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Abstract
The major challenges in traditional color phase hologram generation are the time-consuming iterative procedure and aberration caused by different wavelengths in color holographic display. Based on the original non-iterative phase hologram generation method-optimized random phase (ORAP), combined with the physical limitations of color holographic display, this paper proposes a full-support optimized random phase (FS-ORAP) method for non-iterative color phase hologram generation. FS-ORAP breaks through the limitation of the original ORAP method in the fixed support constraint of the target amplitude in the spatial domain, the full support constraint can be used to generate phase holograms of target amplitudes with arbitrary support size, which fits well with the generation mode of the three-color channel of the color phase hologram. In addition, the color aberration of the reconstructed image is eliminated by scaling the size of the three-color component. At the same time, FS-ORAP is used for the non-iterative fast generation of three-color channel holograms, which can greatly improve the generation speed of color phase holograms and can be adapted to various color holographic display techniques. Experimental results verify the feasibility of our proposed method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
With the rapid development of modern sensor, communication, storage and computing technology, people put forward higher requirements for display technology in medical imaging, geological exploration, entertainment and military affairs. Color holographic display [1] is an important technology that can record and reproduce the color and three-dimensional information of the original object. It can reflect the real information of the object more than monochrome holography and has a very wide application prospect.
With the rapid development of computer science, computer-generated hologram (CGH) [2] can conveniently introduce digital processing methods to eliminate the adverse effects caused by aberrations, noise and nonlinear of the photosensitive characteristic curve of the recording medium, improve the quality of holograms, and also realize the display of three-dimensional objects that do not exist in nature. CGH is used more and more widely in all walks of life. Many researchers have put forward generating color computer-generated holograms [3–5], and these reconstruction methods of color computer-generated holograms are mainly based on the superposition of three primary holograms [6]. The basic idea for generating color holographic display is to decompose the color image into R, G, B components, and then the G-S algorithm can be used to produce a phase-only hologram for each channel. Finally, the produced R, G, B components are synthesized into a color phase hologram to reproduce the color objects.
Usually, classical G-S [7] iterative algorithm is utilized to generate phase-only CGH, and is very time-consuming, especially for color phase holograms generation. In 2018, Alejandro Velez Zea et al. proposed an optimized random phase (ORAP) [8–12] method, which can quickly and non-iteratively generate phase holograms of arbitrary amplitude targets with the same support as the target window, greatly improve the speed of generating multiple phase holograms. The reconstruction quality of ORAP is better than that of single G-S iteration [13,14]. ORAP can be used for non-iterative generation of the phase hologram of any amplitude target with the same support as the target window, but the size and position of the support set of the target and the parameters of the optical system can not be changed. Otherwise, a new ORAP needs to be regenerated by G-S algorithm. In color holographic display, due to the wavelength scaling effect of RGB laser, the channels of different color will be longitudinally scaled and laterally offset. Therefore, when ORAP is directly applied to the non-iterative generation of phase holograms in the color holographic display, ORAP can only be used to generate generated in different channels and then synthesized into color holograms, which increases the total generation time of color phase holograms and lacks flexibility [15].
In this paper, full support based optimized random phase (FS-ORAP) method is proposed to directly apply to generate the color hologram of color holographic display. FS-ORAP method breaks through the limitation of the original ORAP method in the size and position of the fixed support of the target amplitude in the spatial domain. After a single FS-ORAP is iteratively generated by the G-S algorithm, it can be directly applied to the three-color channel to generate color holograms without iteration. Multiple color phase holograms can be quickly generated, and the generation time of color phase holograms can be greatly reduced.
2. Basic principles
2.1 Color holographic display
The color holographic display usually uses RGB three-color laser as light source, and uses time division multiplexing [16], space division multiplexing or spatial division [17] method to establish color holographic display system.
Time division multiplexing uses three-color lasers to illuminate monolithic spatial light modulators [18,19] respectively for optical reproduction. The schematic diagram is shown in Fig. 1. Usually, the unit period T is equally divided into three sub-time periods T1, T2 and T3. In each sub-time period, only a monochromatic light source illuminates the spatial light modulator, and the corresponding component hologram is loaded at this time. The display of RGB three monochrome holographic reconstructed images is completed in each unit time period T. When T is small enough, i.e., the switching frequency of the three-color hologram is high enough, the color holographic reconstructed images are obtained by using the persistence characteristics of human vision. However, this method needs to accurately control the working time of the monochromatic light source and the synchronization of loading the corresponding component hologram, which requires a higher response speed for the hardware loading the three-color information. For the three-color component, there will be energy loss on the time axis.
Fig. 1. Schematic diagram of time division multiplexing and sequence of hologram loaded to Liquid Crystal on Silicon (LCOS)
Space division multiplexing uses three-color lasers or white laser to illuminate three spatial light modulators respectively to synthesis the color images for optical reconstruction in the reconstruction plane. The schematic diagram is depicted in Fig. 2. In holographic reconstruction, the three monochromatic holograms are loaded on the corresponding spatial light modulators at the same time, and the driving circuit drives the three spatial light modulators to work simultaneously. The three-color light sources are respectively incident on the three spatial laser modulators, and the final RGB holographic images are registered and synthesized in space to obtain color holographic reconstruction images. The system based on space division multiplexing has higher optical efficiency and many advantages in color reproduction and resolution enhancement, which is an important development direction of color holographic display.
The spatial division method only uses a single-chip spatial light modulator, the spatial light modulator is divided into three sub-regions. The three-color laser illuminates a sub-region of the spatial light modulator at the same time. Each sub-region is loaded with corresponding red, green and blue holograms. Finally, the diffraction characteristics are used to form a color holographic reconstruction image on the reconstruction plane. The schematic diagram of the spatial light modulator division and the system are shown in Fig. 3. The spatial division uses fewer spatial light modulators like time division multiplexing, but it will lose the resolution of red, green and blue holograms.
Fig. 3. The schematic diagram and the system of spatial partition method using a single-chip spatial light modulator
2.2 Optimized random phase method
In order to improve the generation speed of phase holograms of arbitrary amplitude targets in a given optical system, the optimized random phase (ORAP) method was proposed by Alejandro Velez Zea et al. to quickly generate phase holograms, and its reconstruction quality is better than the single G-S iteration method. In the ORAP method, the traditional G-S algorithm is used to optimize the random phase and generate an optimized random phase, which is used to quickly and non-iteratively generate phase holograms of arbitrary amplitude targets. ORAP method firstly creates a window with the same support size as the target window. A random phase mask multiplies this window amplitude. And then performs the inverse Fourier transform (IFT) of this product, then follows several iterations of the standard G-S algorithm loop, ORAP is obtained on the reconstruction plane. The schematic diagram of ORAP generation is depicted in Fig. 4.
In the process of ORAP generation, for the k-th iteration, the complex amplitudes ${g_k}$ and ${G_k}$ of the SLM plane and the reconstruction plane are respectively, where the $g{^{\prime}_k}$ and $G{^{\prime}_k}$ are the complex amplitude estimation of the SLM plane and the reconstruction plane, and the transformation methods between the two planes are Fourier Transform (FT) and inverse Fourier Transform (IFT):
For a given optical system, the generation of each phase hologram only needs three operations, i.e., 1) a multiplication of the intensity target with ORAP, 2) an IFT, and 3) a phase extraction from the result IFT. After a FT, the target can be rebuilt. For a given optical system, as long as the ORAP is generated, it can be used to quickly generate phase holograms for arbitrary amplitude targets with the same support as the target window, thus avoiding the need for complex iterative algorithms for each new target.
3. Generation of a color phase hologram based on the FS-ORAP method
3.1 Full support based optimized random phase method
The traditional optimized random phase (ORAP) can be used to quickly generate target phase holograms with arbitrary amplitude with the same support size and position as the target window, and the support ratio of the target amplitude is less than 1. Moreover, the size and position of the support set of the target amplitude and the parameters of the optical system cannot be changed, otherwise, a new ORAP needs to be regenerated by the G-S algorithm.
Support ratio is defined as
In color holographic display, due to the different wavelengths of RGB lasers, the channels of different colors will undergo longitudinal size scaling and lateral shift. The size of the imaging area of the three-color component during holographic reproduction is related to the wavelength of the three-color lasers, and the proportional coefficient is:
where, $\Delta {h_r}$, $\Delta {h_g}$ and $\Delta {h_b}$ are the length of the red, green and blue reproduction images in a single dimension, respectively, ${\lambda _r}$, ${\lambda _g}$ and ${\lambda _b}$ correspond to the wavelengths of red, green and blue lasers, respectively. The pixel spacing of LCOS is the same in the x and y dimensions, the schematic diagram of the reconstructed image of the three-color hologram is shown in Fig. 5. ${O_b}$, ${O_g}$ and ${O_r}$ are the centers of the blue, green and red reproduced images, the coordinates are $({{\lambda_b}\textrm{/2}p,{\lambda_b}\textrm{/2}p} )$, $({{\lambda_g}\textrm{/2}p,{\lambda_g}\textrm{/2}p} )$ and $({{\lambda_r}\textrm{/2}p,{\lambda_r}\textrm{/2}p} )$, and p is the pixel spacing of LCOS.Therefore, when ORAP is directly applied to the non-iterative generation of three-color phase holograms in the color holographic display, it can only generate ORAP in different channels and then synthesize color holograms, which increases the total generation time of color phase holograms and lacks flexibility.
In this paper, a full support based optimized random phase (FS-ORAP) method is proposed. This method breaks through the limitation of the original ORAP method in the fixed support constraint of the target amplitude in the spatial domain, and uses the generated FS-ORAP to generate the phase amplitude of any target amplitude with arbitrary support, this FS-ORAP can be applied to any support target amplitude not exceeding the window size.
FS-ORAP method first creates a full support unit amplitude with the same size as the SLM plane in the spatial domain, in order to generate the phase hologram of the target amplitude, the full support unit amplitude is multiplied by the random phase, and then the product is subjected to inverse Fourier transform, and then the result is sent to the G-S loop. Following several standard iterative G-S loops, the full support optimized random phase with better performance can be achieved. The loop continuously alternates between the SLM plane and the reconstruction plane, each time replacing the obtained amplitude with the target amplitude corresponding to each plane. The target amplitude of the SLM plane is the same size as the unit amplitude of the SLM plane, and the amplitude of the reconstruction plane is the full support unit amplitude created at the beginning.
After the FS-ORAP is generated by the G-S algorithm, the product of the obtained FS-ORAP and the target amplitude is transformed into the SLM plane by inverse Fourier transform, and then the phase component of complex amplitude in the SLM plane is extracted to obtain the phase hologram.
The proposed FS-ORAP method can adapt to different sizes of supports and target amplitudes in different positions in the image space. Due to the wavelength scaling effect of the RGB three-color laser, the channels of different colors will be scaled longitudinally and shifted transversely in color holographic display. After FS-ORAP was iteratively generated by the G-S algorithm, it can be directly applied to RGB three-color channels to generate color holograms without iteration, finally multiple color phase holograms can be generated quickly and the generation time of color phase holograms can be greatly reduced.
3.2 Non-iterative generation of color phase holograms by the FS-ORAP method
The proposed FS-ORAP can be directly applied to the generation of color phase holograms in color holographic display based on time division multiplexing, space division multiplexing, spatial division and other methods. FS-ORAP is directly added to RGB three-color channels at the same time to generate monochrome holograms, and finally to synthesize color phase holograms. The FS-ORAP proposed in this paper can directly combine the scaled monochromatic amplitudes of the three channels, only several G-S iterations are required during the generation of FS-ORAP, the product of FS-ORAP and monochromatic amplitude is subjected to inverse Fourier transform (IFT) to obtain a monochromatic phase hologram, and then three-color holograms are synthesized into a color phase hologram. The generation process of color phase hologram is performed in a non-iterative type, and G-S iteration is not required for different channels, which can greatly reduce the calculation time. The flowcharts of FS-ORAP and traditional ORAP for color phase hologram generation are illustrated in Fig. 6.
In order to eliminate the scaling effect caused by wavelengths of different colors channels in color holographic display, the size of red and green components is adjusted according to wavelengths based on blue components. The flowchart of phase-only color holograms generation using FS-ORAP is detailly depicted in Fig. 7.
4. Experiment
To verify the feasibility of our proposed FS-ORAP method for the generation of color phase holograms, this paper designs two groups of experiments. In experiment 1, FS-ORAP is used to generate phase holograms of gray images. The experimental results verify the two advantages of FS-ORAP, which just solves the limitations of ORAP when it is used to generate color phase holograms. In experiment 2, the feasibility and speed of generating color phase holograms by FS-ORAP are tested.
The hardware environment used in this experiment is Core i5 6200U processor, with 8G memory, and the operating system is Windows10. All programs are written by MATLAB R2018b.
4.1 Experiment 1: phase hologram generation of gray-image using the FS-ORAP method
The proposed FS-ORAP method can be adapt to different supports of target amplitudes in image space. In experiment 1, three experiments are presented in terms of different size and different position of target amplitude support in the image plane.
In the first experiment, the size of the full support and FS-ORAP are both 512 ${\times}$ 512. The number of iterations of the G-S algorithm for FS-ORAP generation are 20, and the amplitude of gray target of House, Baboon, Cameraman and Couple are resized to 320 ${\times}$ 240, 256 ${\times}$ 256, 384 ${\times}$ 384 and 512 ${\times}$ 512, respectively. Peak signal-to-noise ratio (PSNR) and correlation coefficient (CC) are used to evaluate the reconstruction quality. The reproduction results using FS-ORAP to generate phase holograms for different target amplitudes with different supports size are depicted in Fig. 8.
Fig. 8. Reconstructs target amplitudes of different support sizes using phase holograms generated by FS-ORAP and ORAP.(The black long line in the last column means ORAP does not support for full-support target, the same meaning are expressed in the following figures.)
In the second experiment, Cameraman is chosen as an example to verify the effectiveness of FS-ORAP for phase holograms generation for the same target amplitude with different support sizes. The size of the full support and FS-ORAP, the number of iterations of the G-S algorithm are the same as the first experiment. The total support ratios of target amplitude are 0.12, 0.20, 0.30, 0.42, 0.56, 0.72, 0.82 and 1.00, respectively. The reproduction results and the CC curve between the target amplitudes and reproduction results under different support ratios are depicted in Fig. 9 and Fig. 10, respectively.
Fig. 10. Phase holograms reconstruction performance curves with different support ratios generated using FS-ORAP and ORAP
As illustrated in Fig. 8, Fig. 9 and Fig. 10, we can find that the FS-ORAP generated by G-S algorithm can be used to generate multiple phase holograms for different amplitudes of the targets with different support ratios. The reproduction quality of FS-ORAP is comparable to that of the ORAP method in terms of PSNR and CC metric, but ORAP does not support for full-support target (e.g., Couple image).
In the third experiment, the Cameraman image is chosen as the target amplitude to verify the effectiveness of FS-ORAP for phase holograms generation with different position and the same support ratio. The size of the full support and FS-ORAP, the number of iterations of the G-S algorithm are the same as the first experiment. The size of gray target is 280×280. The reproduction results using FS-ORAP to generate phase holograms for different positions of target amplitudes with the same supports size are depicted in Fig. 11, which demonstrates that our proposed FS-ORAP is effective for phase holograms generation with different position of the target amplitude.
Fig. 11. Reconstruction results of Cameraman phase holograms at different positions generated by FS-ORAP
4.2 Experiment 2: color phase hologram generation using the FS-ORAP method
The purpose of Experiment 2 is to verify the feasibility and advantage of our proposed FS-ORAP method for color phase holograms generation. In this experiment, the size of the FS-ORAP is 512×512, and the target amplitudes of House, Baboon and Airplane are 320×240×3, 410×410×3 and 512×512×3, respectively. The number of G-S loops for generating ORAP and FS-ORAP are both 20. The reconstruction results of G-S algorithm, ORAP method and our proposed FS-ORAP method are depicted in Fig. 12. The generation speed comparison of G-S, ORAP and FS-ORAP for generating the color holograms are summarized in Table 1.
Table 1. Cost time comparison of generating three sets of color phase holograms by G-S algorithm, ORAP and FS-ORAP methods
As shown in Fig. 12, for the reconstruction quality of color phase holograms generation among the three methods, the classical G-S algorithm is the best, FS-ORAP method and ORAP method are nearly the same, but ORAP does not support for full-support target.
As shown in Table 1, the speed of color phase holograms generation for three target amplitudes using the G-S algorithm requires iterative procedure in the three channels of each target amplitude, with a total time of 17.0782s. Color phase hologram generation using ORAP method, the generation time of ORAP is about 5s. However, due to the need to generate phase holograms of target amplitudes supported by different sizes, ORAP can not be directly adapted, and three channels of ORAP need to be generated three times respectively. However, after ORAP is generated, compared with the G-S algorithm, the hologram generation speed is faster, only needs about 0.2s, and the total time is 11.8109s, but ORAP does not support for full-support target. If the phase hologram of the target amplitude with the same support is generated again, the ORAP method only needs about 0.2s, which is faster than the G-S algorithm.
Color phase holograms generation using FS-ORAP method, the generation time of FS-ORAP is 1.9219s. Once FS-ORAP is generated, the generation time of holograms is only about 0.2s, and the total time is only 2.5782s. Compared with ORAP method, FS-ORAP method can be directly applied to three channels of color target amplitude, which can greatly accelerate the generation speed of color phase hologram. With the increase number of holograms generation, the total calculation time using FS-ORAP method is obviously lower than that using direct G-S algorithm and ORAP method.
For the generation speed of color phase hologram, classical G-S takes too much time for iteratively generating the color phase holograms for each target image. In ORAP method, the ORAP for each color channel requires to be regenerated by G-S algorithm due to the changes of the support size and position. However, our proposed FS-ORAP takes the advantage of flexibility and non-iterative generation for color phase holograms with only one FS-ORAP, and such superiority become more and more obvious when multiple holograms need to be generated.
4.3 Experiment 3: optical experiment
In order to verify the feasibility of the FS-ORAP method proposed in this paper, this experiment compares the three methods through the optical reproduction results of the color phase holograms generated by the G-S algorithm, ORAP method and FS-ORAP method. The LCOS used in this experiment is the MD1280 chip produced by Three-Fivesystems Company,with a resolution of 1280×1024 and pixel spacing of 12um, which is used to load phase holograms. The laser wavelengths used in the laboratory are ${\lambda _r}$ = 671nm, ${\lambda _g}$ = 532nm, ${\lambda _b}$=473nm, respectively. The size of FS-ORAP used in this experiment is 1024×1024.
In this experiment, a simple color letter image (Fig. 14) with a size of 1024×1024 is selected as the first original image, and a complicated color image (Fig. 15) with a size of 768×768 is selected as the second original image. First, the red and green components of the original image are scaled according to Eq. (6), and the number of red and green pixels was expanded to be the same as that of blue by the method of zero filling. Then, converging spherical wave and flared grating were superimposed to reduce the influence of zero-order light and multi-stage diffraction image [1] on the experimental results. Then, the red, green and blue phase holograms are loaded to LCOS and projected onto the receiving plane, and the imaging results of the three-color components are captured by a camera. Finally, the color reconstruction results were obtained by using Matlab for synthesis. The experimental light path is shown in Fig. 13, and the reconstruction results of the three methods are shown in Fig. 14.
Fig. 15. Optical reproductionresults of G-S algorithm, ORAP method and FS-ORAP method (complicated image)
The optical reproduction results of binary objects and complicated images for G-S algorithm, ORAP method and FS-ORAP are depicted in Fig. 14 and Fig. 15. In the two cases, the performances of the three methods are very similar. In a nutshell, the quality of the three methods is comparable, but the advantages of proposed FS-ORAP method are verified in terms of the speed and flexibility for the generation of holograms with different support sizes, especially in color holographical display and multiple groups of color phase holograms needed to be generated in real time.
5. Conclusion
This paper presents a FS-ORAP method, a non-iterative approach for phase hologram generation, which can be directly adapted for various color holographic display setups. Our proposed FS-ORAP method breaks through the limitation of the original ORAP method that only applicable for the fixed support constraint as the target amplitude, and can be used to non-iterative generate phase holograms of target amplitudes with arbitrary support size and position, which fits well for the non-iterative generation of each channels of color phase holograms with only one FS-ORAP to eliminate the color difference of the reconstructed image caused by wavelength scaling in holographic display. Our proposed method takes the strong advantage of low computational cost and high flexibility for color phase hologram generation, especially for multiple holograms generation.
Funding
Natural Science Foundation of Anhui Province (No. 2008085MF209); Major Natural Science Foundation of Higher Education Institutions of Anhui Province (No. KJ2019ZD04, No. KJ2020ZD02); Open Research Fund of Advanced Laser Technology Laboratory of Anhui Province (AHL2020KF05).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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