Abstract
Keywords
Using conventional spatial light modulators (SLMs) for complex modulation of coherent light beams is a challenging task. Because current SLMs cannot perform full complex modulation on a single panel, the complex holograms generated by computers must be converted into amplitude-only holograms or phase-only holograms[
DPHs have attracted significant attention because they can be implemented using phase-only SLMs, such as liquid crystal SLMs[
In the traditional digital encoding methods, two phases can hardly be interweaved completely into a DPH, whose size is identical to that of an original complex field, without information loss. In this work, we propose a spatiotemporal encoded DPH to suppress information loss by using multiple resampled sub-DPHs together with time multiplexing, which can reduce speckles and expand viewing zones in holographic displays[
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From Fig. 1(a), it is clear that the pixel value of a pixelated complex hologram can be expressed by its amplitude and phase as , where and are the pixel indices. The pixels of the complex hologram are then divided into four groups , , , and consisting of non-adjacent pixels. The neighbors of every pixel in each group are excluded from the same group. Therefore, it is convenient to transform pixels in a group into macro-pixels formed by arrays of phase-only pixels, as is shown in Fig. 2.
Figure 1.Processing procedure for a DPH. (a) Transformation of an original object into its complex hologram. (b) Encoding principle for the single-pixel method. (c) Encoding principle for the macro-pixel method. (d) DPH encoded by the macro-pixel method.
Figure 2.Decomposition of a complex hologram into four sub-holograms and time-sequential uploading onto an SLM in the proposed spatiotemporal multiplexing method.
We assume that the desired complex field is encoded by a phase-only SLM with rectangular pixels having a rectangular pixel-active window with dimensions of and pixel distances of and . The rectangular functions that represent the SLM pixels and complex macro-pixels are defined as
The complex wavefront of the group modulated by a spatially quantized element can be expressed as[
To obtain the encoded complex field, every pixel in the complex hologram is transformed into a macro-pixel by using an array of phase-only pixels to perform complex decomposition [Fig. 1(b)]. The decomposed phase functions are
The sub-DPHs are sequentially uploaded onto a phase-only SLM and reconstructed on the back focal plane of a system. The SLM should be equipped with a high frame rate to ensure that the sub-DPHs are played at an adequately short time interval, so that all the macro-pixels from sub-DPHs are combined for time-sequential display, which are captured by human eyes or detectors. Thus, the complex hologram can be fully reconstructed without down-sampling.
The reconstructed image from the sub-DPH encoded by the macro-pixels can be expressed by[
But, the interweaving of two functions makes it unavoidable to allow part of the noise to pass through and lose part of the signal, resulting in reconstruction noise or signal loss. Since the factor varies with different sub-DPHs, impacts of the noise term are weakened by time-averaging of four sub-DPHs. A bandlimit by the requisite filter in the Fourier plane is eased, and the available bandwidth is thus expanded.
To illustrate the working principles of the proposed spatiotemporal multiplexing method, two test images with the same constant phase but different amplitudes are considered. One has mostly low-spatial-frequency components [Fig. 3(a)] and the other has mostly high-spatial-frequency components [Fig. 3(e)]. Each image was encoded using a single-pixel DPH, four separated sub-DPHs, and a spatiotemporal multiplexing DPH, respectively. The images reconstructed from single-pixel DPH [Figs. 3(b) and 3(f)] contain ground noises up to 79.6 in gray value. The standard deviation for the high-intensity section of Fig. 3(b) is 11.36 in gray value, and that of Fig. 3(f) is 37.73 in gray value. They have inaccurate edges and details, indicating the loss of high-spatial-frequency components. In the reconstructions based on sub-DPHs [Figs. 3(c) and 3(g)], the ground noise is alleviated by macro-pixel encoding to almost null. In the high-intensity sections of the images, there are major distortions, representing complementary shapes among the four reconstructions. When time encoding is introduced to coordinate macro-pixel encoding, intensity multiplication enhances the contrast in the image to an accurate level. The standard deviation for the high-intensity section of Fig. 3(d) is 7.10 in gray value, and that of Fig. 3(h) is 32.57 in gray value.
Figure 3.Numerical reconstructions of test images using different methods. (a) Original amplitude with mostly low-spatial-frequency components. (b) Reconstruction of (a) using a single-pixel DPH. (c) Reconstruction of (a) using sub-DPHs. (d) Reconstruction of (a) using spatiotemporal multiplexing DPHs. (e) Original amplitude with mostly high-spatial-frequency components. (f) Reconstruction of (e) using a single-pixel DPH. (g) Reconstruction of (e) using sub-DPHs. (h) Reconstruction of (e) using spatiotemporal multiplexing DPHs. The red curves represent the original image, and other colored curves represent the reconstructed images.
Figure 4 presents the numerical reconstruction of a complex hologram with an image size of , which is generated from an original image in the University of Southern California-Signal and Image Processing Institute (USC-SIPI) Image Database[
Figure 4.Numerically reconstructed images based on different methods. (a) Original image. (b) Reconstruction using a single-pixel DPH. (c) Reconstruction using one sub-DPH. (d) Reconstruction using spatiotemporal multiplexing DPHs. (e) Curves of PSNRs for the reconstructed images changing with the diameter of filter. (f) Curves of SSIMs for the reconstructed images changing with the diameter of the filter.
To demonstrate the effectiveness of the proposed spatiotemporal multiplexing method experimentally, we implemented the optical setup, as shown in Fig. 5. A coherent beam at is emitted from a solid-state laser acting as a light source. It is then attenuated and expanded before passing through a polarizer and onto a reflective liquid crystal on silicon (LCoS) phase-only SLM (Holoeye Gaya). The SLM has a pixel number of , pixel pitch of 3.74 μm, and frame rate of 60 Hz. The desired complex hologram is decomposed and encoded into four sub-DPHs, , , , and , which are then time-sequentially uploaded onto the SLM at a rate of 60 Hz. The beam splitter (BS) allows for both the plane wave illuminating onto the SLM as well as the reflection towards a system with a filter to block unwanted diffraction orders.
Figure 5.Schematic of the optical system for the proposed DPH method.
Figure 6 shows the optically reconstructed images captured by using a complementary metal–oxide–semiconductor (CMOS) detector. It can be seen that the image reconstructed by the spatiotemporal multiplexing DPH method [Fig. 6(d)] retains most of its original features. Compared to the single-pixel method [Fig. 6(b)], it preserves the edges and most of the details of the image with much less blurring and noise. It has the PSNR of 11.44 dB and SSIM of 0.18, while the image reconstructed by the single-pixel method has only PSNR of 11.30 dB and SSIM of 0.14, respectively.
Figure 6.Optically reconstructed images using different methods with partial enlargement. (a) Original image. (b) Reconstruction using a single-pixel DPH. (c) Reconstruction using one sub-DPH. (d) Reconstruction using a spatiotemporal multiplexing DPH.
In summary, the proposed spatiotemporal multiplexing DPH method can deliver high-quality images through SBP-preserved resampling and increased time-bandwidth product. It improves the capability of DPHs to preserve the details of reconstructed images and suppresses the information loss during hologram interweaving. This makes the spatiotemporal DPHs a suitable method for the digital modulation of both static and quasi-static complex fields with existing SLMs. It can be used in a wide range of applications based on complex holograms, from high-quality holographic display to optical field generation.
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