Speckle generation is an inherent problem of holography. A speckle-reduction technique employing a time-multiplexing method is proposed. Object points constituting a reconstructed image are divided into multiple object point groups consisting of sparse object points, and the object point groups are displayed time sequentially. The sparseness and temporal summation enable the suppression of speckle generation. The object point group is decomposed into multiple bit planes to represent the grayscale of object points, and binary holograms are generated from the bit plane patterns by using a half-zone plate technique. The binary holograms are displayed by a high-speed spatial light modulator.
©2011 Optical Society of America
Holography [1, 2] is a three-dimensional (3D) display technique that reconstructs the wavefront of light reflected or scattered by objects. Holography is free from several problems, such as visual fatigue caused by accommodation–vergence conflict, discontinuous motion parallax, the puppet theater effect, and the cardboard effect, that are peculiar to ray reconstruction-type 3D displays including multi-view display and integral imaging display. However, speckles are generated in 3D images reconstructed by holography, because coherent light is used during the recording and reconstruction processes. The speckles significantly degrade the quality of 3D images. Because the coherent light does not exist naturally, we see objects under incoherent light illumination. A wavefront reconstruction technique that does not generate speckles is required. In the present study, we propose a time-multiplexing technique to eliminate speckles in hologram reconstruction
Several techniques have been proposed to reduce speckles in the reconstructed images for both optical and electronic holography [3–10]. Almost all of these techniques decrease the temporal or spatial coherence of the reconstruction light. The temporal coherence is decreased using incoherent light sources such as a light emitting diode , varying the phases of light using a rotating phase plate , and time averaging several reconstructed images [7, 9]. The spatial coherence is decreased by diverging the light using a phase grating  and a diffuser [6, 8]. These techniques average out the speckles temporally or spatially. The spatial techniques add blur to the reconstructed images. The temporal techniques do not necessarily add blur to the reconstructed images. The speckles are not extinguished by decreasing the coherence of light. Holography records the interference pattern of an object wave and a reference wave; speckles are recorded in holograms because the object wave contains the speckles. Therefore, a holographic display technique that does not intrinsically generate speckles is required to avoid blurring of the reconstructed images. In digital holography, researchers have proposed several time averaging techniques to reduce the speckles in reconstructed images calculated by a computer .
In addition to the problem of speckle generation, electronic holography has significant limitations on the viewing zone and display screen size. An ultrafine pixel pitch is required for a spatial light modulator (SLM) to obtain a large viewing zone: the viewing zone angle is inversely proportional to the pixel pitch. An extremely high-resolution is required for an SLM to obtain a large display screen, because the pixel pitch cannot be enlarged to increase the screen size. For instance, a holographic display with a viewing zone angle of 30° and a 10-inch screen requires an SLM with a pixel pitch of 1 μm and a resolution of 203,200 × 152,400 pixels. Several techniques have been proposed to overcome these problems [12–18]. However, their image quality is not satisfactory; the speckles degrade the image quality. Moreover, the grayscale representation of the reconstructed images is unsatisfactory, because a high dynamic range is required for the SLM to display the interference fringes of holograms. The dynamic range required for the SLM to display the interference patterns is higher than that required for the reconstructed images. The grayscale representation of holographic displays should be comparable to that of the modern two-dimensional displays. Several studies have been performed to improve the grayscale representation of hologram reconstruction [19, 20].
In the present study, a time-multiplexing holographic display technique is proposed that not only eliminates speckles from the reconstructed images but also improves the grayscale representation of the reconstructed images.
The speckles treated in this manuscript are limited to those that appear in the reconstructed images of the computer generated holograms. Because the object points constituting a reconstructed image are usually given random phases, the interference among these object points generates speckles in the reconstructed images.
2.1 Speckle elimination
First, we explain speckle elimination in hologram reconstruction using a time-multiplexing technique.
In the present study, we assumed that a reconstructed image consists of an aggregate of object points, as shown in Fig. 1 . Speckle generation can be considered as interference between these object points. To eliminate speckles, the object points are divided into object point groups, denoted by Gt, which consist of sparse object points. The distances between object points in each group are made sufficiently large so as to not interference among the points in each group. All the object point groups are sequentially displayed in a time-multiplexing manner. Thus, interference does not occur between different object point groups because they are displayed at different times. Hence, no interference occurs during hologram reconstruction, and speckle-free hologram reconstruction can be achieved.
A high-speed SLM is used to display the object point groups. As shown in Fig. 2(a) , each group consists of equally spaced object points in the horizontal and vertical directions. The object points are divided into M groups horizontally and N groups vertically. All the object points are divided into M × N object point groups, denoted by Gmn (0 ≤ m ≤ M − 1, 0 ≤ n ≤ N − 1). We use a half-zone plate to generate an object point, because the use of the half-zone plate enables easy removal of the conjugate image from the hologram reconstruction [21–23]. Figure 2(b) shows the way in which the half-zone plates are displayed on the SLM to generate the object point groups Gmn. The SLM’s display screen is divided into many rectangular areas aligned two-dimensionally, and one zone plate is displayed in one rectangular area. The positions of the rectangular areas are shifted both horizontally and vertically to display different object point groups. Figure 2 shows the case in which M = 4 and N = 2.
The resolution of the SLM is represented by X × Y pixels. The size of the rectangular area for displaying the half-zone plate is represented by S × (S/2) pixels. The height of the zone plate is half of its width because the viewing zone is halved in the vertical direction to eliminate the conjugate image . Thus, each object point group consists of (X/S) × (2Y/S) object points. The hologram patterns are shifted by S/M pixels horizontally and S/2N pixels vertically to generate different object point groups. The total number of object points constituting the reconstructed image is (MX/S) × (2NY/S).
2.2 Grayscale representation
The grayscale representation of the reconstructed images is also improved by a time-multiplexing technique.
For the grayscale representation, each object point group is produced by displaying multiple hologram patterns on the high-speed SLM. As shown in Fig. 3 , an object point group with grayscale values is represented by multiple bit patterns. The bit plane decomposition method is used. The binary pattern corresponding to the q-th bit plane is denoted by Gmnq. Binary level object points are generated by displaying the half-zone plates on the high-speed SLM referring to the binary pattern Gmnq, as described in Sec. 2.1. When the binary pattern Gmnq is generated, the SLM is illuminated by a laser light whose intensity is proportional to 2q. When each object point group is decomposed into Q binary patterns (0 ≤ q ≤ Q − 1), the reconstructed images with 2Q gray levels can be produced.
The intensity of the illumination laser light can be modulated using light intensity modulators, such as an acousto-optic modulator and an electro-optic modulator. The intensity can also be modulated by pulse width modulation of a laser, e.g., using direct current modulation of a laser diode.
3. Speckle-free display conditions
A hologram display condition that does not generate speckles is considered.
Each half-zone plate, after passing through the 4f imaging system, generates a spherical wave converging to an object point. The spot size of the object point must be smaller than the interval between the object points in each object point group in order to suppress interference among the object points. When we assume that the half-zone plate provides a perfect spherical wave by neglecting the effects of sampling caused by the pixel structure of the SLM and the effects of the light amplitude quantization caused by the SLM, the spot size is inversely proportional to the size of the half-zone plate. From the sampling theorem, the width and height of the half-zone plate are λz/p and λz/2p, respectively, where λ is the wavelength of the light, z is the depth of the object point, and p is the pixel pitch of the SLM . When the size of the half-zone plate is smaller than the rectangular area, the horizontal and vertical spot sizes are p and 2p, respectively. When the zone plate size is larger than the rectangular area, the spot sizes are inversely proportional to the rectangular area. Because the size of the rectangular area is (Sp) × (Sp/2) and the diffraction pattern of this rectangular area at the distance of z is given by the Fourier transform of this rectangular area, the intensity distribution of the object point is proportional to sinc2(Spx/λz) sinc2(Spy/2λz). Because the width of the main lobe of the sinc function gives the spot size, the horizontal and vertical spot sizes are λz/Sp and 2λz/Sp, respectively. The horizontal and vertical spot sizes are illustrated in Fig. 4 . The spot sizes increase as the depth of the object points increases. Therefore, the interference between object points becomes stronger when the object points are displayed further from the screen. The horizontal and vertical intervals of the object points are equal to the width and the height of the rectangular area, which are given by Sp and Sp/2, respectively. The vertical spot size is twice as large as the horizontal spot size, and the vertical interval of the object points is half of the horizontal interval. Therefore, the vertical interference must be considered. In this study, we assume that the vertical interval should be at least K times larger than the vertical spot size to avoid interference, as illustrated in Fig. 5 , i.e., Sp/2 ≤ K (2λz/Sp). With this assumption, the required depth of the object point is z ≤ S 2 p 2/4Kλ. This means that speckles might not be generated when the object points are displayed within the maximum depth zmax = S 2 p 2/4Kλ. Proper determination of the value of K is important for effective speckle elimination. In this study, the value of K is experimentally determined, as described in Sec. 4.2.
When the object point groups are displayed within the depth zmax, the interval of the object points is at least K times larger than their spot size in the vertical direction in each object point group. Therefore, the number of object point groups aligned in the vertical direction should be K or less from Sparrow’s criterion. In this study, we use N = K. The horizontal interval of the object points is twice the vertical interval in each object point group, and the horizontal spot size of the object points is half the vertical spot size. To make the horizontal interval of the object points equal to the vertical one in the reconstructed images, M must be 2K, although Sparrow’s criterion allows M to be as large as 4K. Therefore, the total number of object points in the reconstructed images is (2KX/S) × (2KY/S).
The frame rate of hologram generation depends on the number of object point groups and the number of bit planes used for the grayscale representation. When the frame rate of the high-speed SLM is denoted by f Hz, the frame rate of hologram generation is given by f/MNQ Hz.
Experiments were conducted to explore the display conditions that enable the elimination of speckles. Speckle-free grayscale hologram reconstruction was verified.
4.1 Experimental system
The experimental system is shown in Fig. 6 . A 4f imaging system was used for hologram reconstruction; it consists of two Fourier-transform lenses and has a single-sideband filter on its Fourier plane. The combination of the 4f imaging system and the use of half-zone plates effectively eliminate the conjugate image and zero-order diffraction light from the hologram reconstruction [21–23]. A high-speed SLM is placed on the object plane, and a hologram that does not generate a conjugate image and zero-order diffraction light is obtained on the image plane.
A digital micromirror device (DMD) (DiscoveryTM 3000, Texas Instruments) was used as the high-speed SLM. The resolution was 1,024 × 768, the pixel pitch was 13.68 μm, and the screen size was 0.69 inches. The number of gray levels represented by the DMD decreases when the frame rate increases. We used binary image mode, which yields a maximum frame rate of 13,333 Hz. Therefore, the half-zone plates were displayed as binary images. A laser diode with a wavelength of 635 nm was used as a light source. The focal length of the two Fourier-transform lenses was 150 mm. Pulse width modulation was used to modulate the intensity of the laser diode. A microcontroller was used to control the pulse width. As the microcontroller, an H8/300 microcontroller (Renesas Technology) running at 2.5 MHz was used. The microcontroller receives an image update signal from a DMD driver and generates pulses to modulate the laser diode. The maximum pulse width corresponds to 128 clock cycles. The time-averaged optical power of the pulse-modulated light was measured by an optical power meter and found to be linear according to the number of clock cycles. The grayscale image was decomposed into eight bit planes (Q = 8).
4.2 Speckle-free display conditions
The value of K, which is the ratio of the interval between the object points and their spot size, is experimentally determined for speckle suppression. In the experiments, binary zone plates are used instead of grayscale zone plates. When the number of zones is large, the intensity distribution on the focal plane generated by the binary zone plate is approximately equal to that generated by a lens that has the same diameter as the binary zone plate . In Sec. 3, we assumed the spot size of the object point generated by the binary zone plate to be equal to that generated by a lens that has the same diameter as the binary zone plate. However, this is not always true. Therefore, the value of K is determined experimentally.
Table 1 shows the maximum depth zmax of the object points calculated for K = 1, 2, 3, 4, and 5. The evaluation was conducted for several sizes of rectangular area in which the zone plate is displayed: S = 16, 32, 48, and 64. The object points were displayed at the five maximum depths zmax, and interference between the object points was evaluated. In an object point group, 13 × 13 object points were generated at the center. The graylevel of the generated object points was 255. The phases of the object points were the same because it was easier to evaluate the constructive interference than the destructive interference. The sensor plane of a cooled CCD camera was placed at the plane where the reconstructed images were generated, i.e., at a distance zmax from the screen. Photographs of the object point groups are shown in Fig. 7 . The photographs show the central 5 × 10 object points. The intensity distributions along the vertical axis are shown in Fig. 8 . Constructive interference was observed when K = 3. For all values of S, no interference was observed when K ≥ 4. Therefore, in the remaining part of this study, K = 4 is used.
4.3 Resolution of reconstructed images
The resolution of 3D images generated by the proposed technique was evaluated. The proposed method can generate a limited number of object points. In this study, the resolution is defined as the number of object points generated in the horizontal and vertical directions.
Table 2 shows the resolution of the reconstructed images for several sizes of the rectangular area for the half-zone plate when K = 4. The allowable maximum depths of the reconstructed images are also shown. To verify the resolution of the reconstructed images, the separation of the object points in the reconstructed images was evaluated. In the reconstructed images, 16 × 16 object points with a gray level of 255 in the central area were generated. Figure 9 shows photographs of the reconstructed images when they were displayed at distances of 75.4 mm (zmax), 56.6 mm (3zmax/4), 37.7 mm (zmax/2), and 18.9 mm (zmax/4) from the screen. The cooled CCD camera, whose sensor plane was placed where the reconstructed images were generated, was also used to capture the images. The reconstructed 16 × 16 object points are shown. The intensity distributions of the reconstructed images along the vertical and horizontal axes are shown in Fig. 10 . The separation between the object points disappeared in the vertical direction when the image was displayed at the maximum depth of 75.4 mm. Under Sparrow’s criterion, the resolution of the reconstructed images was 128 × 96 when they were displayed at distances of less than 75.4 mm.
4.4 Grayscale hologram reconstruction
Next, the grayscale representation by the proposed time-multiplexing technique was verified.
The resolution of the reconstructed images was 128 × 96 (M = 8, N = 4, and S = 64.) The grayscale image was decomposed into eight bit planes (Q = 8), so the number of gray levels was 256. The frame rate for displaying the reconstructed images was 52 Hz.
A test pattern was displayed. Eight filled rectangles consisting of 8 × 8 object points with gray levels of 1, 3, 7, 15, 31, 63, 127, and 255 were generated at a distance of 37.0 mm. The gray levels were chosen so that they were represented by the addition of different bit planes, i.e., 1 = 0000 0001b, 3 = 0000 0011b, 7 = 0000 0111b, 15 = 0000 1111b, 31 = 0001 1111b, 63 = 0011 1111b, 127 = 0111 1111b, and 255 = 1111 1111b. Figure 11 shows the reconstructed image captured by a cooled CCD camera having its sensor plane at the plane where the image was reconstructed. The central 32 × 16 object points are shown. The average intensities in the eight rectangles were measured along the two horizontal lines shown in Fig. 11; the results are shown in Fig. 12 . The results show that the proposed technique yields good linearity in the grayscale representation.
4.5 Speckle-free and grayscale hologram reconstruction
Finally, 3D images were generated by the proposed technique. The display conditions are identical to those in the experiment described in Sec. 4.4. The textures and depths of the 3D images are shown in Fig. 13 . Figure 14 shows photographs of the reconstructed images. The centers of the reconstructed images shown in Figs. 14(a) and 14(b) were located at the distances of 21.9 mm and 23.9 mm from the screen, respectively. In contrast to the previous experiments, which used a cooled CCD camera, a digital camera was used to capture the reconstructed images. The digital camera was placed in the observation space, and its focus was adjusted appropriately for the reconstructed images. No speckles were observed in the reconstructed images. The reconstructed images had good grayscale representation, and no flicker was observed.
We discuss few drawbacks of the proposed technique. The use of the binary zone plate limited the system performance. The value of the parameter K was experimentally determined, because the spot size of the object point is affected by the binary structure of the half-zone plate. Therefore, the interference between the object points varied with S. Assuming that an object point is generated by a spherical wave emitted from the rectangular area S × S/2 and neglecting the binary half-zone plate structure, the object point distribution is given by sinc(Sx/λz) sinc(Sy/2λz), where (x, y) denotes the plane parallel to the display screen located at the depth where the object points are generated. In this case, the main lobes of the sinc function of adjacent object points do not superimpose when K = 2, and the first side lobes do not superimpose when K = 3, so that K = 2 or 3 might be sufficient. However, we experimentally found that K = 4 was required because of the degradation of spot size caused by the binary structure of the half-zone plate.
Another drawback of the proposed technique is that it decreases the light intensity of the reconstructed images. The proposed grayscale representation technique requires the intensity modulation of the illumination laser light. The light source modulation decreases the light intensity of the reconstructed images.
The horizontal and vertical viewing zone angles of the reconstructed images were only 2.7° and 1.3°, respectively, which are given by the Eqs. (2)sin−1(λ/2p) and 2sin−1(λ/4p), respectively. The hologram screen size was only 0.69 inches, because it is identical to the SLM screen size. The technique proposed in this study should be combined with the techniques [12–18] that have been developed to increase the viewing zone angle and the screen size.
In the experimental results shown in Figs. 9 and 11, black areas exist between object points in the horizontal direction, because the horizontal and vertical pitches of the object points were made equal by setting M = 2K. The pitch of the object points was 109 μm, so the black area were not observed in the reconstructed images, as shown in Fig. 14. By setting M = 4K, the horizontal resolution can be doubled.
The proposed technique does not allow overlapping of the half-zone plates. Therefore, each hologram pattern has binary amplitudes. When the high-speed SLM can modulate light with multiple amplitude levels, overlapping of the half-zone plates can be allowed, and the number of object points in the reconstructed images can be increased. Because the DMD can perform pure binary amplitude modulation, the highest frame rate mode was chosen in this study.
A hologram display technique that does not generate speckles and improves the grayscale representation of the reconstructed images was proposed. The object points constituting the reconstructed image were divided into several groups consisting of sparse object points. Then, each object point group was decomposed into multiple bit planes to represent the gray levels of the object points. Finally, binary holograms generated from the bit plane patterns were sequentially displayed using a time-multiplexing technique.
A DMD was used as a high-speed SLM, and the speckle-free reconstructed images with 128 × 96 object points with gray levels of 255 were demonstrated. The frame rate for hologram generation was 52 Hz.
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