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Abstract
The conjugate image and zero-order beam have considerable influences on the optical reconstructions in computer-generated holographic display systems based on the amplitude spatial light modulators. We propose a generalized single-sideband method for suppressing the unwanted terms in computer-generated holography. Computer-generated holograms (CGHs) are calculated based on frequency filtering of the object wave, which redistributes the diffraction wave in spatial frequency domain for spectrum filtering during optical reconstruction. Numerical simulations and optical experiments demonstrate that the proposed method is generally effective for different kinds of CGH algorithms to reconstruct quality three-dimensional scenes that are free of conjugate image and zero-order beam.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Computer-generated holograms (CGHs) can be used to reconstruct three-dimensional (3D) images without optical recording of the interference pattern [1,2]. The developments of spatial light modulators (SLMs) provide the possibility for dynamically uploading the CGHs to modulate the incident wave, which have been widely applied in electronic holographic 3D displays [3]. In recent years, due to the high space-bandwidth product expansion capability of the digital micromirror device (DMD), it has often been used for holographic 3D display to achieve better viewing parameters with the help of mechanic scanning devices [4–12]. However, since DMD only modulates the amplitude of the incident wave, the optical reconstructions would be significantly deteriorated by the conjugate image and zero-order beam.
For phase holograms, the conjugate image can be easily removed by grayscale modulation of the phase levels. Only zero-order beam need to be suppressed in the reconstructions [13,14]. Whereas for amplitude holograms, the appearances of conjugate image and zero-order beam are directly related to the coding process from the complex object wave to the nonnegative amplitude CGH [15,16]. The conjugate image is reconstructed owing to the Hermitian symmetry of a real valued function, whose Fourier transform is symmetric with its complex conjugate [17]. And since the CGH produces nonnegative amplitude modulation of the incident wave, the dc term would be formed during reconstruction due to the dc bias addressed on the amplitude CGH.
Off-axis recording of the hologram could separate different reconstruction terms in the spatial frequency domain by adding a carrier frequency on the interference pattern [18]. This technique has been widely used in digital holography and holographic display for eliminating the conjugate image and zero-order beam [19–22]. While for SLM based holographic 3D display, the limited spatial resolution of the SLM restricts the carrier frequency of the hologram to satisfy the Nyquist sampling criterion, leading to different reconstruction terms in close proximity. Hence the object wave would be easily interrupted by the unwanted terms during optical reconstruction.
Single-sideband holography was proposed to suppress the unwanted terms for in-line optical holography by filtering out half of the spatial frequency in both recording and reconstruction processes [23–25]. In computer-generated holography, single-sideband method was implemented with zone plate processing to demonstrate the effectiveness in CGHs generated by point based algorithm [26–30]. By calculating limited part of the zone plate pattern for each object point, different reconstruction terms are separated in the spatial frequency domain. In optical reconstruction, a knife edge could be used to filter out the unwanted reconstruction terms. Based on the single-sideband technique and half-zone plate processing, band-limited double-step Fresnel diffraction was developed to reconstruct CGHs generated from depth maps [31]. However, the above single-sideband based techniques were specially designed for the corresponding CGH algorithms, hence their universalities in different CGH algorithms were restricted.
In this study, we propose a generalized single-sideband method for suppressing the conjugate image and zero-order beam in 3D CGH reconstruction. The hologram is generated based on frequency filtering of the object wave, which redistributes the reconstruction terms in the spatial frequency domain. The unwanted terms could be blocked by the single-sideband filter in the Fourier plane during optical reconstruction. The generalized single-sideband method is compatible with different kinds of CGH algorithms, including point based, layer based, and polygon based ones, etc. Numerical simulations and optical experiments are performed to demonstrate the effectiveness of the proposed method. And quality 3D scenes are optically reconstructed with the suppressed conjugate image and zero-order beam.
2. Generalized single-sideband method
According to the basic principle of hologram recording, the interference pattern on the hologram plane is given by:
where I denotes the light intensity distribution on the hologram plane, O and R denote the object wave and reference wave, respectively. The object wave is then coded into the interference pattern, where the phase of the object wave is related to the fringe position, and the amplitude of the object wave is related to the fringe contrast. In CGH calculation, the object wave can be coded into different formats according to the modulation type of the SLMs. For amplitude CGH, the first and second terms of Eq. (1) can be substituted by a constant [15,16]. Hence the hologram can be calculated as:where Δ is the constant offset that can keep the intensity distribution nonnegative. Since the reference wave is a uniform plane-wave in in-line holography. H can be given byThe reconstruction wave is the same as the right side of Eq. (3) when illuminated by the reference wave. Fourier transform of Eq. (3) can illustrate the CGH in spatial frequency domain:where δ represents the dc term, which is located at the center of the spatial frequency domain. Of(u, v) denotes the Fourier transform of the object wave F(O). Since there is no carrier frequency addressed in the hologram as the off-axis holography, different reconstruction terms are overlapped in the spatial frequency domain, as shown in Fig. 1(a). The quality of the reconstructed image would be affected by the conjugate image and zero-order beam when the 3D scene is reconstructed from hologram H.
Fig. 1 Amplitude hologram in spatial and spatial frequency domains: (a) unfiltered CGH, and (b) filtered CGH.
In order to suppress the unwanted reconstruction terms in the in-line CGH, single-sideband filter can be implemented to preprocess the object wave in its spatial frequency domain. Let S represent the single-sideband filter:
The filtered object wave Ofiltered can be calculated as:The object wave is modulated in its spatial frequency domain, where half of its spectrum is filtered out. This operation regularizes the object wave in a restricted spectrum. Hence the single-sideband CGH and its Fourier transform can be calculated as: According to the conjugation property of Fourier transform, Eq. (8) can be deduced as:From Eq. (9) we can see that the spectrum of the object wave is located at lower side of the spatial frequency domain, whereas the spectrum of the conjugate wave is located at upper side of the spatial frequency domain. Different with the overlapped spatial frequency of the unfiltered CGH shown in Eq. (4), the spectrum distributions of the object wave and the conjugate wave are separated with the help of the single-sideband filter, as shown in Fig. 1(b). Redistribution of the reconstruction terms in the spatial frequency domain provides the possibility for filtering of the unwanted terms during optical reconstruction.Figure 2(a) illustrates the process for generating the hologram using generalized single-sideband method. First, the object wave on the hologram plane of the 3D scene is calculated. Then the single-sideband filter is addressed numerically after Fourier transform of the object wave. And the filtered object wave can be calculated by inverse Fourier transform of the spectrum. Finally, the filtered object wave can be coded into the amplitude CGH according to Eq. (7). During optical reconstruction, a 4-f system can be used to perform spectrum filtering, as shown in Fig. 2(b). By placing a single-sideband filter in Fourier plane, the conjugate image is filtered during reconstruction. The zero-order beam can be blocked by moving the single-sideband filter to the other side of the Fourier plane for a small distance. Since the proposed method directly processes the complex amplitude distribution of the object wave, it could be applied to various 3D CGH algorithms such as point based, layer based, and polygon based ones, etc.
Fig. 2 The diagram of generalized single-sideband method for 3D CGH: (a) hologram generation, and (b) hologram reconstruction.
3. Experimental results
To demonstrate the effectiveness of the generalized single-sideband method, a series of numerical simulations and optical experiments with different CGH algorithms are performed. A DMD driven by ViLUX V-9501 module is used for optical reconstructions of the CGHs. The DMD has 1920 × 1080 pixels with pixel pitch 10.8 μm, which are addressed with 256 gray-scale levels. The approximate optical efficiency of the micromirror array is over 60%, and the efficiency of the grayscale CGH is about 10%. It can provide amplitude modulation for the illuminating wave with programmable pulse width modulation technique, which can be used for uploading the amplitude CGHs.
Figure 3 illustrates the optical setup of generalized single-sideband CGH reconstructions. The amplitude CGHs are loaded onto the DMD. A He-Ne laser (632.8 nm) is used to produce the collimated beam for illuminating the CGHs. A 4-f system is then implemented to suppress the conjugate image and zero-order beam. The single-sideband filter is placed at the Fourier plane of the 4-f system to eliminate the unwanted terms. And the reconstructed 3D images are captured directly by a Canon 500D camera.
During calculation, a ship model is used as the 3D object to generate the CGH. The 3D object is located at 150-170 mm from the hologram plane. The amplitude CGH used for reconstruction is generated according to Eq. (7). Figure 4 demonstrates the numerical and optical reconstructions of the generalized single-sideband method for a 3D CGH, and the reconstruction distance is 160 mm. The complex amplitude distribution of the 3D scene is generated with layer based method [32]. Figures 4(a) and 4(e) are the 3D model and the CGH generated using generalized single-sideband method. Figures 4(b) and 4(f) are the numerical and optical reconstruction results without filtering of the hologram. We can see clearly that the reconstruction qualities of the CGH are deteriorated seriously by the conjugate images and zero-order beams. Figures 4(c) and 4(g) are the numerical and optical reconstruction results with zero-order filtering. The zero-order filtering can be implemented with the help of a high-pass filter, which blocks the zero-order beam in the Fourier plane. In numerical simulation, the high-pass filter can be implemented by simply eliminating the dc term. While in the optical reconstruction, the filter can be a transparent plate with a small opaque area in the center. Due to the elimination of the zero-order beam, the dc noise is suppressed, providing better reconstruction quality. But the reconstructed images are still affected by the conjugate images, which produce background noises due to the in-line setup of the CGH model. Figures 4(d) and 4(h) are the numerical and optical reconstruction results using the generalized single-sideband method. The reconstruction results are significantly improved due to the successful elimination of the conjugate images and zero-order beams. Mean square error (MSE) is used for numerically analyzing the reconstruction results of Fig. 4. The MSE is calculated by comparing to the reconstructed image from the original complex amplitude distribution of the hologram plane. We can also see the reconstruction quality is improved by the generalized single-sideband method through MSE values. The MSE of the Figs. 4(b), 4(c) and 4(d) are 0.2022, 0.0095 and 0.0042, respectively.
Fig. 4 Numerical and optical reconstructions of the generalized single-sideband CGH: (a), (e) 3D model and CGH, (b), (f) reconstructions without filtering; (c), (g) reconstructions with zero-order filtering; (d), (h) reconstructions with single-sideband filtering.
Figure 5 demonstrates the numerical and optical reconstruction results of the generalized single-sideband CGH with different reconstruction depths. Figures 5(a)-5(d) are the numerical reconstructions when the reconstruction distances are 150 mm, 155 mm, 160 mm, and 165 mm, respectively. Figures 5(e)-5(h) are the optical reconstructions when the reconstruction distances are 150 mm, 155 mm, 160 mm and 165 mm, respectively. It can be seen that the focusing part of the ship changes along with the reconstruction distance, providing accurate depth information of the 3D image.
Fig. 5 Numerical and optical reconstructions of the generalized single-sideband CGH with different reconstruction depths.
To illustrate the generality of the proposed method, different algorithms are implemented to generate the 3D CGHs for reconstructions. Since the generalized single-sideband method processes the complex amplitude distribution of the object wave, it can be directly applied to different 3D CGH algorithms. First, the complex amplitude distribution of object wave is calculated based on the corresponding CGH algorithm. Then the single-sideband filtered object wave is calculated according to Eq. (6). After getting the filtered object wave distribution on the hologram plane, the final CGH can be generated according to Eq. (7). Figure 6 demonstrates the numerical and optical reconstructions of the generalized single-sideband CGHs generated with the layer based [32], point based [33], and polygon based method [21,34], respectively. The left column shows the reconstruction results of the layer based method when the reconstruction planes are placed at different reconstruction distances. The middle and right columns show the reconstruction results at different reconstruction distances of the point based method and the polygon based method, respectively. It can be seen that the generalized single-sideband method can effectively eliminate the conjugate image and zero-order beam for a variety of CGH algorithms and significantly improve the qualities of the reconstructed images. And the reconstruction results at different reconstruction distances demonstrate that the depth information of the 3D scene can be reconstructed successively.
Fig. 6 Numerical simulations and experimental results of the generalized single-sideband method with different CGH algorithms: layer based method (left column), point based method (middle column), and polygon based method (right column).
4. Discussion
The generalized single-sideband method is implemented with the help of frequency filtering of the object wave to suppress the zero-order and conjugate terms. Hence the optical performances of the proposed method would be affected due to the reducing of the space-bandwidth product. Specifically, the point spread function would expand vertically as the single-sideband filter limits the vertical range of the object wave at the Fourier plane, which would sacrifice the vertical resolution of the system. Moreover, due to the limited spectrum range of the object wave, the vertical viewing angle is also reduced by half during reconstruction. Time-alternating method can be used to compensate the vertical viewing angle by synchronizing SLM and the single-sideband filter [26].
Random phases are imposed to the 3D objects in the experiments demonstrated in Sec. 3. In consequence, speckle noise can be observed in the reconstructions. Random phase-free method has been proposed to suppress the speckle noise in the 2D holographic projection system [35]. This method can also be used in 3D CGHs for suppressing the speckle noise. However, the spectrum of the object wave without random phase would be located in a small area of the Fourier plane, as shown in Fig. 7(a). As the object wave is close to the zero-order beam in the Fourier plane, it would be easily affected by the single-sideband filter when filtering the zero-order beam, which would lead artifacts during reconstruction, as shown in Fig. 7(b). Whereas the spectrum of the object wave with random phase spreads evenly in the Fourier plane, as shown in Fig. 7(c). Hence the single-sideband filter has little affection to the object wave, providing better reconstruction quality. The reconstruction result of the CGH with random phase is shown in Fig. 7(d).
Fig. 7 Comparison of the reconstruction results: (a) spectrum and (b) reconstruction without random phase; (c) spectrum and (d) reconstruction with random phase.
The proposed method would not significantly increase the calculation time of the original algorithm for CGH since the generalized single-sideband method includes only three steps: (1) Fourier transform operation, (2) single-sideband filter operation and (3) inverse Fourier transform operation. The CGHs are calculated by using a PC with a CPU of Intel Core i7-7700 (3.60 GHz) and a memory of DELL DDR4 ECC RDIMM (16 GB). The time spent for the generalized single-sideband method is about 0.2 s for CGH with 1920 × 1080 pixels. As a consequence of this, the total calculation time of CGH mainly depends on the original CGH algorithms.
5. Conclusion
In summary, a generalized single-sideband method is proposed to eliminate the conjugate image and zero-order beam in 3D CGH reconstruction. This method provides a general and robust way to suppress the unwanted terms of the amplitude CGH, which is compatible with different 3D CGH algorithms. By redistributing the reconstruction terms in the spatial frequency domain, unwanted terms can be blocked by the single-sideband processing during optical reconstruction. Compared to the current single-sideband methods, the merit of the proposed method is it provides a generalized way for suppressing the conjugate image and zero-order beam in 3D CGH and it can be flexibly applied to a variety of CGH algorithms, such as layer based, point based, and polygon based ones, etc. Numerical simulations and optical experiments demonstrate that the proposed method is effective in reconstructing quality 3D CGHs with the suppressed conjugate image and zero-order beam.
Funding
National Natural Science Foundation of China (NSFC) (61505095, 61875105).
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