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Abstract
In this paper, a holographic display method to suppress the speckle noise is proposed. Firstly, the effective viewing area (EVA) of the reconstructed image is calculated. The object points are separated into groups by pixel separation. Then, the sub-computer-generated holograms (sub-CGHs) which can be reconstructed in the EVA are generated by calculating the principal fringe patterns. Finally, by loading the sub-CGHs on the two spatial light modulators respectively and using spatiotemporal multiplexing method, the reconstructed image can be displayed with lower speckle noise. Moreover, the calculation speed of the hologram is improved. Experimental results demonstrate the feasibility of the proposed method.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, holographic display has become increasingly attractive since it can create the ideal three-dimensional scenes without any special glasses [1]. Therefore, the holographic display has been expected as a desirable way to the current three-dimensional display with drawbacks of eye fatigue and dizziness due to the conflict between the convergence and accommodation of human eyes [2]. Holographic display is generally achieved based on a computer-generated hologram (CGH) with coherent illumination [3,4]. In the holographic reproduction, speckle noise of the reconstructed image severely degrades the quality of the reconstructed image, especially for a phase-only CGH [5]. The formation of the speckle noise in the reconstructed image mainly comes from the highly coherent property of the laser [6]. To solve this problem, numerous methods have been proposed, such as iterative method, light-emitting diode (LED) illumination method, time multiplexing method, down-sampling method, complex amplitude modulation method and pixel separation method [7–14]. Although the iterative method can reduce the speckle noise, the calculation time is long. Complex amplitude modulation method suppresses the speckle noise by modulating both amplitude and phase of the reconstructed image [15]. Nevertheless, the 4f optical filtering structure to rebuild complex amplitude increases the intricacy of the system [16,17]. Pixel separation method carries out speckle noise inhibition by banishing the interference between adjacent image points [18–21]. However, it is still difficult to realize dynamic holographic display at the present time since the calculation speed of the hologram is not fast enough.
In this paper, a method to suppress the speckle noise in the holographic display based on effective utilization of two spatial light modulators (SLMs) is proposed. The effective viewing area (EVA) of the reconstructed image is calculated. Only the sub-CGHs which can be reconstructed in the EVA are generated by calculating the principal fringe patterns (PFP) with shifting and adding operations. By loading the sub-CGHs on two SLMs with spatiotemporal multiplexing method, the reconstructed image can be displayed with lower speckle noise. Moreover, the calculation speed of the hologram is improved. Experimental results demonstrate the feasibility of the proposed method.
2. Principle of the method
Figure 1 shows the schematic diagram of the proposed method, which is composed of three processes. Firstly, the object is separated into object point groups and the EVA is calculated in order to suppress the speckle noise caused by the useless information. Secondly, sub-CGHs are generated by calculating the PFP using shifting and adding operation. Finally, by using spatiotemporal multiplexing method based on the effective utilization of two SLMs, reconstructed image can be displayed with speckle noise suppression.
In the first step, the EVA of the reconstructed image is calculated in order to decrease useless information of the recording process. The object is considered as a set of pixels and is separated into groups of spatial discretization points. p (2 ≤ p ≤ M, N) is assumed as the distance pitch of the pixel separation, where M and N represent the object points numbers of the reconstructed image in the horizontal and vertical directions respectively. Then the recorded object is separated into p2 object point groups. Figure 2 is the connection between the recorded object, the SLM and the reconstructed image. In the traditional holographic recording and reconstruction process using one SLM, the measurement of the CGH is equivalent to that of the SLM. The center of the SLM is assumed to be the origin point of the coordinate frames. H represents the width of the SLM. Z is the distance between recorded object and the SLM, and L is width of the recorded object. A and B are set as two points on the edge of the recorded object. In the recording process, information of each point on the object is recorded by diffraction fringes on the entire hologram plane. The boundaries of the diffraction light of A and B are drawn as blue and green dashed lines respectively. In the reconstruction process, coherent light is modulated by the SLM loaded with CGHs. The image is reconstructed at the diffraction distance of Z. A’ and B’ are the reconstructed images of A and B, respectively.
Fig. 2 Connection between the object, the SLM and the reconstructed image. (a) Holographic recording and reconstruction process; (b) analysis of the EVA.
When light illuminates the SLM, the diffraction angle of the reconstructed image satisfies the following equation:
where λ is the wavelength, and d is the pixel size of the SLM. Since β is very small, Eq. (1) can be expressed by the following equations according to the geometric relationship: The viewing distance is R, and D represents the whole viewing area of the reconstructed image. Da marked in blue is the viewing area of the reconstructed image A’ according to the diffraction theory. The viewing area of the reconstructed image point B’ is marked with Db in green. So, just in the place where Da and Db overlapped, the entire reconstructed image can be observed. The place where Da and Db overlapped is called the EVA and it can be expressed by the following equation according to the geometric relationship:Out of the EVA, the diffraction light cannot reconstruct the entire image, which is the useless information for the observer. However, the useless information which is recorded on the CGH occupies lots of calculation time and causes additional speckle noise. In order to utilize the SLM effectively and decrease the speckle noise further, two SLMs are used in the proposed method. The revised diffraction light boundaries of the points A’ and B’ are described as the yellow and pink dashed lines separately. From geometrical relationship we can see that the width of useless information region is decreased under the premise of keeping the EVA unchanged. In order to obtain revised diffraction lights, the diffraction fringes pattern size of the point needs to be decreased. The diffraction fringes pattern width Wo can be expressed as H-L. So, the information of point A need not to be recorded on the whole SLM.In the second step, hologram patterns for all object point groups are calculated using the novel-look-up-table (NLUT) algorithm. The object is considered as a set of points and each point has an associated realvalued magnitude. The NLUT algorithm can reduce the storage space compared with the LUT, and it contains only the fringe patterns of the object points with unity magnitudes located at each center of the depth-dependent image planes of the object. So, the NLUT algorithm has only one PFP at each image plane and each PFP can be seen as the Fresnel zone pattern computed at each depth [21–23]. Only the PFP for the points located on each center of the depth-dependently sliced image layer are precalculated and stored. Based on the EVA analysis, the PFP is pre-calculated and the resolution of the PFP is smaller than that of the SLM. The size of the PFP is H-L. Then the hologram pattern is calculated by shifting and adding operations of PFPs based on the property of shift-invariance. In the shifting and adding operations of PFPs, the interference patterns of two adjacent pixels are added at a distance of adjacent pixel pitch. The diffraction fringes pattern of one point is calculated. The CGH of the whole object can be simply calculated by shifting and adding operations of PFP. Meanwhile, the interference pattern of each object point has the same size as the PFP based on the EVA calculation. So, p2 sub-CGHs are generated by calculating p2 object point groups. The resolution of the SLM is equal to that of the sub-CGHs.
In the third step, through spatiotemporal multiplexing, the speckle noise of the reconstructed image is reduced by averaging the speckle noise and separating the adjacent image points in space. After the step 2, p2 sub-CGHs is calculated based on p2 object point groups. When these sub-CGHs are loaded on the SLMs, the random speckle noise distributions are different as each sub-CGH records different object group information. Then half of these sub-CGHs are loaded on SLM1, and the other half are changed to mirrored state and loaded on SLM2. The sub-images can be reconstructed and fused together by the two SLMs. Finally, the final image is generated according to time multiplexing of p2 sub-images. When the speckle noise of each sub-image is irrelevant, the speckle contrast C of the final image can be calculated as follows:
where I and σ are the average and standard deviation of the intensity, respectively. According to Eq. (2), we can see that the speckle contrast of the final image is suppressed with the increases of the sub-images.3. Experiments and results
In order to prove the feasibility of the proposed method, the optical experiments are performed. The optical reconstructed system is shown in Fig. 3, which consists of a collimated light source, a beam splitter (BS), two SLMs, a filter and two lenses. The wavelength of the collimated light source is 532nm. The resolution and pixel pitch of the SLMs are 1920 × 1080 and 6.4μm, respectively. The filter and the lenses are used to eliminate high-order diffraction light caused by the SLMs. The SLMs with an addressable gray level of 256 (8 bit) have the frame rate of 60 Hz and provides a phase modulation range of [0, 2π]. The resolution of the recorded objects is 320 × 240. We choose p = 2 to separate the object into four object point groups. The point center of the object is calculated for the PFP by using MATLAB. The resolution of the PFP is 1600 × 840. The CGHs are calculated by shifting and adding operations of the PFP. Then four sub-CGHs with the resolution of 1920 × 1080 are generated. Two of these sub-CGHs are changed to mirrored state and loaded on SLM1, and the rest sub-CGHs are loaded on SLM2. The display time of each sub-CGH is set to 0.033s, which means that the frame rate to reconstruct the sub-CGHs is 30Hz. For each SLM, two sub-images with different speckle noises are reconstructed. The two sub-images are fused together through BS according to the space multiplexing.
The final reconstructed image can be seen on the receiving screen, as shown in Fig. 4(a). Figure 4(b) is the reconstructed image by using the conventional NLUT method. In order to compare the two results more clearly, parts of the reconstructed images are magnified. The magnified images show that the distribution of the reconstructed image using the proposed method seems more uniform. For intuitive comparison, the peak signal to noise ratio (PSNR) and calculation time of the reconstructed images are recorded. In Fig. 4, the resolution of the recorded object is 320 × 240 and p = 2, and the PSNR of the reconstructed image by using the proposed is larger than that of the traditional method. When the object with a larger resolution is used as the recorded object and p is increased, the PSNR can be improved further. The calculation time of the sub-CGHs is reduced by 48.53%. So we can see that the reconstructed image can be displayed with lower speckle noise by using the proposed method. Moreover, the calculation time is greatly shortened.
As shown in Fig. 4, the sample for the experiment is simple (the resolution is 320 × 240, p = 2), so the PSNR of the reconstructed image does not seem to be improved a lot. Besides, more samples are used for the experiment. Figure 5 is the reconstructed image of the letters “AC”, where Fig. 5(a) is the result by using the proposed method and Fig. 5(b) is the result by using the traditional NLUT method. The partial reconstructed image of “A” is enlarged for further comparison of the results. It can be seen clearly that by using the proposed method, the speckle noise of the reconstructed image can be suppressed effectively. Moreover, the letters “BH” are also used as the 3D object to verify the feasibility of the proposed method, where “B” and “H” are located at different depths. The results are shown in Fig. 6. The speckle contrast C is calculated according to Eq. (5) in order to analyze the reconstructed image more intuitively. The speckle contrasts of the letters “AC” by using the proposed method and the conventional NLUT are 0.3872 and 0.4937, respectively. So compared with the traditional NLUT method, the speckle contrast can be reduced by ~22%, when the proposed method is used, as shown in Table. 1. When we choose p = 4 to separate the object into 16 object point groups, the speckle contrast can be reduced to ~60%.
Fig. 5 Reconstructed images of the letters “AC” by using (a) the proposed method and (b) the conventional NLUT method.
Figure 6 Reconstructed images of the 3D object. (a) Result of the proposed method when “B” is focused; (b) result of the proposed method when “H” is focused; (c) result with the conventional NLUT method when “B” is focused; (d) result with the conventional NLUT method when “H” is focused.
Table 1. Speckle contrast and calculation time of the reconstructed images.
In the experiment, the number of the object points is small. When the object points increase, the advantage of the proposed method will become more obvious. Here, we record the relationship between the calculation time of the sub-CGHs and the object points, as shown in Fig. 7. The red line represents the relationship by using the conventional NLUT method, and the black line represents the relationship by using the proposed method. When the number of the object points is relatively small, the advantage of the proposed method does not seem obvious. However, when the number of the object points increases, the difference of the calculation speed between the two methods will become much larger. Therefore, the proposed method has a great advantage in calculating three-dimensional objects.
In the proposed method, the algorithm used in the experiment is NLUT. In order to verify the versatility of the proposed method, Gerchberg-Saxton (GS) algorithm is used for the simulation experiment. The letters “BH” and a car are used as the recorded objects. The number of the iterations is set to 20 in the GS algorithm. All recorded objects have the resolution of 320 × 240. The wavelength of the collimated light is 532nm. The resolution and pixel pitch of the SLM are 1920 × 1080 and 6.4μm, respectively. The results of the simulation experiment are shown in Fig. 8, where Figs. 8(a)-(b) are the results of the reconstructed image of GS algorithm without using the proposed method, while Figs. 8(c)-(d) are the results of the reconstructed image by using the proposed method. Besides, the intensities of the reconstructed images are given, as shown in Fig. 9. It can be clear that the quality of using the proposed method is better than that of using the conventional GS algorithm.
Fig. 8 Results of simulation experiment based on GS algorithm. (a) Reconstructed image of the letters without using the proposed method; (b) reconstructed image of the car without using the proposed method; (c) reconstructed image of the letters by using the proposed method; (d) reconstructed image of the car by using the proposed method.
Fig. 9 Intensity of the reconstructed images. (a) Intensity of Fig. 8 (a); (b) intensity of Fig. 8 (c).
In the experiment, we choose p = 2 to separate the object into four object point groups. When p increases, the speckle noise of the reconstructed image will be suppressed further, as shown in Fig. 10. However, the calculation time will be longer accordingly. In the holographic reconstruction, the calculation speed and the quality of the reconstructed image are the two major problems that limit the further development of holography. The proposed method in this paper is mainly to suppress the speckle noise in holographic reconstructed image. By utilizing two SLMs effectively, the calculation speed is also improved. However, the current speed is still difficult to meet the requirement of holographic dynamic display. In the next work, we will try to build a CUDA parallel computing platform to achieve fast calculation of holography. Besides, the viewing angle of the holographic reconstructed image is very small. The proposed method uses two SLMs for spatiotemporal multiplexing. Maybe next we can make full use of these two SLMs to expand the viewing angle while suppressing speckle noise. Faced with these problems in holographic research, we hope that our research can make a modest contribution to the development of holography.
Fig. 10 Result of the reconstructed image when p changes. (a) Result by using the traditional NLUT method; (b) result by using the proposed method when p = 2; (c) result by using the proposed method when p = 3.
4. Conclusion
In this paper, a method to suppress the speckle noise based on the effective utilization of two SLMs is proposed. The object is separated into groups by pixel separation to decrease the coherence place of the reconstructed image points. Then speckle noise is suppressed by decreasing the recording of useless information. Finally, the speckle noise is equalized by spatiotemporal multiplexing. Experimental results reveal that when we separate the recorded object into p2 = 4 points groups, the PSNR of the final image is increased compared with the conventional NLUT method. Furthermore, the calculation time of the sub-CGHs is reduced by 48.53%. With the increase of effective object points, the advantages of the proposed method will become more apparent. The proposed method can also be used in other holographic algorithms.
Funding
National Natural Science Foundation of China under Grant No. 61805130, 61805169 and 61535007.
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