## Abstract

Faithful reconstruction is a key and crucial point of polarization holography research, and has significant research interests in the field of holographic storage and image display. In order to realize faithful reconstruction of the reconstructed wave, the interference angle and dielectric tensor should be controlled. In contrast, we used the polarization holography theory described by tensor, and obtained the faithful reconstruction condition of linear polarization wave under arbitrary interference angle without any dielectric tensor constraint, which is further verified by experimental analysis.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In 1948, D. Gabor proposed holography technology [1]. This technology used the interference of two coherent beams to record the amplitude and phase information of the light field. With the advent of highly coherent light source devices such as laser, holography has developed rapidly. However, the conventional holography technology is incomplete, which only focuses on the amplitude and phase of the light field, but neglects the polarization characteristics of the light field. In 1972, Sh. D. Kakichashvili experimentally discovered and theoretically described the phenomena of polarization state reconstruction in light field, and opened up a new direction of coherent optics-polarization holography [2]. Sh. D. Kakichashvili first introduced Jones matrix to polarization holography, and then established the theory of polarization holography of full polarization light field and partial polarization light field [3–5]. Later on, driven by many researchers, polarization holography developed rapidly [6–8]. In 2011, K. Kuroda reported the polarization holography described by tensor theory [9], in which the influence of light field on optical properties of polarization-sensitive media is described by the change of dielectric tensor during exposure. In the previous studies, based on the tensor polarization holography theory, the faithful reconstruction [10–18], null reconstruction [16–21], and inverse reconstruction [22], etc. [23] are realized. In polarization holography, faithful reconstruction means that the polarization states of the reconstructed wave and the signal wave must be same, which indicates that the polarization information of the signal wave can be accurately recorded. Considering the derivation of tensor polarization holography theory, the polarization states of the reconstructed wave and the signal wave are different under the general conditions. Controlling the recording and reconstructing of polarization holography to realize faithful reconstruction, which is the key point to be solved in polarization holography.

In polarization holography theory based on tensor, the analysis of related studies showed that there are two methods to realize faithful reconstruction. One is the faithful reconstruction can be realized by controlling the exposure energy, so that the dielectric tensor coefficients can meet a special condition [10–13,18]. However, the limitation of this method that the dielectric tensor changes with the exposure energy and faithful reconstruction depends on a specific dielectric tensor, which can only exist for a short time. The another one is to fix the interference angle at 90° to achieve faithful reconstruction without dielectric tensor constraint [14–17]. Nevertheless, this law does not satisfy other interference angles. Due to the influence of refractive index of PQ-PMMA, only bulk material can be used at 90° interference angle, which reduces the amplitude of reconstructed wave.

By summarizing the past investigations of faithful reconstruction, we observed that the faithful reconstruction strongly depends on special interference angle or dielectric tensor. But other situations under general interference angles and without dielectric tensor constraint have not been demonstrated. Based on the tensor polarization holography theory, we proposed a method for realizing faithful reconstruction of linear polarization wave under arbitrary interference angle and without dielectric tensor constraint. In this paper, we employed the linear polarization holography model for theoretical analysis followed by experimental verification.

## 2. Theoretical analysis

The linear polarization holography is divided into recording stage and reconstructing stage, as shown in Fig. 1. To simplify the description, we define $\theta$ as the angle between the propagation directions of two recording beams inside the material, i.e., the interference angle $\theta = {\theta _ + } + {\theta _ - }$. The *s*-polarization direction is parallel to the *y*-axis of the coordinate system, the *p*-polarization direction is in the *x*-*z* plane and is vertical to that of the propagation direction of beam. ${{\boldsymbol G}_ + }$, ${{\boldsymbol G}_ - }$, ${{\boldsymbol F}_ - }$ and ${{\boldsymbol G}_\textrm{F}}$ represent signal wave, reference wave, reading wave, and reconstructed wave, respectively.

First, the two-unit vectors of orthogonal polarization are

Here, ${\boldsymbol s}$ and ${{\boldsymbol p}_j}$($j ={+} , -$) represent unit vectors of *s*-polarization and *p*-polarization, the subscripts ‘$+$’ and ‘$-$’ represent signal wave and reference wave, respectively.

The recording stage is shown in Fig. 1(a), where the signal wave ${{\boldsymbol G}_ + }$ with incident angle ${\theta _ + }$ and the reference wave ${{\boldsymbol G}_ - }$ with incident angle ${\theta _ - }$ are symmetric about the *z*-axis, and the information is stored in the recording material. The interference light field is composed of signal wave and reference wave is

Where ${\boldsymbol r}$, ${{\boldsymbol k}_ + }$ and ${{\boldsymbol k}_ - }$ are the position vector, signal wave propagation vector, and reference wave propagation vector, respectively.

The reconstructing stage is shown in Fig. 1(b), where the recorded hologram is illuminated by a reading wave satisfying Bragg condition. Based on the tensor and coupling wave theory [7,9], we obtained the reconstructed wave as

In Eq. (3), *B* is the dielectric tensor coefficient of polarization holography, *A* is the dielectric tensor coefficient of intensity holography, and the value of dielectric tensor is determined by the exposure energy absorbed by the material [9]. We use the signal wave ${{\boldsymbol G}_ + }$, reference wave ${{\boldsymbol G}_ - }$, and reading wave ${{\boldsymbol F}_ - }$ with arbitrary polarization angles to perform recording and reconstructing. The signal wave, reference wave, and reading wave with arbitrary polarization angles are

Where ${G_ + }$ is the initial complex amplitude of signal wave, ${G_ - }$ is the initial complex amplitude of reference wave and reading wave. Since we aim to investigate the variation of the polarization state of the reconstructed wave, the initial complex amplitudes of signal wave, reference wave, and reading wave are set to 1. If we consider the polarization angle of linear polarization wave to be the angle between their polarization state and *p*-polarization direction, then the $\gamma$, $\alpha$ and $\beta$ are respectively the polarization angles of signal wave, reference wave, and reading wave. By substituting the signal wave, reference wave, and reading wave in Eqs. (4a), (4b), and (4c) into Eq. (3), we obtained the expression of the reconstructed wave is

The reconstructed wave is still linearly polarized as can be seen from Eq. (5). In general situation, the polarization states of reconstructed and signal waves are different. The polarization state of the reconstructed wave is modulated by the interference angle $\theta$, the dielectric tensor coefficients *A* and *B*, and the polarization angles of the signal wave, reference wave and reading wave.

By solving Eq. (5), we obtain the method for realizing faithful reconstruction of linear polarization wave without dielectric tensor constraint as shown in Table 1.

As shown in the Table 1, under special conditions, such as $\theta$=90° and $\alpha$=0°, the signal wave of any polarization angle can be reconstructed faithfully without dielectric tensor constraint, which has been proved to be correct [14–15]. Generally, the realization of faithful reconstruction for the linear polarization wave without the constraint of dielectric tensor must have same polarization angles of reference and reading waves. Besides, the interference angle and the polarization angles of signal wave and reference wave should satisfy the conditions shown in Table 1. Comparative analysis the conditions led us to conclude that at a fixed interference angle, when $\alpha$ and $\gamma$ are fixed, two different interference angles can be selected to realize the faithful reconstruction of linear polarization wave without dielectric tensor constraint. Alternatively, at a fixed interference angle, there will always be two reference waves with different polarization angles, that can realize faithful reconstruction without dielectric tensor constraint. Furthermore, we verified this phenomenon from the experimental analysis. We choose a fixed 40° interference angle. The theoretical value is shown in Table 2, when the polarization angle of the signal wave is 60°, and the interference angle is 40°, then the reference wave and reading wave with the same polarization angle will have two solutions. Experimental setup of linear polarization wave is shown in Fig. 2.

## 3. Experimental methodology

The experimental setup is shown in Fig. 2. The diameter of the laser with a wavelength of 532 nm is extended to 8 mm along with the beam extending system. Then, the beam goes through the PBS1 and splits into signal wave (** s** branch path) and reference path (

**branch path). We controlled the HWP1 to allow a similar value (40 mW/cm**

*p*^{2}) of the light intensity of the signal wave path and reference wave path. In order to observe the response of PQ-PMMA material with the polarization wave field in real-time, we need to observe the short time recording degree of the holographic gratings in the recording stage. The experiment controls the holographic recording and reading time by setting periodic SH1 and SH2. In one period of each shutter, the exposure time is 5s, where SH1 permits the light path to pass while SH2 blocks the light path. The reading time is 0.5s, in which SH1 blocks the light path while SH2 permits the light path.

On controlling HWP3 and HWP2, the polarization angle of signal wave and reference wave can be achieved 60° and 0°, respectively. The 1 mm thick PQ-PMMA sheets are prepared in our lab and used as polarization-sensitive material in the recording stage [24], as shown in Fig. 3. The refractive index of PQ-PMMA material is 1.492. The interference angle of two incident light beams outside the material is 61.4° while inside the material is 40°. In the reading stage, the polarization angle of the reading wave is controlled by HWP2. The bulk holographic gratings, recorded inside the material are irradiated by the reading wave having a same polarization angle to that of the reference wave. After this, the generated reconstructed wave beam is split by PBS2 and captured by PMs. The HWP4 has been employed to test the polarization state of the reconstructed wave. Afterward, the polarization angles of reference and reading wave are simultaneously changed every 10° and the experiment is repeated several times. When the exposure energy is 30J/cm^{2}, we stop recording and calculate the polarization angle of the reconstructed wave according to the light intensity captured by PMs. The obtained experimental data is shown in Fig. 4.

## 4. Results and analysis

From the analysis of Fig. 4, we can see that the polarization angle of reconstructed wave varies along with the polarization angle of reference and reading waves. When the polarization angle of reference and reading waves is between 50-60° or 150-160°, then the polarization angle of reconstructed wave is close to 60°. To obtain an accurate value, the sampling range of reference and reading wave is narrowed, and observed the accurate polarization angle of the required reference and reading wave such as 55° and 155.5°, respectively. From these angles, the faithful reconstruction of the reconstructed wave is realized. Moreover, it is found that both the experimental and theoretical results have a good correlation as can be seen from Table 2. The above results and discussion verify that normal interference angles have two reference waves with different polarization states. This allows the linear polarization wave to realize the faithful reconstruction. Next, we will investigate how to get rid of the constraint of dielectric tensor.

When the polarization angle of the signal wave is 60°, and the polarization angles of reference and reading waves are 55° and 155.5°, respectively, then the *s* and *p*-polarization of the reconstructed wave is captured by PMs, the data are shown in Fig. 5(a) and 6(a). Comparative analysis of the data of Fig. 5(a) and 6(a) led us to conclude that when the exposure energy increases, then the intensity of the reconstructed wave also increases. The reason behind this is the gradual formation of bulk holographic gratings, under the regulation of polarization wave. After that, we tested the polarization angle of the reconstructed wave, and the method is rotated 180° from the horizontal direction of the fast axis of HWP4, the rotation direction is shown in Fig. 2 while the theoretical value is given in Fig. 7, and the experimental value is depicted in the upper right corner of Fig. 5(a) and 6(a). When the normalized diffraction efficiency (NDE) of the *s*-polarization component is maximum then its *p*-polarization component becomes opposite, which indicates that the *s* and *p*-polarization component of the reconstructed wave has zero phase difference. Thus, the reconstructed wave can be determined as a linear polarization wave. The amplitude ratio of the *s*-polarization component to the *p*-polarization component in the reconstructed wave is the tangent value of the polarization angle of the reconstructed wave. The polarization angle of the reconstructed wave varies with the exposure energy as shown in Fig. 5(b) and 6(b). Under such recording and reconstructing conditions, only a small amount of exposure energy is required to keep consistent of the polarization angles of the reconstructed and the signal waves. According to tensor polarization holography theory, the polarization-sensitive material is isotropic before exposure. During the exposure stage, the polarization wave interacts with the polarization-sensitive material and record the polarization information, the material gradually changes to anisotropy with the passage of time. The dielectric tensor of the material varies along with the increase in exposure energy. However, in our experiments the experimental results are shown in Fig. 5(b) and 6(b), the polarization angle of the reconstructed wave can always be stabilized near the signal wave, realizing faithful reconstruction at any exposure energy. In contrast, the faithful reconstruction can only be achieved at a specific exposure energy as reported elsewhere [10–13,18], and cannot be maintained for a long time. These experimental phenomena can indicate that after the formation of stable bulk holographic gratings inside the material, the faithful reconstruction gradually gets rid of the constraint of dielectric tensor which effectively prove the correctness of this method.

## 5. Conclusion

In this paper, the tensor theory has been employed to investigate the faithful reconstruction of linear polarization wave under arbitrary interference angle and without dielectric tensor constraint. When the interference angle and the polarization state of signal and reference waves satisfy $\cos \theta = \tan \alpha \cot \gamma$ or $\cos \theta ={-} \tan \alpha \tan \gamma$, then the reading wave with a same polarization state to that of the reference wave can be used to realize the faithful reconstruction without dielectric tensor constraint. At an interference angle of 40°, we find a strong correlation between the experimental results and the theoretical values, which consequently verifies the correctness of our proposed method. Finally, this work suggests further improvement of holographic storage density [25,26], using polarization multiplexing holography technology.

## Funding

National Key Research and Development Program of China (2018YFA0701800); National Natural Science Foundation of China (11804051).

## Disclosures

The authors declare no conflicts of interest.

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