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Abstract
A dynamic sampling iterative phase retrieval method, which dynamically samples the Fourier intensity distribution of the reconstruction beam captured by the detector, is proposed to shorten the iterative number and decrease the phase error rate of phase retrieval in the phase-modulated holographic data storage. By the dynamic sampling method, that keeping relatively low frequency component of Fourier intensity spectrum at the beginning of iteration and gradually releasing more high frequency component at the subsequent iterations, we shortened the iterative number by 2 times and decreased the phase error rate to some extent because our method provided a better convergent path to the phase retrieval. We also believe the thought of our method can be used in more image retrieval fields.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Rising demand of data storage density and data transfer rate is increasingly straining [1,2]. Holographic data storage technology as a promising approach for long-term data storage has been researched a lot [3–6]. Holographic data storage is characterized by the use of three-dimensional volume storage mode [7]. In addition, holographic data storage can greatly improve storage density by different multiplexing technologies such as shift multiplexing [8], wavelength multiplexing [9], phase coded multiplexing [10], angular multiplexing [11] and potential polarization-multiplexing [12–16]. Traditional amplitude-modulated holographic data storage is hard to improve the data storage density [17,18]. Phase-modulated holographic data storage due to its higher code rate and higher signal-noise-ratio (SNR) has been paid more attention and research interests recently [19–22].
Though there is a problem that the phase distribution cannot be detected by the detector directly, we can solve this problem by using interferometric method or non-interferometric method. However, interferometry is not an ideal decoding method for the phase-modulated holographic data storage system due to the instability of the interference system and usually multiple operations [23,24]. Non-interferometric is to retrieve the phase distribution from the intensity distribution captured by the detector directly, therefore the system is stable. Some non-interferometric algorithms such as the ptychographical iterative engine (PIE) and the transport of intensity equation (TIE) need multiple operations in the intensity images capture leading to slower data transfer rate [25–27]. Single-shot algorithm such as the iterative Fourier transform (IFT) owns more simple operation, however, usually it needs more iterative computing [28].
In our previous work, we have decreased the iterative numbers by using embedded data in the IFT algorithm which makes the iterative number acceptable relatively [29]. However, we believe there must be a more optimized phase retrieval path. In this paper, we propose a method that dynamically samples the Fourier intensity distribution of the reconstruction beam captured by the detector to shorten the iterative number of the phase retrieval by 2 times. In the Fourier intensity image, the low frequency component and high frequency component own core information with high intensity and marginal detail information with low intensity respectively. In the first few iterations, the low frequency component is dominant. Therefore, we kept relatively low frequency component of Fourier intensity spectrum at the beginning of iteration and gradually released more high frequency component at the subsequent iterations. At every iteration, there is a best region separating the low frequency component and high frequency component. By accumulating enough data, we can get an experience curve about separating frequency that is the optimized phase retrieval path. Once system parameters are decided, we can always get a certain path to shorten the iterative number. The source for phase retrieval in our method is only one single-shot Fourier intensity image which is same to previous work, therefore there is no extra cost both in data collection and computing.
2. Theory and methods
A scheme of non-interferometric system for phase retrieval is shown in Fig. 1. In the encoding process, we encode a certain proportion of embedded data, where the position and phase values are known, into the original phase data page and upload it on the spatial light modulator (SLM). The embedded data work as a strong constraint in the object domain. Higher proportion of embedded data is more conducive to fast and accurate phase retrieval. In the writing process, the signal beam will interfere with a reference beam and form a hologram in the media. In the reading process, the hologram in the material is illuminated by the same reference beam to form a reconstructed beam. Our aim is to retrieve the phase information of the reconstructed beam. We used a non-interferometric system to make the reconstructed beam pass through a Fourier lens and capture the intensity distribution of the Fourier transform of the reconstructed beam by the CMOS.
In the phase decoding calculation process, the Fourier intensity distribution and the embedded data are used to retrieve the phase information of the reconstructed beam by the IFT algorithm. The IFT algorithm combined with our dynamic sampling is as follows. First, an initial guess phase distribution ${\boldsymbol{\varphi}_0}$ is set, then assign ${\boldsymbol{\varphi}_0}$ to ${\boldsymbol{\varphi}_n}$, so we can get an initial guess complex amplitude distribution ${\boldsymbol{U}_n}$ in the object domain as shown in Eq. (1).
After Fourier transform, a complex amplitude distribution ${\boldsymbol{V}_n}$ in the Fourier domain is obtained as shown in Eq. (2).
Then, dynamic sampling is carried out for the intensity distribution ${\boldsymbol{I}_0}$ captured by the CMOS in every iteration. Consider the intensity distribution represented as a two-dimensional gray scale image with columns s and rows t, every pixel in the intensity distribution can be denoted by ${\boldsymbol{I}_0}({x,y} ),x \in [{1,s} ],y \in [{1,t} ]$. We can get sampled distributions ${\boldsymbol{I}_n}({x,y} )$ in different iterations as shown in Eq. (3).
We use the square root of the sampled intensity distribution $\sqrt {{\boldsymbol{I}_n}}$ to replace the amplitude ${\boldsymbol{A}_n}$ thus we get the distribution ${\boldsymbol{V^{\prime}_n}}$ as shown in Eq. (4).
After inverse Fourier Transform, we can get the distribution ${\boldsymbol{U^{\prime}_n}}$ in the object domain as shown in Eq. (5).
Since we use phase-only code, ${\boldsymbol{A^{\prime}}_n}$ should be set to 1. In addition, the phase of the embedded data should replace the phase of corresponding positions. Therefore, a new complex amplitude distribution ${\boldsymbol{U^{\prime\prime}}_n}$ can be got as shown in Eq. (6).
The phase ${\boldsymbol{\varPhi}^{\prime\prime}_n}$ is assigned to the initial phase for the next iteration until convergence and stagnation. We can get the distribution ${\boldsymbol{V}^{\prime\prime}_n}$ as shown in Eq. (7). Then we can compute the intensity error rate ${\boldsymbol{E}_k}$ as shown in Eq. (8) and the difference $\Delta \boldsymbol{E}$ between two adjacent intensity error rates is used as the convergence condition as shown in Eq. (9).
When $\Delta \textrm{E}$ falls below 10−3, we think the iteration is convergent and we should stop the iteration. Otherwise, ${\boldsymbol{\varPhi}^{\prime\prime}_n}$ will be the new guess ${\boldsymbol{\varPhi}_{n + 1}}$.
In the paper, the input on the object domain is random 4-level phase patterns (0, π/2, π, 3π/2) with a size of 32 × 32 data. Every data will be displayed by a block of 4×4 pixels on the SLM as shown in Fig. 2(a). The intensity distribution of the Fourier spectrum and its three-dimensional intensity distribution are shown in Fig. 2(b) and Fig. 2(c) respectively. Figure 2(c) shows the low frequencies with high power and the high frequencies with low power. The red square in Fig. 2(c) means there are gray values in the red square. In the process of iteration, we set a series of thresholds in different iterations to discard parts of the high-frequency information and parts of low-intensity low-frequency information. Because the intensity of the high-frequency is lower, the thresholds owns higher effect on the high-frequency. Therefore, when the threshold is small, the threshold works as a low-pass filter approximately. The core of our method is to find suitable thresholds to discard some gray values where these discarded low intensity points usually own lower SNR and lower weight of phase retrieval. The spectrum after discarding some low intensity points owns simpler distribution with higher SNR. That is helpful for speeding up phase retrieval in the first few iterations. As an example, the Fourier spectrum of discarded gray value is shown in Fig. 2(d), and its three-dimensional intensity distribution is shown in Fig. 2(e). The size of the red square in Fig. 2(e) is same to that in Fig. 2(c). The yellow square in Fig. 2(e) means there are gray values in the yellow square.
Fig. 2. (a) The phase pattern uploaded on SLM, (b) intensity distribution of the Fourier spectrum, (c) three-dimensional intensity distribution of image (b), (d) the Fourier spectrum of discarded gray value, (e) three-dimensional intensity distribution of image (d).
3. Simulation and experimental results
We simulated the non-interferometric phase retrieval according to real parameters of devices. In order to verify the feasibility of the dynamic sampling method, first we do not consider the effect of noise. The dynamic sampling of the Fourier spectrum can be realized by discarding the gray values of the Fourier spectrum.
We use the first iteration as an example. We retrieved phase by using different sampled Fourier intensity which is discarded different gray values and calculated the phase error rate in these situations. The curve between discarded gray value and phase error rate is shown in Fig. 3. We picked the gray value as the suitable threshold at the minimum of the phase error rate.
In subsequent iterations, the thresholds of discarding gray values tend to decrease with the increase of iteration numbers. Figure 4 shows the scatter plot of the thresholds of discarding gray value with different iteration numbers. Therefore, the discarded high frequency information is released gradually with the increase of iteration numbers. The intensity distribution of the original Fourier spectrum with two Nyquist size is shown in Fig. 5(a), and we increased the image display intensity for clearer understanding of dynamic sampling process as following. The first to third sampling intensity distribution (T1=9, T2=7, T3=5) of the Fourier spectrum are shown in Figs. 5(b)–5(d). In the simulation without noise, the comparison of phase retrieval result between the dynamic sampling iterative calculation and the conventional iterative calculation directly using the whole captured Fourier spectrum is shown in Fig. 6. It shows that dynamic sampling method can get lower phase error rate at the same iterative number.
Fig. 5. (a)The intensity distribution of the original Fourier spectrum, (b)-(d) the first to third sampling intensity distribution (T1=9, T2=7, T3=5) of the original Fourier spectrum respectively.
Fig. 6. In the simulation without noise, the comparison of phase retrieval result between the dynamic sampling iterative calculation and the conventional iterative calculation directly using the whole captured Fourier spectrum.
Noise are certain to arise in the real system. Therefore, a white Gaussian noise was added resulting in different signal-to-noise ratio (SNR) to demonstrate the ant-noise performance of the dynamic sampling method. It should be noted that the gray value is normalized to 0-255. Thus, if the gray value is smaller than 0 or larger than 255, we must set it to 0 or 255 respectively.
We set SNR=5, 6, 7, 8, 9 respectively and calculated the training curves of the thresholds of discarding gray value as shown in Fig. 7. It shows that even if the SNR is different, the training curve is similar. We get the average training curve by taking the average of the actual training curves of different SNR.
In Table 1, we compared the phase retrieval simulation results for the dynamic sampling method and the conventional method under different SNR. The smaller the SNR is, the stronger the noise is. In the conventional method, the phase error rates still exist after 20 iterations. In our dynamic sampling method, the phase error rates achieve 0 after only 10 iterations.
Table 1. Simulation results by the dynamic sampling method and the conventional method in different SNR
Figure 8 shows the comparison of convergent curves between the dynamic sampling method and the conventional method. Our method can effectively shorten the iteration number and reduce the phase error rate compared to the conventional method.
Fig. 8. In the simulation (SNR=5), convergent curves of dynamic sampling method and conventional method.
An off-axis holographic data storage system used in the experiment is shown in Fig. 9. The power of the laser was 300 mW with a wavelength of 532 nm. The SLM is X10468-04 by HAMAMATSU with a pixel pitch of 20µm and the resolutions 792×600. The media is Irgacure 784-doped PMMA photopolymer [30]. The CMOS is DCC3260M by Thorlabs with a pixel pitch of 5.86 µm and the resolutions 1936×1216. The encoded phase pattern includes 32×32 phase data. Every phase data is displayed by a block of 4×4 pixels on the SLM.
Fig. 9. The optical setup of off-axis holographic data storage system. HWP: half wave plate, BS: beam splitter, L1 ∼ L5: lens (L1 ∼ L4=150 mm, L5=50 mm), SLM: spatial light modulator.
In the experiment, the average training curve of the thresholds of discarding gray value is shown in Fig. 10, is used to dynamically sample the experimental Fourier spectrum. The intensity distribution of the experimental Fourier spectrum is shown in Fig. 11(a). Because the noise of the captured image in the experiment cause the phase data error rate larger than 5% which is not acceptable before correction in holographic data storage, we used a convolutional neural network model to denoise the captured image and got the denoised image shown in Fig. 11(b). This denoising convolutional neural network model [31,32] learns the relationship between the captured intensity images and the simulation truth images by using 5000 pairs of images, so it can well grasp the complex noise pattern in the captured image. Compared with the traditional denoising method, the convolutional neural network method can reduce the noise and retain the texture information of the original image better.
Fig. 11. (a) The intensity distribution of the experimental Fourier spectrum, (b) the denoised image of image (a).
In Fig. 12, when we used the captured original image to retrieve the phase, the conventional method requires 20 iterations to achieve 6.84% phase data error rate and our dynamic sampling method only requires 10 iterations to achieve 6.54 phase data error rate. However, the phase error rates both exceed 5%, so denoising is necessary. Figure 13 shows the denoising result, the conventional method requires 20 iterations to achieve 1.37% phase data error rate, however, our dynamic sampling method only requires 10 iterations to achieve same phase data error rate. The phase retrieval and phase data error rate results by using the denoising image are shown in Fig. 14. Figure 14(a) is the original phase pattern that we uploaded on the SLM. Figures 14(b) and 14(c) are retrieved phase pattern and phase error distribution after 10 iterations by the conventional method. Figures 14(d) and 14(e) are retrieved phase pattern and phase error distribution after 10 iterations by the dynamic sampling method. The red rectangles in Figs. 14(a)-(e) mean the unknown phase code positions and the other sides are the embedded data positions. It effectively proves that our method can provide a better convergent path to the phase retrieval.
Fig. 12. In the experiment, convergent curves of dynamic sampling method and conventional method by using captured original image to retrieve phase.
Fig. 13. In the experiment, convergent curves of dynamic sampling method and conventional method by using denoising image to retrieve phase.
Fig. 14. (a) The original phase pattern that we uploaded on the SLM, (b) the retrieved phase data after 10 iterations by the conventional method, (c) phase error distribution corresponding to the retrieved phase (b), (c) the retrieved phase data after 10 iterations by the dynamic sampling method, (e) phase error distribution corresponding to the retrieved phase (d).
4. Conclusions
In this paper, we proposed a method of dynamic sampling iterative phase retrieval for holographic data storage. We verified the proposed method own higher efficiency of phase retrieval than conventional method both in the simulation and experiment. The dynamic sampling method can provide a better convergent path to retrieve phase, therefore, it can shorten the iterative number by 2 times and decrease the phase error rate to some extent. We also believe the thought of our method can be used in more image retrieval fields but not only in the holographic data storage.
Funding
National Key Research and Development Program of China (2018YFA0701800); Wuhan National Laboratory for Optoelectronics (2019WNLOKF007).
Disclosures
The authors declare no conflicts of interest.
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