 |
Description  |
|
|
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.
BRIEF DESCRIPTION OF THE DRAWINGS
These and other objects and features of the present invention will become
clear from the following description taken in conjunction with the
preferred embodiment thereof with reference to the accompanying drawings,
in which:
FIG. 1 is s schematic perspective view of a liquid crystal spatial light
modulator and the driving part thereof.
FIG. 2 is a schematic sectional view of a pixel of the liquid crystal
spatial light modulator.
FIGS. 3(a), 3(b) and 3(c) are schematic diagrams of the change in the
alignment of liquid crystal molecules according to the applied driving
voltage.
FIG. 4 is a graph of the transmittance plotted against the applied electric
voltage.
FIG. 5 is a graph of the phase distortion plotted against the applied
electric voltage.
FIG. 6 is a diagram of a Lohmann type computer generated hologram.
FIG. 7 is a diagram of a computer generated hologram of Example 1 corrected
for the phase distortion.
FIG. 8 is a diagram of a Lee type computer generated hologram.
FIG. 9 is a graph of a complex transmittance function g(a,b) of a computer
generated hologram in a complex plane.
FIG. 10 is a graph of a complex transmittance function g'(a,b) including
the phase distortion of a computer generated hologram.
FIG. 11 is a flowchart of producing a computer generated hologram of
Example 2.
FIG. 12 is a plan view of an original figure to be reproduced by a computer
generated hologram.
FIG. 13 is a plan view of a reproduced image of the original image of FIG.
12 without correcting the phase distortion.
FIG. 14 is a plan view of a computer generated hologram of Example 2 of the
original image of FIG. 12.
FIG. 15 is a plan view of a reproduced image with use of the computer
generated hologram of FIG. 14.
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:
.PSI.=2.pi.(.DELTA.P/m) (1)
wherein -.pi..ltoreq..PSI.<.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:
.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:
.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).
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:
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
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
.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.
* * * * *
|
|
 |
|
 |
Description |
|
