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Abstract
Polarization holography has attracted considerable attention in recent years, due to its capability of recording the polarization information in polarization-sensitive material. Particularly, the faithful reconstruction (FR) can retrieve the polarization information of the recorded signal. To date, studies referring to these topics mainly concentrate on the interference between the same type of polarization such as linearly, circularly, and elliptically polarized light. In addition, most of the reading wave is strictly limited to some specified polarization state to achieve the FR. Here, we apply the linearly polarized light as the reference wave to record the circularly polarized light, and then the circular polarization state would be faithfully reconstructed by the arbitrarily polarized reading wave. We theoretically analyze its polarization characteristic based on the tensor theory and experimentally verified the analytical results. This result further extending the FR in polarization holography, and provides a practicable way to generate circular polarization which is easily fabricated. Moreover, the work would lay a favorable theoretical foundation for the future preparation of circular polarization generator and discloses a new insight in polarization manipulation for tailoring the optical field.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Polarization holography, compared with conventional holography, not only records the amplitude and phase information, but also records the polarization information of the signal wave [1]. The information is recorded by the interference between two polarized waves in polarization-sensitive material.
Under the paraxial approximation, the polarization holography based on the Jones matrix description is proposed and widely utilized [2]. However, due to the limitation of the small interference angle, it is incapable for us to conduct an insight study and obtain a more comprehensive understanding of the polarization holography. Most recently, Kuroda proposed the novel polarization holography theory based on the description of the dielectric tensor theory which improved the original polarization holography theory to a large interference angle [3]. The new polarization holography theory breaks through the limitation of the paraxial approximation, and it is easily accessible for us to explain and understand the recording mechanism of polarization holography [4,5].
Based on the tensor theory, numerous of interesting polarization holography properties have been reported such as the null reconstruction (NR) of linear polarization [6,7], the NR of orthogonal circular polarization [8,9], the NR of orthogonal elliptical polarization [10,11], the inverse polarizing effect [12], the faithful reconstruction (FR) of linear polarization [13], the FR of circular polarization [14], the FR of orthogonal elliptical polarization [15,16]. The FR in polarization holography, indicating the polarization state of the diffracted wave is consistent with that of the signal. It can be achieved in the exposure process when the intensity and polarization holographic gratings attained a balance. Among these characteristics, the FR is one of the most significantly important phenomena. As it will have a great potential in imaging information encoding [17,18], high-definition holographic display [19,20], increasing the data storage intensity [21,22].
Until now, there are fruitful researches concerning the characteristic of FR and its applications. Zang and his co-workers investigated the properties of polarization holography for linearly polarized waves. They found that the signal wave can be faithfully reconstructed in the special exposure condition [13]. Hong et al. by using a smart method with the well-designed reference and reading wave to faithfully reconstruct the signal wave at arbitrary exposure energy [14]. Huang et al. studied the prerequisite for the FR of orthogonal elliptical polarization holography with the reading wave different from the recording reference wave [15,16]. These previous research work focus on the FR which are limited to the interference cases between the same type of polarization states, such as two linearly, two circularly, and two elliptically polarized waves. Motivated by the above researches, it is interesting to explore the polarization interference properties of the linearly and circularly polarized wave. Hence, in this work, we theoretically and experimentally study the faithful reconstruction characteristic for linear and circular polarization states in polarization holography, and find the FR of circular polarization could be obtained while reading with arbitrarily polarized waves. The objective of the present work is to study the polarization holography properties of linear and circular polarization, with particular emphasis on the arbitrary reading wave to achieve the FR phenomena and further applied in circular polarization generator.
The existed techniques for generating the circularly polarized wave, which requires the additional quarter-wave plate [23], and specially designed layer or structure, such as stacked chiral polymer films [24], metasurfaces [25,26], dielectric plate arrays [27], plasmonic nanomaterials [28], frequency selective surface [29]. Compared with the previous methods, our proposed circular polarization generator based on the polarization holography exhibits to be more convenient, easily fabricated, and compatible with an arbitrary incident wave.
Our research would facilitate the understanding and applications of polarization characteristic referring to the interference between the linearly and circularly polarization states, which are absent in the present polarization hologram. Moreover, our work would lay very favorable theoretical and experimental foundations for the preparation of circular polarization generators with arbitrary reading polarization.
2. Theory
The vector polarization holography theory is based on the description of the dielectric tensor theory, which reveals a concise analysis of the polarization holography under non-paraxial approximation. The polarization information is written by the photo-induced anisotropic polarization-sensitive materials. Before exposure, the photo-sensitive material exhibit to be isotropy. After exposure, the periodic birefringent grating is formed and the optical wave information is recorded in the material. The molecules distribution of the material would be modulated to be anisotropic by the polarization interference pattern, which can be expressed in the form of a dielectric tensor as
Considering the recording stage of the thick volume holography, the schematic of the new polarization holography is shown in Fig. 1. The x-y axis represents the surface of the material, and the z-axis is the thickness direction of the material. We define the linearly polarized wave parallel to the y-axis of the coordinate system as s-polarization. The linearly polarized wave in the x-z plane and perpendicular to the wave propagation direction is defined as p-polarization. The unit polarization vector of two orthogonal can be written as
In the reconstruction stage, as shown in Fig. 1(b), when the incident angle of the reading wave F- is strictly satisfied with the Bragg diffraction condition. The reconstruction wave can be obtained by the dielectric tensor method and coupled-wave theory [3], which can be expressed as
We define some of the concepts and notations that will be used throughout the rest of this paper. Since we aim to investigate the interference characteristics of the linear and circular polarization. In the recording stage, we employ the circular polarization as the signal wave, which can be written as
In the reconstruction stage, we employ arbitrary polarization as the reading reference wave and denote it as
where α and β are arbitrary coefficients with real or imaginary numbers. By substituting the corresponding expression of interference waves into Eq. (4), the reconstructed signal wave read by the arbitrarily polarized wave can be expressed asFollowing the same derivation approach, we have calculated the rest of the cases about the right- and left- handed circular polarization signal recorded by s- or p- polarization reference wave. We have deduced the reconstructed waves and summarized all cases in Table 1.
It is clear that when the circular polarization signal is recorded by the linear polarization reference wave in polarization holography, the circular polarization would be faithfully reconstructed by the reading wave F- with arbitrary polarization when A + B=0. In experiment, the requirement of A + B=0 can be achieved at proper exposure dose, when the polarization and intensity hologram reached a balance during the exposure process. The above derivations and analysis indicate the reconstructed wave will always be a circularly polarized beam independent of the polarization of the reading wave. Noteworthily, it breaks the limitation that the FR should be obtained only when the polarization of the reading wave is the same as that of the recording reference wave.
Table 1. Polarization State of Waves in Recording and Reconstruction Stage
3. Experiments and results
The experimental setup is shown in Fig. 2. We build a well-designed light path of recording and reading system. The 532nm laser is employed and divided by polarization beam splitter (PBS1) into two parts for recording and reconstructing the signal separately. In the recording stage, the beam reflected from the PBS1 serves as a signal beam, and its circular polarization state is adjusted by the polarizer (P1) and the quarter-wave plate (QWP1). Likewise, the beam transmitted from PBS2 is utilized as the recording reference beam whose linear polarization state is adjusted by the half-wave plate (HWP2). In the reconstruction stage, the beam reflected from the PBS2 act as a reading wave, and by adjusting HWP3 and QWP2, the beam would set to be an arbitrarily polarized wave.
Fig. 2. The schematic of the verification experiment. Sig. and Ref. represent the signal and the reference light path, respectively. At, Attenuator; M, Mirror; BE, Beam Expander; A, Aperture; PBS, Polarization Beam Splitter; BS, Beam Splitter; P, Polarizer; HWP, Half Wave Plate; QWP, Quarter Wave Plate; SH, Shutter; PM, Power Meter.
In our work, the recording polarization-sensitive material is the home-made phenanthrenequinone-doped polymethyl methacrylate photopolymer (PQ/PMMA) [30,31], following the process [32]. The polarization characteristics investigated in our work are universal in polarization hologram and not limited to the polarization sensitive materials. The commonly used hologram materials with polarization sensitivity can be applied to record and reconstruct the polarization in our work as well. Here, the crossing angle between the signal and reference wave is 48° outside the recording material which is chosen randomly. Due to the refractive index of our recording material is 1.51, the interference angle of the two recording waves inside the material is 31.2°. The incident beam diameter on the material is around 5mm, and the power intensity of that is 20mW.
In the recording stage, since the FR would be obtained under the condition of A + B=0, we should modulate the exposure dose precisely to reach a balance between the intensity hologram and polarization hologram. Here, we schedule the on and off states of the electronic-controlled shutters to manage the period of recording and reading. Firstly, we set the shutter (SH1) and SH2 to be opened, SH3 and SH4 be closed for recording. Then, we reverse the setting in the reading stage. The scheduling period of the experiment is 6s and 0.3s for recording and reading, respectively. The transient reading time is in avoid of the erase of the recorded grating. During the reading stage, the reconstruction wave is split into s- and p- polarization states by PBS3. The power intensities of s- and p- polarization components are measured by power meters (PM1) and PM2, which will help us determine the acquiring of A + B=0.
In the reconstruction stage, we further detect the handedness of the reconstruction wave after obtaining the state of A + B=0. The handedness of polarization should be verified to be consistent with the original signal wave. The handedness of the reconstructed signal is determined by the handedness verification system, consisting of QWP3, PBS4, PM3, and PM4. On account of the circularly polarized signal wave, its polarization would be reconstructed, only when the optical power ratio of s- and p- polarization components of the reconstructed wave reaches 1:1. When the power ratio of PM1 to PM2 is 1:1, the QWP3 with its fast axis in the horizontal direction rotated 360 degrees counterclockwise. Here, we will explain the verification principle of rotation direction in detail.
For a QWP with α with respect to the x-axis, the Jones matrix is
Considering an incident beam Iin with right-handed circular polarization, which passes through the QWP3, the output beam can be described as
Theoretically, when α changes from 0 to 360 degrees, the power intensity of s- and p- polarization components of the right-handed circular polarization would vary periodically as shown in Fig. 3. When α changes from 0 to 360 degrees (counterclockwise), the p-polarization component first rises up to its maximum and then goes down to its minimum, while the s-polarization component changes in the opposite way to the p-polarization component. In the reconstruction stage, if the intensity trends of the reconstructed waves are consistent with the original signal wave, that verify the handedness of the reconstructed waves are identical to the signal waves. Thus, the FR of circular polarization can be confirmed by two necessary conditions. The first is the power ratio of s- and p- polarization components of the reconstructed wave reached 1:1; the second is the intensity variation of s- and p- polarization components after QWP3 are consistent with that of the signal [14].
Fig. 3. Intensities of s- and p- polarization components of the reconstructed signal wave change with angle α.
Here, we design the experiments to testify the derivations and theoretical predictions in Table 1. We simplify the situations into two groups with the same recording signal wave of right-handed circular polarization, but with different reference wave (s- or p- polarization wave). As the s- or p- polarized waves are the eigenstates which form the constituent components of any linear polarized wave. We simplify the situations into two categories exhibit in Table 2 and Table 3. Besides, we generalize the reading wave as linear (s, p), circular (l, r), and elliptical (ms + nip) polarization, respectively. The coefficient m and n are s- and p- components of the elliptically polarized waves. The elliptical polarization represents more universal states and will help us to testify our analytical predictions.
Table 2. FR characteristics of right-handed circular polarization recorded by the p-polarization wave
Table 3. FR characteristics of right-handed circular polarization recorded by the s-polarization wave
We carry out the cases summarized in Table 2, where the right-handed circular polarization is recorded by p-polarization. Figure 4 (a)-(e) exhibits the normalized intensities of the s- and p- polarization components of the reconstructed waves read by different polarization waves. The intensity ratio of the s-polarization (PM1) component and p-polarization (PM2) component is varying and approaching each other over time. When the diffraction intensity ratios of the s- and p-polarization components of the reconstructed waves are 1:1. Thus, two polarization components overlap with each other, indicating the theoretical condition of A + B=0 is satisfied. At this point, we check the polarization handedness of the reconstructed wave by the rotation QWP3, with its fast axis was parallel to the horizontal direction before the PBS4. Figure 4(f) is the original signal passing through the verification experimental set for polarization handedness. During the rotation of QWP3, the detected power intensity will fluctuate as a sin2 function. The verification of polarization handedness for different reconstructed waves is shown in the upper right corner of Figs. 4(a)-(e), respectively. The intensity trends of the s- and p- polarization components of the diffraction signals are consistent with that in Fig. 4(f), indicating the right-circularly polarized beam has been reconstructed.
Fig. 4. Circular polarization is recorded by the p-polarization wave and reconstructed by different reading waves. The blue and red lines represent the intensities of the s- and p- components of the reconstructed wave, respectively. (a) reading wave, s; (b) reading wave, p; (c) reading wave, s + ip; (d) reading wave, s-ip; (e) reading wave, 1.76s + ip. The upper right inserted figures are the verification of polarization handedness at the point of FR. (f): The intensities variation of the signal wave after the rotated QWP3.
Obviously, according to the previous investigations, the circularly polarized wave can be read by the wave identical to the polarization of the original reference wave, as shown in Fig. 4(b). Nevertheless, as shown in Fig. 4(a)-(e), we notice that after the exposure process, the FR can be obtained with the reference wave different from the recording reference as well. These experimental results agree well with our theoretical calculations summarized in Table 2.
We further conduct the cases summarized in Table 3, where the right-handed circular polarization is recorded by s-polarization. The experiments are conducted following the same approaches as demonstrated in the abovementioned cases of Table 2. Likewise, the results shown in Fig. 5 indicate the circular polarization signal can be faithfully reconstructed by various polarization waves, such as linear, circular, and elliptical polarization. Considering the results of Fig. 4 and Fig. 5, we can generalize that the circular polarization record by the linear polarization can be faithfully reconstructed by arbitrary reading waves.
Fig. 5. Circular polarization recorded by the s-polarization wave and reconstructed by different reading waves. The blue and red lines represent the intensities of the s- and p- components of the reconstructed wave, respectively. (a) reading wave, s; (b) reading wave, p; (c) reading wave, s + ip; (d) reading wave, s-ip; (e) reading wave, 1.76s + ip. The upper right inserted figures are the verification of polarization handedness at the point of FR. (f): The intensities variation of the signal wave after the rotated QWP3.
However, we can notice that the handedness verification in the upper right corner of Fig. 4(a)-(e) and Fig. 5(a)-(e) does not precisely coincide with the Fig. 4(f) and Fig. 5(f) accordingly, where Fig. 4(f) and Fig. 5(f) are the original signals passing through the verification experimental set for polarization handedness. There exist some subtle differences between experiment and theory. We assume these differences result from the anisotropy of the polarization-sensitive material PQ/PMMA.
To support our assumption and explain the above differences, we have conducted a comparative verification experiment. In the case of Fig. 5(a), we record and read right-handed circular polarization by the s-polarization wave. While the exposure reaches A + B=0 and the FR is realized, we make the original signal instead of the diffracted wave pass through the handedness verification system. Figure 6 shows the variation tendency when the original signal is transmitted through the already formatted polarization grating. It is observed to be distorted in contrast to the beam without passing through PQ/PMMA as exhibited in Fig. 5(f), nevertheless is identical with Fig. 5(a). The comparison experiments intuitively reveal that the anisotropy of PQ/PMMA will lead to the subtle fluctuation difference in the polarization handedness verification process.
Fig. 6. Intensities variation of the transmitted circular polarization signal after the rotation of the QWP3 (counterclockwise).
As depicted in Fig. 4 or Fig. 5, we notice an interesting phenomenon, the variation curves of s- and p- components of the reconstructed wave are distinct from each other by the different polarized reading waves. We observe that in (a)(b)(c)(e), the curve of s- and p- polarization components of the reconstructed wave are approaching mutually during exposure. However, for the polarization hologram in (d) with its reading polarization (left-handed polarization) orthogonal to the signal (right-handed polarization), the variation curves of s- and p- polarization components are increasing simultaneously. The trend in (d) is completely different from other situations in (a)(b)(c)(e) of Fig. 4 and Fig. 5. We assume the differences come from the ${\pi}$ phase differences between the signal and reconstruction wave. More detailed analysis and experimental verification is the subject of a future paper, it will help us to elucidate how the phase influences the formation of polarization hologram.
4. Conclusion
In summary, we have theoretically and experimentally studied the FR characteristic for the interference between the linear and circular polarization in polarization holography. Compared with most of the previous studies, our recorded circularly polarized wave can be acquired with arbitrarily polarized reading waves, such as linear, circular, or elliptical polarization. Present results have may considerably extend the study of characteristics of volume polarization holography and will have an important implication in understanding the interaction of each polarization during exposure. Furthermore, our research may provide a new method for circular polarization generator and may find applications in domains such as depolarizer devices, polarization-sensitive spectrometers, optical field manipulation, and so on.
Funding
National Key Research and Development Program of China (2018YFA0701800); The Special Funds of the Central Government Guiding Local Science and Technology Development (No. 2020L3008).
Disclosures
The authors declare no conflicts of interest.
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