Open Access
Add to BibSonomy
Abstract
Diffraction characteristics of polarization holograms have important research significance and application prospects. In this paper, the Tensor theory was used to investigate the diffraction characteristics of a linear polarization hologram in a coaxial recording. The results show that, when the signal and reference waves are coaxial aligned, the sum of the polarized direction angles of two waves in the reconstruction process is equal to the sum of these in the recording process under the condition that, the gratings of recorded intensity response and the polarization response are in balance. The study on the diffraction characteristics of linear polarization holograms based on the Tensor theory in the coaxial recording may help us with a deeper insight into the polarization holography theory. Again, our results helped in the use of a proper design for the polarizer to change the polarization direction of the incident linear polarization waves.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Sh. D. Kakichashvili published the first paper in polarization holography in 1972. He experimentally discovered and theoretically described the phenomenon of an electromagnetic field vector wave pattern reconstruction and created a new direction of coherent optics - polarization holography [1]. Compared with the traditional holography, the polarization holography not only records the amplitude and phase information of the light wave, but also records information of polarization at the same time [2,3]. Sh. D. kakichashvili used for the first time Jones vector-matrix method in the theory polarization holography. On this basis Kakichashvili created the detailed theory of polarization holography both at completely polarized and at partial polarized field of electromagnetic waves [4–6]. P. Cai successfully recorded the gratings of pure polarization modulation using two orthogonal linear polarization waves in the azobenzene liquid crystal polymer in the paraxial approximation based on the Jones theory, and also found that the polarization gratings can switch the polarization state of the incident wave, which function is similar to a half-wave plate having an optical axis of 45° to the horizontal direction [7].
In 2011, K. Kuroda of Tokyo University introduced dielectric tensor to describe the refractive index change of photoinduced anisotropic media during exposure, and combining with Maxwell's equations, the general rule of polarization holography has been found by solving the coupled wave equation (Tensor theory for short) [8]. Tensor theory can examine the property of polarization hologram under arbitrary interference angle.” And in the first paragraph of the introduction, these two papers are cited. It should be noted that the gratings of recorded intensity response and the polarization response are in a balance. This special balance condition of Tensor theory is different from Jones theory.
In recent years, the research based on Tensor theory has made a lot of new progress. A. Wu and J. Wang verified the faithfully and null reconstruction of polarization hologram by orthogonal circular-polarization waves [9,10]. They pointed out that these reconstructions just under the balance condition is satisfied. Based on the Tensor theory, J. Zang examined the diffraction characteristics of linear polarization hologram in the recording condition with the interference angle of 38.8° [11]. His paper pointed out that the reconstructed wave is still linear polarization wave in the orthogonal recording, if linear polarization wave is used to reconstruct polarization hologram. Moreover, the polarization state of the reconstructed wave varies with the changes of the reconstructing reference wave. However, the specific variation law between the polarization direction of the reconstructed wave and the reconstructing reference wave was not established in this paper.
Based on the Tensor theory, the diffraction characteristics of the linear polarization hologram that the signal wave is arbitrary linear polarization waves in coaxial recording are studied in this paper. Our study shows that the polarization state of the reconstructed wave varies regularly with the polarization state of the signal wave, the recording and reconstructing reference waves. Our findings have been verified by the experiment. The theoretical and experimental study indicates that the polarization state of the reconstructed wave could be modulated by controlling the polarization state of the reconstructing reference wave in reconstructing process. Finally, the summarization and the potential applications based on the theory are pointed out.
2. Theoretical analysis
The recording and reconstructing processes of polarization hologram are shown in Fig. 1. To simplify the description, we defined $\theta$ as the angle between two propagation directions of recording waves, which is the interference angle. The s polarization is parallel to the y-axis of coordinate system, and the p polarization is in the x-z plane, vertical to the propagation direction of the light wave. The signal wave, the recording reference wave, the reconstructing reference wave and the reconstructed wave are described by ${\boldsymbol{G}_ + }$, ${\boldsymbol{G}_ - }$, ${\boldsymbol{F}_ - }$, $\boldsymbol{G}_{\textbf{F}}$ respectively.
Fig. 1. Schematic diagram of the polarization hologram. (a) The recording process. (b) The reconstructing process.
Based on the Tensor theory, at the recording process, the signal wave(${\boldsymbol{G}_ + }$)interferes with the recording reference wave(${\boldsymbol{G}_ - }$) by the angle $\theta$, and the polarization interference field is recorded inside the polarization-sensitive media. At the same time, after exposure the expression for material’s dielectric tensor is [12]:
At the reconstructing process, we used the reconstructing reference wave(${\boldsymbol{F}_ - }$)which strictly satisfied the Bragg condition to reconstruct the polarization hologram. By placing the dielectric tensor of Eq. (1) after exposure into Maxwell’s wave equation and solving the coupled wave equation, we can obtain the expression for reconstructed wave($\boldsymbol{G}_{\textbf{F}}$) as [13]:
At the recording process, as shown in Fig. 1(a), the signal and the recording reference waves of arbitrary linear polarization wave can be written as:
As shown in Eq. (7), the reconstructed wave is still linear polarization wave in the coaxial recording, Whose polarization state depends on value of $\alpha$, $\beta$, $\gamma$, $A$ and $B$. But we can't obtain the variation property of polarization state of the reconstructed wave further. However, when $A + B = 0$ (i.e. the gratings of recorded intensity response and the polarization response are in balance), Eq. (7) can be simplified as:
From Eq. (8), we can further conclude:
Therefore, the following conclusions can be drawn: in the coaxial recording, the sum of the polarized direction angles of two waves in the reconstruction process is equal to the sum of these in the recording process when $A + B = 0$. And at the reconstructing process, diffracted intensity of the reconstructed wave is maintained unchanged and does not change with the polarization state of reconstructing reference wave.
It should be noted that, when polarization state of the signal and the recording reference waves are orthogonal to each other, we can then use $\gamma = \beta - {90^ \circ }$ in Eq. (7) and obtain the reconstructed wave expression as:
Equation (10) is the same as the Eq. (8) when $\gamma = \beta - {90^ \circ }$, which indicates that when the signal and the recording reference waves are orthogonal to each other, the constraint condition $A + B = 0$ can be removed and the reconstructed wave still can be expressed by Eq. (8).
The above result, as a special example of Eq. (9), which is consistent with Cai's result using the signal wave of s polarization and the recording reference wave of p polarization in the paraxial approximation situation on the basis of the Jones theory. After using the paraxial approximation, the two theories predict the same diffraction characteristics in orthogonal recording. Therefore, we can see that by comparing with the Jones theory, the polarization holography based on the Tensor theory demonstrates more generality and comprehension.
3. Experiment
The experiment equipment is shown in Fig. 2. The laser wavelength is 532 nm with linear polarization. A spatial filter and Lens are used to expand the wave and a polarization beam splitter (PBS1) is used to split the wave into the signal and reference waves. The half-wave plate (HWP1) is used to adjust powers ratio of signal and reference waves, and the powers of signal and reference waves are 117.2mW/cm2 equally. The polarization state of the signal wave can be changed by the half-wave plate (HWP2). In the experiment, the recording material is PQ/PMMA, which is prepared in our laboratory [14–16]. The preparation procedure consists of two steps. In the first step, we dissolve phenanthraquinone (PQ) and 2,2-Azobisisobutyronitrile (AIBN) in glass bottle filled with methyl methacrylate (MMA), where PQ is photo sensitizer, AIBN is thermo-initiator, and MMA is liquid monomer. The solution is mixed evenly via using an ultrasonic water bath and the impurities are filtered out by a filter mesh. The glass bottle is then placed in Magnetic Stirrers and kept at a constant temperature until the solution becomes homogeneously viscoid. In the second step, the syrup is poured into a glass mold. Then the mold has been heated to 60°C for 1 day to solidify the mixture. Figure 3 shows the photo of the obtained PQ/PMMA material, which is 5.1 cm × 5.4 cm in size and 1 mm in thickness, and the concentration of PQ is 1 wt.%. Obviously, the size and thickness may be changed when the different glass mold is used.”
Fig. 2. Optical setup for recording and reconstructing polarization hologram in PQ/PMMA. SF is spatial filter, M is mirror, HWP is half wave plate, PBS is polarization beam splitter, SH is shutter and PM is power meter.
In the experiment, the real external interference angle of the material is 5°. The refractive index of the bonded material is 1.51, so the internal interference angle of the material is 3.312°. Since $\sin {3.31^ \circ } \approx 0.058 \approx \sin {0^ \circ }$, the optical path in the experiment can be approximately regarded as a coaxial optical path.
At the recording process, since the $\boldsymbol{G}_{\textbf{F}}$ would be obtained under the condition of $A + B = 0$, we should modulate the exposure dose precisely to reach a balance between gratings of intensity response and polarization response. Here, we schedule the on and off states of SH to manage the period of recording and reading. Firstly, we set the shutter SHI and SH2 to be opened, SH3 be closed for recording. Then, we set the shutter SHI and SH3 to be opened, SH2 be closed for reading. The scheduling period of the experiment is 5s and 0.5s for recording and reading, respectively [17]. For simplicity, the recording reference wave was fixed to p polarization at the experimental recording process. And, the polarized direction angle $\beta$ of the signal wave was chosen to be 35.26°(the intensity ratio of s polarization to p polarization is 1:2 in this polarization state), 45°(the intensity ratio of s polarization to p polarization is 1:1 in this polarization state), and 90°(that is s polarization), respectively. As shown in Figs. 4(a) and 4(b), when the $\beta$ is 35.26° or 45°, the recording is stopped if the reconstructed wave can accurately reconstruct the polarization state of signal wave, which is the condition of $A + B = 0$ is satisfied for the specific time. When the $\beta$ is 90°, since the $A + B = 0$ does not need to be satisfied. As shown in Fig. 4(c), the recording is stopped when the intensity of the reconstructed wave does not obviously tend for increasing, and is maintained relatively unchanged with the increase in exposure time. In Fig. 4, the blue curve and red curve represent the experimentally measured reconstructed wave p polarization component and s polarization component intensities, respectively. In the experiment, the diffraction efficiency is 0.97% to 1.39%.
Fig. 4. (a) the signal wave is the linear polarization wave of $\beta = {35.26^ \circ }$; (b) the signal wave is the linear polarization wave of $\beta = {45^ \circ }$; (c) the signal wave is s polarization. All of the recording and reconstructing reference waves are p polarization. Note, that in each figure the corresponding values are not to scale.
At the reconstructing process, SH2 was shut down and SH1 and SH3 were periodically switched synchronously to control the time of short-time observing the reconstructed wave, with the time of observing diffraction set as 0.6 s. PQ/PMMA does not change during the reading time, and the response time of PQ/PMMA for used power density is more than 0.3s [17,18]. In the situation, where SH2 is shut down, we used the half-wave plate (HWP3) to change the polarization state of reconstructing reference wave, and observe the variation of polarization state of the reconstructed wave. The reconstructed wave is split by PBS2 into p polarization component and s polarization component, and then the powers variation for two waves was measured by rear power meter which can be used to analyze polarization state of the reconstructed wave.
4. Results and analysis
From Eqs. (8) and (9), the sum of the polarized direction angles of two waves in the reconstruction process is equal to the sum of these in the recording process (i.e. $\alpha ^{\prime}\textrm{ + }\alpha = \beta + \gamma$), we know polarization state of the reconstructed wave shown in Table 1.
Table 1. Polarization state of hologram read by linear polarization wave when reconstructing reference wave was fixed to p polarization
In the experiment, we used the p polarization as the recording reference wave, the signal and reconstructing reference waves as shown in Table 1, whereas the obtained polarization state of reconstructed wave is expressed in the polar coordinates. In rectangular coordinate system, we use formula $E({\cos \alpha {p_\textrm{ + }} + \sin \alpha \hat{s}} )$ to describe linear polarization waves with arbitrary polarization direction, where E is the amplitude of linear polarization wave, $\alpha$ is the angle between the polarization direction and p polarization, p+ is the unit polarization vector of p polarization, $\hat{s}$ is the unit polarization vector of s polarization. It can be seen from the formula that $\cos \alpha$ represents the amplitude of p-polarization component of linear polarization wave and $\sin \alpha$ represents the amplitude of s-polarization component of linear polarization wave. $\cos \alpha$ in polar coordinates, as shown by the red dot line in Fig. 5(a). And $\sin \alpha$ in polar coordinates, as shown by the red dot line in Fig. 5(b). By using the polar representation to express diffraction characteristics of the reconstructed wave, we able not only vividly demonstrate polarization state of the reconstructed wave, but also the area surrounded by the curve in the polar representation that may characterize the diffracted intensity of reconstructed wave after normalizing the data.
The experimental results are shown in Figs. 6–8, where at the upper right of each subgraph the angle $\alpha$ represents polarized direction angle of the reconstructing reference wave. In the figure, the black solid line represents the measured value of the reconstructing reference wave in use, the red solid line represents the reconstructed wave value according to the theoretical derivation, and the blue dashed line represents the experimentally measured value of the reconstructed wave.
Fig. 6. Diffraction characteristics of reconstructed wave when the signal wave is linear polarization wave of $\beta = {35.26^ \circ }$.
Fig. 7. Diffraction characteristics of reconstructed when the signal wave is linear polarization wave of $\beta = {45^ \circ }$.
From Figs. 6–8 we can see that the experimental results are in perfect agreement with the theoretical prediction in Table 2. That is, when the signal and reference waves are coaxial aligned, the sum of the polarized direction angles of two waves in the reconstruction process is equal to the sum of these in the recording process under the condition that $A + B = 0$. Since the recording reference wave is p polarization (i.e. $\gamma = {0^ \circ }$), we can get $\alpha ^{\prime}\textrm{ + }\alpha = \beta$. And diffracted intensity of the reconstructed wave does not change with polarization state of the reconstructing reference wave. When polarization state of the signal and recording reference waves are orthogonal to each other (i.e. the signal wave is s polarization and the recording reference wave is p polarization), the state of the reconstructed wave can always satisfy the above results even if $A + B \ne 0$.
5. Conclusion
In this work, the diffraction characteristics of linear polarization hologram in the coaxial recording are given based on the Tensor theory. It has been shown that when the balance is attained between gratings of intensity response and polarization response or polarization state of the signal and recording reference waves are orthogonal to each other, the sum of the polarized direction angles of two waves in the reconstruction process is equal to the sum of these in the recording process (i.e. $\alpha ^{\prime} + \alpha = \beta + \gamma$). At the reconstructing process, the polarization state of reconstructed wave could be modulated by controlling the polarization state of reconstructing reference wave. Previous studies verified that the polarization holography has the potential for applications in multi-functional optical devices such as bifocal-polarization holographic lens [19,20] and holographic data storage [21,22]. The results of our work also indicate that the polarization holography has the potential for polarizing optical elements. Based on this conclusion, we can manufacture polarization holographic gratings according to specific requirements and realize changing the polarization state of the incident linear polarization waves.
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.
References
1. S. D. Kakichashvili, “On polarization recording of holograms,” J Opt. Spectrosc 33(2), 324–327 (1972).
2. A. W. Lohmann, “Reconstruction of VectorialWavefronts,” Appl. Opt. 4(12), 1667–1668 (1965). [CrossRef]
3. M. E. Fourney, K. V. Mate, and A. P. Waggoner, “Recording Polarization Effects via Holography,” J. Opt. Soc. Am. 58(5), 701–702 (1968). [CrossRef]
4. Sh. D. Kakichashvili and B. Kilosanidze, “On nonparaxial generalization of polarization holography,” Optics and Spectroscopy 71(6), 1050–1055 (1991). [CrossRef]
5. S. D. Kakichashvili and B. Kilosanidze, “On the theory of polarization holography in three-dimensional photoanisotropic media,” Opt. Spectrosc. 65(2), 409–414 (1988).
6. Sh. D. Kakichashvili, Polarization Holography (Nauka, 1989), p. 144.
7. P. Cai, C. Wang, F. Zhao, P. Zeng, and M. Qing “Diffractive Waveplates Based on Polarization Holography,” Shanghai: Shanghai Jiaotong University, 42–47 (2016).
8. K. Kuroda, Y. Matsuhashi, and T. Shimura, “ Reconstruction characteristics of polarization holograms,” In Proceeding of IEEE 2012 11th Euro-American Workshop on Information Optics, 1–2 (2012).
9. A. Wu, G. Kang, J. Zang, Y. Liu, X. Tan, T. Shimura, and K. Kuroda, “Null reconstruction of orthogonal circular polarization hologram with large recording angle,” Opt. Express 23(7), 8880–8887 (2015). [CrossRef]
10. J. Wang, G. Kang, A. Wu, Y. Liu, J. Zang, P. Li, X. Tan, T. Shimura, and K. Kuroda, “Investigation of the extraordinary null reconstruction phenomenon in polarization volume hologram,” Opt. Express 24(2), 1641–1647 (2016). [CrossRef]
11. J. Zang, “Fundamental Research on Polarization Holography Based on Tensor theory,” Beijing: Beijing Institute of Technology, 54–58 (2017).
12. K. Kuroda, Y. Matsuhashi, R. Fujimura, and T. Shimura, “Theory of polarization holography,” Opt. Rev. 18(5), 374–382 (2011). [CrossRef]
13. J. Zang, A. Wu, Y. Liu, J. Wang, X. Lin, X. Tan, T. Shimura, and K. Kuroda, “Characteristics of volume polarization holography with linear polarization light,” Opt. Rev. 22(5), 829–831 (2015). [CrossRef]
14. S. H. Lin, J. H. Lin, Y. N. Shiao, and K. Y. Shu, “Doped poly (methyl methacrylate) photopolymers for holographic data storage,” J. Nonlinear Opt. Phys. Mater. 15(02), 239–252 (2006). [CrossRef]
15. Z. Huang, C. Wu, Y. Chen, X. Lin, and X. Tan, “Faithful reconstruction in orthogonal elliptical polarization holography read by different polarized waves,” Opt. Express 28(16), 23679–23689 (2020). [CrossRef]
16. Y. Liu, F. Fan, Y. Hong, J. Zang, and X. Tan, “Volume holographic recording in irgacure 784-doped pmma photopolymer,” Opt. Express 25(17), 20654–20662 (2017). [CrossRef]
17. X. Xu, Y. Zhang, H. Song, and X. Tan, “Generation of circular polarization with an arbitrarily polarized reading wave,” Opt. Express 29(2), 2613–2623 (2021). [CrossRef]
18. L. Shao, J. Zang, F. Fan, Y. Liu, and X. Tan, “Investigation of the null reconstruction effect of an orthogonal elliptical polarization hologram at a large recording angle,” Appl. Opt. 58(36), 9983–9989 (2019). [CrossRef]
19. G. Martinezponce, T. Petrova, N. Tomova, V. Dragostinova, T. Todorov, and L. N. Udmila, “Bifocal-polarization holographic lens,” Opt. Lett. 29(9), 1001–1003 (2004). [CrossRef]
20. R. Wei, J. Zang, Y. Liu, F. Fan, Z. Huang, L. Zhu, and X. Tan, “Review on polarization holography for high density storage,” Opto-Electron. Eng. 46(3), 180598 (2009). [CrossRef]
21. X. Lin, J. Liu, J. Hao, K. Wang, Y. Zhang, H. Li, H. Horimai, and X. Tan, “Collinear holographic data storage technologies,” Opto-Electron. Adv. 3(3), 19000401–19000408 (2020). [CrossRef]
22. H. Horimai, X. Tan, and J. Li, “Collinear holography,” Appl. Opt. 44(13), 2575–2579 (2005). [CrossRef]
