Reducing the Crosstalk in Collinear Holographic Data Storage Systems Based on Random Position Orthogonal Phase-Coding Reference

Abstract

Previous studies have shown that orthogonal phase-coding multiplexing performs well with low crosstalk in conventional off-axis systems. However, noticeable crosstalk occurs when applying the orthogonal phase-coding multiplexing to collinear holographic data storage systems. This paper demonstrates the crosstalk generation mechanism, features, and elimination methods. The crosstalk is caused by an inconsistency in the intensity reconstruction from the orthogonal phase-coded reference wave. The intensity fluctuation range was approximately 40%. Moreover, the more concentrated the distribution of pixels with the same phase key, the more pronounced the crosstalk. We propose an effective random orthogonal phase-coding reference wave method to reduce the crosstalk. The orthogonal phase-coded reference wave is randomly distributed over the entire reference wave. These disordered orthogonal phase-coded reference waves achieve consistent reconstruction intensities exhibiting the desired low-crosstalk storage effect. The average correlation coefficient between pages decreased by 73%, and the similarity decreased by 85%. This orthogonal phase-coding multiplexing method can be applied to encrypted holographic data storage. The low-crosstalk nature of this technique will make the encryption system more secure.

1. Introduction

Holographic data storage (HDS), deoxyribonucleic acid (DNA) storage, and glass storage are the three giants of the new generation of data storage. DNA storage technology [1,2] is a potential storage technology with large capacity and long storage life. Glass storage technology [3,4] is a simple mechanism, with strong environmental adaptability and practical value. HDS technology with rich encryption has a high read and write speed, parallel read and write ability, and a good competitive advantage [5,6,7]. The multi-dimensional modulation capability of HDS has received extensive attention. Parameter multiplexing increases the capacity of data, including wavelength, angle, phase, amplitude, polarization, etc. [8,9,10,11,12,13]. Moreover, the phase-coding multiplexing holographic data storage system has drawn much attention because of its light energy efficiency, configuration stability, and the ease with which it is modulated by a phase spatial light modulator (P-SLM) [14,15,16,17]. Double random-phase encoding based on phase-coding multiplexing has been developed to encrypt the signal [18,19,20,21]. Moreover, orthogonal phase-coding multiplexing is a promising approach due to its advantages, such as low crosstalk performance and minimal code storage requirements. Several studies have shown that various types of crosstalk in HDS arise in a practical system due to the influence of signal–noise ratio (SNR) characteristics, encoding methods, and system devices [22,23,24]. This crosstalk will affect the performance of reconstruction holograms, such as the signal–noise ratio (SNR) reduction, the bit error rate (BER) increases of reconstruction data pages, and the storage capacity reduction of the system [25,26].

To increase the storage capacity of HDS, researchers have proposed techniques such as orthogonal phase-coding multiplexing and novel phase modulation. Different orthogonal phase keys record holograms in holographic storage materials. When a wrong phase key is used in the reconstruction, the crosstalk leads to signal leakage [27,28,29]. Such crosstalk significantly reduces the quality of HDS. In conventional off-axis configurations, crosstalk in orthogonal phase-coding multiplexing HDS can be caused by pixel fitness and system stabilization imperfections. A signal light and reference wave with the same optical axis in collinear holographic data storage systems (CHDSSs) provide high system stability and tolerance to environmental perturbations [30,31,32]. The collinear system synchronizes the relative vibration between the reference wave and the signal light. However, crosstalk still exists in CHDSSs even though the system configuration has changed. The crosstalk phenomenon of orthogonal phase-coding multiplexing in CHDSS has not yet been adequately described. This crosstalk may limit the further development of CHDSSs.

In this paper, we demonstrate the mechanism and characteristics of crosstalk in orthogonal phase-coding multiplexing CHDSSs. Based on the above characteristic, we propose a “random orthogonal phase-coding reference wave” as an efficient crosstalk suppression. The “random” here emphasizes that the coding unit’s location is randomly distributed in the reference wave. In Section 2, the signal light is encoded using the sector orthogonal phase-coding reference wave (SOPCR) method. A CHDSS is used for optical recording and the reconstruction of signal light. In Section 3, the hologram intensities read out separately from the different encoded units disagree. The crosstalk is exacerbated by the concentration of coding units with the same phase difference. Residual powers of phase cancelations can induce crosstalk. Next, we propose a random orthogonal phase-coding reference wave method (ROPCR) to reduce this type of crosstalk. The encoded units of the ROPCR are relatively scattered, and the crosstalk brought by the scattered encoded units is also scattered. The purpose of crosstalk is weakened by dispersing the distribution of crosstalk. Experimental results show that the ROPCR can reduce the incongruous diffraction intensity of the encoded unit from the root and, hence, reduce the crosstalk superposition effect of the concentrated diffraction intensity of the encoded unit. Section 4 describes the methodology and justification for using the ROPCR.

2. Configurations and Methods

2.1. Configurations

Orthogonal codes are a widely used communication technique and have been shown to have good anti-crosstalk performance and improve data accuracy [33]. The SOPCR method is an orthogonal phase-coding method that uses a ring-shaped reference wave and splits it into n parts. Each part is considered a coding unit, and the reference wave is divided into Unit₁–Unit₆₄ in clockwise order (This section uses n = 64 as an example.). The initial signal light and reference wave phase are set at 0, as shown in Figure 1a. The SOPCR method involves adding an orthogonal degree to the reference wave, as shown in Figure 1b. The choice of different phase keys determines the upload reference phase map on the SLM during recording and reconstruction.

The Hadamard matrix generates orthogonal phases [34]. Both rows and columns of the Hadamard matrix can form orthogonal phases. The selected columns of the Hadamard matrix determine the phase value of the reference wave. Each coding unit will add a phase value of π or 0 according to the Hadamard matrix value 1 or −1. A one-to-one correspondence between matrix elements and encoding units is established. The coding matrix is shown in Figure 1c. For example, when choosing key₂ to record the signal light, upload the corresponding phase value of the second column of Figure 1c to the coding units (as shown in Figure 1b).

The schematic of a phase-coding multiplexing CHDSS is shown in Figure 1d and expanded a single longitudinal mode green laser into a uniform beam of 78.5 mm². In the recording process, the amplitude spatial light modulator (A-SLM) modulates the signal light and reference wave with consistent intensity. The signal light and reference wave are then imaged on a P-SLM. The P-SLM uploads phases according to the modulation of the coding matrix. The oversampling rate of the A-SLM and P-SLM is 3 and 2, respectively. Oversampling is a cost-effective procedure in which the input signal is sampled at a rate significantly higher than the Nyquist frequency, improving the SNR and resolution [35]. L2 converts the light into the forward focal plane of the OBJ. During the signal recording process, the P-polarized signal light and the reference wave pass through the QWP to become circularly polarized light, which then passes through the OBJ to record the signal on the RHSM and form a hologram. During reconstruction, the A-SLM and P-SLM generate the reference frame’s light field shape and phase in succession. The reference wave in the hologram diffracts the signal light and then passes through the OBJ and QWP again to become S-polarized light. The reflected light from the PBS passes through L3 and is finally captured by the CCD.

The hologram is formed by the interference of the signal light and the reference wave in the back focal plane of the OBJ in the RHSM. Figure 1e shows the picture of RHSM. The material is a three-layer structure. The first is glass, the second is a recording material surrounded by flanges, and the third is a reflector. The recording material was Phenanthrenequinone-doped polymethyl methacrylate (PQ-PMMA) photosensitive polymer [36], which has many advantages of high sensitivity, high SNR, and low cost.

2.2. Methods

It is crucial to clarify the crosstalk mechanism for the further development of the orthogonal phase-coded multiplexed CHDSS. Figure 1a is the original ring-shape reference wave. To observe the crosstalk mechanism more clearly, we use a simplified reference wave and a signal light to study the recording procedure. The spatial relationship between the signal light and reference wave cannot be ignored. The angle between the two interfering beams determines the grating parameters and will further affect the diffraction properties.

$${\mathit{k}}_s=k_0\left(-\frac{x_s}{f},-\frac{y_s}{f},\sqrt{n^2-{\left(\frac{x_s}{f}\right)}^2-{\left(\frac{y_s}{f}\right)}^2},{\phi }_s\right),$$

(1)

$${\mathit{k}}_r=k_0\left(-\frac{x_r}{f},-\frac{y_r}{f},\sqrt{n^2-{\left(\frac{x_r}{f}\right)}^2-{\left(\frac{y_r}{f}\right)}^2},{\phi }_r\right),$$

(2)

$${\mathit{k}}_G={\mathit{k}}_s-{\mathit{k}}_r,$$

(3)

${\mathit{k}}_s$ is the vector of the signal light. ${\mathit{k}}_r$ is the vector of the reference wave. ${\mathit{k}}_G$ is the vector of grating. $k_0$ is the wave vector. $f$ is the focal length of OBJ. $n$ is the refractive index of the material. $\left(x_s,y_s\right)\left(x_r,y_r\right)$ is the coordinates of the signal light and reference wave. ${\phi }_s,{\phi }_r$ is the phase of the signal light and reference wave. According to the coupled wave theory [37], the larger the angle between the reference wave and the signal light during recording, the larger the grating vector and the higher the diffraction intensity. In the reading process, the diffraction intensity of the signal point from coding unit V is higher than coding unit Ⅰ, as shown in Figure 2a,b. According to the coupled wave theory and the principle of orthogonal phase coding, when the orthogonal key is used for reconstruction, the diffractive light of two orthogonal pixels will be canceled out because of the orthogonal phase. The difference in diffraction angle causes the diffraction intensity of pixels with orthogonal phase to be unbalanced. In the reconstruction process, the parts that the orthogonal phase cannot offset will retain part of the diffraction intensity, which will become stronger as the number of pixels/coding units increases. This is the cause of crosstalk.

This paper introduces the random position orthogonal coding reference method (ROPCR). This approach perturbs the distribution location of the encoding unit such that the crosstalk generated by the encoding unit is not superimposed at a particular location, thus reducing the crosstalk. The diffraction intensities of the signal light from different coding units are consistent. When the reconstructed key is the same as the recorded key, the key can eventually reconstruct the signal light because the diffraction intensities of each encoding unit are consistent. When reconstructed with orthogonal keys, half of the encoded units have the same phase as the recorded key. The other half of the coding units have π phase differences so that the reconstruction signal light can be diffracted into two components by the two parts. Two signal components have π phase differences and can be offset completely.

3. Results

3.1. Crosstalk in Orthogonal Phase-Coding Multiplexing CHDSS with SOPCR

This paper presents the SOPCR to an encoding that utilizes a 64th-order orthogonal phase Key₂ to record the signal light alphabet ‘FJNU’ and symbol φ. The system then reconstructs the signal light using Key₁–Key₆₄. One of the key features of this system is its ability to entirely reconstruct the signal light with the correct Key₂, as shown in Figure 3a; the holograms depicted in Figure 3a–e show some representative examples. Figure 3f–j show the reconstructed phase key corresponding to Figure 3a–e, while Figure 3h–l show the difference between the reconstructed and recorded phase keys. Blue indicates phase 0, and yellow indicates phase π.

In most cases, the orthogonal phase offsets the diffraction intensity, leading to only dark and blurry-like random crosstalk, as shown in Figure 3b,c, which means that each signal pattern has only one fixed key that corresponds to the function of realizing signal coding. It is important to note that crosstalk can occasionally significantly reduce the security of orthogonal phase encoding, as seen in the case of individual readings, such as those depicted in Figure 3d,e, where Key₁₈ consists of some crosstalk. Through comparing the phase key of Figure 3f–j, the orthogonal phase is randomly distributed on the reference wave, and there is no formal difference between the reconstruct phase key with crosstalk and the normal reconstruct phase key. However, through comparing the phase difference between the reconstruct phase key and the record phase key in Figure 3k–o, there is an obvious rule in the part with crosstalk. In Figure 3n,o, the phase difference distribution is concentrated in the semicircular region, but Figure 3l,m do not show obvious crosstalk.

Moreover, it is worth noting that the curves have similar characteristics, as shown in Figure 4. Regardless of whether the graph is recorded by key₂, key₉, or key₃₂, the distance between the position of the crosstalk and the serial number of the registered phase key is always eight keys, which is caused by a certain law in the Hadamard matrix generation process. Through comparing the structure similarity index measure (SSIM) value for the 8th- and 64th-order keys, we find this phenomenon is stronger in higher-order orthogonal phase codes. The SSIM is a perceptual model, which is more in line with the intuitive feeling of human eyes. This means that the higher the orthogonal order used, the stronger the crosstalk becomes. Therefore, the choice of orthogonal order is an important consideration in SOPCR coding.

Overall, these observations provide important insights into the behavior of crosstalk in the SOPCR and can guide the development of more robust and secure signal coding methods. Through understanding the characteristics of crosstalk in SOPCR coding, we can develop strategies to mitigate its effects and improve the quality and security of the encoded signal.

3.2. Use ROPCR to Suppress Crosstalk

We propose to use ROPCR to reduce crosstalk in phase-coded multiplexing, where randomization emphasizes the random position of the encoded units since an orthogonal matrix determines the phase. Thus, the phase value cannot be random. We can permute only by means of combining the positions of the encoded units. The “random” coding unit makes the distribution of diffraction intensity uniform, and the crosstalk noise is superimposed with the diffraction intensity of the coding unit, as shown in Figure 5.

For ease of calculation, we can provide more details on the experiment conducted to test the diffraction intensity of the I–VIII coding unit for signal point under the SOPCR and ROPCR methods. Each coding unit records and reconstructs signal points separately. Figure 2 and Figure 5 show the signal point and coding Unit II of the SOPCR and ROPCR, respectively. It is important to note that only the A-SLM controls the intensity, and all encoded units have phase zero. To analyze the results, we examined the experimental results’ standard deviation and maximum difference for both the SOPCR and ROPCR methods. Here, the standard deviation is computed as the normalized intensity. As shown in Figure 6, the standard deviation of the ROPCR’s experimental results is 0.037, and the maximum difference is not more than 0.124. For the SOPCR, on the other hand, the standard deviation of the experimental results is 0.597 and the maximum difference is no more than 0.403. These results suggest that the diffraction intensity of each encoded unit of the SOPCR is not uniform for signal points. The SOPCR is essentially limited by the intensity inconsistency of the reconstructed intensity based on phase coding. In contrast, in the ROPCR, there is no clear trend in the diffraction intensity with respect to the mean value. The pixels contained in each encoded unit of the ROPCR are discrete. This indicates that the diffraction intensities of each coding unit are approximately equal in the ROPCR mode and do not differ significantly.

In the next experiment, we study the variation of the diffraction intensity at the signal point when the phase is attached to reference waves I to VIII. We record and reconstruct the signal points separately using the encoded units. As mentioned earlier, each encoded unit generates a diffracted signal light, and the intensities of the different diffracted lights are superimposed to reconstruct the signal light. We recorded the signal point in Figure 5 with phase address A₀ in Figure 7b using all coding units with phase 0. Then, we reconstructed the hologram with phase address A₀–A₈ in series, with the π phase difference in proportion to the recording phase key. We found that the minimum intensity value occurs at the reference address A₄ because the two signal components have similar intensity and π phase differences, which enables them to offset each other in the ROPCR. However, this phenomenon is quite different from the SOPCR approach. The intensity of the two holograms differed when reconstruction with π phase caused differences in coding units. Although there is a π phase difference between the two signal components, the signal still cannot be offset totally due to the unequal intensity of the two signal components. The ROPCR has a significantly lower minimum than the SOPCR, but some systematic noise prevents it from reaching zero.

We already know from Figure 6 that the diffraction intensities of different encoded units are not uniform. Therefore, in the next study, we seek to investigate the crosstalk that occurs under orthogonal coding conditions. For the effect of having cells with the same phase difference concentrated in one area, we used Key₂ for recording and Key₁–Key₈ for reconstruction; Figure 7d lists the corresponding distributions. Figure 7c shows the highest diffraction intensity was located at Key₂ because the reconstruction key was the same as the record key. For SOPCR, the diffraction intensities of the two signal components, zero and π, are different due to the varying signal intensities of each coding unit. This results in a larger residual signal intensity, which leads to signal crosstalk. Thus, the signal strength of each orthogonal bond is higher than that of the ROPCR. However, since the recording angles from the signal point to each encoding unit are scattered in the ROPCR, the diffraction intensities are consistent for each encoded unit. Therefore, each orthogonal phase key in the red line can better cancel the central signal.

4. Discussion

This section uses the SSIM and correlation coefficient (COEF) to evaluate this crosstalk. The SSIM can compare two images by brightness, contrast, and structure. The COEF can statistically describe the relationship and correlation between two images. When reading with the wrong key, we expect the crosstalk to be signal-independent. Figure 8 compares all read images with the original recorded maps. The mean SIMM of the ROPCR was reduced by 85% compared with the SOPCR. The small peaks of Key₁₈ and Key₂₆ that appear in the SOPCR almost disappear. The mean COEF of the ROPCR was also reduced by 73% compared to the SOPCR. Moreover, we can see that in both the SOPCR and ROPCR, the crosstalk of the first key will be higher than that of the other keys, which is normal. The literature of Kim and Lee mentions this key [38], a unique key with high crosstalk. The overall diffraction efficiency enhancement was brought about by the uniform phase reading.

Figure 9 clearly shows that the crosstalk of Key₁₈, Key₂₆, Key₃₄, and Key₅₀ is much smaller in ROPCR mode than in SOPCR mode. This significant crosstalk reduction can be attributed to the unique features of ROPCR coding. As can be seen from the orthogonal phase distribution diagram of the reconstruction keys in Figure 9f–j, the phase distribution is discrete, and the phase difference between them and the record key is also discrete, as shown in Figure 9k–o. Compared with the crosstalk in Figure 3d,e, Figure 9d,e demonstrate a significant reduction. Meanwhile, the white noise in Figure 9b,c is also significantly lower than that in Figure 3b,c.

Overall, the comparison of the SOPCR and ROPCR coding methods highlights the importance of considering the phase difference distribution of recorded and reconstructed phase keys in signal coding methods. It also demonstrates the effectiveness of ROPCR coding in reducing crosstalk and improving the coherence of diffraction intensities across different coding units. Through understanding the characteristics of crosstalk in other coding methods, we can develop strategies to mitigate its effects and improve the quality and security of encoded signals.

5. Conclusions

In this paper, we study the mechanism and characteristics of signal crosstalk in an orthogonal phase-coded multiplexed CHDSS. This study shows that the reconstructed signal intensity and phase are modulated simultaneously across the crosstalk via a stepwise superposition. The signal intensities diffracted by each coding unit are not consistent. Orthogonal phase-coding multiplexing fails to fully cancel the reconstructed signal due to residual signals and eventually induces crosstalk. Moreover, more severe inconsistencies in the signals generated by each coding unit lead to more severe crosstalk. For the first time, to our knowledge, the high-crosstalk key index can be predicted with the knowledge of this paper when the recorded key has been determined through looking at the characteristics of the phase difference distribution between the recorded reference wave and the reconstructed one. To reduce crosstalk, a novel coding scheme, ROPCR, is proposed to minimize crosstalk. The encoded units are randomly distributed on the reference wave, and the diffraction intensities of the encoded units are scattered and superimposed at the same diffraction position. The diffraction intensity is approximately uncorrelated with the white noise at the wrong phase key. Experiments show that the SSIM and COEF curves of the ROPCR are more stable and lower than those of the SOPCR, implying lower crosstalk and more favorable storage. The ROPCR has a clear weakening effect at locations where the SOPCR has a sharp crosstalk peak. In this paper, we extend our understanding of the phenomenon of signal crosstalk in orthogonal phase-coded multiplexing CHDSSs.