1 Introduction
With the development of information technology, extensive data are being generated at all times. Traditional data storage technology has difficulty coping with the increasing demand for data growth [1,2]. In 1963, Van Heerden formally proposed the concept of holographic data storage (HDS) [3]. HDS technology, as a three-dimensional volume storage technology, has garnered renewed attention because of its ultra-high storage density and ultra-fast data transfer rate [4–6].
Traditional HDS technology uses amplitude modulation, but the code rate is relatively low. In contrast, phase-modulated HDS can increase the code rate [7,8] and the recording signal-to-noise ratio (SNR), which is conducive to further increasing the storage density; however, the camera cannot read the phase information directly. Typically, interferometric and non-interferometric phase-retrieval methods are used to decode the phase information. However, the interferometric phase decoding method suffers from problems, such as system complexity and instability of detection results [9–11], whereas non-interferometric phase-retrieval methods usually have a simpler and more stable detection system [12]. There exist many non-interferometric phase-retrieval methods, such as the ptycholographic iterative engine (PIE) algorithm [13,14], transport-of-intensity equation (TIE) [15,16], and iterative Fourier transform (IFT) algorithms [17,18]. This study employs the IFT algorithm to retrieve phase owing to its simpler and more stable system and single-shot operation.
The traditional IFT algorithm requires a multitude of iterations; thus, to reduce the number of iterations, the addition of available embedded data is proposed to enhance the constraint condition [19]. In actual experiments, iterations are typically more than simple simulations because the noise generated by the optical system and electrical components affect the Fourier intensity distribution [20]. The Fourier intensity distribution includes high and low-frequency information. Although the high-frequency information of the Fourier intensity aids in more accurate and faster phase retrieval, its intensity is weaker, and the noise immunity is worse than low-frequency information. Therefore, this paper proposes a method to encode embedded patterns to improve the SNR of high-frequency information in the Fourier intensity distribution and thus further increase the efficiency of phase retrieval.
2 Theory and methods
The model of the HDS and phase-retrieval system with embedded data is shown in Fig. 1.
In Fig. 1, the signal phase pattern is composed of unknown and embedded phase data. Unknown data refers to the valid information that needs to be recorded. Embedded data are known data, which can accelerate the phase reconstruction efficiency. In the recording process, the signal phase pattern was uploaded on the spatial light modulator (SLM) as the signal beam interferes with the reference beam, thereby recording a hologram in the media. Thereafter, in the reading process, the same reference beam illuminates the hologram, and the reconstructed beam is diffracted. Furthermore, to read out the phase information of the reconstructed beam, a single lens was used, and the charge-coupled device (CCD) captured the Fourier intensity distribution of the reconstructed beam. Finally, by using the IFT algorithm [12], the unknown phase data in the signal phase pattern was retrieved.
From previous studies, we discussed the proportion of embedded data, where 50% of the embedded data are appropriate [12]. Therefore, we used the 4-level phase data shown in Fig. 2(a) as the signal phase pattern, which is composed of 50% of unknown and embedded phase data each, as shown in Figs. 2(b) and 2(c), respectively. The Fourier intensity distribution captured by the CCD and its three-dimensional distribution is shown in Figs. 2(d) and 2(e), respectively. The red curve in Fig. 2(e) represents the intensity envelope in the Fourier plane.
In Fig. 2, it is intuitively evident that the high-frequency information intensity is relatively low compared to the low-frequency information intensity. The experiments performed indicated that the noise in Fourier intensity gradually decays from low to high-frequency, and thus a similar noise was added to the simulation, as shown in Fig. 3(a). The Fourier intensities before and after the addition of noise are shown in Figs. 3(b) and 3(c), and the corresponding three-dimensional envelope distributions are shown in Figs. 3(d) and 3(e). Furthermore, the calculated value of the SNR of the low-frequency part (in the red frame) is 7.52, and that of the high-frequency part (outside the red frame) is 0.38, which renders the high-frequency information susceptible when there is noise. High-frequency information can aid in retrieving the phase quicker with high accuracy; therefore, the aim of our method is to increase the intensity of the high-frequency information by encoding the embedded phase pattern.
The corresponding relationship between the phase pattern and the envelope intensity distribution is shown in Fig. 4.
Figures 4(a1)−4(a3) represent the unknown, previous embedded, and combined overall signal phase patterns, respectively. The envelope intensity distributions corresponding to the three-phase patterns are the red, blue, and solid yellow lines in Fig. 4(a), respectively. Figures 4(b1) and 4(b2) represent the new embedded and combined overall new signal phase patterns, respectively, and their corresponding envelope intensity distributions are represented by the blue and yellow dotted lines in Fig. 4(b), respectively.
In the actual experiment, we used a 4-level phase (0, π/2, π, 3π/2) containing 50% embedded data. Moreover, oversampling should be used to avoid the phase gap effect of adjacent pixels on the SLM. Typically, one phase of data should be displayed by 4 × 4 pixels on the SLM, such that the oversampling rate is 4. The spatial frequency on the SLM plane is the same for the unknown and previous embedded phase patterns. Therefore, their envelope intensity distributions are identical, as shown in Fig. 4(a), for the red and blue solid lines. We aim to design an envelope with a high-frequency, such as the blue dotted line shown in Fig. 4(b), and determine a new embedded phase pattern corresponding to it, as shown in Fig. 4(b1). The space-frequency spectrum of embedded data we designed needs to be higher than that of the phase data page to improve the high-frequency component of the overall Fourier spectrum.
Because the Fourier intensity envelope is related to the phase pattern, after designing the intensity envelope about the embedded phase pattern, the embedded phase pattern itself can be obtained by using the IFT algorithm. Therefore, in our method, the embedded phase pattern is not assigned directly but is generated by iterative computing according to the designed intensity envelope of the embedded data.
The method to design a new embedded data phase pattern is as follows.
We used MATLAB software to generate a three-dimensional embedded data envelope intensity distribution with higher high-frequency as the constraint condition, as shown in Fig. 5(a).
Through the IFT algorithm, the previous embedded envelope intensity distribution, as shown in Fig. 5(b), was constrained and fitted to the designed three-dimensional envelope intensity distribution gradually, as shown in Fig. 5(a). Fitting is an iterative process; thus, typically, 20 iterations were required to complete the fitting, and in each iteration, a new embedded data envelope intensity distribution was obtained, as shown in Figs. 5(c) and 5(d), and new embedded data pattern corresponding to the envelope intensity distribution was recorded, to finally obtain 20 different embedded data patterns, as shown in Figs. 5(f) and 5(g). Our purpose was to determine the embedded phase pattern corresponding to the optimal envelope intensity distribution. By combining different new embedded data patterns with the same unknown data pattern, a series of different Fourier intensity distributions can be obtained, and thereafter, using the IFT algorithm [12], the phase retrieval results of different new embedded data patterns can be compared to determine the best new embedded data pattern.
Fig.5 Schematic diagram of generating new embedded data. (a) Constrained envelope distribution. (b) Previous embedded data envelope and its corresponding (e) embedded data pattern. (c) and (d) Envelope distribution under different iteration times and its corresponding (f) and (g) new embedded data patterns |
Different new embedded data patterns affect the efficiency of phase retrieval. The specific algorithm for generating new embedded phase patterns is as follows:
In the first step, we consider an initial guess phase distribution of the embedded data , n = 1, 2, 3,..., which represents the number of iterations, and the position of the embedded data remains unchanged during the iteration. This results in a complex amplitude distribution in the object domain, as shown in Eq. (1).
This is followed by the second step, wherein we obtain a complex amplitude distribution on the Fourier domain after the Fourier transform, as shown in Eq. (2).
where represents the Fourier transform operator, and and represent the amplitude and phase, respectively.
Further, in the third step, we use the square root of the designed envelope intensity to replace the amplitude ; thus, a new distribution can be obtained, as shown in Eq. (3).
Thereafter, in the fourth step, we continue to go through the inverse Fourier transform to obtain the new complex amplitude distribution on the object domain, as shown in Eq. (4).
where represents the inverse Fourier transform operator, and and represent the new amplitude and phase, respectively.
Finally, in the fifth step, phase-only is used as a constraint condition to rectify the complex amplitude distribution. Therefore, the amplitude was normalized to 1, resulting in a new distribution as shown in Eq. (5).
where is the new embedded phase pattern.
The afore described steps were repeated several times, and after each iteration, a new embedded data phase pattern was obtained; therefore, after several iterations, a series of different new embedded data phase patterns can be obtained.
3 Simulation and experimental results
We used a 32 × 32 signal phase pattern randomly with 4-level phase (0, π/2, π, 3π/2), which contains 50% embedded data shown in Fig. 6(a), and each data was displayed in a 4 × 4 pixel block on the SLM with a pixel pitch of 20 μm. The pixel size of the CCD is 5.86 μm. According to the Nyquist sampling theorem, we chose two Nyquist size frequencies in the Fourier intensity shown in Fig. 6(b) for phase retrieval.
According to the flow shown in Fig. 5, we obtained new embedded data. To distinguish the previously embedded data from the new embedded data, they were named as the assigned embedded data and the iterative embedded data, respectively. The difference in the phase patterns between these two embedded data is shown in Fig. 7. Compared with the assigned embedded data, the iterative embedded data are more complex and finer because space complexity provides more high-frequency components.
Figure 8 shows the Fourier intensity distributions corresponding to the assigned embedded and iterative embedded data. It is evident that the details of the high-frequency information in Fig. 8(d) are more abundant than those in Fig. 8(b). Following this, we also evaluated the intensity of the high- and low-frequency signals in the Fourier plane from two perspectives. It was found that when the high-frequency intensity increased, the ratio of the high-frequency intensity to the total intensity increased; whereas, when the low-frequency intensity decreased, it resulted in a decrease in the ratio of the low-frequency intensity to the total intensity.
One Nyquist size-frequency is denoted as , and we obtained two Nyquist size frequencies in the Fourier intensity, as shown in Fig. 9(a). Further, we calculated the average intensity of the two Nyquist size frequencies to be .
Furthermore, we calculated the intensity of the high-frequency information. The size of the yellow box gradually reduced from 1.9 to 1.5, and to express our choice of ideas more vividly; we only chose three yellow boxes of different sizes as the explanation. The yellow boxes in Figs. 9(b)–9(d) show three different size windows: 1.9, 1.7, and 1.5. Only the values outside these windows were retained, and thereafter the average intensity was calculated.
In contrast, we calculated the intensity of the low-frequency information. The size of the red box was gradually increased from 0.1 to 0.5, and similarly, three windows were selected as an illustration. The red boxes in Figs. 9(e)–9(g) are three different size windows of 0.1, 0.3, and 0.5. Similarly, only the values inside these windows were retained, followed by the calculation of the average intensity .
We calculated the proportional intensity coefficients and to evaluate the intensity distribution of the high- and low-frequency regions according to Eq. (6).
In the afore described embedded data algorithm, by controlling the number of iterations, we can obtain different embedded data envelope distributions corresponding to different iterative embedded data patterns, and thereby obtain a series of overall Fourier intensity distributions composed of different iterative embedded data patterns along with the same unknown data pattern. In Fig. 10, we selected three cases as illustrations.
In case 1, iterative embedded data are used to reduce the high-frequency Fourier intensity. Whereas in case 2, the high-frequency intensity was slightly greater than that in case 1 by using the assigned embedded data. Finally, in case 3, the iterative embedded data was changed to render the high-frequency of the Fourier intensity slightly greater than that in case 2. Figures 10(a) and 10(b) show the proportional intensity coefficient curves corresponding to the calculation results of and in Eq. (6), respectively. A high value of along with a low value of indicates greater high-frequency information intensity, and vice-versa. and represent the same conclusions. However, the differences among the values were small, whereas the differences among the values were larger. The latter is more conducive to distinguishing different Fourier intensities; thus, we chose under 0.1 times the Nyquist size and used it to express the following calculation.
In the absence of noise, the assigned embedded and iterative embedded data are approximately the same for the phase retrieval because the SNR of high-frequency intensity does not decrease without noise, therefore strengthening of the high-frequency intensity is not required; thus, we only discuss the impact of the two forms of embedded data on phase retrieval in the presence of noise.
Noise exists in every process in the HDS system and deteriorates the phase retrieval directly. To simulate the influence of the experimental noise in the simulation, we added a certain amount of Gaussian model noise to the Fourier intensities formed by the assigned embedded data and iterative embedded data methods. The Gaussian model noise is shown in Fig. 3(a).
Thereafter, we used the IFT algorithm to calculate the bit error rate (BER) after 10 iterations with different Fourier intensities and determined the value corresponding to the minimum BER as shown in Fig. 11, and thus we determined the corresponding embedded data pattern at this time. Smaller the value, lower the low-frequency and higher the high-frequency. In Fig. 11, the black curve and red dot represent the phase-retrieval results corresponding to different iterative embedded and assigned embedded data patterns, respectively. When was 6.00, the BER of the iterative embedded data method was 0.033, while for of 8.31, and the BER was 0.064 in the case of the assigned embedded data method. It can be seen that phase retrieval is better when using the iterative embedded data method.
In Fig. 12, we select two cases where the value of the assigned embedded data is 8.31, and the value of the iterative embedded data is 6.00, and compare the corresponding BER curves. When we use the iterative embedded data method, the phase retrieval is better because the value of the iterative embedded data method is lower than that of the assigned embedded method, which implies that the high-frequency intensity in the iterative embedded data method is higher. Under the same noise, the SNR of the assigned embedded data method was 6.0, while that of the iterative embedded data method was 6.6. Therefore, the convergence speed of the BER using the iterative embedded data method was faster.
The experimental setup of the non-interference phase-retrieval system is shown in Fig. 13.
We used the same parameters as those in the simulation. The wavelength of the laser was 532 nm, and its power was 300 mW. The SLM is a HAMAMATSU X10468-04 with a pixel pitch of 20 μm and a resolution of 792 × 600 pixels. The CCD is Thorlabs' DCC3260M, with pixel pitch of 5.86 μm, and resolution of 1936 × 1216 pixels. The media used was a PMMA photopolymer. The phase pattern was 32 × 32 phase data based on the combination of half embedded and half unknown phase data, and we used a block of 4 × 4 pixels to denote one-phase data. Finally, a 128 × 128 pixel phase pattern was loaded onto the SLM.
In the experiment, the Fourier intensity distributions corresponding to the assigned embedded data method and the iterative embedded data method are shown in Fig. 14.
It was experimentally verified that different intensity distributions were captured on the CCD. It is evident that the high-frequency detail information in Fig. 14(c) is more abundant. The phase-retrieval results corresponding to the assigned embedded data and the iterative embedded data are shown in Fig. 15.
In Fig. 15, after ten iterations, the iterative embedded data method can reduce the BER by approximately a factor of one. This proves that our proposed method of using embedded data to enhance the high-frequency intensity aids in improving phase retrieval.
4 Conclusions
In this study, we used the iterative embedded data method to increase the Fourier high-frequency intensity compared to the assigned embedded data method. When considering noise, the iterative embedded data method can reduce the BER by a factor of one, and the function of the known embedded data is further expanded, thus achieving faster retrieval of phase information.