A fast and non-iterative method for generating a phase-only hologram, hereafter referred to as the patterned-phase-only hologram (PPOH), is reported in this paper. Briefly, a phase mask with a periodic phase pattern is added to the source image, and converted into a hologram. Subsequently, only the phase component is retained as a phase-only hologram. Experimental evaluation reveals that the visual quality of the reconstructed images of the PPOH generated with our proposed method is favorable, and superior to that obtained with existing methods.
© 2017 Optical Society of America
In the past 2 decades, numerous efforts have been conducted in the generation and processing of digital holograms. Despite the success of these methods, they will not be applicable in practice unless the holograms can be displayed and observed visually. Although a digital hologram can be printed on a physical medium with a fringe printer, the content is static and cannot be changed afterwards. For digital holograms representing animated video signal, they are usually displayed with high-resolution devices such as a Spatial Light Modulator (SLM). However, a hologram is a complex-valued image, but existing SLMs are only capable of presenting either the magnitude, or the phase component of a 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 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. On the downside, 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) , that by adding a random phase mask to the source image prior to the generation of the hologram, the latter can be reconstructed with the phase component alone. The method is faster than traditional approach that requires numerous rounds of iterations to derive a phase-only or an amplitude-only hologram for a given target source image (for example [10–12]). On the downside, the lines and edges are fragmented, and the reconstructed images are rather noisy. Degradation on the lines and edges can be alleviated by confining the addition of the phase noise to the smooth region of the source image . However, for the noise contamination, multiple hologram frames, each representing the same source image and which has been added with different random phase noise, have to be generated and presented in rapid succession to reduce the noisy appearance of the reconstructed image. As such, the computation loading on the hologram generation process could be overwhelming. Recently, Tsang et al. have proposed the Bi-directional Error Diffusion (BERD) method [14,15] which enables a complex Fresnel hologram to be converted swiftly into a POH through the Floyd-Steinberg error diffusion . They have shown that a BERD hologram is capable of preserving high fidelity on the object image. Despite the success of this method, error diffusion is a recursive process that is rather time-consuming, and the diffraction efficiency of the reconstructed image is quite low. In another method, known as the sampled-phase-only hologram (SPOH) [17,18], the intensity profile of the object is down-sampled with a uniform lattice prior to the generation of the complex-valued hologram. Subsequently, only the phase component is retained to be the phase-only hologram, and the magnitude is forced to a constant value. This method is capable of preserving good visual quality on the object scene, but resulted in a sparse reconstructed image (i.e., lots of empty holes) that is masked with the texture of the down-sampling lattice. The quality of the reconstructed image can be enhanced by rapidly displaying 2 SPOHs, each deriving from a different down-sampling lattice , a method known as the complementary phase-only hologram (CSPOH). The method, though effective, doubles the computational loading and requires a fast SLM to switch the pair of holograms at higher frame-rate. In this paper, we propose a method to overcome the above mentioned problems. Details of our proposed method and experimental evaluation will be provided in Sections 2 and 3, respectively.
2. Proposed method for generating patterned-phase-only hologram (PPOH)
The concept of our proposed method in generating a PPOH of a source image is outlined as follows. To begin with, a phase mask that is comprising of repetitive blocks of phase pattern is added to the source image. The repetitive phase mask effectuates a diffuser action that is capable of smoothing the magnitude of the diffracted waves to a near uniform distribution. However, as identical phase pattern is added to uniformly spaced local regions of the source image, the reconstructed image of the phase hologram will bear a fine pixelated rather than a noisy appearance. In the rest of this section, we shall describe the realization of our proposed method. Consider an intensity source image that is consisting of X columns and Y rows. Our proposed method for generating a PPOH of the source image can be divided into 4 steps and described as follows.
- Stage 1: The intensity image is partitioned into non-overlapping image blocks of size as shown in Fig. 1.
- Stage 3: For each image block, its pixel value is multiplied with the corresponding phase term in the phase mask to give a modified image given by
where , , and is the modulus operator. Note that the addition of the phase mask does not change the magnitude of the source image.
- Step 4: The intensity image is assumed to be parallel to, and at an axial distance from the hologram plane. A complex-valued hologram is generated as given by
Subsequently, the patterned-phase-only hologram is obtained by retaining the phase component of the hologram:
3. Experimental evaluation
To demonstrate our proposed method, we apply it to generate the PPOHs for the source image “Lenna” in Fig. 2(a), based on , , and M = 4,8, and 12. The source image and the hologram, which are separated at an axial distance of , have sizes of and . The numerical reconstruction of the three PPOHs at the focused plane at 0.03m is shown in Figs. 2(b)-2(d). The peak-signal-to-noise-ratios (PSNR) of the reconstructed images based on M = 4,8, and 12 are 23.97dB, 25.62dB, and 24.7dB, respectively. A larger value of M effectuates a stronger diffuser that smooth out the magnitude to a more uniform distribution, but also cause more distortion to the reconstructed image. Although the visual quality of the 3 reconstructed images are similar, the PSNR is highest for M = 8. As such, we have employed M = 8 in our subsequent evaluation. Next we compare our proposed method with 3 existing non-iterative methods, namely the magnitude removal, noise addition , and sampling [17,18] methods, as well as the Gerchberg-Saxton algorithm (GSA). For each method, we generate the phase-only holograms for axial distances ranging from 0.02m to 0.1m in step of 0.01m. In the magnitude removal method, only the phase component of the hologram is retained as a phase-only hologram. For the sampling method, the image is down-sampled prior to generating the phase-only hologram, and the sampling factor is set to 8 to provide good visual quality. In the GSA method, 5 rounds of back and forth Fresnel transform  are conducted so that the computation time is about 10 times of, and not too far-off from the non-iterative methods. The plots of the PSNRs (based on the source image “Lenna” as the reference image) versus the axial distance of the reconstructed images of the phase-only holograms obtained from each method are shown in Fig. 3. We have observed that the PSNRs of our proposed method is at least 2dB higher than that of the magnitude removal, sampling, and the GSA methods. The PSNR of the noise addition method is rather similar to that of our proposed method. However, as we shall show later, the reconstructed image of a phase-only hologram derived from our proposed method is less noisy than the one obtained with noise addition. Next in Figs. 4(a)-4(e), we display the reconstructed image of the phase-only holograms obtained with different methods. The axial distance is selected to be 0.05m, at which medium performance in terms of PSNR is noted for each method. It can be seen that the visual quality of the reconstructed image corresponding to our proposed method is obvious superior than those of considered existing methods. In comparison with the noise addition method, the reconstructed image of our proposed method has a pixelated appearance, but the noise contamination is less severe.
To further demonstrate our proposed method, we apply it to generate the phase-only hologram of the image “USAF” in Fig. 5(a). The reconstructed images at 0.03m and 0.06m are shown in Figs. 5(b) and 5(c), respectively. We observe that the reconstructed images are similar to the original one and all the fine bar patterns are clearly resolved. Finally, we would like to show the optical reconstructed image of the PPOHs prepared by our proposed method, representing the “Lenna” and the “USAF” image at axial distance of 0.03m, on a phase-only SLM. The SLM has a size of , and a pixel size of . The pair of optical reconstructed images are shown in Figs. 6(a) and 6(b). We can see that the quality of the reconstructed images is favorable, and the images resemble the numerical reconstructed images shown in Figs. 4(a) and 5(b).
In this paper, we have proposed a non-iterative method for generating a patterned phase-only hologram (PPOH). By illuminating the PPOH with a coherent beam, the source image can be reconstructed with favorable quality, and without the need of optical filters or other optical accessories. In comparison with existing methods, our proposed method has three major advantages. First, the additional computation load (on top of the generation of the hologram based on Fresnel diffraction) is negligible. This is an obvious advantage over methods that require multiple rounds of iterations for deriving the phase-only hologram. Second, as the reconstructed image is obtained with a single PPOH, it is unnecessary, as in the case of the OSPR and the CSPOH methods, to display 2 or more holograms in rapid succession on the SLM. Third, the visual quality of the reconstructed image of the PPOH is higher than that of the magnitude removal, sampling, and GSA methods, and also less noisy than the noise addition method.
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