The resolution redistribution (RR) technique can increase the horizontal viewing-zone angle and screen size of electronic holographic display. The present study developed a zone plate method that would reduce hologram calculation time for the RR technique. This method enables calculation of an image displayed on a spatial light modulator by performing additions of the zone plates, while the previous calculation method required performing the Fourier transform twice. The derivation and modeling of the zone plate are shown. In addition, the look-up table approach was introduced for further reduction in computation time. Experimental verification using a holographic display module based on the RR technique is presented.
©2011 Optical Society of America
The resolution redistribution (RR) technique can alter the ratio of the horizontal and vertical resolutions of a spatial light modulator (SLM) [1,2]. The RR technique was developed for electronic holographic display to increase the horizontal viewing-zone angle by increasing the horizontal resolution. A modified RR technique  also allows increasing the screen size. This study presents a computation method for electronic holographic display by using an RR technique that is based on the zone plate method and enables reduction in the hologram calculation time.
The viewing-zone angle and screen size of conventional electronic holography using an SLM are limited by pixel pitch and resolution of the SLM. An ultrafine pixel pitch is required for an SLM to obtain a large viewing-zone angle. The viewing-zone angle is given by 2sin−1(λ/2p), where p is the pixel pitch of the SLM and λ is the wavelength of light. An extremely high resolution is required for an SLM to obtain a large screen size; the screen size is given by Nxp × Nyp, where Nx × Ny is the resolution of the SLM. Unlike two-dimensional (2D) displays, pixel pitch cannot be enlarged in order to increase the screen size. For instance, a holographic display with a viewing-zone angle of 30° and a screen size of 10 inches requires an SLM with a pixel pitch of 1 μm and a resolution of 203,000 × 152,000 pixels.
Several techniques have been proposed to enlarge the viewing-zone angle and the screen size of electronic holography, none of which depend solely on the development of ultrahigh-resolution SLMs. In the holographic display system proposed by MIT [4,5], a high-resolution one-dimensional (1D) hologram distribution generated by an acousto-optic modulator is scanned two dimensionally by a mechanical scanner to reduce pixel pitch and increase pixel count. A technique for doubling the viewing-zone angle by using a time-multiplexing technique was proposed [6,7]. An active tiling technique  was also proposed in which demagnified images generated by a high-speed SLM are tiled onto an optically addressed SLM in a time-sequential manner. The use of multiple SLMs was proposed by several authors [9–11]. Another technique is a horizontally scanning holographic display [12–14] in which a series of elementary holograms generated by a high-speed SLM are aligned horizontally by a mechanical scanner. The use of a photorefractive polymer as a rewritable hologram recording material was developed [15,16]. The combination of an eye tracking system and a viewing zone steering system was developed to realize a large hologram with the viewing zone enlarged by an eye tracking technique [17,18].
The hologram display system using the RR technique increases the horizontal viewing-zone angle of a hologram by increasing the horizontal resolution by several times . This technique requires neither mechanical scanning nor time multiplexing. A technique for eliminating a conjugate image was also proposed . The RR optical system was modified to allow the increase in screen size, enabling construction of a hologram display module with a frameless display screen. The 2D arrangement of frameless display modules enables further increase in the screen size .
In order to calculate the distribution displayed on the SLM used in the RR optical system, Fourier transform must be performed twice [1,2]. Therefore, the frame rate of updating the reconstructed images was limited. In the present study, a zone plate method specialized for the RR technique is developed. This technique allows the direct calculation of the distribution displayed on the SLM from the information about object points (i.e., position, amplitude, and phase) that constitute a reconstructed image.
The zone plate method was proposed by Waters  in order to synthesize a computer-generated hologram (CGH). This method was inspired by Rogers’ study , reporting that a Fresnel zone plate can be considered as a hologram. Siemens–Wapniarski et al. demonstrated the synthesis of a hologram by superimposing optically generated zone plates . A hologram produced by the zone plate method generates both a reconstructed image and a conjugate image, similar to an optically produced hologram. Thereafter, it was shown that the use of a half-zone plate for hologram generation and the reconstruction using a 4f optical system with a single-sideband filter made it possible to eliminate the conjugate image [6,22,23]. The conjugate image appears because a complex-conjugate wavefront is added to the wavefront to obtain an amplitude distribution, that can be recorded on a film or displayed on an amplitude modulation SLM. The use of phase-only devices does not generate the conjugate image. However, the use of the phase distribution only, except the amplitude distribution, will degrade the reconstructed image.
In the present paper, after an explanation of the RR technique and the previous calculation method for the SLM display pattern, a zone plate appropriate for the RR technique is derived and modeled. Finally, experimental verification of the developed method is presented.
2. RR technique for holographic display
The modified RR optical system  is illustrated in Fig. 1 . An imaging lens images the display area of an SLM onto a screen lens that has been placed in the image plane. The focal plane of the imaging lens coincides with that of the screen lens so that a plane wave that illuminates the SLM is converged on the common focal plane and is then converted back to a plane wave by the screen lens. A spherical wave emitted from a point light source is converted to a plane wave by a condenser lens to illuminate the SLM. A Fourier transformed image of an image displayed on the SLM is obtained on the common focal plane, i.e., the Fourier plane. The use of mutually coherent multiple point light sources generates multiple Fourier transformed images in the Fourier plane.
Figure 2 shows the arrangement of the multiple Fourier transformed images in the Fourier plane. Because the Fourier plane and the plane where the point light sources are arranged are conjugate, when the point light sources are aligned in a diagonal direction, the multiple Fourier transformed images are also aligned in a diagonal direction. The number of point light sources is denoted by K, and the size of the Fourier transformed images is denoted by w × w. The multiple point light sources are arranged such that the Fourier transformed images are aligned with a horizontal pitch of w and a vertical pitch of w/2K, as shown in Fig. 2. A horizontal slit with a height of w/2K is placed in the Fourier plane. The distribution passed through the horizontal slit has a width of Kw and a height of w/2K. The reshaped distribution consists of different regions of the original Fourier transformed image so that the complex-amplitude of the reshaped distribution can be controlled arbitrarily. The width of the Fourier transformed image increases by K times, and its height decreases by 2K times in the Fourier plane. Therefore, the horizontal resolution in the image plane increases by K times, and the vertical resolution decreases by 2K times. When the pixel pitch of the SLM is denoted by p and the magnification of the imaging system is denoted by M, the horizontal pixel pitch in the image plane is given by Mp/K and the vertical pixel pitch is given by 2KMp. By proper determination of the parameters K and M, both horizontal pixel pitch reduction and image size enlargement can be achieved. Therefore, both the horizontal viewing-zone angle and the screen size of electronic holographic display can be increased. Because the vertical pixel pitch increases, the RR technique can be applied to horizontal-parallax-only (HPO) holography . A vertical diffuser is placed in the imaging plane of the RR optical system to enlarge the vertical viewing zone.
The calculation method  for determining the SLM display pattern is illustrated in Fig. 3 . First, an object wave is calculated with a horizontal pitch of Mp/K and a vertical pitch of 2KMp. Then, the inverse Fourier transform is performed. The inverse-Fourier transformed image, which has a width of Kw and a height of w/2K, is split into K regions in the horizontal direction. The split distributions are rearranged in the vertical direction and the complex conjugate and symmetric distribution of the rearranged distribution is added. The synthesized distribution, with size w × w, is the Fourier transform of the distribution displayed on the SLM. Finally, the second inverse Fourier transform is performed to obtain the distribution displayed on the SLM. The addition of the complex conjugate and symmetric distribution in the Fourier plane makes the SLM image a real-valued distribution. A constant distribution is added to the obtained SLM image to make it positive real so that the image can be displayed on the amplitude-modulated SLM. For electronic holographic displays that use the RR technique, Fourier transform must be performed twice to calculate the distribution displayed on the SLM, in addition to calculating the object wave.
3.1. Zone plate method
The zone plate method is one of several methods for synthesizing CGHs [19,21]. A three-dimensional (3D) object to be reconstructed is represented by an aggregate of object points. Because a zone plate has the ability to converge light to generate a light spot, i.e., an object point, the superposition of the zone plates provides a hologram distribution that produces a reconstructed image.
The conventional zone plate generates both a converging spherical wave and a diverging spherical wave, as shown in Fig. 4(a) . When the converging spherical wave is used to generate the reconstructed image, the diverging spherical wave generates a conjugate image, and vice versa. Therefore, a CGH synthesized by the zone plate method produces both the reconstructed image and the conjugate image, similar to an optically fabricated hologram.
The conjugate image can be eliminated by the single-sideband technique, which employs a 4f optical system with a single-sideband filter . A half-zone plate is used to synthesize a CGH [6,7]. As shown in Fig. 4(b), a spherical wave and a conjugate spherical wave generated by the half-zone plate spatially separate in the Fourier plane because the two waves proceed in opposite vertical directions. The conjugate spherical wave generated by the half-zone plate is blocked by the single-sideband filter placed in the Fourier plane.
In previous studies of the holographic display using the RR technique [1–3], an object wave was calculated by the superposition of spherical waves that generate the object points. Then, Fourier transform was performed twice to generate the SLM display pattern, as described in the previous section. In this study, we derive a zone plate used for the RR technique. The SLM display pattern can be directly calculated by superposition of the derived zone plates, and no Fourier transform is required.
Many object modeling methods are used for hologram calculation. In this study, a 3D object is represented by an aggregate of object points. A 3D object can also be represented by an aggregate of polygons . With the polygon model, an object wave is calculated by superposing wavefronts generated by the polygons. Because the Fourier transform is required to calculate each wavefront generated by each polygon, the polygon method requires more than two Fourier transforms. When the number of object points is small, the object point method requires less calculation time. However, the calculation time increases as the number of object points increases. When the polygon method is used to calculate an object wave for the RR technique, the two Fourier transforms still must be performed to obtain the distribution displayed on the SLM. The distribution that generates one polygon should be derived to avoid performing the two Fourier transforms.
3.2. Derivation of zone plate
The zone plate used for the RR technique is derived on the basis of the calculation method described in Sec. 2.
A spherical wave generating an object point is represented by f(xi, yi), where xi and yi are the coordinates of the image plane. The inverse-Fourier transformed image of f(xi, yi) is represented by F(u, v), where u and v are the spatial frequencies in the Fourier plane. As shown in Fig. 3, the inverse-Fourier transformed image is horizontally split into K rectangular regions, which are then rearranged vertically. Figure 5 depicts the realignment of the j-th rectangular region (−K/2 ≤ j ≤ K/2 − 1). The positions of the rectangular regions before and after the realignment are designated by us = (j + 1/2) w and vd = (K/2 − j − 1/2)(w/2K). The distribution after the realignment of the j-th rectangular region is given by
where the symbol * represents the complex conjugate. The inverse Fourier transform of Gj(u, v) gives the SLM display pattern gj(x, y), which generates the j-th rectangular region of the Fourier plane, where x = xi/M and y = yi/M are the coordinates of the SLM plane:
Because w = 1/p, sinc[w (x−ξ)] = sinc[(x−ξ)/p], which becomes zero with an interval of p except at x = ξ. Therefore, sinc[w(x−ξ)] ≃p δ(x−ξ) at the SLM pixel positions. With this approximation, Eq. (2) becomes
The function f(xi, yi) is a 1D spherical wave because the RR technique is applied to HPO holography. Therefore, the 1D spherical wave that generates an object point at the distance of z is given by
This equation is substituted into Eq. (3):
Finally, the spatial frequencies us and vd are represented by using the region number j:
This distribution has a phase distribution of a spherical wave in the x-direction and that of an inclined plane wave in the y-direction. The horizontal phase distribution ϕx = −π [Mx + λz(j + 1/2)/Mp]2/λz needs to satisfy the sampling theorem, i.e., . The sampling theorem requires that the distribution generating the j-th region of the Fourier plane has a finite width in the SLM display plane.
The width of each distribution generating each rectangular region of the Fourier plane is λz/M 2 p. To generate all rectangular regions of the Fourier plane, K distributions with a width of λz/M 2 p are aligned with a pitch of λz/M 2 p in the horizontal direction in the SLM display plane. Finally, the zone plate for the RR technique is given by the sum of all distributions:
The derived zone plate is illustrated in Fig. 6(a) . The zone plate consists of K regions. Each region has a spherical phase distribution ϕx in the horizontal direction, whose center of curvature lies at the center of each region. Each region has a phase distribution of an inclined plane wave in the vertical direction, ϕy = π(K/2−j−1/2)y/Kp, whose inclination angle depends on the region number j. The amplitude of the derived zone plate, represented by sinc (y/2Kp), varies in the vertical direction. Figure 7(a) shows an example of the derived zone plate when K = 4. A constant amplitude was added to make the distribution non-negative.
3.3. Modeling of zone plate
The zone plate derived in Sec. 3.2 has infinite distribution in the vertical direction and finite distribution in the horizontal direction. Here the vertical distribution of the zone plate is limited in order to reduce the calculation time.
As shown in Fig. 6(b), the zone plate is modeled by restricting its vertical distribution within the main lobe of the sinc function. The maximum amplitude of the first side lobe is 2/3π times that of the main lobe. The height of the modeled zone plate becomes 4Kp. Figure 7(b) shows an example of the modeled zone plate when K = 4.
When the number of zone plates increases, the calculation time increases. If all possible depths of the object points are determined, all distributions of the zone plates can be calculated in advance and the precalculated distributions can be stored in a look-up table. The distribution displayed on the SLM can be calculated by adding the precalculated distributions stored in the look-up table. This look-up table approach considerably reduces the computation time because the distribution is calculated by performing only additions. In order to provide random phases to the object points, zone plates with several initial phases must be prepared. When the number of initial phases is denoted by R, the size of the look-up table becomes R times larger.
4.1. Newly developed hologram display module
A newly developed hologram display module was used in the experiments. This module is briefly explained in this section.
Figure 8 shows the photograph of the module. An optical fiber array consisting of four optical fibers was used as point light sources (K = 4). A polarization beam splitter (PBS) reflects light from the optical fiber array. A combination lens works as a condenser lens as well as an imaging lens. Multiple plane waves illuminate a reflection-type SLM. A liquid-crystal-on-silicon SLM with a resolution of 4,096 × 2,400 and a pixel pitch of 4.8 μm was used. The SLM performed amplitude modulation. The light reflected and modulated by the SLM passes through the combined lens and the PBS to form an image on the screen lens. The focal length of the combined lens was 38.4 mm and the magnification of the imaging system was M = −2.88. The central 3,200 × 1,800 pixels of the SLM were used considering the imaging properties of the combined lens so that the screen size of the module was 2.0 inches. The Fourier plane coincided with one surface of the PBS, to which a horizontal slit with a height of 0.60 mm was attached. A plano-convex lens with a focal length of 100.0 mm was used as the screen lens and placed in the image plane. A lenticular lens was used as the vertical diffuser and attached to the screen lens. A fiber-coupled laser diode with a wavelength of 635 nm was used as the light source. A fiber coupler was used to split the laser light into four optical fibers, which were connected to the optical fiber array. The RR technique changed the resolution to 12,800 × 225 in the image plane. The horizontal pixel pitch became 3.46 μm and the horizontal viewing-zone angle was enlarged to 10.5°.
In the new module, 4k2k SLM was used instead of 2k1k SLM, which was used in the previous module  to increase the horizontal viewing-zone angle. A polarization beam splitter was used instead of the non-polarizing beam splitter that was used in the previous module to increase the light intensity of reconstructed images. A mirror was added to the optical fiber array to attach the optical fiber array to the surface of the polarization beam splitter.
4.2. Experimental results
The developed calculation method was used to generate the distribution displayed on the SLM. The 3D images were reconstructed by the newly developed holographic display module.
A 3D object consisting of 710 object points was reconstructed. The SLM distributions calculated by the previous method, the method using the unmodeled zone plate, and that using the modeled zone plate are shown in Figs. 9(a) , 9(b), and 9(c), respectively. The reconstructed images generated by the holographic display module are shown in Fig. 10 . The reconstructed image was captured from four different directions corresponding to the four times enlargement of the horizontal viewing-zone angle. Obvious differences were not observed among the reconstructed images. The calculation time for the previous method was 1.853 s, including 0.042 s for the object wave calculation, 0.628 s for the first fast Fourier transform (FFT), 0.030 s for the rearrangement in the Fourier plane, and 1.153 s for the second FFT. The calculation was performed using a PC with CPU of Intel Core(TM) i7 870 (3.0 GHz). The calculation using the unmodeled zone plate required 11.438 s. Because the height of each zone plate is equal to that of the SLM, the calculation time was longer than that of the previous calculation method. The calculation method using the modeled zone plate required 0.080 s. The restriction of the height of the zone plates reduced the calculation time. When the look-up table approach was used, the calculation time was reduced even more, i.e., to 0.0075 s. The number of depths was 256 and the number of random phases was 16 so that 4,096 zone plates were calculated in advance. The generated SLM pattern is shown in Fig. 9(d).
The calculation time depends on the number of object points constituting a 3D object. A 3D object consisting of 1,460 object points was reconstructed. The calculation time for the previous method was 1.853 s, including 0.074 s for object wave calculation, 0.593 s for the first FFT, 0.030 s for the rearrangement in the Fourier plane, and 1.156 s for the second FFT. The calculation using the modeled zone plate required 0.141 s. The look-up table approach required a calculation time of 0.0134 s. The reconstructed image generated by the SLM image calculated by the look-up table method is shown in Fig. 11 .
The calculation methods were applied to the case of a 3D object represented by the combination of a texture image and a depth image. The resolution of the texture and the depth images was 400 × 225 pixels. Because the object points were not generated for the black pixels in the texture image, the total number of object points was 15,958. The calculation time for the previous method was 2.694 s, including 0.989 s for object wave calculation, 0.520 s for the first FFT, 0.030 s for the rearrangement in the Fourier plane, and 1.155 s for the second FFT. The calculation using the modeled zone plate required 1.955 s. Because the number of zone plates increased much more than that in the previous two examples, the zone plate method still required a longer calculation time. The look-up table approach reduced the calculation time to 0.0717 s. The reconstructed image obtained by using the look-up table method is shown in Fig. 12 .
From Fig. 9, the distribution generated by the previous method looks identical to that generated using the unmodeled zone plate. The distribution generated using the modeled zone plate is similar to that generated by the look-up table method except for the random phases of the zone plates because the method of adding random phases to the object points differed for the two methods. For the look-up table method, the phases were randomly chosen from 16 phases.
The modeling of the zone plate, i.e., restriction of the zone plate in the vertical direction, might affect the vertical distribution of the object point generated by the zone plate. However, the difference between the object points generated by the two methods was not observable, as shown in Fig. 10. This is because light intensity exists mostly within the main lobe of the sinc function in the vertical direction and light is diffused in the vertical direction by the vertical diffuser placed on the screen.
The look-up table method considerably reduced the calculation time. In the experiments, the calculation was performed by a single-thread program. If the program was implemented as a multithread program, the computation time could be reduced even more. The use of GPU for the calculation can also reduce the computation time.
The look-up table method requires the preparation of memory space to store the precalculated zone plates. The memory size depends on the maximum depth of the reconstructed image because the width of the zone plate depends on its depth. In the experiments described in the previous section, the maximum depth was 15 mm from the screen and was quantized into 256 levels. The number of random phases was 16. The total memory size required for the look-up table was 12.4 MB. Hence, the calculation time and the memory size increase with the maximum depth of 3D images.
For the look-up table method, the calculation time did not increase proportionally with the number of object points. The L3 cache memory size of the CPU was 8 MB. The cache memory might have contributed to the reduction in the calculation time.
In the reconstructed images shown in Figs. 10–12, stray light, caused by the reflection on the surfaces of the lenses and the prism in the module, affected the image quality. Careful anti-reflection coating can improve the image quality.
In the reconstructed images, higher order reconstructed images were not clearly observed. The intensities of these higher order reconstructed images decrease when the aperture ratio of the SLM pixel structure increases. The LCOS-type SLM used in the experiments had a high aperture ratio.
The SLM was illuminated by multiple plane waves having different incidence angles. The maximum incidence angle was 8.4°. The oblique illumination causes a decrease in the contrast of the amplitude modulation of the SLM, thus resulting in a decrease in the diffraction efficiency of the reconstructed image. Therefore, the light intensities are different for four reconstructed images generated by four illumination plane waves. The difference can be corrected by adjusting the optical powers of the illumination light. It can also be corrected by modifying the hologram calculation program. In this study, the corrections were not performed because the difference was not relatively obvious.
The zone plate method appropriate for HPO holography using the RR technique was developed to reduce the time required for hologram calculation. The zone plate was theoretically derived and its modeling was then carried out to reduce the computation time. The use of the look-up table method further decreased the computation time. The developed method was verified using a holographic display module based on the RR technique. The look-up table method required 0.0717 s for a 3D image consisting of 15,958 object points, while the conventional calculation method, which performs the Fourier transform twice, required 2.694 s.
This research is partly supported by the National Institute of Information and Communications Technology (NICT), Japan.
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