# Combination Compensation Method to Improve the Tolerance of Recording Medium Shrinkage in Collinear Holographic Storage

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. The Compensation for Medium Dimensional Change

_{1}and θ

_{2}, respectively. The original grating period was Λ. The dimension change in the recording medium would cause Λ and the fringe surface inclination angle θ

_{i}to change. The change in the fringe angle was mainly caused by the change in thickness, which is called the vertical dimension change in this paper, and correspondingly, the lateral dimension change refers to the changes in the x–y plane. Though the change in the grating period is related to both the lateral dimension change and the vertical dimension change, the lateral one would play an important role.

**is equal to 2π/Λ. Furthermore, the change in the grating period indicates the changes in the length of the grating vector, as illustrated in Figure 1b, in which K**

_{0}**becomes K**

_{0}**. In addition, if the inclination angle of the grating fringe surface changes either, the direction of the grating vector would be rotated as well, which is illustrated as K**

_{1}_{1}’ in Figure 2b. In the condition that the dimension of the recording medium is changed, the symbol of the grating vector after the change is denoted as K’, then, if the light of reading is incident at the original incident angle θ

_{1}with the original wavelength λ (i.e., λ’ = λ), the three vectors (i.e., the read light vector k

_{p}, the grating vector K’ and the reconstructed light vector k

_{d}) cannot form a closed triangle, which means the Bragg condition cannot be matched, and the diffraction efficiency would be sharply reduced. As shown in Figure 1b, the incident angle of the readout light or the wavelength of the reading light should be adjusted so as to make the three vectors re-form a closed triangle and match the Bragg condition, then reacquire a high diffraction efficiency. The wavelength adjustment (Δλ ≠ 0) is similar to that in the compensation for the off-axis system, whereas, due to the fact that hologram writing and reading are performed in a single optical path in a collinear system, the incident angle of the reading light cannot be directly adjusted.

_{1}corresponds to θ

_{1}, and f

_{2}corresponds to θ

_{2}). Then, based on increasing or decreasing the focal length of the lens, the angles of the reading light could be adjusted, and the inferior reconstructed hologram caused by the material dimension change could be repaired and compensated. To clarify, Figure 2 only illustrates the schematic diagrams; the real holographic storage system is more complex, as fine-tuning the focal length could be realized by shifting the relative positions of the relay lens module in the system, and the achromatic lens would be used in the real system.

_{i}, y

_{j}) on the SLM would be a plane wave incident into the recording medium after propagating through the focusing objective lens. In the writing procedure, the

**k**vector of the plane wave is given by Equation (1) [35], where k

_{0}is equal to 2π/λ, f is the focal length of the lens and n is the refractive index of the recording medium. The grating vector written by the plane waves from two different pixels on the SLM can be calculated by the subtraction of the two

**k**vectors [35].

**formed by writing is denoted as K**

_{0}**= [K**

_{0}_{0x}, K

_{0y}, K

_{0z}], which is the original grating vector. The original length and the thickness of the recording medium are L

_{0}and T

_{0}, respectively, and L’ and T’ represent the corresponding dimensions after the volume change of the medium. The rate of dimension change in the lateral plane (σ

_{L}) is assumed to be isotropic, which is defined by Equation (2), and the vertical dimension change rate (σ

_{V}) is defined by Equation (3). When the dimension of the recording medium has been changed, the corresponding grating vector K’ is described by Equation (4).

_{rk},y

_{rl}) in the reading light pattern on the SLM, is given as Equation (5), where k’

_{0}= 2π/λ’ = 2π/(λ + Δλ) and Δf is the amount of change in focal length (the focal length after adjustment is f’, which is equal to f + Δf).

_{error}represents the number of the symbol units with errors in the reconstructed signal light pattern, and N

_{total}is the total number of symbols in the original signal pattern, and each symbol corresponds to a block of 4 × 4 pixels on the SLM.

#### 2.2. Parameters in the Simulation

^{−4}.

## 3. Simulation Results and Discussion

_{V}≡ 0) and the vertical dimension change condition (σ

_{L}≡ 0) were analyzed respectively, so as to clearly investigate and compare the compensation effects.

_{L}and σ

_{V}were as high as 1.5%, where the BER for the lateral dimension change increased to 0.7153, and the BER for the vertical dimension change was 0.5417.

_{L}= 0 and 1.5%, respectively. The pictures denoted as Figure 8(a2–c2), etc., immediately to the right of the reconstructed patterns, are the corresponding symbol error diagrams. In these diagrams, each of the tiny bright squares, denoted as ‘An error symbol’ in Figure 8(d2), represents error within the symbol located there, and the bright space is the composition of these error symbols. The compensation results in Figure 8e–h directly show the effectiveness of this combination compensation method.

_{L}= 1.5%, the intensity distribution of the dark and the bright pixels became consecutive, as shown in Figure 9b. In Figure 9c,d,i, when the wavelength decreased by 9 nm, the focal length was not sufficient to ensure an effective compensation. When the focal length adjustment increased to the range of (30, 80) µm, as shown in Figure 9e–h, the dark and the bright pixels were separated again without overlapping along the abscissa. The results also suggest that the combination compensation method of the focal length and the wavelength adjustments can maintain a BER of 0 in a larger selection range of focal length.

_{L}= 1.5%) and the vertical dimension change condition (σ

_{V}= 1.5%), we systematically analyzed the compensation effects by varying the wavelength and the focal length of the objective lens in a large scope. The compensation effects are shown in Figure 10; it can be seen that for both of the lateral dimension change and the vertical dimension change, the scopes when adjusting the wavelength and the focal length for realizing zero BER compensation are relatively large. There are plenty of combinations of wavelength and focal length that result in a BER value of zero, for which one-to-one correspondence between the wavelength and the focal length is not required, whereas under the different wavelength values, the range of options for the focal length adjustment would be different. It is suggested that the wavelength compensation be scanned first, as under a wavelength which leads to a smaller BER, the range for a successful focal length compensation would be larger. Compared with the compensation method, which only adjusts the wavelength or the focal length, the combination compensation method can enlarge the compensation scope and thus increase the robustness of the collinear system.

## 4. Conclusions

_{L}= 1.5%, σ

_{V}= 1.5%). The combination compensation method can enlarge the adjustment scope, and the two adjustment parameters (the wavelength of the reading light and the focal length of the lens) do not need to be in one-to-one correspondence to each other. The large compensation scope would facilitate the experiment, and the simulation results can be used to instruct the implementation of collinear holographic storage systems. Based on results achieved for the combination compensation method, the tolerance of the volume change in the recording medium in a collinear holographic storage system could be improved as well.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The schematic diagram for the model of the dimension change in the recording medium; (

**b**) the illustration of the vector diagram in the Ewald sphere.

**Figure 2.**The illustrations of (

**a**) the principle for the focal length compensation in the collinear holographic storage system, and (

**b**) the simplified optical path of the collinear holographic storage system.

**Figure 3.**(

**a**) The reference light pattern and the signal light pattern uploaded on the SLM; (

**b**) the reference light pattern, which is also the reading light pattern.

**Figure 4.**Without any compensation, the BER of the reconstructed signal pattern for different lateral dimension change rates σ

_{L}and vertical dimension change rates σ

_{V}. (σ = σ

_{L,}σ= σ

_{V}).

**Figure 5.**The BER with different wavelength adjustments of the reading light for the lateral dimension change condition (σ

_{L}= 1.5%) and the vertical dimension change condition (σ

_{V}= 1.5%) when the focal length of the objective lens was unchanged.

**Figure 6.**The BER with different objective focal length adjustments for (

**a**) the lateral dimension change condition (σ

_{L}= 1.5%) and (

**b**) the vertical dimension change condition (σ

_{V}= 1.5%) when the wavelength of the reading light was unchanged.

**Figure 7.**The BER with different focal length adjustments when the wavelength of the reading light was reduced by 9 nm and σ

_{L}was 1.5%.

**Figure 8.**(

**a**) The reconstructed signal pattern without the dimension change of the recording medium; (

**b**) the reconstructed signal pattern with σ

_{L}= 1.5% and without any compensation. When σ

_{L}= 1.5%, for the reconstructed signal patterns with the combination compensation method, the reading light wavelength was decreased by 9 nm and the focal length was changed by (

**c**) 0 μm, (

**d**) 10 μm, (

**e**) 30 μm, (

**f**) 50 μm, (

**g**) 60 μm, (

**h**) 80 μm and (

**i**) 110 μm. The corresponding symbol error diagrams are depicted to the right of the correlated reconstructed signal patterns.

**Figure 9.**The normalized pixel distribution histograms of (

**a**) the original reconstructed signal pattern; (

**b**) the reconstructed signal pattern under σ

_{L}was 1.5% and without any compensation; the reconstructed signal patterns under σ

_{L}were 1.5% with the combination compensation method, in which the reading light wavelength was decreased by 9 nm and the focal length was changed by (

**c**) 0 μm, (

**d**) 10 μm, (

**e**) 30 μm, (

**f**) 50 μm, (

**g**) 60 μm, (

**h**) 80 μm and (

**i**) 110 μm.

**Figure 10.**The compensation effects of the combination compensation method for (

**a**) the lateral dimension change condition (σ

_{L}= 1.5%) and (

**b**) the vertical dimension change condition (σ

_{V}= 1.5%).

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**MDPI and ACS Style**

Qiu, X.; Wang, K.; Lin, X.; Hao, J.; Lin, D.; Zheng, Q.; Chen, R.; Wang, S.; Tan, X.
Combination Compensation Method to Improve the Tolerance of Recording Medium Shrinkage in Collinear Holographic Storage. *Photonics* **2022**, *9*, 149.
https://doi.org/10.3390/photonics9030149

**AMA Style**

Qiu X, Wang K, Lin X, Hao J, Lin D, Zheng Q, Chen R, Wang S, Tan X.
Combination Compensation Method to Improve the Tolerance of Recording Medium Shrinkage in Collinear Holographic Storage. *Photonics*. 2022; 9(3):149.
https://doi.org/10.3390/photonics9030149

**Chicago/Turabian Style**

Qiu, Xianying, Kun Wang, Xiao Lin, Jianying Hao, Dakui Lin, Qijing Zheng, Ruixian Chen, Suping Wang, and Xiaodi Tan.
2022. "Combination Compensation Method to Improve the Tolerance of Recording Medium Shrinkage in Collinear Holographic Storage" *Photonics* 9, no. 3: 149.
https://doi.org/10.3390/photonics9030149