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Abstract
Polarization is a natural property of a lightwave and makes a significant contribution to various scientific and technological applications, due to the different states of polarization (SoP) of a lightwave that may manifest distinct behaviors. Hence, it is important to determine the SoP of the lightwave. Generally, the SoP of a lightwave can be recognized by the Stokes parameters. In this paper, we proposed a novel method to simultaneously characterize the Stokes parameters of a lightwave, by employing the tensor polarization holography theory. This is done through merely a piece of polarization-sensitive material. Compared with the traditional method, this method requires only one measurement to obtain all the Stokes parameters, without using additional polarizing elements. The experimental result shows excellent agreement with the theoretical one, which confirmed the reliability and accuracy of the proposed method. We believe that this work may broaden the application field of polarization holography.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Polarization is a natural property of lightwave [1]. In scientific and technological applications, polarization has been widely used in imaging [2], sensing [3], navigation [4], and astronomy [5]. Since the state of polarization (SoP) of lightwave cannot be measured directly, it is meaningful to characterize the SoP of lightwave in an easily accessible way. The most common SoPs include: linear polarization wave, circular polarization wave, elliptical polarization wave, partial polarization wave, and non-polarization wave. Generally, it is feasible to describe the SoP of lightwave by the amplitude parameters of lightwave, through four Stokes parameters [6]. Traditionally, it is enabled by properly arranged polarizers and a quarter-wave plate (QWP), four Stoke parameters can be calculated by the measured light intensities that the lightwave to be measured penetrates these polarizing elements. However, the traditional Stokes measurement method involves several measurement steps, resulting in a relatively troublesome measurement process [7]. Furthermore, the combined Jones-Stokes polarimetry can obtain the Stokes parameter through two shots, however, the setup is too bulky, due to it being designed by combining Sagnac and Mach-Zehnder interferometers. And it can only measure the Stokes parameter of the polarization waves [8].
On the other side, it has been proven that polarization holography is capable of manipulating the SoP of lightwave [9,10]. Polarization holography is an improvement on traditional holography, which can only manipulate the amplitude and phase of lightwave [11–14]. Notably, the developing tensor polarization holography can be used to clarify the properties of polarization holograms at any interference angle [15,16]. In recent years, much progress has been made in the study of tensor polarization holography. When the Bragg condition is satisfied, some phenomena have been reported, such as the faithful reconstruction, the null reconstruction, the inverse polarizing effect reconstruction, etc [17–19]. This makes it possible to realize simultaneous measurements of the Stokes parameters of lightwave in our work utilizing angular multiplexing techniques, while the angular multiplexing technique requires different interference angles for recording polarization holograms. The advantage of utilizing tensor polarization holography to achieve these features is that the polarization-sensitive material (phenanthrenequinone-doped poly (methyl methacrylate), PQ/PMMA) can perform the target function after exposure, by adapting the signal wave and the reference wave to the appropriate SoP for the material. In other words, the preparation process is simple, and the after-exposure polarization-sensitive material will become a target element with a compact size. Although previous researchers have attempted to measure Stokes parameters of lightwave utilizing jones polarization holography, however, this method inevitably requires additional polarizing elements [20].
In this paper, a novel method is proposed to simultaneously characterize the Stokes parameters of lightwave utilizing tensor polarization holography theory. Here, we utilize tensor polarization holography to make the polarization holograms within the polarization-sensitive material resemble the features of polarizers and QWPs, this enables the implementation of Stokes parameters measurements through the after-exposure polarization-sensitive material. Firstly, four specifically designed polarization holograms with different recording conditions are recorded onto the same area of the polarization-sensitive material. Next, when the lightwave to be measured illuminates the after-exposure polarization-sensitive material, four reconstructed waves with different directions will be observed and captured in the meantime, due to angular multiplexing technology originating from the Bragg match condition in coupled wave theory [21]. Finally, the Stokes parameters of the lightwave to be measured are determined by measuring the four reconstructed wave’s diffraction efficiency. And the experimental result shows the average error in the measurement of the normalized Stokes parameters is 0.090, which is nearly like then that reported during detection using metasurface, Pancharatnam-Berry metahologram, and silicon photonic circuits technologies in recent years [7,22,23], in addition, because the polarization-sensitive material (PQ/PMMA) is cheap, this method allows for accurate measurements at a lower cost compared with the above-mentioned method. Meanwhile, this method requires only one measurement to obtain all the Stokes parameters without using additional polarizing elements, featuring remarkable simplicity and convenience. Since the proposed method requires only a small area of the polarization-sensitive material (the material size is determined according to the spot area of the lightwave to be measured) to measure Stokes parameters, indicating the possible usage of the method for miniaturization scenarios in the future.
2. Theoretical analysis
2.1. Stokes parameters and Poincaré sphere
Stokes parameters of lightwave, S0, S1, S2, and S3, completely characterize its SoP and light intensity, as shown in Eq. (1).
We established a rectangular coordinate system with (S1, S2, S3) as the coordinate axis and constructed a ball with the origin as the center and S0 as the radius, as shown in Fig. 1(a). This mathematical model is based on the Stokes parameters and used to represent the SoP of lightwave in a Poincaré sphere [25].
Fig. 1. Two schematics to describe the identical SoP: (a) Poincaré sphere. (b) Elliptical polarization wave.
As shown in Fig. 1(a), since any point on the Poincaré sphere can represent a polarization wave, the SoP of the polarization wave can be described by the longitude and latitude coordinates (2ψ, 2χ) on the sphere. At the center of the sphere, is the non-polarization wave. Inside the sphere, it represents a partial polarization wave. According to the geometric relationship, the longitude and latitude are 2ψ = arctan(S2/S1) and 2χ = arcsin(S3/S0), respectively. And in Fig. 1(b), S0 and S1 can frame the shape of a rectangle circumscribed by the ellipse. Ψ and χ are ellipsometric parameters. ψ represents elliptical orientation (0≤ψ≤π). χ represents ellipticity and ellipse rotation direction (-π/4≤χ≤π/4). The rotation direction of the polarization wave can be determined based on whether S3 is positive or negative. When S3 is positive (δ>0), the lightwave is a right-handed polarization wave, and when it is negative (δ<0), the lightwave is a left-handed polarization wave [1].
2.2. Tensor polarization holography theory
Generally, tensor polarization holography can be divided into recording and reconstructing processes which show in Fig. 2, respectively. To simplify the description, we defined θ as the angle between the two propagation directions of two recording waves, i.e., the interference angle. The s-polarization is parallel to the y-axis of the coordinate system, while the p-polarization is in the x-z plane and perpendicular to the propagation direction of the lightwave. s is the unit vector of s-polarization, p+ and p- are the unit vectors of p-polarization for the signal, reference, and reading waves respectively, as shown in Eq. (2).
Fig. 2. Schematic diagram of polarization hologram: (a) recording process and (b) reconstructing process.
The signal wave, reference wave, reading wave, and reconstructed wave are labeled as G+, G-, F-, and GF, respectively [26,27].
According to tensor polarization holography, during the recording process, the signal wave (G+) interferes with the reference wave (G−) at angle θ, and the polarization interference field is recorded within the polarization-sensitive material in the form of intensity grating and polarization grating. For the convenience of description, the polarization interference field recorded in the polarization-sensitive material is generally referred to as a polarization hologram. When it comes to the reconstruction process, we used the reading wave (F−), whose propagation direction is the same as the reference wave, and strictly satisfies the Bragg condition, to reconstruct the recorded polarization hologram. By substituting the dielectric tensor in Maxwell’s wave equation after exposure and solving the coupled wave equation, we obtained the expression for the reconstructed wave (GF), as Eq. (3) [15,28]:
A and B represent the response degrees of the polarization-sensitive material to the intensity grating and polarization grating, respectively; these are determined by the properties of the material itself. The values of A and B change with the exposure energy [15]. Superscript * represents the complex conjugate. k+ is the propagation vector of the signal wave, and k- is the propagation vector of the reference or reading wave.
From Eq. (4), one can see that the SoP and diffraction efficiency of the reconstructed wave is related to the SoP of the signal, reference, and reading waves, and interference angle θ as well as the exposure response coefficient, A/B of the polarization-sensitive material [29]. By carefully controlling these parameters, we can effectively control the SoP and hence the diffraction efficiency of the reconstructed wave.
2.3. Two type polarization holograms for the measurement of Stokes parameters
As shown in Eq. (1), to obtain the Stokes parameters of the lightwave, the values of I0°, I90°, I45°, and Iλ/4,0° need to be determined by three polarizers and a QWP. Therefore, we use tensor polarization holography to record four designed polarization holograms in polarization-sensitive material, and the four designed polarization holograms can play a similar role as three polarizers and a QWP to determine the values of I0°, I90°, I45°, and Iλ/4,0° for the reading wave. The first type of polarization holograms resembles the polarizers, which are used to characterize the linear polarization components of the lightwave. In other words, we can use the first type of polarization holograms to measure I0°, I45°, and I90°, and then determine the values of Stokes parameters S0, S1, and S2. Specifically, When the lightwave with different SoPs passes through a polarizer, the light intensity of the transmitted lightwave will change accordingly. This variation is the same as the light intensity change of the reconstructed waves generated by the first type of polarization holograms, while the lightwave is regarded as a reading wave. Concerning the polarization holograms in the second type, using them would be similar to simultaneously using a QWP and a polarizer, that is, measuring Iλ/4,0° to determine the values of Stokes parameter S3. Specifically, when the lightwave with different SoP pass through a QWP (the angle between the fast axis and the x-axis is 45°) and a polarizer (the transmission axis is the x-axis) successively, the light intensity of the transmitted lightwave will change accordingly. This variation is the same as the light intensity change of the reconstructed waves generated by the second type of polarization holograms, while the lightwave is regarded as a reading wave.
The design strategy of the first type of polarization holograms is depicted as follows: by using the Jones matrix, the electric field (Ex, Ey) of an arbitrary polarization wave passing through a polarizer can be expressed as Eq. (4):
Here, the arbitrary polarization wave is described as [a1 a2eiδ]T, where ϑ is the angle between the light transmission axis of the polarizer and the horizontal direction. The relationship between the light intensity of the transmitted lightwave and the incident wave with arbitrary polarization can be formulated as Eq. (5):
The signal wave with linear polarization (G+), reference wave with linear polarization (G-), and reading wave with arbitrary polarization (F-) are defined as Eq. (6):
From Eq. (7), we can get that the reconstructed wave is a linear polarization wave. So we define the polarization angle of the reconstructed wave as χ. That is, the amplitude ratio of s- and p-polarization components in the reconstructed wave is tanχ. Therefore, the light intensity of the reconstructed wave can be expressed as Eq. (8):
This system of equations consists of three equations. The three known quantities are A/B, ϑ, and θ. The value of the exposure response coefficient, A/B, in the initial stage of exposure is not affected by the recording conditions (i.e., α, β, and θ). The initial A/B value and range of exposure energy corresponding to the initial value are only related to the material used, and its value can be measured in advance [29]. The three unknowns are α, β, and m.
We solve the equations by substituting the value of ϑ (to be determined), that of θ (to be determined), and that of A/B (known). Then get the required values of α and β for the designed polarization holograms by numerical calculation. The designed polarization holograms using this method meet the requirements of the first type.
Next, we design the second type of polarization holograms. The Jones vector expression of a wave that an arbitrary polarization wave passing through a QWP (the angle between the fast axis of QWP and the x-axis is 45°) and a polarizer (the angle between the light transmission axis and the x-axis of the polarizer is 0°) successively is as Eq. (10):
The expression for the light intensity of the transmitted lightwave can be simplified to Eq. (11):
To obtain the second type of polarization holograms, the signal wave with left-hand elliptical polarization (G+), reference wave with right-hand circular polarization (G-), and reading wave with arbitrary polarization (F-) can be defined as Eq. (12) [31]:
Substituting Eq. (12) in Eq. (3) yields the expression for the reconstructed wave:
From Eq. (13), we can see that the SoP of the reconstructed wave is not affected by the SoP of the reading wave. In addition, the light intensity of the reconstructed wave can be expressed as Eq. (14):
On comparing IF with I4/λ,0° that given in Eq. (11), we can see that the designed polarization holograms using this method meet the requirements of the second type.
3. Experimental setup and procedure
The experimental setup is shown in Fig. 3(a). The wavelength and coherence length of both lasers 1 and 2 are 532nm and 50m, respectively. Firstly, the laser beam emitted from laser 1 (dark green) is expanded into a circular uniform intensity distribution beam with a diameter of 5 mm through the BE1. After the expanded beam passes through PBS1, the reflected lightwave is used as the signal wave, and the transmitted lightwave is used as the reference or reading wave. Besides, the HWP1 is used to control the optical power ratio between the signal wave and reference or reading wave, at last, the powers of the signal wave and reference or reading wave are 101.9 mW/cm2 equally. The SoP of the signal wave can be changed by using a combination of HWP2 and QWP1, while that of the reference and reading waves can also be changed by using HWP3 and QWP2. Finally, the reconstructed wave is divided into the p- and s-polarization components by PBS2. Subsequently, the ratio of the p- and s-polarization components of the reconstructed wave can be captured and analyzed, with PM1 and PM2. Figure 4 shows a photograph of the PQ/PMMA sample used in the experiment; its dimensions are 51 mm × 54 mm × 1 mm (thickness), and the concentration of PQ is 1 wt%. According to our previous experimental experience, when PQ is 1 wt%, the material performance is relatively stable [32]. When only laser 1 is turned on, the reading wave is the polarization wave, which is regarded as the polarization wave to be measured during reconstructing process. If laser 1 and 2 are both turned on, the reading wave is the non-polarization wave as these two beams emitted from laser1 and laser2 are non-coherent; the reading wave is formed of the beam emitted by laser 1 and that emitted by laser 2. We use a QWP and polarizers to test and confirm that the incoherent reading wave is the non-polarization wave. In summary, once the recording process is complete, we turn on laser 1 to measure the Stokes parameter of the polarization wave and turn on laser 1 and laser 2 to measure the Stokes parameter of the non-polarization wave, during the reconstructing process.
Fig. 3. (a) Experimental setup. BE is the beam expander, M is a mirror, HWP is a half-wave plate, QWP is a quarter-wave plate, PBS is the polarization beam splitter, BS is the beam splitter, POL is a Polarizer, BT is the beam terminator, SH is shutter and PM is a power meter. (b) Recording and (c) reconstruction processes.
During the recording process, the propagation direction of the signal wave is changed by varying the position and rotation direction of mirror M3, for purpose of the signal wave and the reference wave to form four specifically designed polarization holograms at different interference angles, as depicted in Fig. 3(a). Four polarization holograms are successively recorded at the same position of the PQ/PMMA sample. According to Kogelnik’s coupled wave theory, to ensure that the deviation in the interference angle between each signal wave and reference wave is greater than the Bragg angular selectivity [21], we set the interference angle, θ, values of the four polarization holograms to θ1 = 6.65°, θ2 = 9.93°, θ3 = 13.18°, and θ4 = 16.36°, afterward, in the reconstructing process, the four reconstructed waves can be independently and non-interferingly generated in different propagation directions. As shown in Fig. 3(b).
The exposure response coefficient of the PQ/PMMA sample used in the experiment is approximately 5.01 in the initial stage of exposure (i.e., when the exposure energy is less than 30 J/cm2) [29]. The experimental recording conditions of the first type of polarization holograms (the first three polarization holograms) are set as follows: The interference angles are θ1 = 6.65°, θ2 = 9.93°, and θ3 = 13.18°, respectively. And substituted the desired ϑ1 = 0°, ϑ2 = 45°, and ϑ3 = 90° values in Eq. (9) to determine the values of α and β. The numerical solutions are listed in columns 2, 3, and 4 of Table 1.
Table 1. Conditions used for recording polarization holograms
Subsequently, the recording condition of the second type of polarization holograms (the fourth polarization hologram), which is used to measure Iλ/4,0°, is set as follow: The interference angle, θ4 = 16.36°, is substituted in Eq. (12), and the SoP of the signal and reference waves, needed to record the polarization hologram, is determined, as shown in column 5 of Table 1.
The recording process is divided into four steps, and a polarization hologram is recorded in each step. During the recording step of each polarization hologram, SH2 is opened and SH3 is closed. At the same time, we controlled the time of polarization holographic recording and briefly observed the reconstructed wave by cyclically switching SH1 and SH4 alternately. In each cycle, the recording time is 5 s, and the observation time of the reconstructed wave is 0.5 s. The brief observation of the reconstructed wave did not damage the hologram [26]. When we observe χ of the reconstructed wave with linear polarization and amplitude ratio of the p- and s-polarization component of the reconstructed wave with elliptical polarization are consistent with that listed in lines 9 and 10 of Table 1, the polarization hologram recording at this step is stopped. The exposure energy required to record each polarization hologram is approximated at 6.11 J/cm2 (exposure time of 30 s) in this step. We record polarization holograms 1-4 in the order shown in Table 1, as shown in Fig. 5.
Fig. 5. Polarization holograms recording process. (a) Polarization hologram 1 resembles a polarizer of ϑ1 = 0°. (b) Polarization hologram 2 resembles a polarizer of ϑ2 = 45°. (c) Polarization hologram 3 resembles a polarizer of ϑ3 = 90°. (d) Polarization hologram 4 resembles the combination of a QWP of the fast axis is 45° and a polarizer of ϑ1 = 0°.
During the reconstructing process, when the reading wave is a polarization wave with an optical power of 101.9 mW/cm2, SH1 and SH3 are closed and SH2 and SH4 are periodically synchronized to control the observation time for the reconstructed wave to 0.5 s. By rotating HWP3 and QWP2, we can control the SoP of the reading wave. Firstly, four holograms are read using a p-polarization wave, and the displayed power value of PM1 receiving GF1 is calibrated unity. Next, the linear polarization wave of polarization angle is 45°, s-polarization wave and right-hand circular polarization wave read polarization holograms successively. Use the same method to calibrate the power of GF2-4 received by PM2-4 at this time as unity. Then, polarization holograms are read using the polarization waves with unknown SoP. And we processed the power value of PM1-4 receiving reconstructed waves to obtain the normalized diffraction efficiency of the GF2-4.
When the reading wave is the non-polarization wave, SH1 is closed, and SH3 is opened. SH2 and SH4 are periodically synchronized to control the observation time for the reconstructed wave to 0.5 s. The beam emitted by laser 1 and that emitted by laser 2 are combined at BS1. The combined beam is a non-polarization wave with an optical power of 101.9 mW/cm2. Hence, the proposed measurement method, which uses a non-polarization wave as the reading wave, is like those that use a polarization wave as the reading wave. Finally, we substituted the four normalized diffraction efficiencies of GF2-4 for the light intensities in Eq. (1) to obtain the normalized Stokes parameters of the reading wave.
4. Experimental results and discussion
Once the recording process is completed, the reading waves are regarded as the polarization or non-polarization waves to be measured in the experiment. When the reading wave illuminates the four polarization holograms in the after-exposure material, four reconstructed waves appear simultaneously and their normalized diffraction efficiencies, are captured by PM1-4. Then the corresponding Stokes parameter is calculated by E. (1). In this experiment, we present four sets of reading waves of different SoPs. The first three sets of reading waves are polarization waves, and the last set of reading waves is a non-polarization wave. For a clear illustration, part of the first three sets of reading waves is charted, as shown in Figs. 6(a)–8(a). Here, the SoP of polarization waves is expressed as the function of ellipsometric parameters (Ψ and χ) as shown in Fig. 1(b). e. g. As for Fig. 6(a), χ is equal to 0° and ψ varies from 0° to 180°, we can see that the reading waves are linear polarization waves all the time, and the orientation of reading waves with linear polarization varies with the change of ψ. In Figs. 6(b)–8(b), the abscissa in Figs. 6(b)–8(b) shows the values of ellipsometric parameters ψ and χ of the reading wave, the red solid line and red triangle points represent the theoretical and experimental values of the normalized diffraction efficiency of the p-polarization component in the reconstructed wave, respectively. The green solid line and green circular points represent the theoretical and experimental values of the normalized diffraction efficiency of the linear polarization component at an angle of +45° to the x-axis in the reconstructed wave, respectively. The blue solid line and blue square points represent the theoretical and experimental values of the normalized diffraction efficiency of the s-polarization component in the reconstructed wave, respectively. The black solid line and black rhombic dots represent the theoretical and experimental values of the normalized diffraction efficiency of the right-handed circular polarization component in the reconstructed wave, respectively. Finally, the corresponding Stokes parameters are shown in Figs. 6(c)–8(c); The black and red dots are the distributions of the theoretical and experimental values of the reading waves’ Stokes parameters on the Poincaré sphere. From all these results, it is obvious that the experimental values have very similar values to the theoretical ones, indicating that it is very accurate the proposed method.
Fig. 6. (a) The SoP of partial reading waves is expressed by the function of ellipsometric parameters (Ψ and χ) in the first set of the experiment. (b) The reading waves used in the first set of the experiment are χ = 0°, Ψ starts from 0° and rotates clockwise for 10° each time, with a total rotation of 360°. That is, there are 18 different reading waves used in the first set of the experiment. (c) The first set of the experimental and theoretical values of the reading waves’ Stokes parameters on the Poincaré sphere.
Fig. 7. (a) The SoP of partial reading waves is expressed by the function of ellipsometric parameters (Ψ and χ) in the second set of the experiment. (b) The reading waves used in the second set of the experiment are Ψ = 0°, χ starts from 0° and rotates clockwise each time, with a total rotation of 360°. That is, there are 20 different reading waves used in the second set of experiments. (c) The second set of the experimental and theoretical values of the reading waves’ Stokes parameters on the Poincaré sphere.
Fig. 8. (a) The SoP of partial reading waves is expressed by the function of ellipsometric parameters (Ψ and χ) in the third set of the experiment. (b) The reading waves used in the third set of the experiment are χ = 20.01°, Ψ starts from 0° and rotates clockwise for 10° each time, with a total rotation of 360°. That is, there are 18 different reading waves used in the third set of experiments. (c) The third set of the experimental and theoretical values of the reading waves’ Stokes parameters on the Poincaré sphere.
The errors between the experimental and theoretical values of the Stokes parameters of several special SoPs of polarization waves are summarized in Table 2. Notably, the first to fourth rows depict p-polarization, s-polarization, right-handed polarization, and left-handed polarization waves, respectively. Column 1 lists the theoretical value of the normalized Stokes parameter of this reading wave while columns 2-5 list experimental and average values of the normalized Stokes parameter of this reading wave. Column 6 lists 2-norm error ||Stheoretical-Saverage(Meas)||2, where Stheoretical = (S1, S2, S3)/S0 is theoretical normalized Stokes vector and Saverage(Meas) is average of experimental Stokes vector. The average value of all errors is 0.089 in the first three sets of the experiment (A total of 54 polarization waves).
Table 2. The theoretical and experimental values of the Stokes parameters of special polarization waves
When we turn on both laser 1 and laser 2, the reading wave becomes a non-polarization wave which is the last set of reading waves in the experiment. Its Stokes parameter (S1, S2, S3) equals (0, 0, 0), which is at the center of the Poincaré sphere. The experimental results are shown in Fig. 9. The red and black dots in the figure represent the distributions of the experimental and theoretical values of the reading waves’ Stokes parameters on the Poincaré sphere, respectively. The error between the experimental and theoretical values of the Stokes parameters in this state is shown in Table 3. This error is calculated in the same way as in column 6 of Table 2. A total of five measurements are made, and the average value of all errors is 0.100.
Table 3. The theoretical and experimental values of the Stokes parameters of non-polarization wave
Overall, according to the results of Figs. 6–9 and Tables 2–3, we can conclude that whatever lightwave with arbitrary SoP can be accurately measured by the proposed method based on the tensor polarization holography. Therefore, the Stokes parameters calculated based on the normalized diffraction efficiencies of the reconstructed waves relatively accurately reflect the SoP of the reading waves to be measured. The error between the experimental and theoretical values may be caused by two factors. The first is the modulation of the photoinduced anisotropic refractive index. According to polarization holography theory, polarization-sensitive materials are isotropic before exposure but become anisotropic after the polarization hologram has been formed. Thus, the polarization hologram recorded later will be affected since the material becomes anisotropic rather than isotropic after the polarization hologram recorded earlier [33]. Another possible factor is dark diffusional enhancement [34]. Because the diffraction efficiency of the reconstructed waves is observed every 5 s during the reconstruction process, the material would exhibit dark diffusional enhancement for 5 s without exposure, and the diffraction efficiency of the subsequent reconstructed waves would change, thus affecting the final Stokes parameters. In conclusion, the proposed method can accurately characterize the Stokes parameters of both polarization and non-polarization waves. Notably, Since the partial polarization wave is a combination of polarization and non-polarization waves, it is obvious can be characterized by this method [35].
5. Conclusions
Utilizing tensor polarization holography, four polarization holograms are recorded by angular multiplexing technology at the same position on a PQ/PMMA sample. As a result, the recorded PQ/PMMA with four specially designed polarization holograms can measure SoPs of any lightwave. The lightwave to be measured is regarded as a reading wave to illuminate the after-exposure polarization-sensitive material, four reconstructed waves appeared at the back of the material simultaneously. By measuring the diffraction efficiencies of the four reconstructed waves, the Stokes parameters of any lightwave can be determined, hence, the SoP of the lightwave to be measured is determined. Thus, this method is suitable for determining the SoP of monochromatic light through a single measurement without any additional polarizing elements. The SoP of an unknown lightwave can be determined quickly, simply, and accurately by this method. During the experiment, we found that the experimentally recorded polarization holograms are stable for about 3 hours. It is worth noting that owing to the existence of the Bragg condition, polarization holograms designed by this method can only measure the Stokes parameters of a lightwave whose wavelength is similar to that of the waves used for the recording process [21]. And the response range of the PQ/PMMA sample to wavelength is about 470-550nm. In general, as long as the wavelength of signal wave, reference wave, and reading wave are identical and within the response range of the recording material, the approach we proposed is feasible [36]. The experimental results showed that there is an error of approximately 0.090 between the experimental and theoretical values of the normalized Stokes parameters.
In recent years, several applications utilizing tensor polarization holography have been reported, such as planar bifocal lens [31], circular polarization wave generators [37], and vector and vortex light generators [38,39]. The method for measuring the Stokes parameters of lightwave utilizing tensor polarization holography proposed in this study should deepen our understanding of polarization holography and broaden its applicability, particularly in the areas of polarization detection and polarization imaging [40]. Although polarization holography is capable of completely manipulating the lightwave, however, the application of polarization holography theory has not been fully explored yet. The work in this paper convinces us that polarization holography theory can also be exploited for a wide range of applications. In prospect, it is possible to record spatially distributed polarization holograms in a single recording process, with the help of spatial light modulator (SLM) [41], and to replace the existing conventional micro-nano technology [42] and metamaterials [43].
Funding
National Key Research and Development Program of China (2018YFA0701800); Project of Fujian Province Major Science and Technology (2020HZ01012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the authors upon reasonable request.
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