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Abstract
A decision-free downsampling method (DFDS) assisted by channel-transfer information for phase-modulated holographic data storage is proposed. DFDS is used to address the issue of the accumulation of decision errors induced by traditional downsampling. The issue degrades the downsampling accuracy. DFDS comprises two functional segments: acquiring the channel-transfer information offline and performing decision-free downsampling online. With the assistance of the channel-transfer information, DFDS uses Bayesian posterior probabilities instead of traditional decision results to avoid the accumulation of decision errors and achieve more accurate downsampling. The simulation and experimental results show that DFDS reduces the phase error rate, thereby improving the reliability of the holographic data storage system.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
To cater to the rising demand for data storage, traditional means of optical storage, such as Blu-ray discs, have been investigated to achieve smaller spot sizes and a narrower bit spacing to improve the storage density and capacity [1]. However, owing to diffraction limits, the development of traditional optical-storage systems has been severely constrained. Multiplexing technology has become an effective way to increase optical storage capacity, and it has been extensively investigated [2–5]. As a new optical storage technology, holographic storage can also use multiplexing technologies to increase storage density and capacity, and it has become one of the most promising storage technologies in the era of big data [6–8]. Holographic storage uses the amplitude or phase information of light to record data. Depending on the optical parameter employed, holographic data storage comprises amplitude-modulated holographic storage [9–13] and phase-modulated holographic storage [14–18]. Phase-modulated holographic storage offers a higher encoding rate and diffraction efficiency, thereby attracting considerable attention [19]. Nevertheless, the data reliability of phase-modulated holographic storage is severely hindered by the limitations of phase-reconstruction methods and the effects of multiple complex noises. [20,21]. The reduced data reliability greatly impedes the practicality of phase-modulated holographic storage.
To realize actual applications, the different aspects of phase-modulated holographic storage have been investigated in various studies. Xu et al. proposed an unequally spaced phase pair-coding method to address the phase ambiguity problem [22]. Moreover, by analyzing the spectrum distribution of the embedded and signal phases, Yu et al. revealed that optimizing the embedded-phase coding can reduce the phase error rate [23]. Furthermore, Hao et al. investigated the relationship between the magnitudes of the reference beam and the information beam, and they reported that the phase error can be reduced when the amplitudes of the reference and the information beams are in a specific ratio [24]. Chen et al. proposed a dynamic spectrum-selection technique that can dynamically eliminate high-frequency noise during phase retrieval [25]. In particular, a phase-integration technique for the stored signal along the shear direction was investigated to reduce the phase errors caused by the rotation or displacement of holographic storage discs during the reading process [26]. Additionally, we proposed a phase-distribution-aware adaptive decision scheme to mitigate the inaccurate decisions caused by the implementation of a uniform threshold [27] and proposed a reliable bit-aware low-density parity-check code that can correct more erroneous bits [28]. Although these studies managed to reduce the number of errors, phase errors persist. To address this limitation, new optimization techniques should be investigated. Notably, the optimization of phase downsampling is yet to be characterized. In particular, the accuracy of phase downsampling is closely related to the accuracy of the read data. Unfortunately, the traditional downsampling method neither fully utilizes the phase-transfer information of the holographic storage channel nor comprehensively integrates the phase distribution of all elements in the same phase group; thus, it is unable to yield accurate results. Considering these limitations, an optimized downsampling scheme should be designed to improve the accuracy of the read data.
In this study, we analyzed the reasons for the inaccuracy of traditional downsampling methods. Traditional downsampling methods rely on the decision results of each reconstructed phase within a phase group. The process of traditional downsampling involves initially judging multiple reconstructed phases of the analog form in the phase group into multiple modulated phases of the digital form and subsequently converting multiple digital quantities into a single digital quantity. However, as each reconstructed phase may be derived from any of all modulated phases, the manner in which each reconstructed phase is directly converted into a certain modulated phase is inaccurate and results in the loss of useful information. Moreover, because the decision of a single reconstructed phase is performed independently and does not consider the phase distribution of other elements in an integrated manner, the decision results contain certain errors. Hence, the accumulation of the decision error from each reconstructed phase within a phase group impacts the downsampling results of the phase group. To overcome this limitation, we propose a decision-free downsampling method (DFDS) that consists of two functional segments: acquiring the channel-transfer information offline and performing decision-free downsampling online. Based on the offline channel-transfer information, DFDS can utilize the Bayesian formula to yield a set of posterior probabilities for each reconstructed phase within a phase group. These posterior probabilities include all possible phase-transfer cases, thereby yielding more comprehensive and accurate information for downsampling. In the online reading process, DFDS synthetically considers the posterior probabilities of each reconstructed phase based on the phase distribution of the phase group and selects the modulated phase with the highest average posterior probability as the result of downsampling. We conducted simulations and experiments to validate the effectiveness of DFDS. Simulation results revealed that DFDS can reduce the occurrence of noise dots in the read gray image. Furthermore, the experimental results demonstrated that DFDS can reduce the average phase error rate by up to 70% compared with the traditional downsampling method. Consequently, DFDS can downsample the phase group to a more accurate modulation phase with the assistance of channel-transfer information, thereby improving the data reliability of the holographic storage system.
2. Background
2.1 Phase modulated holographic data storage
To record and retrieve information, phase-modulated holographic storage spatially modulates the phase of the beam using unique data patterns. Figure 1 depicts the principle of data recording and reading in a phase-modulated holographic storage system. The process of data recording consists of four steps: 1) The user data are modulated into signal phases; 2) the signal phase and reference phase of known values are combined to form the phase data page and are distributed on the left and right sides of the page [24]; 3) the phase data page is uploaded onto the spatial light modulator (SLM), and a signal beam and a reference beam are then generated via laser irradiation [15]; 4) the signal beam and the reference beam interfere in the holographic medium to form a stable grating that carries the recording information, and the red dot on the medium, as illustrated in Fig. 1, represents the recorded grating information. Notably, the reference phase with known values does not provide user information; it just generates the reference beam necessary for data recording. This reference beam provides the information necessary for the recording process, which is determined by the recording principle of holographic data storage.
The process of data reading includes six steps: 1) The reference phase is uploaded to the right half of the SLM; 2) the reference beam generated via SLM modulation is illuminated onto the medium to diffract the reconstructed beam (i.e., signal beam); 3) the Fourier lens converts the reconstructed beam to the Fourier domain; 4) a complementary metal-oxide-semiconductor (CMOS) camera records the Fourier intensity information of the reconstructed beam; 5) the reconstructed phase is recovered from the Fourier intensity information by using an iterative Fourier-transform algorithm [29]; 6) the reconstructed phase is utilized to acquire user data. Specifically, the CMOS camera cannot receive the phase information of the beam directly; therefore, a Fourier optical lens is required to conduct a Fourier transform on the reconstructed beam. Phase information can be recreated using the transformed Fourier information.
2.2 Oversampling and downsampling
To efficiently recover phase data during the aforementioned reading procedure, the Fourier intensity data acquired by the CMOS camera cannot be smaller than the two Nyquist sizes. To ensure that the Fourier intensity distribution is sampled appropriately by the CMOS camera, the real system often uses multiple pixels on the SLM to represent the phase data of the phase data page [24,25]. The phase data page obtained directly by user data is referred to as the original data page, while the data page uploaded onto the SLM is referred to as the uploaded data page. The method of getting the uploaded data page from the original data page is called oversampling. To balance the storage capacity and data reliability, phase-modulated holographic data storage usually uses ${\rm {4}} \times {\rm {4}}$ oversampling. In other words, ${\rm {4}} \times {\rm {4}}$ pixels on the SLM are used to represent one-phase data. As shown in Fig. 2, the original data page with a size of ${\rm {m}} \times {\rm {n}}$ is transformed into an uploaded data page with a size of ${\rm {4m}} \times {\rm {4n}}$ after oversampling. The phase $P{H_{ij}}$ in the original data page corresponds to the phases $P{H_{ij}}\_1 \sim P{H_{ij}}\_16$ in the uploaded data page. We regarded $P{H_{ij}}\_1 \sim P{H_{ij}}\_16$ as a phase group. Each element of $P{H_{ij}}\_1 \sim P{H_{ij}}\_16$ in the uploaded data page is equal to $P{H_{ij}}$ of the original data page after the oversampling is performed.
Because the uploaded data page of $4m$ rows and $4n$ columns is used to record data, the reconstructed data page during the reading process also comprises $4m$ rows and $4n$ columns. To recover the original phase data, it is necessary to convert the reconstructed data page into identical $m$-row and $n$-column data pages. The converted data page is called a read-data page. The process of obtaining read-data pages from reconstructed data pages is called downsampling. The traditional downsampling method comprises two steps. The first step involves performing a phase decision for the reconstructed phase. Phase-modulated holographic data storage typically employs 4-level phase modulation with phase values of $0$, $0.5\pi$, $\pi$, and $1.5\pi$. Because of noise in the storage channel, each phase value in the reconstructed data page is not the same as the modulation phase value in the uploaded data page. Instead, each phase value in the reconstructed data page is a series of values within the range of $\left [ {{\rm {0}}, {\rm {\ 2}}\pi } \right ]$. This implies that the phase values change from four digital quantities to a series of analog quantities after the storage channel. The phase decision aims to convert analog quantities into digital quantities. The common practice is that the phase values belonging to $(-0.25\pi,0.25\pi ]$, $(0.25\pi, 0.75\pi ]$, $(0.75\pi,1.25\pi ]$, and $(1.25\pi, 1.75\pi ]$ are considered modulated phases $0$, $0.5\pi$, $\pi$, and $1.5\pi$, respectively. Notably, because the phase has a period of 2$\pi$, $-0.25\pi$ is equal to $1.75\pi$. In the second step, the number of each modulated phase within a phase group is compared. The value of the read phase equals the modulated phase with the highest number.
3. Motivation
The traditional downsampling method initially converts multiple reconstructed phases of the analog form within the phase group into multiple modulated phases of the digital form and subsequently converts multiple digital quantities into a single digital quantity. However, converting each analog quantity to one digital quantity may yield errors, and the accuracy of traditional downsampling can be affected. We investigated an example wherein an error occurred in the reading phase, as depicted in Fig. 3. The results in Fig. 3 demonstrate the complete process of oversampling and downsampling of one-phase data. The value of the original phase data was $1.5\pi$. A ${\rm {4}} \times {\rm {4}}$ recorded phase group was obtained after oversampling, and each phase within the recorded phase group ($PH\_1 \sim PH\_16$) also possessed the value $1.5\pi$. Owing to the noise of the storage channel, each phase in the reconstructed phase group does not maintain the value $1.5\pi$, but rather some analog quantities distributed in the range of $\left [ {{\rm {0}}, {\rm {\ 2}}\pi } \right ]$. The traditional downsampling method must perform phase decisions for these analog quantities. As illustrated in Fig. 3, some reconstructed phases are correctly judged as $1.5\pi$, while others are incorrectly judged as $\pi$. The accumulation errors of reconstructed phases in the phase decision cause a result of downsampling to be a value $\pi$, which finally leads to the read-phase error.
We believe that directly adjudicating each reconstructed phase within the phase group as a modulated phase is an inaccurate approach. Thus, we conducted experiments using 100 data pages to prove this. The experimental setup was identical to that detailed in Section 5.2. We counted the number of four recorded phase types when the reconstructed phases were ${\rm {1}}{\rm {.5}}\pi$, ${\rm {1}}{\rm {.45}}\pi$, ${\rm {1}}{\rm {.4}}\pi$, ${\rm {1}}{\rm {.35}}\pi$, ${\rm {1}}{\rm {.3}}\pi$, and ${\rm {1}}{\rm {.25}}\pi$, respectively. As depicted in Fig. 4, when the reconstructed phases were ${\rm {1}}{\rm {.45}}\pi$, the numbers of recorded phases as $0$, $0.5\pi$, $\pi$, and $1.5\pi$ were 58, 13, 98, and 772. In other words, when the reconstructed phase is ${\rm {1}}{\rm {.45}}\pi$, the recorded phase is not necessarily $1.5\pi$ but may be a value of $0$, $0.5\pi$, or $\pi$. Consequently, directly judging the reconstructed phase with a phase value of ${\rm {1}}{\rm {.45}}\pi$ as $1.5\pi$ is inaccurate. In addition, as the value of the reconstructed phase gradually decreases from ${\rm {1}}{\rm {.5}}\pi$ to ${\rm {1}}{\rm {.25}}\pi$, the number of recorded phase $1.5\pi$ gradually decreases. This indicates that the probability of $1.5\pi$ being transferred to each phase is not identical after passing through the holograph storage channel. Therefore, in the traditional downsampling method, the phase decision process that indiscriminately treats the reconstructed phase with different values is inaccurate. An inaccurate decision process may cause decision errors in some reconstructed phases, and the accumulation of these phase decision errors can affect the results of downsampling.
Based on the aforementioned analysis, we further observed the reconstructed phase group depicted in Fig. 3. Figure 5 demonstrates the phase distribution of the phase group in polar coordinates. The polar angles represent the phase values of each element in the phase group. For observation, we set different values for the polar radii of each phase. In Fig. 5, the number of reconstructed phases located in the interval $(0.75\pi, 1.25\pi ]$ is 9, and the number of reconstructed phases located in the interval $(1.25\pi,1.75\pi ]$ is 7. In the traditional downsampling method, the number of phases $\pi$ is greater than the number of phases $1.5\pi$ after the phase decision; therefore, the reading phase is $\pi$. However, the nine reconstructed phases located in the interval $(0.75\pi, 1.25\pi ]$ are mostly closer to the decision threshold $1.25\pi$; the seven reconstructed phases located in the interval $(1.25\pi, 1.75\pi ]$ are far from the threshold $1.25\pi$ and close to the modulation phase $1.5\pi$. We considered that the reconstructed phases close to the decision threshold $1.25\pi$ have an approximately equal probability of originating from recorded phases $\pi$ and $1.5\pi$. The reconstructed phases that were located far from the threshold $1.25\pi$ and closer to the modulation phase $1.5\pi$ have a higher probability of originating from the recorded phase $1.5\pi$ and a lower probability of originating from the recorded phase $\pi$. Therefore, although the number of reconstructed phases in the interval $(0.75\pi, 1.25\pi ]$ is larger, the probability of the recorded phase being $1.5\pi$ is greater. The conventional downsampling method does not consider the phase distribution of each phase within the phase group as a whole and thus is not sufficiently accurate. An optimized downsampling method should be investigated to improve the accuracy of the read data.
4. Method
Traditional downsampling methods rely on the results of phase decisions. However, converting the reconstructed phase in an analog form directly into the modulated phase in a digital form yields inaccurate results and introduces a certain amount of cumulative error. In this study, we chose to consider each reconstructed phase within the phase group in the downsampling process instead of performing the phase decision for each reconstructed phase. Therefore, we proposed DFDS. As illustrated in Fig. 6, unlike the traditional downsampling method, DFDS yields a single digital quantity directly from multiple analog quantities with the aid of the channel-transfer information. DFDS can prevent the accumulation of errors associated with phase decisions in each reconstructed phase. Moreover, it allows for a comprehensive consideration of the phase distribution of each element within the phase group. DFDS comprises two steps: acquiring the channel-transfer information offline and performing decision-free downsampling online. These two steps are described hereafter.
4.1 Acquiring channel transfer information offline
As characterized in Section 3, each reconstructed phase may originate from any one of the modulated phases, and the probability that the different reconstructed phases originate from each modulated phase is distinct. This is because each modulated phase can be transferred to any of the phases in the range of $\left [ {{\rm {0}}, {\rm {\ 2}}\pi } \right ]$, and the channel transfer probabilities are unequal. The phase decision of traditional downsampling methods does not fully consider channel-transfer information and directly judges a single reconstructed phase as a certain modulated phase. However, the result of the phase decision is a local optimal solution for a single reconstructed phase. In particular, the phase decision results in the loss of useful information for the entire phase group. A more suitable method involves obtaining the probability of each reconstructed phase originating from each modulated phase rather than directly adjudicating each reconstructed phase into one modulated phase. Obtaining the aforementioned probability ensures that each reconstructed phase provides comprehensive information to its phase group.
In essence, the aforementioned method involves obtaining the posterior probability of the record phase through the reconstructed phase. In a phase-modulated holographic storage system, the posterior probability of the record phase cannot be obtained directly. However, this posterior probability can be obtained via Bayesian formula conversion. We employed $t$ and $r$ to denote the recorded and reconstructed phases, respectively. $P(t|r)$ represents the posterior probability of the recorded phase. In accordance with the Bayesian formula, $P(t|r)$ can be computed as
where $P\left ( r \right )$ represents the probability of the reconstructed phase $r$, $P\left ( t \right )$ denotes the prior probability of the recorded phase $t$, and $P\left ( {r|t} \right )$ depicts the transfer probability of a phase-modulated holographic storage channel. Each reconstructed data page possesses a different $P\left ( r \right )$ that can be calculated using the number of reconstructed phases in the reconstructed data page. Because user data are randomly scrambled before phase modulation, the probability of each modulation phase in the original data page is approximately identical. Specifically, $P\left ( t \right )$ on each data page is identical. For K-level phase modulation, the prior probability of each modulated phase can be expressed as: where $i$ represents an integer in the range of $[1,K]$ and ${k_i} = \frac {{2\pi }}{K}(i - 1)$. In addition, as phase-modulated holographic storage systems generally employ 4-level modulation, $P(t = {k_i})$ is assumed to be 0.25.Based on the aforementioned analysis, the key to calculating the posterior probability involves obtaining the transfer probability of the holographic storage channel. After the construction of a holographic storage system was finished, the parameters of each part remained unchanged. Thus, the channel transfer probability is determined. In particular, the channel-transfer probability can be determined offline. Specifically, after establishing a holographic storage system, we initially generated random data pages and recorded the phase values of each location in these data pages. These data pages were subsequently placed on the system and processed by the holographic storage channel. Finally, the data page was read, and the transfer situation of each modulated phase was counted according to the recorded and reconstructed phases. Figure 7 depicts the phase-transfer situation of each modulation phase using 1000 data pages under the 4-level phase modulation, wherein Fig. 7(a), 7(b), 7(c), and 7(d) indicate the number of different reconstructed phases transferred by modulation phases $0$, $0.5\pi$, $\pi$, and $1.5\pi$, respectively. The phase transfer of each modulation phase was used to calculate the channel-transfer probability. For statistical convenience as well as to simplify the calculation, we segmented the phase of the range $\left [ {{\rm {0}}, {\rm {\ 2}}\pi } \right ]$ into $0.5\times m \times n$ phase intervals, where $m$ and $n$ are the rows and columns of the original data page, respectively. Each phase interval was defined as $sg(l) = (\frac {{4\pi }}{{mn}}(l - 1),{\rm {\ }}\frac {{4\pi }}{{mn}}l]$, where $l$ is an integer in the range $[1,0.5\times m \times n]$. Thus, the transfer probability of each modulation phase can be expressed as:
where ${{N_1}(x)}$ represents the number of cases that satisfies condition $x$ represented in Fig. 7. After calculating the transfer probabilities from each modulated phase to all reconstructed phases in accordance with Eq. (3), all transfer probabilities were recorded as a lookup table. The probability table was used directly in the online reading process.
Fig. 7. Phase transfer situation of each modulation phase. (a) Transfer situation of modulation phase $0$; (b) transfer situation of modulation phase $0.5\pi$; (c) transfer situation of modulation phase $\pi$; (d) transfer situation of modulation phase $1.5\pi$.
4.2 Performing decision-free downsampling online
The traditional downsampling method initially converts multiple analog quantities within the phase group into multiple digital quantities and subsequently converts the multiple digital quantities into a single digital quantity. Compared with the phase decision process of traditional downsampling methods, DFDS maintains the analog properties of each reconstructed phase and yields multiple posterior probabilities for each reconstructed phase of the analog form. These posterior probabilities correspond to all possible phase transfer situations, thereby yielding more comprehensive and accurate information for downsampling. With the posterior probabilities of each modulated phase obtained by each reconstructed phase within the phase group, DFDS can obtain a single digital quantity directly using multiple analog quantities. Specifically, this method first obtains the posterior probability of each modulation phase for a single element and thereafter calculates the average of the posterior probabilities of each modulation phase within the phase group. The modulation phase with the largest average posterior probability was the result of downsampling. The online execution of the decision-free downsampling is described in detail hereafter.
As $P\left ( t \right )$ and $P\left ( {r|t} \right )$ have already been obtained in the offline stage, the posterior probability could be calculated by acquiring $P\left ( r \right )$ in the reading process online. The phase value of each element within the phase group is denoted as $PHG = \{ PH\_1,PH\_2,PH\_3, \ldots,PH\_j\}$, where $j$ is an integer in the range of $\left [ {1,{\rm {\ }}{{\rm {d}}^2}} \right ]$ and the $d$ indicates that the system uses an oversampling of $d \times d$, and the d is usually assumed to be 4. For a single element $PH\_j$ in the phase group, its phase interval must first be determined by
Notably, ${P((r \in sg(l){|_{r = PH\_j}})|t = {k_i})}$ can be obtained from the table, as the table of channel transfer probabilities was already obtained in the offline stage, as detailed in Section 4.1. The average posterior probability of each element within the phase group can be subsequently obtained, as depicted in Eq. (7).
Finally, the average posterior probability ${P_{ave}}$ was calculated using Eq. (7), when $i$ assumes different values. If $i$ corresponding to the largest ${P_{ave}}$ is ${i_{\max }}$, then the result of downsampling is ${k_{{i_{\max }}}} = \frac {{2\pi }}{K}({i_{\max }} - 1)$.
5. Results and discussion
5.1 Simulation
We verified the effectiveness of DFDS via simulation, and the results are presented in this section. The parameters involved in the simulation were identical to those of the real system described in Section 5.2. A grey image with $128\times 128$ pixels was used as the data source for the simulation, as illustrated in Fig. 8. Because the gray value of each pixel can be represented via eight bits, a gray image with $128\times 128$ pixels can be converted into a bit stream of 131,072 bits. Using 4-level phase modulation, every two bits in the bit stream was sequentially mapped to one phase, resulting in 128 signal-phase pages with a size of $32\times 16$. Each signal phase page was combined with a known-value reference phase page to form an original phase data page. Thereafter, the original phase data page was converted into an upload data page by $4\times 4$ oversampling and uploaded onto the SLM to perform the recording process and transport computing. The CMOS camera was used to capture the Fourier intensity information of the upload data page. Moreover, Gaussian noise was added to the Fourier intensity information to simulate the effect of noise. Subsequently, phase reconstruction was performed using an iterative Fourier-transform algorithm with a maximum iteration of 30. After obtaining the reconstructed data page, the read-data page could be acquired by downsampling the reconstructed data page. We employed both traditional downsampling and DFDS on the reconstructed data page to observe the read outcomes of the two methods.
First, we observed the results of the downsampling in one-phase group using these two methods, as listed in Table 1. Both the reconstructed phases in Table 1 and Fig. 3 were identical. In particular, the traditional downsampling method initially performed a phase decision for each reconstructed phase within the phase group. Because the number of $\pi$ was 9 and the number of $1.5\pi$ was 7 after the phase decision, the traditional downsampling result was $\pi$. Consequently, the traditional downsampling method resulted in reading errors. Instead of performing the phase decision, DFDS utilized the average posterior probability as the foundation for downsampling. As listed in Table 1, the result of DFDS was $1.5\pi$ because the average posterior probability ${P_{ave}}(t = {k_4}|r = PHG)$ possesses the highest values. Consequently, DFDS can correct certain incorrect phase groups of traditional downsampling methods.
Table 1. Comparison of the results of the two downsampling methods
The results associated with the one-phase data page are illustrated in Fig. 9. Figure 9(a) is the original data page. Figure 9(b) illustrates the Fourier intensity information of Fig. 9(a) after the simulated holographic storage channel. Figure 9(c) shows the read-data page obtained from Fig. 9(b) using the traditional downsampling method. Figure 9(d) depicts the error distribution of Fig. 9(c) with respect to Fig. 9(a). Figure 9(e) portrays the read-data page obtained from Fig. 9(b) using DFDS. Figure 9(f) illustrates the error distribution of Fig. 9(e) relative to Fig. 9(a). As can be seen from Fig. 9, the read data pages obtained using DFDS have fewer phase errors compared to the traditional downsampling method.
Fig. 9. Read results of one phase data page. (a) The original data page; (b) the Fourier intensity information of (a) after simulating the holographic storage channel; (c) the read-data page obtained from (b) using the traditional downsampling method; (d) the error distribution of (c) with respect to (a); (e) the read-data page obtained from (b) using DFDS; (f) the error distribution of (e) relative to (a).
Finally, we observed the read results for the entire grey image. After obtaining all 128 read-data pages via downsampling, the bit stream could be acquired by de-mapping each phase of all read-data pages. Subsequently, each 8-bit unit of the bit stream was converted to a gray value, and the recorded gray image could be recovered. Figure 10 illustrates the gray images read using two downsampling methods, where Fig. 10(a) and Fig. 10(b) are the gray images obtained via the traditional downsampling and DFDS, respectively. Observably, Fig. 10(b) exhibits fewer noise dots than Fig. 10(a). To provide a more accurate depiction of the effect of the two methods, we also computed the pixel error rate of the two images. Figure 10(a) has a pixel error rate of 0.0473, while Fig. 10(b) has a pixel error rate of 0.0148. The pixel error rate of Fig. 10(b) is reduced by 68.71% compared with that of Fig. 10(a). Therefore, DFDS can effectively improve the data reliability of phase-modulated holographic storage.
Fig. 10. Gray image reading results. (a) The gray image obtained via the traditional downsampling; (b) the gray image obtained via DFDS.
5.2 Experiment
We conducted experiments using a real system to evaluate the effects of DFDS, and the experimental details are presented in this section. A phase-modulated coaxial holographic storage system was employed, as shown in Fig. 11. Along the optical path, the experimental devices comprised a laser, an attenuator, a beam expander, aperture control components, a polarizer, a aperture, lens 1, lens 2, a half-wave plate (HWP), a beam splitter, a SLM, lens 3, holographic medium, lens 4, a mirror, lens 5, and a CMOS camera. The wavelength of the laser was 532nm, and the polarization of the laser was linear polarization light. During the experiment, the laser irradiated the SLM in a positive incidence manner. The HWP could adjust the polarization to be suitable for phase modulation of the SLM. The aperture comprised two rectangular frames: left and right rectangular frame. The left rectangular frame corresponds to the signal data on the left half of the phase-data page illustrated in Fig. 8 and was used to control the transmission of the signal beam through the optical path; the right rectangular frame corresponds to the reference data on the right half of the phase data page depicted in Fig. 8 and was utilized to control the transmission of the reference beam through the optical path. The aperture control components are used to precisely control whether the reference and signal beams pass through the optical path. They consist of two parts. The first part consists of a beam splitter BS1, a shutter, an aperture A1, and a beam splitter BS2, being used to control the signal beam. The second part consists of the BS1, a mirror, an aperture A2, a mirror, and the BS2, being utilized to control the reference beam. In the recording process, both the rectangular frames of the aperture and shutter were opened simultaneously, and the laser irradiated the SLM, thereby generating the signal beam and the reference beam. The signal beam and the reference beam interfered with each other in the holographic medium, thus generating a grating to record the data. During the reading process, the shutter was closed, and solely the right rectangular frame of the aperture was switched on to block the signal beam and generate the reference beam. The reference beam illuminated the grating in the holographic medium to diffract the reconstructed beam. Subsequently, the reconstructed beam passed through the Fourier lens (i.e., lens 5) to be converted into the Fourier domain, and the CMOS camera captured the Fourier intensity information of the reconstructed beam. The received Fourier intensity information was first processed via noise reduction, and phase reconstruction was thereafter performed using an iterative Fourier-transform algorithm. After reconstructing the phase-data page, the read-data page was obtained via downsampling. The parameters of each device in the experimental system are as follows: The resolution and pixel pitch of the pure-phase SLM (X10468-04, Hamamatsu) were $792 \times 600$ and 20 $\mu {\rm {m}}$, respectively. The resolution and pixel pitch of the CMOS camera (DCC3260M, Thorlabs) were $1936 \times 1216$ and 5.86 $\mu {\rm {m}}$, respectively. The holographic medium was a photopolymer with a thickness of 1.5 mm [30]. The focal lengths of Lens1-Lens4 in the experiment were 150mm, and the focal length of Lens5 was 300mm. In the simulation and experiments, we employed 4-level phase modulation and $4\times 4$ oversampling.
We randomly generated 1000 phase-data pages and performed recording and reading operations on the aforementioned experimental system. After obtaining the reconstructed phase data pages, we processed the reconstructed data pages using two downsampling methods to obtain the read-data pages. The phase error rate of 1000 data pages with a maximum number of 30 iterations is illustrated in Fig. 12. Evidently, the phase error rate using DFDS was lower than that using the conventional downsampling method for most of the data pages. Specifically, 849 data pages yielded a lower phase error rate after using DFDS than after using the traditional downsampling method; 12 data pages yielded an equal phase error rate after using the DFDS and traditional downsampling methods; and 139 data pages yielded a higher phase error rate after using DFDS than after using the traditional downsampling method. Compared with the traditional downsampling method, the data pages with increased phase error rate in DFDS solely accounted for 13.9% of all data pages. Moreover, DFDS can reduce the average phase error rate of all data pages by 60% compared with the traditional downsampling method. Therefore, DFDS is also effective in a real system. This is because unlike traditional downsampling methods, DFDS does not depend on the result of the phase decision, thereby circumventing the undesirable effects caused by the accumulation of decision errors. Furthermore, DFDS employs the average posterior probability as the basis for downsampling, which fully considers the phase distribution characteristics of each element within the phase group.
Finally, to evaluate the sensitivity of DFDS, we counted the average phase error rates of the two downsampling methods for different numbers of iterations. As illustrated in Fig. 13, when the number of iterations is 10, the average phase error rates of the traditional downsampling and DFDS methods are 0.02 and 0.014, respectively. As the number of iterations increases, the average phase error rates of both methods decrease, and the average phase error rate of DFDS decreases rapidly. When the number of iterations is increased to 30, the average phase error rates of the traditional downsampling and DFDS methods are 0.0065 and 0.0021, respectively. With a further increase in the number of iterations, the change in the average phase error rates in both methods is considerably small. In addition, we normalized the average phase error rate for each number of iterations with respect to the phase error rate of traditional downsampling to obtain the results depicted in the upper-right corner of Fig. 13. Compared with the traditional downsampling method, DFDS reduced the average phase error rate by 30%, 60%, 68%, 69%, and 70% when the number of iterations was 10, 20, 30, 40, and 50, respectively. This result suggests that DFDS is more accurate than the traditional downsampling methods at all iteration numbers. Thus, DFDS can improve the reliability of phase-modulated holographic storage systems.
Fig. 13. The average phase error rate of the two downsampling methods at different iteration numbers.
We note that DFDS performs better with high iterations in the results shown in Fig. 13. This is because the number of phase groups in favor of DFDS is larger at higher iterations. Specifically, when the number of iterations is low, the phase distribution of the individual elements in a phase group is too spread out, which hinders the accurate execution of DFDS. As the number of iterations increases, the phase values of each element will be closer to the correct phase, making the posterior probability of the correct phase gradually increase while the posterior probabilities of other modulated phases gradually decrease. When the posterior probability of the correct phase increases to a certain value, DFDS can perform correct downsampling, while the traditional downsampling method does not work well yet. Thus, in the higher iterations, DFDS achieves greater performance improvement than the conventional downsampling technique.
6. Conclusion
In this study, we observed and characterized phase errors occurring in the reading process and analyzed the reasons for inaccurate phase downsampling. We discovered that traditional downsampling methods rely on the results of phase decisions and that the accumulation of decision errors can result in downsampling errors. To mitigate the adverse effects of phase decisions, we proposed a decision-free downsampling method, DFDS, which comprises two steps: acquiring the channel-transfer information offline and performing decision-free downsampling online. By incorporating these two steps, DFDS utilizes the Bayesian formula to calculate a set of posterior probabilities for each reconstructed phase within the phase group based on the offline channel-transfer information. Those posterior probabilities can fully leverage the phase-transfer characteristics of the holographic storage channel, and the phase distribution of all elements in the same phase group can be considered comprehensively. Consequently, DFDS mitigated the effect of decision errors and facilitated the use of more comprehensive and accurate information in the downsampling process. The experimental results revealed that DFDS could reduce the average phase error rate by 70% compared with the traditional downsampling method. Therefore, DFDS increases the downsampling accuracy and improves the reliability of the phase-modulated holographic storage system, thereby accelerating the realization of holographic storage systems. Furthermore, we believe that our work can be utilized in several areas involving phase downsampling in addition to holographic data storage.
Funding
National Key Research and Development Program of China (2018YFA0701800); Project of Fujian Province Major Science and Technology (2020HZ01012).
Acknowledgments
This work is supported by Key Laboratory of Information Storage System, Ministry of Education of China, and the Engineering Research Center of Data Storage Systems and Technology.
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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