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Abstract
A phase retrieval method based on deep learning with bandpass filtering in holographic data storage is proposed. The relationship between the known encoded data pages and their near-field diffraction intensity patterns is established by an end-to-end convolutional neural network, which is used to predict the unknown phase data page. We found the training efficiency of phase retrieval by deep learning is mainly determined by the edge details of the adjacent phase codes, which are the high-frequency components of the phase code. Therefore, we can attenuate the low-frequency components to reduce material consumption. Besides, we also filter out the high-order frequency over twice Nyquist size, which is redundant information with poor anti-noise performance. Compared with full-frequency recording, the consumption of storage media is reduced by 2.94 times, thus improving the storage density.
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1. Introduction
With the advent of the information era, people's demand for data storage capacity and data transfer rate is becoming more and more intense [1,2]. Holographic data storage (HDS) technology is a form of three-dimensional data storage technology, which has the characteristics of high storage density and long storage life, so it is considered as one of the most promising storage technologies [3–7]. Phase-modulated HDS technology has become a research hotspot in recent years because of its higher code rate and higher signal-to-noise ratio (SNR) [8–10]. However, efficient phase decoding is a tough problem.
Since the image detector cannot detect the phase information directly, the interferometric method is commonly utilized to transform phase information into the detectable intensity information, which is captured by the charge-coupled device (CCD) or complementary metal-oxide-semiconductor (CMOS) [11,12]. The interferometric method is unsuitable for HDS because the system is complex and unstable, which is easily affected by environmental disturbance [13–15]. Conversely, the non-interferometric method is more ideal in HDS because of its stability. There are many non-interferometric methods such as PIE, TIE, IFTA and so on [16–19]. Considering the demand of HDS to simple operation and stable system, the single-shot phase retrieval method is more suitable. Lin et al. proposed a non-interferometric method combining embedded phase data and iterative Fourier transform algorithm, which realized accurate and quick phase retrieval in the holographic data storage system [20]. Chen et al. proposed a dynamic sampling iterative phase retrieval method, which can shorten the iteration number and reduce the phase error rate of phase retrieval [21]. Even so, the traditional iterative method still hinders the improvement of data transfer rate in holographic storage systems. In addition, the iterative method needs to retrieve the phase through the captured the Fourier intensity spectrum distribution. But the noise can introduce additional spectral components, which can affect the phase retrieval results. Therefore, a phase reconstruction method with stronger sampling noise resistance is needed. Extensive research has demonstrated that deep learning can attain favorable retrieval results for both phase and complex amplitude images [22–30]. In previous studies, we proposed a lensless phase retrieval method for HDS based on deep learning and embedded data which can not only improve the data transfer rate and the noise resistance performance, but also improve the storage density by reducing the proportion of embedded phase data [31]. The relationship between the known encoded data page and its near-field diffraction intensity image can be established by the end-to-end convolutional neural network, so as to predict the unknown phase data page.
In the phase pattern, the low frequency component represents the smooth region with high intensity, and the high frequency component represents the edge details with low intensity. Then, we found that the training efficiency of the network model is mainly determined by the edge of the phase pattern, which means that we can selectively record only those frequencies that have a gain for model training. In this paper, we propose a deep learning combined with bandpass filtering for phase retrieval method in holographic data storage, which can realize phase retrieval by only recording and using partial frequency components of phase pattern for deep learning training. We attenuate the low frequency component of the phase pattern to explore the frequency range that can be filtered. Besides, in order to eliminate noise caused by high-order frequencies, we filter out the frequency above twice the Nyquist interval to achieve bandpass filtering of the phase pattern. The experimental results show that our proposed method can reduce the consumption of media and improve the storage density.
2. Theory and methods
The illustration of the data storage and phase retrieval system is shown in Fig. 1. In the recording process, the encoded phase data page is loaded on the spatial light modulator (SLM) as the information beam, which interferes with another reference beam and forms a hologram in the storage media. In the reading process, only the reference beam illuminates the holographic media, and a diffractive beam is generated to realize the reproduction of the information beam, which is called the reconstruction beam. Aperture is used as a spatial filter to limit the size of frequency distribution and regulate the information beam. The calculation method of Nyquist size w as shown in Eq. (1).
where $\lambda$ is wavelength of laser, f is focal length of lens and l is the spatial size of one phase data. In the experiment, one phase data is described by 4 × 4 pixels block on the SLM to show and the pixel pitch is 12.5 µm. So $l$ should be 50 µm. The size parameters of aperture are calculated as shown in Eq. (2) and Eq. (3). where $\tau$ is the high pass filter factor. The setting of $\tau$ determines the size of the attenuation region (lower stopband) of the central part of the aperture. The band-pass filtering method is that the frequencies above ${D_1}$(upper stopband) and below ${D_2}$(lower stopband) are filtered. The frequencies between ${D_1}$ and ${D_2}$ is passband.First, considering the pure phase modulation on SLM, the optical field and its Fourier transform of the phase data page as shown in Eq. (4) and Eq. (5) respectively.
where $\varphi$ is the phase data page, $\mathrm{{\cal F}}\{ \} $ denotes Fourier transform operator, the position of the reconstructed beam plane (the back focal of L4) is defined as $z = 0$.Second, we can regulate the information beam by designing different aperture transmission functions $T(u,v)$ to change the spatial frequency of the information beam. Therefore, real recorded Fourier frequency of the phase-only input as shown in Eq. (6).
Then, if the influence of system and material noise is ignored, the light field of the reconstructed beam as shown in Eq. (7).
Last, the CMOS is placed at a distance d from the reconstructed beam plane to capture the diffraction intensity $I({x,y,z = d} )$. the propagating and imaging process is modeled as shown in Eq. (8).
where $\varphi $ is the phase data page, ${U_d}(x,y,z = d)$ is the complex amplitude distribution of the reconstructed beam at the L4 back focal plane by the near-field diffraction distance d toward the detector plane.$H({\cdot} )$ is a function of the propagating and imaging system, denoting the mapping relationship between phase data page and diffraction intensity captured by the detector.The phase retrieval is to solve the inverse function ${H^{ - 1}}({\cdot} )$. In the experiment, the transmission and imaging system is usually nonlinear, so the estimate of phase data page as shown in Eq. (9).
Convolutional neural networks are good at solving inverse problems in imaging [32]. In this study, we trained the end-to-end convolutional neural network U-net to obtain the relationship between the diffraction intensity pattern and the phase data page, that is, let the neural network solve ${H^{ - 1}}({\cdot} )$[33,34]. Once the model training is completed, the phase data page can be reconstructed directly from the diffraction intensity pattern.
The architecture of the U-net convolutional neural network used in this paper is shown in Fig. 2. Its structure is symmetrical. The left part is the contracting path, the diffraction intensity image is down-sampled and feature extraction is carried out through the 2*2 max pooling operation, which can be regarded as an encoder. The right part is the expanding path, which is the up-sampling process, realizing the stitching and retrieval of the features of the intensity image, and mapping to the phase data page. The activation function following the convolutional layer is Rectified Linear Unit (ReLU), and Sigmoid as the activation function of the output layer [35]. We use the diffraction intensity image captured by the CMOS and the phase data page of the same size as the input and output of the neural network respectively.
The loss function of the neural network is the mean square error (MSE) as shown in Eq. (10).
The training super parameters of the neural network are set as shown in Table 1. The program is implemented in Python 3.6 based on PyTorch framework and used GeForce RTX 3090 for accelerated calculation. In the all simulations, we used 6000 pairs of images as a training data set, 1000 pairs of images as a test data set, and 10 intensity images as a validation data set to calculate the average phase data error rate.
Table 1. Super Parameters for the Neural Network Training
3. Simulation and experimental results
In the simulation, we set the parameters according to the experimental devices. In order to verify the feasibility of the bandpass filtering method, no noise is added in the simulation. Although higher-level encodings can improve the encoding rate, they are more sensitive to noise. The four-level phase encoding may be preferred due to its balance of these factors. For conventional four-level equal interval phase encoding (0, π/2, π, 3π/2), each encoding value may produce the same phase differences, which is not suitable for deep learning methods [38]. Therefore, all the phase data was randomly encoded by the four-level phase (π/6, 2π/3, π, 3π/2) with the different intervals in this paper.
Considering the sequence of propagating and imaging processes, we set the diffraction distance d = 2 mm. We first analyzed the influence of the aperture function $T(u,v)$ on the phase retrieval according to Eq. (6). In the phase-modulated HDS system, the evaluation criterion of phase retrieval is bit error rate (BER), which is the ratio of error retrieved phase data pixels to total phase data pixels [31].
In order to evaluate the shape and size of the filter, we designed circular and square bandpass filters, whose aperture functions as shown in Eq. (11) and Eq. (12) respectively.
The intensity distribution of the Fourier spectrum of the phase pattern is shown in Fig. 3(a). The schematic diagram of the circular filter is shown in Fig. 3(b). The schematic diagram of the square filter is shown in Fig. 3(d). The filtered intensity distribution of the Fourier spectrum of the phase pattern is shown in Fig. 3(c) and Fig. 3(e).
Fig. 3. (a) The intensity distribution of the Fourier spectrum of the phase pattern, (b) the schematic diagram of the circular bandpass filter, (c) intensity distribution of the Fourier spectrum after the circular bandpass filter, (d) the schematic diagram of the square bandpass filter, (e) intensity distribution of the Fourier spectrum after the square bandpass filter.
The high pass filter factor $\tau$ is changed from 0.1 to 2 (when the high pass filter factor of the square filter is set to 2, all frequencies are cut off, which is meaningless). The phase data error rate of the two filtering functions at different high pass filter factors is shown in Fig. 4. When the high pass filter factor value is greater than 1.1, the phase retrieval effect of the circular bandpass filter is better than that of the square bandpass filter because the former frequency response is smoother than the latter, which can better retain edge information in the image and avoid sharpening effect and edge blurring. Therefore, only circular bandpass filter is used in subsequent simulations.
Fig. 4. The phase data error rate of the two filtering functions at different high pass filter factors.
Due to the printing process of ultraviolet curing, it is difficult to completely attenuate the spectrum intensity of the lower stopband. Therefore, we set the lower stopband transmittance from 0.01 to 0.05, and obtained the curve relationship between the high pass filter factor and the phase error rate is shown in Fig. 5(a). It can be seen that due to the high intensity of low-frequency information, when the high pass filter factor value is large, the higher stopband transmittance has a more significant effect on the gain of phase retrieval.
Fig. 5. BER with (a) high pass filter factor under different lower stopband transmittance and (b) passband transmittance under different high pass filtering factors.
In addition, since the transmittance of common transparent optical materials is about 0.8-1, we set $\tau $= 0.5, 1, 1.5, 2 respectively and changed the passband transmittance of the filter from 0.81 to 0.99. The result is shown in Fig. 5(b). Because the passband is high-frequency information with relatively low intensity, the transmittance of the passband has little influence on phase retrieval. This means that selecting the base material of the filter in the experiment has a higher degree of freedom and does not have to be restricted to materials with high transmittance.
In the previous work, we found that different diffraction distances d would produce different intensity images, is shown in Fig. 6, which would affect the results of phase retrieval. Therefore, we investigated the influence of diffraction distance on the phase retrieval effect with different high pass filter factors. We compared the BER with $\tau $= 0.5, 1, 1.5, 2 at the diffraction distance d from 0 mm to 3 mm. The result is shown in Fig. 7. As the diffraction distance increases within a given range, the edge details of the intensity image become progressively blurred but the diffraction features are gradually enriched. Because high-frequency information above the 2 times Nyquist interval is filtered out, the partial details of the image are lost. In a certain range of diffraction distances, the phase retrieval effect of this method is almost unaffected.
Before and after bandpass filtering, intensity images captured by the CMOS are shown in Fig. 8(b) and Fig. 8(d) respectively. Their three-dimensional intensity distributions of the Fourier spectrum are shown in Fig. 8(c) and Fig. 8(e) respectively. It can be seen that after filtering, the light intensity recorded in the media is significantly reduced. According to the parameters measured by experiment, we set the lower stopband transmittance to 0.02, and the passband transmittance to 0.91. Compared with all frequency records, we calculate the multiples by which the bandpass filtering method saves the storage material. The result is shown in Fig. 9.
Fig. 8. (a) The phase pattern uploaded on SLM, (b) intensity image captured by the CMOS before filtering, (c) three-dimensional intensity distributions of the Fourier spectrum of image (b), (d) intensity image captured by the CMOS after filtering and (e) three-dimensional intensity distributions of the Fourier spectrum of image (d).
We also considered the case of filter position shift. In the experiment, it is difficult to achieve that the position of the aperture is exactly in the center of the spectrum, and the shift can be ensured below $0.1w$ usually. Taking the high pass filter factor $\tau = 1$ as an example, the aperture was shift by 1-6 pixels to the upper left, lower left, upper right and lower right respectively (6 pixels corresponded to $0.1w$ in the experiment). The curve relationship between BER and the number of shift pixels in the simulation is shown in Fig. 10. Due to the symmetry of the spectrum, the direction of shift has little impact on the phase retrieval results. In addition, within the shift of 6 pixels, the fluctuation range of BER is below 0.1%, which proves the tolerance of the bandpass filtering method to aperture position shift is large.
The experimental setup of the off-axis holographic data storage system is shown in Fig. 11. In this study, considering that media will cause uneven data samples, we did not use media for data capture. The wavelength of the laser is 532 nm. When the laser wavelength is 532 nm, the holographic storage media is more sensitive and has good diffraction efficiency. The SLM is X15213-16 by HAMAMATSU, which has the resolutions of 1272 × 1024 and a pixel spacing of 12.5 µm. The aperture is placed on the back focal plane of L1 to perform spatial filtering on the information beam. We used polymethyl methacrylate (the transmittance is about 0.91) as the substrate material for the filter and coated the attenuation area of the filter (the transmittance is about 0.02). The CMOS1 is DCC3260 M by Thorlabs, which has the resolutions of 1936 × 1216 and a pixel spacing of 5.86 µm. It was placed 1 mm from the back focal plane of the L4 and used to capture the diffraction intensity of the reconstructed beam. The CMOS2 is DCC1545M-GL by Thorlabs, which has the resolutions of 1280 × 1024 and a pixel spacing of 5.2 µm and was used to capture the Fourier spectrum intensity of the reconstructed beam. By observing the images captured by the CMOS2, we could determine whether aperture is located at the center of the spectrum.
Fig. 11. The experimental setup of the off-axis holographic data storage system. BE: beam expander, BS: beam splitter, HWP: half wave plate, SLM: spatial light modulator, L1-L5: lens (150 mm).
The phase data page uploaded on the SLM and its diffraction intensity image are shown in Fig. 12. The encoded phase pattern is a 32 ∗ 32 data matrix and every phase data is displayed by a block of 4 × 4 pixels on the SLM. Therefore, the pixel matrix of the phase data page uploaded to the SLM is 128*128. According to the parameters of the SLM and the CMOS, the diffraction intensity pattern pixel size of data page captured in the CMOS1 is approximately 273*273. To capture all the diffraction features, we took 288*288 of the intensity pattern, shrunken it to 256∗256 by the bicubic interpolation algorithm, and padded zero to 288 ∗ 288. Meanwhile, the phase data page was expanded to 256*256 by 2 times upsampling, also padded zero to 288∗288 to maintain the same size as the intensity image. The diffraction intensity patterns captured by the CMOS for each high pass filter factor in the experiment are shown in Fig. 13.
Fig. 12. (a) The encoded phase data page uploaded on the SLM, (b) the phase data page was expanded to 256*256 by 2 times upsampling and then padded zero to 288∗288, (c) The diffraction intensity image captured by the CMOS with a high pass filter factor of 0.5 at a diffraction distance of 1 mm.
Fig. 13. Diffraction intensity captured by the CMOS with different high pass filter factors in the experiment.
For each size high pass filter factor, 7000 randomly generated phase data pages are uploaded to SLM and corresponding intensity images are captured by the CMOS as the dataset. The proportions of each dataset were consistent with the simulation. In order to suppress the impact of the dynamic noise, we shuffled the experimental images. When the $\tau $= 1.9, 2.0, the system noise is almost consistent with the intensity of the reconstructed beam, resulting in no corresponding intensity images captured by the CMOS. The curve relationship between high pass filter factor and phase error rate in the experiment and simulation is shown in Fig. 14. As the high-pass filter factor increases, the diffraction intensity captured by the CMOS decreases consequently in the experiment, which amplifies the impact of the dynamic noise of the CMOS on the retrieval effect. On the contrary, since no noise is added to the simulation, the diffraction intensity pattern is normalized to enhance its darker high-frequency details, which makes the phase retrieval relatively stable. When no check code is added, we think that BER of around 0.1% is acceptable in the phase-modulated HDS system. Therefore, the optimal high pass filter factor is determined to be 0.9, and the BER is 0.1%. After bandpass filtering, the recorded light intensity is only about 33.97% of the unfiltered one, which can save about 2.94 times the media.
Fig. 14. The curve relationship between high pass filter factor and phase data error rate in the experiment and simulation.
4. Conclusions
In this paper, we proposed a bandpass filtering combined with deep learning for phase retrieval method in holographic data storage. The parameter settings of this method were discussed through simulation, and the feasibility was verified by experiments. This method could reduce the consumption of storage media and increase the storage density by 2.94 times by bandpass filtering on the information beam. Besides, the method filters out frequencies above twice the Nyquist interval, improves the noise resistance of the system and is insensitive to the diffraction distance. Meanwhile, this method has a higher tolerance for the transmittance and placement of the aperture in the experimental system, which is quite suitable for the phase-modulated HDS system. In the future, we will use storage media for data capture to further investigate the influence of sample inhomogeneity caused by media on bandpass filtering methods.
Funding
National Natural Science Foundation of China (62005048, U22A2080); National Key Research and Development Program of China (2018YFA0701800); Project of Fujian Province Major Science and Technology (2020HZ01012).
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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