Method for forming a computer generated hologram

Abstract: New methods for producing computer generated holograms to be used for a liquid crystal spatial light modulator (1) having a plurality of pixels (6), 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 spatial light modulator.

Description: 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 spacial light modulator. 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 as the reference light from the same laser. A computer generated hologram is optical information on 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 spacial light modulator. In other words, an electric voltage applied to each pixel of the liquid crystal spacial light modulator is controlled according to the data of the computer generated hologram so as to modulate spacially the transmittance or the reflectance of pixels. In a layer of twisted nematic type liquid crystal of a spacial light modulator, the longer molecular axes of liquid crystal molecules are twisted by 90° 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 spacially 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 spacial light modulator. It is an object of the present invention to provide a method for producing a computer generated hologram for display with a liquid crystal spacial light modulator, which computer generated hologram not being affected by the phase distortion. A method according to the present invention is producing a computer generated hologram to be recorded in a liquid crystal light modulator having a plurality of cells, each cell comprising a plurality of pixels to be switched between a transmissive and a nontransmissive state, wherein in each cell amplitude and phase information is recorded, the amplitude component being represented by the area of an aperture formed by adjacent transmissive pixels and the phase component being represented by the displacement of the center of the aperture with respect to the center of the cell, the method comprising the steps of: (a) calculating the amplitude component and the phase component for each cell; (b) adding a phase distortion component to said calculated phase component by using previously derived experimental data relating to the phase distortion in a pixel so as to compensate a change in the optical length when the pixels forming the aperture are switched; and (c) modifying the location of the aperture in each cell according to the result of said adding step. It is an advantage of the present invention that a computer generated hologram for display with a liquid crystal spacial light modulator without the effect of the phase distortion can be produced. 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 a schematic perspective view of a liquid crystal spacial light modulator and the driving part thereof: Fig. 2 is a schematic sectional view of a pixel of the liquid crystal spacial 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. 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 spacial light modulator will be explained. Fig. 1 shows an example of a liquid crystal light modulator and its driver. A liquid crystal light modulator 1 is composed of a plurality of cells 21, each cell 21 comprising a plurality of pixels 6 arranged as a two-dimensional matrix. A driver of the liquid crystal 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 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° from the incident side to the outgoing side. The liquid crystal 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 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° in the layer 14, the linearly polarized light in the X 1 direction is converted to the linearly polarized light in the X 2 direction in the outgoing side if no driving voltage is applied to the pixel 6. On the other hand, because the analyzer 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 analyzer 8. Next, the driving method of the pixels 6 of the liquid crystal light modulator 1 is explained below. 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 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 to that of the outgoing light because liquid crystal molecules are twisted by about 90° in the layer 14, as shown in Fig. 3(a). If the signal voltage V s is increased, the twist of the polarizing direction decreases, as shown in Fig. 3(b). If the maximum electric voltage V 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 spacial light modulator 1 with the signal voltage V s for driving (hereinafter referred to as V-T characteristic). The transmittance T can be modulated spacially for each cell by controlling the magnitude of the signal voltage to be applied to a pixel 6 of the liquid crystal light modulator 1. However, the alignment of liquid crystal molecules changes by changing the applied electric voltage V 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 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. 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 as Lohmann type. A cell 21 consists of m * n (8 * 8 in this example) of pixels 6 of a liquid crystal 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 ψ is expressed as a distance ΔP of the center of the aperture 22 from the center of a cell 21 in the horizontal direction. Thus, the phase component ψ is expressed by the following equation: ψ = 2π (ΔP/m) wherein -π ≦ ψ < π 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 ΔP' is expressed by the following equation: ΔP' = mψ / 2π 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 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 represented by the displacement of the center of the aperture 22 with respect to the center of cell 21 and 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 ψ obtained only by the calculation of wave optics, and such a computer generated hologram is displayed in the liquid crystal 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 Φ 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 ψ. That is, the distance ΔP for the phase component is calculated as follows: ΔP = m (ψ + Φ) / 2 π The phase distortion Φ is π/2 as shown in Fig. 5. Thus, the position of the pixels for displaying the phase component is changed according to the corrected distance ΔP of Equation (3). By using such a computer generated hologram for a liquid crystal 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. A method for forming a computer generated hologram called in general as 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 1 + b 1 ) + i(b 1 + b 2 ), wherein g is a complex transmittance function of the Lee type computer generated hologram in a coherent optical system, a 1 is the positive real part of g, a 2 is the negative real part of g, b 1 is the positive imaginary part of g, and b 2 is the negative imaginary part of g. In other words, a 1 is the amplitude component at phase ψ = 0, a 2 is the amplitude component at phase ψ = π, b 1 is the amplitude component at phase ψ = π/2, and b 2 is the amplitude component at phase ψ = 3π/2. Then, the display of this Lee type computer generated hologram with the liquid crystal 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 ψ (= 0, 2/π, π, 3π/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 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 1 , Ta 2 , Tb 1 and Tb 2 defined as the projections of the complex transmittance function to the a 1 , a 2 , b 1 and b 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 1 - axis (ψ = 0) and b 1 -axis (ψ = π/2), and the components, Ta 1 and Ta 2 , along the two axes are defined as the amplitude transmittance in correspondence with a 1 and with a 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 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 Φa 1 and Φb 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) = ∫∫ G(x,y) exp (-2πi(ax + by)) dxdy. Then, g(a,b) is expanded in a form of g(a,b) = (a 1 + a 2 ) + i(b 1 + b 2 ), wherein a 1 = |g(a,b)| cos ψ, if cos ψ > 0 a 1 = 0, otherwise a 2 = |g(a,b)| cos ψ - a 1 , b 1 = |g(a,b)| sin ψ, if sin ψ > 0 b 1 = 0, otherwise and b 2 = |g(a,b)| sin ψ - b 1 . Next, the amplitude transmittances Ta 1 , Ta 2 , Tb 1 and Tb 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 Φa 1 , Φa 2 , Φb 1 and Φb 2 of the pixels 31, 32, 33 and 34 in a cell (array) 31 are obtained from a phase distortion function P(V, Φ) against the applied voitage V 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 snown 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 |g(a,b) - g'(a,b)| / |g(a,b)| ≦ K; 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 |g(a,b) - g'(a,b)|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 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 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 light modulator can be used similarly. It is also possible to produce similarly a computer generated hologram for a reflection type liquid crystal 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.

Claims: A method of producing a computer generated hologram to be recorded in a liquid crystal light modulator having a plurality of cells (21), each cell comprising a plurality of pixels to be switched between a transmissive and a nontransmissive state, wherein in each cell amplitude and phase information is recorded, the amplitude component being represented by the area of an aperture (22) formed by adjacent transmissive pixels and the phase component being represented by the displacement of the center of the aperture with respect to the center of the cell, the method comprising the steps of: (a) calculating the amplitude component and the phase component for each cell; (b) adding a phase distortion component to said calculated phase component by using previously derived experimental data relating to the phase distortion in a pixel so as to compensate a change in the optical length when the pixels forming the aperture are switched; and (c) modifying the location of the aperture in each cell according to the result of said adding step.

Inventors: FUKUI ATSUSHI, ITOH MASAMI, KAWAMURA HIROYUKI, NISHII KANJI, FUKUI, ATSUSHI, ITOH, MASAMI, KAWAMURA, HIROYUKI, NISHII, KANJI
Applicants: MATSUSHITA ELECTRIC IND CO LTD, MATSUSHITA ELECTRIC INDUSTRIAL CO., LTD.
Application year: 1991
Main classification: G02B5
IPC codes: G02B5, G03H1, G06K7, G06E3