We present a novel non-iterative method for generating phase-only Fresnel holograms. The intensity image of the source object scene is first down-sampled with uniform grid-cross lattices. A Fresnel hologram is then generated from the intensity and the depth information of the sampled object points. Subsequently, only the phase component of the hologram is preserved, resulting in a pure phase hologram that we call the sampled-phase-only hologram (SPOH). Experimental evaluation reveals that the numerical, as well as the optical reconstructed images of the proposed phase-only hologram derived with our method are of high visual quality. Moreover, the reconstructed optical image is brighter, and less affected by phase noise contamination on the hologram as compared with those generated with existing error-diffusion approaches.
© 2014 Optical Society of America
In the past 2 decades, numerous efforts have been conducted in the generation and processing of digital holograms. However, albeit the success of these methods, they will not be applicable in practice unless the holograms can be displayed and observed visually. At present, the main bottleneck in the realization of holographic display is that existing devices such as the Spatial Light Modulator (SLM) or the Liquid Crystal-on-Silicon (LCoS), are only capable of presenting either the magnitude, or the phase component of a complex hologram. A straightforward to address the problem is through the optical integration of a pair of SLMs to the real and imaginary components, or the magnitude and phase components of a hologram [1–3]. It is also possible to convert a complex hologram into a double phase hologram, and display it with a 2 phase-only SLM . Despite the effectiveness of these approaches, the optical setup is rather cumbersome and precise alignment between different elements in the display system is needed. The use of a single SLM for displaying a complex hologram, has been studied and reported in [5–8]. In these methods, the orthogonal components of a hologram are displayed in non-overlapping sections in a single SLM. The reconstructed images are combined with a high-resolution grating. Despite the success, the usable area of the display is reduced and precise alignment between different elements in the optical setup is required. A more promising solution is to convert the complex hologram into a phase-only hologram (POH), and display it with a phase-only SLM. Notably Buckley and his co-authors have proposed a method, which is referred to as One-Step-Phase-Retrieval (OSPR) [9–11], that by adding random phase perpetuation to the source image prior to the generation of the hologram, the hologram can be reconstructed with the phase component alone. The method is faster than traditional approach that requires numerous rounds of iterations to derive the POH for a given target source image [12,13]. On the downside, the reconstructed images are rather noisy, and multiple hologram frames, each representing the same source image but added with different random phase noise, have to be presented in rapid succession to reduce the noise contamination. Recently, Tsang et al. have proposed the uni-direction Error Diffusion (UERD) and the Bi-directional Error Diffusion (BERD) methods  which enable a complex Fresnel hologram to be converted swiftly into a POH through the Floyd Steinberg error diffusion . They have shown that the UERD and the BERD holograms are capable of preserving high fidelity on the object image. However, as we shall illustrate in the later part of this paper, the brightness and quality of the reconstructed image are inferior in optical reconstruction. In this paper, we shall propose a novel method for the generation of POHs, so that the above mentioned problems can be alleviated. Our rationale is based on the fact that lines and edges can be reconstructed favorably with the phase component of the hologram alone. As such, we propose to convert, through a simple sampling process, the intensity profile of the object into line patterns prior to the generation of the hologram. Subsequently, only the phase component is retained to be the hologram, and the magnitude is forced to a constant value. Our method, which is referred to as the Sampled-Phase-Only hologram (SPOH), is capable of preserving good visual quality on the object scene. Details on the proposed scheme will be described in the subsequent sections of the paper in the following manner. In section 2, we shall briefly describe the principles of computer-generated holography, and illustrate the effect on the reconstructed image when the magnitude component of the hologram is removed. Next, our proposed method for the generation of POHs will be presented in section 3. Experimental evaluation on our method will be provided in section 4. Finally, we conclude the essential findings in section 5.
2. Effect of removing the magnitude component of a complex hologram
In computer-generated holography, a three-dimensional (3-D) object can be converted into a complex Fresnel hologram with the Fresnel diffraction formula. Consider an object scene which is defined in a 3-D space shown in Fig. 1, with each point represented the intensity image and the depth map . The hologram can be derived as 
The object can be reconstructed from the complex hologram by illuminating the hologram with a coherent optical beam. According to past research , it is well known that reconstruction of phase-only information gives an edge image of the original object, hence leading to heavy distortion if the magnitude component is discarded. To illustrate this issue, a complex hologram is generated from the planar source image shown in Fig. 2(a) with the optical settings: , , and 2048 pixels along the x and y directions. The source image is parallel to, and located at an axial distance of from the hologram. As the numerical reconstructed image of the hologram is almost identical to the source image, it is not shown in here. However, we would like to show the reconstructed image based on the phase component of the hologram (Fig. 2(b)) alone. From the phase component of the hologram, the numerical reconstructed image is obtained with Eq. (2) and shown in Fig. 2(c).Eq. (2), denotes the free-space impulse response given by , and is the convolution operator. From Fig. 2 (c), it can be seen that smooth shaded regions of the image are heavily attenuated. On the other hand, the edges which correspond to stronger diffraction, are retained.
3. Proposed sampled-phase-only hologram (SPOH)
Our novel idea of generating the sampled-phase-only hologram is based on physical intuition that sub-sampling along various directions of the object corresponds to overlaying uniform gratings along the sampled lines of the object. As a grating tends to spread out light in a preferred direction, multiple gratings in various directions will tend to disperse light over many directions, thereby having the effect like a diffuser which results in a more or less constant energy distribution on the reconstruction plane. The method can be divided into 2 stages, and described as follows. In the first stage, the intensity image is down-sampled with uniform grid-cross lattices, resulting in a down-sampled image . The process is equivalent to the union of four down-sampled versions of the intensity image as given byEq. (3) for , with the dark pixels denoted the sampled points, is shown in Fig. 3. As we mentioned earlier, phase-only holographic reconstruction results in energy concentrated along the edge of the image, the down-sampling process in Eq. (2) is equivalent to overlaying a sampling gratings on the object scene which would spread out such concentrated energy to have the reconstructed image looking more appealing visually. In the second stage, the down-sampled image is taken to generate the Fresnel hologram based on Eq. (1), replacing the term with . Subsequently, the magnitude component is discarded and resulted in the SPOH.
4. Experimental results
To demonstrate our proposed method, we have applied it to generate the SPOHs for the binary image “CTU” in Fig. 4(a), based on the previous optical settings, and down-sampling factors of M = 10,12, and 14. The numerical reconstructed images at the focused plane, derived from Eq. (2) with , are shown in Figs. 4(a)-4(c). The results show that good preservation on the shaded area is attained with M = 12, with which the reconstructed images are imposed with a mild, less observable grid-cross pattern. A slight non-uniformality on the intensity, which is quite acceptable visually, is observed as a grating like structure is imposed by the down-sampling. In addition, a high correlation factor of 0.87 is noted between the reconstructed image of the SPOH, and the complex hologram. As such, we have adopted M = 12 in generating the SPOH in the rest of our evaluation. Subsequently, the hologram is inputted to LC-R1080 reflective SLM from HOLOEYE (which has a display area of pixels with pixel pitch of , and phase shift within the range ), and illuminated with a collimated Green laser beam. The optical reconstructed images of the hologram is shown in Figs. 5(a). We have observed that the visual quality of the reconstructed images is favorable, and consistent to the ones obtained with numerical reconstruction. However, the result is degraded as compared with that obtained with numerical reconstruction due to the limited size of the SLM, as well as the imperfection of the optical setup. Next, we illustrate the advantage of our method as compared with phase holograms that are obtained by simply discarding the magnitude component of the hologram, and the 2 recently proposed error diffusion methods in . The optical reconstruction from the phase component of the holograms alone is shown in Figs. 5(b). It can be seen that the reconstructed images are heavily distorted in the shaded areas and energies are concentrated on the edges. The optical reconstructed images of the BERD and the UERD holograms are shown in Figs. 5(c) to 5(d). We have observed that in both methods, the reconstructed images are rather similar and better than those in Figs. 5(b). However, comparing with the reconstructed images of the proposed SPOH holograms, they are significantly dimmer, and lower in visual quality. We perform similar evaluation with the image “Lena’s eye” in Fig. 6(a). The numerical reconstructed image of the SPOH of the image “Lena’s eye” is shown in Fig. 6(b), we observed that favorable visual quality is attained with a high correlation factor of 0.91 as compared with the reconstructed image of the complex hologram. The optical reconstructed image of the hologram for the SPOH, magnitude discarded hologram, BERD, and the UERD holograms are shown in Figs. 7(a)-7(d), respectively. The findings are identical to those obtained with the “CTU” image, with the reconstructed image of the SPOH exhibiting the best visual quality. Finally we would like to show, in Table 1, the maximum intensity (normalized to the range of the numerical reconstructed images of the holograms for the 2 sample images that are generated with our proposed and the other 3 methods. We note that for both cases the maximum intensities of the reconstructed images of the SPOHs are much higher than that of the BERD and the UERD holograms, which result in brighter images. Although the reconstructed images of the magnitude discarded phase-only holograms have the highest intensity, the quality is poor as only the edge contents are preserved.
In this paper, we report a method for generating digital, phase-only Fresnel holograms. The rationale of our method is based on the general phenomena that lines and edge patterns can be preserved with only the phase component of a hologram. As such, we have proposed to down-sample the intensity image with uniform grid-cross lattices prior to the generation of the phase hologram. The hologram generated with our proposed method is referred to as the Sampled-Phase-Only hologram (SPOH). Experimental results demonstrate that the optical reconstructed images of the SPOH have good visual quality, and significantly better than the ones obtained with only the phase component of the complex hologram. We also show that the performance of our method is also superior to the uni-direction error diffusion and the bi-directional error diffusion algorithms reported in  when the holograms are optically reconstructed. Another merit of the proposed SPOH is that it is not susceptible to phase errors presented in the SLM (such as the non-uniformity of the phase over the surface of the SLM) as error diffusion process was not used in generating the phase-only holograms, thereby less affected by phase noise contamination. Although the down-sampling process will impose a pattern on the source image, it is not prominent in the reconstructed image optically. Furthermore the down-sampling process, which only involves the selection of pixels from an intensity image, is near computation-free and only impose negligible overhead on the hologram generation process. Although our proposed method may result in reconstructed image which is lower in quality than phase-only holograms that are derived with iterative methods, it gains significantly in the computation efficiency. As such, our proposed method is more suitable for applications where phase-only holograms have to be computed at high frame rate.
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