calculation of wave 
optics, and if such a computer generated hologram is displayed in the 
liquid crystal spatial light modulator 1, the phase distortion explained 
above with reference to FIG. 5 arises owing to the modulation of the 
transmittance of the apertures 32-35 as in Example 1. Because the 
transmittance is controlled to express gradation in a Lee type computer 
generated hologram, the phase distortion also changes the gradation. FIG. 
10 shows the complex transmittance function g'(a,b) including the phase 
distortions .phi.a.sub.1 and .phi.b.sub.1 which arise in the transmission 
through liquid crystal molecules. 
FIG. 11 shows a flow of calculating the complex transmittance function 
g'(a,b) for constructing a computer generated hologram. First, a complex 
transmittance function g(a,b) is calculated (step S1), from a desired 
image G(x,y) to be reproduced from the computer generated hologram: 
EQU g(a,b)=.intg..intg.G(x,y) exp (-2.pi.i(ax+by))dxdy. (5) 
Then, g(a,b) is expanded in a form of 
EQU g(a,b)=(a.sub.1 +a.sub.2)+i(b.sub.1 +b.sub.2), (6) 
wherein 
##EQU1## 
Next, the amplitude transmittances Ta.sub.1, Ta.sub.2, Tb.sub.1 and 
Tb.sub.2 for pixels 32, 33, 34 and 35 in a cell (array) 31 are obtained 
from the complex transmittance function by the method shown in FIG. 8 
(step S2). 
Then, the phase distortions .phi.a.sub.1, .phi.a.sub.2, .phi.b.sub.1 and 
.phi.b.sub.2 of the pixels 31, 32, 33 and 34 in a cell (array) 31 are 
obtained from a phase distortion function P(V, .phi.) against the applied 
voltage V.sub.s shown in FIG. 5, which function has been obtained for 
example by fitting to a polynomial expansion. Next, a complex 
transmittance function g'(a,b) including the phase distortion is 
calculated with the obtained phase distortions by using the method shown 
in FIG. 10 (step S3). 
Then, it is decided if the difference between g'(a,b) and g(a,b) is 
sufficiently small or not (step S4). For example, it is decided if 
EQU .vertline.g(a,b)-g'(a,b).vertline./.vertline.g(a,b).vertline..ltoreq.K;(8) 
and K is taken for example as 0.1. 
If the difference is sufficiently small, the initial g(a,b) is regarded as 
a final value (step S5). If the difference is decided not to be 
sufficiently small, the obtained g'(a,b) is taken as the initial g(a,b) 
(step S6), and the flow returns to step S2 to carry out similar 
calculations successively. 
Though not shown in FIG. 11, an appropriate weight function may be used to 
accelerate the conversion and to decrease 
.vertline.g(a,b)-g'(a,b).vertline. further. 
Next, an example of a reproduction of an image will be shown. FIG. 12 shows 
a desired reproduction image G(x,y), that is, a character "F". 
When, the phase distortion is not corrected for a computer generated 
hologram, a distorted image is reproduced with a liquid crystal spatial 
light modulator 1, as shown in FIG. 13. 
FIG. 14 is a computer generated hologram wherein the phase distortion is 
corrected by the procedure explained above. The size of a dot displays the 
transmittance of a pixel. FIG. 15 is a reproduction image obtained with 
use of this computer generated hologram, wherein the value of K in the 
convergence condition (8) is set to be 0.1. It is clear that the effect of 
phase distortion is compensated largely. 
In the examples explained above, a normally black type liquid crystal 
spatial light modulator is used for the display of computer generated 
holograms. In other words, the modulator normally shades the incident 
light. However, computer generated hologram for a normally white type 
liquid crystal spatial light modulator can be used similarly. 
It is also possible to produce similarly a computer generated hologram for 
a reflection type liquid crystal spatial light modulator. In this case, 
the phase distortion due to the reflecting light is taken into account. 
For example, in a method explained in Example 2, the reflectance is used 
instead of transmittance. 
It is understood that various other modifications will be apparent to and 
can be readily made by those skilled in the art without departing from the 
scope and spirit of the present invention Accordingly, it is nor intended 
that the scope of the claims appended hereto be limited to the description 
as set forth herein, but rather that the claims be construed as 
encompassing all the features of patentable novelty that reside in the 
present invention, including all features that would be treated as 
equivalents thereof by those skilled in the art to which the present 
invention pertains.