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Abstract
Computer-generated hologram (CGH) is an evolving field that facilitates three-dimensional displays, with speckle noise reduction being a pivotal challenge. In hologram synthesis, complex data with random phase distributions are typically employed as carrier waves for wide viewing angles and a shallow depth of focus (DOF). However, these carrier waves are a source of speckle noise, which can significantly degrade image quality. In this paper, we propose a novel technique for speckle reduction for single sideband (SSB)-encoded holograms, applicable to any arbitrary 3D object. The proposed method focuses on optimizing the random carrier wave used in the hologram synthesis to achieve a uniform amplitude distribution at the object's location. This optimization results in a carrier wave that consistently exhibits uniform amplitude at specific depth planes, leading to a significant reduction of the speckle occurring from the carrier wave. The proposed method has been validated through simulations and optical experiments.
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1. Introduction
Computer-generated holograms (CGHs), which enable the display of three-dimensional (3D) objects, have been advancing continuously in recent years [1]. CGHs are synthesized for amplitude 3D objects with various carrier waves which determine the phase distribution on the object surface. In order to achieve wide viewing angle and shallow depth of focus (DOF), a random phase complex field is usually used in synthesizing CGHs.
CGHs can be complex data representing amplitude and phase distribution of the object complex field in the hologram plane. In simulations, one can obtain noise-free object reconstruction by propagating the complex-valued CGH to the object plane numerically. For optical reconstruction of the CGH, however, its complex values should be encoded into amplitude-only or phase-only data due to the limited modulation property of currently available spatial light modulators (SLMs). This encoding process generally leads to various noises in the reconstruction, due to the data loss from the original complex data. One of the major noises is the speckle.
Speckle noise is caused by the integration of random phase light within a resolvable spot of the observation system, as depicted in Fig. 1. In the synthesis of a CGH, the carrier wave typically has random phase distribution, which results in speckle noise in the reconstruction of the encoded CGH. By adjusting the randomness of the carrier wave phase, it is possible to control the level of speckle noise at the expense of the viewing angle [2,3].
To achieve both low speckle noise and wide viewing angle in encoded CGHs, various methods have been explored [4–8]. Time multiplexing of multiple CGHs with different random phase carrier waves is a traditional method. In this approach, the speckle noise is reduced as the multiple reconstructions with non-correlated speckle noise patterns are accumulated in the object plane in their intensities [4,5]. Although the speckle noise can be reduced while keeping the wide viewing angle, this time-multiplexing approach is limited by the frame rate of the SLM. The Gerchberg-Saxton (GS) algorithm has also been studied for creating higher quality reconstructions of the CGHs [6,7]. This method optimizes the phase of the CGH by iteratively calculating the difference between the numerical reconstruction and target image. While it is effective, the GS algorithm requires the object-wise iteration which is time-consuming. To avoid such object-wise iteration, a method for optimizing random phase for Fourier hologram, which forms object in the Fourier plane, has been proposed [8]. In this method, the random phase is first optimized to have uniform intensity in the Fourier plane and the optimized random phase is used for synthesis of the Fourier hologram. However, this method was only for two-dimensional (2D) image in the Fourier plane and the hologram synthesis for 3D objects with multiple depths were not addressed. The wavelength diversity approach which uses multiple wavelengths in the optical reconstruction was also proposed [9,10]. However, this method usually results in blur in the object reconstruction.
Recently, learning-based methods for synthesizing high-quality CGHs have been proposed [11–14]. D. Wang et al. proposed a full-color grating-based holographic 3D display for achieving wide viewing angle with low speckle noise [11]. In this method, wide viewing angle is achieved by using a liquid crystal grating, and they optimized the phase-only hologram to reduce the speckle noise. S. Choi et al. proposed a deep learning-based phase-only CGH approach for immersive holographic near-eye displays, utilizing the stochastic gradient descent (SGD) method [12]. Time-multiplexing, optimization of the propagation kernel, and deep neural networks are utilized to achieve vivid optical reconstructions with low speckle noise. Similarly, a binary CGH optimization method was also proposed [13] which achieves high-contrast, speckle-free 3D holograms by optimizing binarized CGH and employing time-multiplexing. However, these methods require time-consuming object-wise training and optimization processes, hindering their direct application to arbitrary objects. Meanwhile, the virtual splicing of SLM and the hologram generation using a physical model-driven network was also proposed to achieve wide viewing angle and low speckle noise [14]. The temporal division multiplexing and beam deflection elements are used to expand the viewing angle. Although the wide viewing angle is achieved, the DOF of individual sub-hologram generated by the network is extended to suppress the speckle noise.
In this paper, we propose a novel contents-independent speckle reduction method that gives a wide viewing angle and shallow DOF for an encoded CGH. The proposed method first finds an optimal carrier wave that gives uniform intensity distribution in target depth planes considering the amplitude-only encoding of the complex CGH, and then applies it to arbitrary objects in the CGH synthesis. As the optimal carrier wave acquisition is independent from the target objects, it can be performed offline before the actual CGH synthesis for specific objects.
Figure 2 illustrates the proposed method. In the first step, the proposed method finds the optimized carrier wave by performing single sideband (SSB) filtering, numerical propagation to the object plane [15,16], loss calculation based on target uniform intensity, and update of the carrier wave. Through this first step, the carrier wave is adjusted to have uniform intensities at the desired object distances. Note that the SSB filtering, i.e., the band rejection of the half of the entire angular spectrum is considered in this step because it is commonly employed in the optical experiments to separate the desired reconstruction from DC and conjugate noise arising from limited modulation property of the SLMs. Also note that even though the SSB filtering during the numerical propagation from the hologram plane to the object plane is considered in this step, the optimized carrier wave itself is still complex field. The second step is the hologram synthesis of 3D objects using the optimized carrier wave. The 3D objects can be represented using either a light field or multiple layers. Finally, the synthesized complex hologram is encoded into amplitude only hologram using the SSB encoding. Thanks to the application of the optimized carrier wave, the reconstruction of the final amplitude only hologram achieves uniform intensity, effectively suppressing the speckle noise at the designated object distances. The main contributions of the proposed method can be summarized as follows.
- 1) Enhanced Quality of SSB-encoded CGH: By optimizing the carrier wave for specific object distances, the synthesized CGH using the optimized carrier wave exhibits lower speckle noise and higher signal contrast than traditional random phase carrier wave.
- 2) Content-Independent Synthesis: The optimized carrier wave is used in the synthesis of CGHs for any 3D object without content dependency. Despite the optimization process is conducted without specific object information, the optimized carrier wave is universally applicable to any object.
- 3) Utilization of the Angular Spectrum in SLM: The proposed method enables the design and utilization of the carrier wave's angular spectrum with the desired size and shape within the angular frequency domain provided by the SLM. While the proposed method is limited to half area due to the SSB, optimization enables the use of the entire area corresponding to half of the domain, allowing for the synthesis of holograms with a wide viewing angle and shallow DOF.
The validity of the proposed method is verified through numerical simulations and optical experiments. The following sections present the details of the proposed method along with the experimental results.
2. Proposed method
2.1 Carrier wave
The carrier wave is an auxiliary complex field used in the CGH synthesis of a given object. The object is defined by the amplitude distribution of the object points in a 2D plane or 3D space. The phase distribution of the object points is, therefore, left to be assigned in the CGH synthesis, giving a degree of freedom. The phase of the object points can be assigned individually or determined by assuming a wave illuminating the object. The carrier wave is the virtual wave illuminating the object, and thus determining the phase distribution of the object points.
The phase distribution of the object defines the angular spectrum of the object complex field which determines the angular properties of the object surface reconstruction including the diffusiveness, viewing angle, and DOF. Figure 3 shows an example of the holograms synthesized with different carrier waves using the Wigner-inverse transform-based light field to hologram (WLFH) method [17,18]. When the uniform phase plane carrier wave is used, the surfaces of the reconstructed object mimic specular ones. The angular spectrum is concentrated around the center, giving highly limited viewing angle and deep DOF of the reconstruction as shown in Fig. 3(a). On the other hand, the random phase carrier wave results in diffusive object surfaces. The hologram has a wide angular spectrum covering entire spatial frequency domain, giving wide viewing angle only limited by the spatial sampling pitch of the hologram and the shallow DOF of the reconstruction as shown in Fig. 3(b).
Fig. 3. Reconstruction results with different carrier waves. (a) Uniform phase plane carrier wave, (b) random phase carrier wave.
The speckle noise of the reconstruction is also affected by the phase distribution of the object points. The uniform or slow-varying well-structured phase distribution gives less intensity fluctuations in the spatial integration of the reconstructed complex field within the resolvable spot of the observation system, resulting in less speckle noise. To the contrary, the fast-varying random phase distribution gives strong intensity fluctuations, resulting in high speckle noise. The clean reconstruction in Fig. 3(a) and the speckled reconstruction in Fig. 3(b) also demonstrate the dependency of the speckle noise on the phase distribution of the object.
In short, the object phase distribution imposed by the carrier wave during the CGH synthesis affects the viewing angle, DOF, and the speckle noise of the reconstruction. Uniform phase distribution gives speckle-less clean reconstruction but suffers from small viewing angle and deep DOF. The random phase distribution makes the best use of the viewing angle and the shallow DOF but the reconstruction is contaminated by the speckle noise. Therefore, there is a tradeoff between them. Since achieving shallow DOF while suppressing the speckle noise is important for a vivid 3D object reconstruction, it is important to find the optimal carrier wave that balances the tradeoff between DOF and speckle noise. The conventional approaches try to optimize the carrier wave for the given objects, requiring new optimizations for new objects. To the contrary, the proposed method aims to find the carrier wave optimized for certain object distances. The carrier wave optimized by the proposed method is not specific to the object, thus it can be universally applied to any objects at the same distances.
2.2 Optimization of carrier wave
The proposed method is designed to ensure that the carrier wave exhibits reduced intensity fluctuations at the object reconstruction distance. As depicted in Fig. 2, the proposed method begins by initializing an arbitrary random complex field as the carrier wave. This initialized random carrier wave is then encoded using the SSB method considering the limited modulation property of the actual SLM. The SSB-encoded carrier wave is propagated to specific distances where the target object is located. Given that the initial carrier wave utilizes an arbitrary random complex field, the amplitude at this step exhibits random perturbations. We calculate the difference between the amplitude of the carrier wave at these specific distances and the desired uniform amplitude, setting the mean square error as the loss function. This loss function is then utilized to update the carrier wave using the ADAM optimizer [19]. Finally, the process is repeated iteratively to obtain the optimized carrier wave.
Figure 4 illustrates the reconstruction examples of the holograms synthesized with various carrier waves. An arbitrary random carrier wave inherently has random amplitude perturbations in the object plane. As a result, holograms synthesized with this type of carrier wave generate speckle noise, as demonstrated in Fig. 4(a). Conversely, employing the carrier wave optimized by the proposed method, results in significantly less speckled outcomes, as illustrated in Fig. 4(b). The proposed method is designed to minimize amplitude fluctuations at the specific distance plane where the object is located. By using this optimized carrier wave, therefore, we maintain the randomness inherent to the carrier wave while achieving cleaner reconstruction image with notably reduced speckle noise at the object distance.
Fig. 4. Comparison of the (a) random phase carrier wave, and (b) carrier wave optimized by the proposed method.
2.3 Hologram synthesis
In order to verify the proposed method, the CGHs are synthesized using two methods, i.e., multi-layer-based method, and WLFH method. In the muti-layer-based method, the 3D object O is partitioned into several discrete layers Ll according to their depths as
where the l is the layer index. These discrete layers are propagated to the hologram plane by the band-limited angular spectrum method (BLAS) after multiplying the carrier wave [11], which givesThe WLFH method is also adopted to demonstrate that the proposed method is applicable not only to the multi-layer-based approach but also to the light field-based CGH methods. The WLFH technique applies the Wigner inverse transform to the light field data, i.e., an array of the different views of the 3D object to synthesize the corresponding hologram [17,18]. Suppose that the light field is given by L(tx,ty,ux,uy) with spatial coordinates (tx,ty) in the hologram plane. Note that the spatial frequency (ux,uy) is related to the ray angle (θx,θy) = (sin-1λux, sin-1λuy) and thus the light field L(tx,ty,ux,uy) can be thought as the collection of orthographic views L(tx,ty,ux,uy)|ux = uxo, uy = uyo at various observation angles (uxo, uyo). In the WLFH method, the hologram is obtained by [17,18]
where W(xc,yc) is an arbitrary carrier wave of the synthesized hologram. The elementary hologram H(x,y;xc,yc) in Eq. (4) is calculated from the given light field L(tx,ty,ux,uy) byIn this paper, we conduct numerical simulations and optical experiments by applying the proposed method to both multi-layer-based and WLFH CGH methods. In both CGH methods, the carrier wave W optimized by the proposed technique is used instead of the conventional random phase carrier wave to verify the enhanced reconstruction quality of the proposed method.
2.4 Stochastic gradient descent-based (SGD) method
The proposed method is closely related to the SGD CGH technique. The SGD method also uses an iterative method to obtain the phase only hologram for target objects. As depicted in Fig. 5, the SGD method for SSB-encoded holograms involves the following steps: 1) Defining the initial phase as a random complex field. 2) Encoding the random field with the SSB technique. 3) Propagating the encoded complex field to a specified target distance. 4) Calculating the loss value between the amplitudes of the target and the propagated complex field. 5) Updating the initial random phase with the obtained loss value. These steps are repeated until the loss value meets the set constraints. Consequently, the optimized phase only hologram contains the target information with reduced speckle noise. However, this method necessitates target-wise optimization. In the following sections, we present a comparative analysis between the proposed method and the SGD method to evaluate the quality of the proposed method, noting that the proposed method provides high quality reconstruction similar to the SGD technique without any specific 3D object information.
3. Numerical simulations
In order to verify the proposed method, we conducted numerical simulations using two types of 3D objects for the hologram: a ‘socket’ light field and a multi-layered resolution target, as illustrated in Fig. 6. For the optimization of the carrier wave, the Adam optimizer was employed with a learning rate of 0.1 and the iteration of 500 times.
Fig. 6. The 3D objects for hologram synthesis. (a) The socket light field is used for the WLFH method. Each socket is located at 7 mm and -10 mm, respectively. (b) The resolution targets are used to multi-layer-based method. The distances of the resolution targets are 3 cm and 1 cm, respectively.
3.1 WLFH method
The WLFH method for hologram synthesis was simulated using a ‘socket’ light field. This light field has a resolution of 300 × 200 with 40 × 40 views. The depths for two sockets are 7 mm and -10 mm, respectively, as shown in Fig. 6(a). Employing the WLFH method with this light field, the CGH was synthesized with a resolution of 3000 × 2000 and a pixel pitch of 3.74 µm. The SSB technique was applied for the reconstruction of the CGH. For the proposed method, the carrier wave was optimized to have uniform intensities at the same distances. Figure 7 presents the simulation results comparing the conventional general random phase as carrier wave with the proposed method. In the conventional random phase case, significant speckle noise is observed in the reconstruction as expected. With the application of the proposed method, however, the reconstruction is clean with much less speckle noise as highlighted in the red boxed region in Fig. 7. The peak signal-to-noise ratio (PSNR) values of the focused reconstruction parts at 10 mm and -7 mm depths in case of the proposed method are 21.4 dB and 21.3 dB, respectively, while they are only 14.9 dB and 14.8 dB in case of the conventional random phase carrier wave. This confirms that the proposed method results in enhanced reconstruction quality with much higher PSNR values. Figure 8 shows the amplitude distribution of the carrier wave at the optimized distances. While the carrier wave at these distances does not exhibit perfect uniformity, it shows reduced fluctuations compared to the usual random phase.
Fig. 7. Simulation results of the WLFH synthesized with the conventional random phase carrier wave and proposed optimal carrier wave. The red boxed region shows the magnified image.
3.2 Multi-layer-based method
The proposed method can also be applied to the multi-layer-based hologram synthesis approach. We conducted a simulation for a multi-layer-based hologram case. Figure 9 shows the reconstruction results from the multi-layer-based holograms. In this simulation, two USAF-1951 resolution targets were used as the 3D objects, positioned at distances of 1 cm and 3 cm from the hologram plane, as illustrated in Fig. 6(b). The carrier wave for the proposed method was optimized with the same resolution and distances. The synthesized hologram has a resolution of 3840 × 2160 with a pixel pitch of 3.74 µm. The real part of the original complex hologram is extracted, and SSB filtering is conducted for reconstruction of our synthesized hologram. The magnified images of the center of the resolution target are highlighted in the red boxed regions in Fig. 9. As indicated in Fig. 9, the proposed method demonstrates superior performance compared to the conventional method, delivering higher quality and improved signal contrast. The PSNR values of the focused regions at 1 cm and 3 cm are 22.3 dB for both distances in case of the proposed hologram while they are only 14.1 dB and 13.7 dB, respectively in case of the hologram synthesized using conventional random phase carrier wave.
Fig. 9. Simulation results of the conventional random phase carrier wave and proposed optimal carrier wave with multi-layer-based method. The red boxed region shows the magnified image.
A major advantage of the proposed method is that once the carrier wave optimized for given distances is obtained, it can be applied to arbitrary objects at the same distances. Figures 10 and 11 show the reconstruction results of the holograms for different objects. In each object scene shown in Figs. 10 and 11, left object is located at 1 cm and the right one is located at 3 cm. The carrier wave used in the hologram synthesis in case of the proposed method is the same for all object scenes. The red boxed focused regions in Figs. 10 and 11 are cropped to measure the PSNR values as before. As shown in Figs. 10 and 11, the proposed method shows superior quality with higher PSNR values than the conventional method across all tested object scenes even though the same carrier wave optimized for the distance set 1 cm and 3 cm is applied. This result verifies that the proposed method allows the application of the same carrier wave to arbitrary objects at the same distances once the carrier wave is optimized for those distances.
Fig. 10. Numerical reconstructions at 1 cm distance of the multi-layer-based holograms synthesized by using conventional random phase carrier wave and proposed optimal carrier wave. In each row, left and right objects are located at 1 cm and 3 cm, respectively. The red boxed focused region is used for the PSNR measurement.
Fig. 11. Numerical reconstructions at 3 cm distance of the multi-layer-based holograms synthesized by using conventional random phase carrier wave and proposed optimal carrier wave. In each row, left and right objects are located at 1 cm and 3 cm, respectively. The red boxed focused region is used for the PSNR measurement.
3.3 Comparison with SGD hologram
The proposed method offers a significant advantage in which the optimized carrier wave can be applied to any arbitrary object while the SGD hologram method optimizes the hologram itself to a specific object. Figure 12 presents the reconstruction results of the SGD hologram, proposed hologram, and conventional random phase hologram. All holograms used in this comparative simulation were created with a single resolution target, located at 1 cm. As illustrated in Fig. 12, the results of the contents-specific SGD hologram and the contents-independent proposed hologram show similar quality which is much higher than the conventional random phase carrier wave hologram. Although the PSNR of the contents-specific SGD hologram was measured to be higher than the contents-independent proposed hologram due to its higher brightness bias, the actual resolution and the speckle noise have negligible difference as can be observed in the red rectangle part in Fig. 12. This verifies that the proposed method is content-independent and offers comparable quality to the conventional SGD method.
Fig. 12. Simulation results of the SGD hologram, proposed hologram, and conventional random phase with multi-layer-based method.
One notable factor is the computation time. The SGD method requires the iterative optimization for every object. To the contrary, the proposed method uses the same optimized carrier wave to all objects at the same distance without any further object-specific optimization. Therefore, once the optimized carrier wave is prepared by the pre-processing, the actual hologram computation time for each object is fast. In our simulation for Fig. 12, the proposed hologram synthesis using the pre-optimized carrier wave took only 1 sec which is the same as the conventional random phase hologram synthesis, while the SGD hologram synthesis took 50 sec in our Intel core i7-10700 CPU, 64 GB RAM, and NVIDIA GeForce RTX4090 system.
3.4 Detailed analysis of the simulation results
To verify the superiority of the proposed method more clearly, we conducted additional comparative analysis of the simulation results. Figure 13 shows the WLFH results in Fig. 7 with more details. Figure 13(a) and (d) are target images generated directly from the light field and Figs. 13(b) and (e) are reconstructions of the proposed and conventional holograms. Figure 13(c) and (f) shows the amplitude distribution along the yellow lines in Fig. 13(b) and (e). As shown in Fig. 13(c) and (f), the amplitude of the proposed method is well matched to the ground truth while there is a large gap when the proposed method was not employed. For more accurate analysis, we calculated the L2 distance between these distributions using Eq. (7).
where x1 and x2 are the data under comparison and the N is the total number of the sampling points. The results show that the L2 distance of the proposed method from the ground truth is approximately 2.5 times (for 10 mm distance), and 10 times (for -7 mm distance) smaller than that of the conventional method.Fig. 13. Comparative analysis when the proposed method is applied to the WLFH CGH. The refocused images in (a), (d) are synthesized using original light field. (b) and (e) are the numerical reconstruction results of the proposed method (above) and conventional random phase (below), which are the same as the ones in Fig. 7. The red boxed regions are used for comparison between the ground truth and the reconstructions. (c) and (f) show the amplitude distribution along the yellow line in (a), (b), (d), (e).
Figure 14 shows the results for the multi-layer-based hologram method. Similar to the above, Fig. 14(a) and (d) represent the ground truth images used for comparison. Figure 14(b) and (e) are the same ones with Fig. 9. Figure 14(c) and (f) show the amplitude along the yellow lines in Fig. 14(a), (b), (d), and (e). Again, it can be observed that the proposed method enables much closer amplitude distribution to the ground truth than the conventional method, giving higher reconstruction quality.
Fig. 14. Comparative analysis when the proposed method is applied to the multi-layer-based CGH. The ground truth images in (a), (d) are the original 2D images used for the hologram synthesis. (b) and (e) are the numerical reconstruction results of the proposed method (above) and conventional random phase (below), which are the same as the ones in Fig. 9. The red boxed regions are used for comparison between the ground truth and the reconstructions. (c) and (f) show the amplitude distribution along the yellow line in (a), (b), (d), (e).
4. Optical experiments
Optical experiments were also conducted to validate the proposed method. Figures 15 and 16 depict our schematic diagram and a photo of the experimental setup, respectively. We adopted a 4-f system for applying the SSB technique. The aperture used in the 4-f system was made using a 3D printer. The lenses have focal lengths of 10 cm for collimation and 15 cm for the 4-f system. For the optical experiment, we utilized the Holoeye GAEA-2-VIS as the SLM and the FISBA READY BEAM as the laser source. Although the READY BEAM can emit R, G, B lasers, only the 520 nm green laser was used for the experiment. The carrier wave was optimized to have uniform intensities at distances of 1 cm and 3 cm for the multi-layer-based method, and 7 mm and -10 mm for the WLFH method. The optimized carrier waves were restricted to a smaller angular frequency domain to avoid defects such as higher-order terms and unwanted reflections in actual optical experiments. For the optical experiments, the CGHs with the carrier waves optimized by the proposed method and of the usual random phase were synthesized, and the resolution targets and sockets were once again employed as the 3D objects.
4.1 WLFH method
The light field of the two sockets located at 7 mm and -10 mm was used to synthesize the CGH as in Fig. 6. (a). The CGH has 3000 × 2000 resolution with a pixel pitch of 3.74 µm. In order to compare the proposed method with the conventional method, the normally distributed random phase was used as the carrier wave for the conventional method while the optimized phase was used for the proposed method. For optical reconstruction, as we did in our simulation, we extracted only the real parts of each hologram to obtain an amplitude-only hologram, which we then uploaded to the SLM. Figure 17 shows the optical experimental results of the CGH synthesized with WLFH method. As expected, the conventional random phase carrier wave creates more speckled reconstruction due to their random distribution in the object location. Note that the speckle noise in the proposed method is not fully suppressed in the optical reconstruction shown in Fig. 17 unlike the numerical simulation shown in Fig. 7. It is caused by the non-ideal laser illumination and the propagation kernel in the optical setup as will be discussed in Section 5. Nevertheless, the reconstructed image still has better quality with higher PSNR value in the proposed method, by removing the speckle factor occurring from the intensity fluctuation of the carrier wave itself.
4.2 Multi-layer-based method
The two resolution targets located at 1 cm and 3 cm, respectively, are utilized to synthesize the CGH of the 3840 × 2160 resolution with a pixel pitch of 3.74 µm in the multi-layer-based method. The normal random phase is also used for the comparison again. The carrier wave for the proposed method is optimized to have uniform intensity at 1 cm, 3 cm simultaneously. Figure 18 shows the optical experimental results when the camera is focused at the corresponding distances. In Figs. 18(a) and (b), the upper row is the reconstruction using the proposed method, and the lower row is the result of using the conventional random phase. As can be seen in the yellow, red, and gray enlarged sections of Fig. 18, the results obtained using the proposed method successfully achieved superior reconstruction outcomes in multiple layers compared to those using the conventional method. Additionally, for a clear comparison, we measured the PSNR between each cropped section and the original object used in the synthesis of the hologram. The results show that, although not as high as in the simulation, the use of the proposed method yields the higher PSNR values.
Fig. 18. Optical experimental results of the multi-layer-based holograms with (a) 1 cm, and (b) 3 cm distance.
A comparison with the conventional SGD hologram was also conducted. In this case, we placed the resolution target at 1 cm distance in the CGH synthesis following each method. Figure 19 shows the results of the optical experiments. Similar to the results in Fig. 12, it is verified that the proposed method and the SGD method exhibits comparable quality even though the proposed method does not require the object specific optimization while both holograms have superior quality compared to the conventional random phase hologram.
5. Discussion
The proposed method optimizes the carrier wave for the object distances, not specific to the object shape. Once the carrier wave is optimized, the carrier wave can be applied to any objects in the similar depth, showing superior reconstruction quality. The speckle-reduced and signal contrast-enhanced reconstruction of the proposed method was verified successfully by numerical simulations as shown in section 3. The optical experiment results in section 4 also show better reconstruction quality of the proposed method than usual random phase carrier wave. However, it is noted that the reconstruction quality of the optical experiment is worse than the simulation results. While the speckle noise is significantly suppressed in the numerical simulation, it persists in the optical reconstruction results. We believe that this speckle noise comes from the non-ideal conditions in the actual optical implementation. Various factors including the pixel-to-pixel variation of the SLM modulation property, the non-uniform illumination of the laser source on the SLM, and the dust in the optical path contribute to the speckle noise. Additionally, we used the SLM with two polarizers, assuming that it operates as the ideal amplitude modulator with negligible phase modulation [20–22]. However, unwanted residual phase modulation could occur, and it may affect the quality of our optical experiment. Our current carrier wave optimization scheme does not consider these factors, creating the background noise which degrades the optical reconstruction quality. Note that the recently proposed feedback technique, i.e., the camera in the loop (CITL) [23] could be applied to the proposed method. The CITL captures the optically reconstructed images and update the hologram accordingly. The non-ideal propagation kernel is also found during this process. Once the optical setup is characterized by the CITL, the obtained propagation kernel can be considered in the carrier wave optimization process in the proposed method. We believe that this application of the CITL would enhance the optical reconstruction quality of the proposed method, showing its superiority more clearly in the optical experiment.
Another limitation of the proposed method is that the carrier wave is optimized at a few object distances, not addressing entire space where the objects can be located. Achieving uniform amplitude distribution over the entire object space is formidable due to the wide angular spectrum bandwidth of the carrier wave which is desired for the shallow DOF. We believe that the proposed method can be extended to time-multiplexing of the multiple carrier waves optimized for different set of the object distances, giving superior reconstruction quality over wide object depth range.
6. Conclusion
In this paper, we proposed the speckle reduction method for the SSB-encoded CGH in which the carrier wave used for the synthesis of the CGH is optimized to have a uniform intensity at the object distance. The optimized carrier wave can be applied to any arbitrary 3D object, showing the reduced speckle noise and improved signal contrast at the optimized distance regardless of the object type. Unlike previous iterative methods, the proposed method is applicable to the different object contents at the optimized distances and retains the shallow DOF and wide viewing angle of the conventional random phase. Numerical simulations and optical experiments show that the proposed method demonstrates superior reconstruction quality compared to the conventional methods using the random phase carrier waves, validating its effectiveness. Further research is anticipated to reduce the additional speckle and background noise caused by the optical implementation.
Funding
Samsung Research Funding Center of Samsung Electronics (SRFC-IT1702-54); Institute of Information & Communications Technology Planning & Evaluation (2020-0-00548); National Research Foundation of Korea (2022R1A2C2013455).
Acknowledgments
This research was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2022R1A2C2013455); Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.2020-0-00548, (Sub3) Development of technology for deep learning-based real-time acquisition and pre-processing of hologram for 5 G service); Samsung Research Funding Center of Samsung Electronics (SRFC-IT1702-54).
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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