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Abstract
Aiming at the major demand for polarization information gap in earth observation and space exploration, we proposed a four-quadrant retarder array imaging spectropolarimeter (FQRAISP) in view of the existing technical problem of the spectral resolution degradation along with spectral aliasing crosstalk. The optical schematic diagram of the FQRAISP together with its interference model was conceptually described, and the effectiveness of the scheme was validated through the experimental simulation, which demonstrated the competitive efficiency and accuracy in the proposed FQRAISP. The FQRAISP could restore the incident Stokes vector spectrum without any errors, and the inversion accuracy was increased by seven times, avoiding the spectrum aliasing and channel filtering in the channel modulation. In order to evaluate the influences of the alignment deviation of four-partition phase retarder component, together with its thickness deviation on the reconstructed Stokes parameters, the numerical simulations were carried out, and the results showed that the alignment deviations had a relatively weak effect on the reconstructed Stokes spectra, while the thickness deviations had an obvious influence. Therefore, the alignment deviations controlled in a range of $[{ - 0.43^\circ , + 0.43^\circ } ]$ and $[{ - 0.22^\circ ,\textrm{ + }0.22^\circ } ]$ together with the thickness deviations in a range of [ − 0.03µm, + 0.03µm] were an optimal choice for the engineering implementation of the FQRAISP. This research provided a novel method for the hardware realization of the accurate acquisition of all-optical information, having broad application prospects in remote sensing (deep space exploration), biomedicine and other fields.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the exploration, development and utilization of earth resource and survival environment, as well as the remote sensing and monitoring on the land surface, atmosphere, ocean and spatial target, the research and development of high-tech detection technology with high spatial resolution, high spectral resolution and polarization information have been paid more and more attention by governments, scientists and military.
As a cutting-edge and high-tech remote sensing technology, the interference imaging spectropolarimetry (IISP) integrates the camera technology [1,2], spectral technology [3] and polarization measurement technology [4] to reflect the essence of the target from multiple perspectives [5–8]. The IISP covers a wide application in the fields of the atmospheric sounding, haze monitoring, astronomical observation, earth resource survey, biomedical diagnosis and ocean remote sensing monitoring [9–14]. The imaging spectropolarimeter (ISP) is a novel photoelectric instrument combining the imaging spectrometer and imaging polarimeter, and it is divided into the acoustooptic tunable filter ISP [15], the liquid crystal tunable filter ISP [16], the polarization grating ISP [17], along with the ISPs based on the polarization components added in the optical paths of the dispersive [18], tomographic [19] and interferometric imaging spectrometers [20]. It extends three-dimensional information to four-dimensional information, including two-dimensional spatial image, one-dimensional spectral information and one-dimensional polarization information.
Due to the polarization switching element generally combined with the imaging spectrometer for the conventional imaging spectropolarimeter, the jitter noise and beam drift from the polarization element must be minimized [21–23]. On basis of the channeled spectrum, the novel imaging spectropolarimeter is rapidly developed, and it is formed by adding a polarizer and two thick retarders into the imaging spectrometer or imaging interferometer, arousing great interest of the researchers because of the rapid imaging capability [24,25]. Such imaging spectropolarimeter can obtain the Stokes vector parameters by only a single spectrum. Typically, there are more than three channels for the demodulated interferogram, which results in a lower spectral resolution for each Stokes parameter than that of the system [26,27]. On basis of the amplitude modulation and interferometric imaging spectroscopy, a static imaging spectropolarimeter was presented, which can modulate the input Stokes component into different channels by using the achromatic quarter-wave plate together with the multistage phase retarder and linear polarizer [28]. The modulation interferogram at one single exposure is obtained, and the spectral polarization information of the incident light is decomposed by the Fourier transform [29]. Adopting two polarization arrays with four different partitions, four Stokes parameters of the incident light are modulated into four partitions in space, and four interferograms are formed then demodulated to obtain the Stokes parameters [30]. Combining the channeled spectropolarimetry with the polarization interference imaging spectroscopy, the interference imaging spectropolarimeter was introduced to realize the simultaneous acquisition of the image, spectrum and polarization information for the target [31–34]. In recent years, significant progress has been made in the static, compact and robust performances for the interference imaging spectropolarimeter with a birefringent interferometer, particularly the preservation ability of polarization [35–37]. However, there is a lower spectral resolution for such imaging spectropolarimeter, and the restoration accuracy is greatly reduced due to the spectral aliasing.
Aiming at the international high resolution multi-dimensional information acquisition technology, in view of the major demand for the polarization information gap in the earth observation and space exploration, breaking through the technical bottleneck of the spectral resolution degradation and spectral aliasing crosstalk, we proposed a four-quadrant retarder array imaging spectropolarimeter (FQRAISP) that could realize the real-time and no-error acquisition of the high-quality image, high-precision spectrum and full polarization information for the targets. The optical layout and interference model for the FQRAISP were demonstrated in the section 2, and the mathematical simulation of the effectiveness validation for the FQRAISP scheme was implemented in the section 3, where the design parameter simulation in the section 3.1, the calibration of four-partition modulation module in the section 3.2 and the influences of the alignment deviation of the FPR along with its thickness deviation on the input Stokes parameters in the section 3.3, respectively. Our conclusion was contained in the section 4.
2. Optical layout and interference model
The optical layout of a four-quadrant retarder array imaging spectropolarimeter (FQRAISP) is depicted in Fig. 1, and it consists of a four-partition phase retarder component (FPR) with the thickness satisfying a certain relationship and the fast axis angular $0^\circ $ and $45^\circ $ to the x axis, respectively. A linear polarizer ${P_1}$ with its dielectric direction along x axis is placed behind the FPR, followed by the tetrahedral prism $PP$. A Savart polariscope $SP$ with its bottom oriented at $45^\circ $ with respect to the $yz$ plane, composed of two identical uniaxial crystal plates with orthogonally oriented principal sections, is positioned behind the $PP$. The optical axes of two plates are oriented at $45^\circ $ relative to the z axis. A linear analyzer ${P_2}$ follows the group with its transmission axis at $0^\circ $. A single charge coupled device CCD is placed on the back focal plane of the imaging lens ${L_3}$.
Fig. 1. Optical schematic diagram of the FQRAISP, where the positive x and z axes are pointing in the vertical and horizontal directions, respectively, while the positive y axis forms an reverse right hand spiral relative to the $xz$ plane.
The light from a scene is collected and collimated by the front telescopic system (${L_1}$ and ${L_2}$), and then passes through the FPR. The light emerging from the FPR is modulated to different wave numbers, which is divided into four beams of linear polarization light with equal intensities by the ${P_1}$ and $PP$. The $SP$ splits each of the incoming light into two equal amplitude, orthogonally polarized components, and the identical linear polarization components are extracted by the ${P_2}$. Each pair of the equal-amplitude polarization components are reunited on the four parts of the CCD camera, and four interference images in the spatial domain can be recorded simultaneously. Meanwhile, the full Stokes parameters of the incident light are obtained by solving the equations.
According to the definition of the Stokes parameters as expressed Eq. (1), six polarimetric intensities of a scene should be measured for the determination of full Stokes parameters [38].
After passing through the FPR and ${P_1}$, the output Stokes vector for each polarimetric channel was derived from the Stokes-Muller algorithm.
Because of the detector responding to the total intensity, only the first parameter ${S^{\prime}_0}$ of the Stokes vector could be measured. Hence, after the polarization modulation module, the incident light became
If the phases of four partitions for ${R_1}$ and ${R_2}$ satisfied the relationship in Eq. (6), four intensities were measured.
After passing through the polarization modulation module, the light ray was divided into two beams $oe$ and $eo$ by the $SP$ whose vibration directions were perpendicular to each other. Afterwards, they became the linear polarization light by the ${P_2}$ whose vibration directions were $45^\circ $ with respect to the x axis. Finally, these two beams were concentrated on the focal plane of the imaging lens ${L_3}$, then the interferograms of the modulation spectra ${S^{\prime}_0}(1 )$, ${S^{\prime}_0}(2 )$, ${S^{\prime}_0}(3 )$ and ${S^{\prime}_0}(4 )$ were obtained in different regions of the CCD detector. Herein, the obtained intensity pattern, measured by the FQRAISP, was given by
The phase term is presented as ${\varphi _{sp}}({\sigma ,\Delta } )= 2\pi \Delta \sigma $, and the optical path difference $\Delta $ produced by the $SP$ is expressed as follows
3. Effectiveness validation for the FQRAISP scheme
3.1 Design parameter simulation
In order to demonstrate the versatility of the FQRAISP system, a mathematical model for the simulation and reconstruction was performed, and only the reconstruction of the polarization spectrum data from the extended source was considered in this section. The spectral region of the incident light was set as $[{480\textrm{nm},960\textrm{nm}} ]$, and the CCD detector is a 16-bit monochrome camera with the resolution of $512 \times 512$ and the pixel size of $18{\mathrm{\mu} \mathrm{m}} \times 18{\mathrm{\mu} \mathrm{m}}$, together with each interferogram occupying $256 \times 256$ pixels. On basis of the Nyquist sampling theorem, the sampling interval of the interferogram is no more than $\delta \Delta \textrm{ = }{{{\lambda _{\min }}} / 2}\textrm{ = 0}\mathrm{.24\mu m}$ to avoid spectrum aliasing. The maximum OPD of the $SP$ was effectively limited by the Nyquist criterion, which required two data points at least per fringe period. Hence, the maximum OPD was ${\Delta _{\max }} = 128 \times \delta \Delta = 30.72{\mathrm{\mu} \mathrm{m}}$ if the interferogram was symmetrically recorded about the zero OPD. In the developed scheme, the thicknesses of two retarders ${R_1}$ and ${R_2}$ for the FPR were assumed as ${d_\alpha } = 3.5\textrm{mm}$ and ${d_\beta } = 7.0\textrm{mm}$, together with $|{{n_e} - {n_o}} |= 0.009$.
The input Stokes vector spectra for the target along with its modulated spectra were shown in Figs. 2 and 3, respectively, while the four-partition interferogram and point interferogram extracted in line were displayed in Fig. 4 and 5. Adopting the Fourier transformation with the hamming apodization, the reconstructed spectra of the Stokes parameters were demonstrated in Fig. 6, and the reconstructed data was in good agreement with the input data, mainly due to the fact that the noise was not taken into account in the simulation. However, the noise would cause deviations, especially at the edges of the interferogram where the signal was small. This result proved the performance principle of the FQRAISP scheme.
The spectral resolution of the Stokes vector spectra obtained by the FQRAISP was seven times higher than that of single-channel imaging spectropolarimeter. Not only the original spectral resolution of the imaging spectrometer was maintained, but it also integrated better the advantages of the imaging spectrometer and imaging polarimeter. In addition, the Stokes vector spectra recovered by the FQRAISP eliminated the spectral distortion caused by spectrum aliasing.
3.2 Four-partition modulation module calibration
In generally, the polarization direction of the polarizer ${P_1}$ can be determined by the linearly polarized light, while the fast axis directions of the retarders ${R_1}$ and ${R_2}$ are marked directly by the crystal manufacturer, with the accuracy only reaching the order of degree. The FPR is made of quartz crystal material, and its dispersion effect is very weak in the operating band of $[{480\textrm{nm},960\textrm{nm}} ]$. Hence, we considered the alignment errors, ${\varepsilon _1}$ and ${\varepsilon _2}$, of the fast axis directions of ${R_1}$ and ${R_2}$ with respect to the ideal circumstance, together with their phase delay errors ${\delta _1} = 2\pi \sigma |{{n_e} - {n_o}} |\delta {d_\alpha }$ and ${\delta _2} = 2\pi \sigma |{{n_e} - {n_o}} |\delta {d_\beta }$, where $\delta {d_\alpha }$ and $\delta {d_\beta }$ represented the thickness deviations of ${R_1}$ and ${R_2}$. By substituting the error terms into Eq. (4), the Mueller matrix of the FPR was rewritten as
Based on Eq. (6), four intensities measured by the FQRAISP became
Here
Similarly, the full Stokes parameters of the target were recovered by four modulation spectra ${S^{\prime\prime}_0}(1 )$ and ${S^{\prime\prime}_0}(2 )$ together with ${S^{\prime\prime}_0}(3 )$ and ${S^{\prime\prime}_0}(4 )$.
3.3 Stokes parameter errors due to the alignment deviation of the FPR together with its thickness deviation
In order to evaluate the influences of the alignment deviations of the FPR along with its thickness deviations on the reconstructed Stokes vector spectra, the simulation of a design example was carried out in detail. During the simulation process, two parameters were variable while other parameters fixed. Firstly, the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$ remained a constant, and three-dimensional surfaces of the absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ for the Stokes parameters changing with the alignment deviations ${\varepsilon _1}$ and ${\varepsilon _2}$ were shown in Figs. 7 –9. With the condition of $\delta {d_\alpha } = \delta {d_\beta } = 0\textrm{ mm}$, the distributions of the absolute errors for the Stokes parameters had strong regularity and little clutter, and the $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ values increased with the increasing ${\varepsilon _1}$ while decreasing with various ${\varepsilon _2}$. However, the variation of $\Delta {S_1}$ presented an opposite characteristic.
Fig. 7. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters as a function of the alignment deviations ${\varepsilon _1}$ and ${\varepsilon _2}$ under the condition of $\delta {d_\alpha } = \delta {d_\beta } = 0\textrm{ mm}$.
Fig. 8. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters as a function of the alignment deviations ${\varepsilon _1}$ and ${\varepsilon _2}$ under the condition of $\delta {d_\alpha } = \delta {d_\beta } = 0.5\textrm{ mm}$.
Fig. 9. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters as a function of the alignment deviations ${\varepsilon _1}$ and ${\varepsilon _2}$ under the condition of $\delta {d_\alpha } = \delta {d_\beta } = \textrm{ - }0.5\textrm{ mm}$.
On basis of Figs. 8, 9, the $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ values decreased with ${\varepsilon _1}$ increasing when ${\varepsilon _2}$ was fixed, while $\Delta {S_1}$ increased with various ${\varepsilon _1}$. However, for $\delta {d_\alpha } = \delta {d_\beta } = 0.5\textrm{ mm}$, $\Delta {S_0}$ and $\Delta {S_3}$ first decreased slowly, then increased and reached a maximum with the increasing of ${\varepsilon _2}$, while $\Delta {S_1}$ and $\Delta {S_2}$ exhibited an opposite behavior. For $\delta {d_\alpha } = \delta {d_\beta } ={-} 0.5\textrm{ mm}$, $\Delta {S_0}$ and $\Delta {S_2}$ showed a similar distribution, and they first decreased gradually then increased with the increasing of ${\varepsilon _2}$, while the $\Delta {S_1}$ value first increased slowly, afterward, tended stable then increased slowly. Meanwhile, the value of $\Delta {S_3}$ first increased then decreased with ${\varepsilon _2}$ increasing for ${\varepsilon _1} \in [{ - 2^\circ ,0.42^\circ } ]$, while increasing with the various ${\varepsilon _2}$ for $i \in ({0.42^\circ ,2^\circ } ]$. In addition, the rangeability of $\Delta {S_0}$ and $\Delta {S_1}$ together with $\Delta {S_2}$ and $\Delta {S_3}$ with ${\varepsilon _1}$ were larger than that with ${\varepsilon _2}$, namely, the Stokes parameter errors caused by the alignment deviation ${\varepsilon _1}$ were more obvious.
To further identify the effects of the alignment deviation ${\varepsilon _1}$ on the Stokes parameters ${S_0}$, ${S_1}$, ${S_2}$ and ${S_3}$, two-dimensional simulation was performed for the case of $\delta {d_\alpha } \ne \delta {d_\beta }$, where the numerical range of the coordinate was narrowed down to $[{ - 1^\circ ,1^\circ } ]$.
From Figs. 10 and 11, it is easy to find that, the values of $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ presented similar variation curves, and they increased linearly with ${\varepsilon _1}$ increasing no matter what the value of ${\varepsilon _2}$ was, while $\Delta {S_1}$ decreasing with various ${\varepsilon _1}$. The values of $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ appeared the highest jump for ${\varepsilon _2} ={-} 1^\circ $ and the lowest jump for ${\varepsilon _2} = 1^\circ $ when ${\varepsilon _1}$ was changed by $2^\circ $, but $\Delta {S_1}$ was just the opposite. With $\delta {d_\alpha } = 0.025\textrm{ mm}$ and $\delta {d_\beta } = 0.05\textrm{ mm}$, the range ability for $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ arrived at the maximums of 0.0243, 0.0203, 1.1086 and 0.3299, along with the minimums of 0.0153, 0.0147, 1.0409 and 0.3206. With $\delta {d_\alpha } = 0.05\textrm{ mm}$ and $\delta {d_\beta } = 0.025\textrm{ mm}$, the range ability of $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ reached the maximums 0.0279, 0.0251, 0.8651 and 0.3236, together with the minimums 0.0199, 0.0140, 0.8107 and 0.2912. Besides, the values of $\Delta {S_2}$ and $\Delta {S_3}$ had little changes with various ${\varepsilon _2}$ when ${\varepsilon _1}$ took a certain value.
Fig. 10. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the alignment deviation ${\varepsilon _1}$ when $\delta {d_\alpha } = 0.025\textrm{ mm}$ and $\delta {d_\beta } = 0.05\textrm{ mm}$.
Fig. 11. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the alignment deviation ${\varepsilon _1}$ when $\delta {d_\alpha } = 0.05\textrm{ mm}$ and $\delta {d_\beta } = 0.025\textrm{ mm}$.
Based on Fig. 12, it is seen obviously that, $\Delta {S_0}$ presented an overall downward while $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ showing a upward trend, and the range ability of $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ appeared the maximums 0.0167, 0.2100 and 0.0593 at ${\varepsilon _2} = 1^\circ $, together with the minimums of 0.0088, 0.0321 and 0.0204 at ${\varepsilon _2} ={-} 1^\circ $. However, the range ability of $\Delta {S_1}$ had the maximum 0.0234 at ${\varepsilon _2} ={-} 1^\circ $ and the minimum at ${\varepsilon _2} = 1^\circ $, respectively. By Fig. 13, $\Delta {S_0}$ and $\Delta {S_2}$ decreased while $\Delta {S_3}$ increased with the increasing of ${\varepsilon _1}$. If ${\varepsilon _2}$ took the values of $- 1^\circ $ and $- 0.5^\circ $, $\Delta {S_1}$ decreased with the increasing ${\varepsilon _1}$ while increased for ${\varepsilon _2}$ equal to $0.5^\circ $ and $1^\circ $. Moreover, the maximal and minimal range ability for $\Delta {S_0}$ and $\Delta {S_3}$ occurred at ${\varepsilon _2} ={\pm} 1^\circ $, and they were equal to 0.0091 and 0.0056 for $\Delta {S_0}$, together with 0.0154 and 0.008 for $\Delta {S_3}$. The range ability of $\Delta {S_1}$ and $\Delta {S_2}$ had the maximum 0.0074 and 0.1075 at ${\varepsilon _2} ={-} 1^\circ $, and the minimum 0.0031 and 0.0764 at ${\varepsilon _2} = 0.5^\circ $ and ${\varepsilon _2} = 1^\circ $, respectively.
Fig. 12. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the alignment deviation ${\varepsilon _1}$ when $\delta {d_\alpha } = 0\textrm{ mm}$ and $\delta {d_\beta } = 0.025\textrm{ mm}$.
Fig. 13. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the alignment deviation ${\varepsilon _1}$ when $\delta {d_\alpha } = 0.025\textrm{ mm}$ and $\delta {d_\beta } = 0\textrm{ mm}$.
Secondly, the influences of the thickness deviation of the phase delay on the Stokes parameters were concerned, and the distribution functions of the Stokes parameters errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ with the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$, for the selected ${\varepsilon _1} = {\varepsilon _2} = 0^\circ $ and ${\varepsilon _1} = {\varepsilon _2} ={\pm} 1^\circ $, were displayed in Figs. 14 –16. Where the horizontal and vertical axes represented the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$, respectively, while the color distribution stood for the values of $\Delta {S_0}$ and $\Delta {S_1}$ together with $\Delta {S_2}$ and $\Delta {S_3}$. With ${\varepsilon _1} = {\varepsilon _2} = 0^\circ $, the variations of $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ behaved as a rolling mountain, and periodically distributed with respect to $\delta {d_\alpha }$ and $\delta {d_\beta }$, where $\Delta {S_0}$ and $\Delta {S_2}$ exhibited the similar behaviors while $\Delta {S_3}$ just the opposite. However, the $\Delta {S_1}$ variation was even symmetric to the axis $({\delta {d_\alpha } = 0\textrm{ mm,}\delta {d_\beta } \in [{ - 0.06\textrm{mm},0.06\textrm{mm}} ]} )$, and it increased with the increasing $\delta {d_\alpha }$ for $\delta {d_\alpha } \in [{ - 0.0350\textrm{mm},0\textrm{mm}} )$ and decreased for $\delta {d_\alpha } \in [{0\textrm{mm},\textrm{ }0.0350\textrm{mm }} )$. In one period, the values of $\Delta {S_0}$ and $\Delta {S_2}$ first decreased then increased with the increasing $|{\delta {d_\alpha }} |$, afterward, rising again for the top part, but they were just the opposite for the bottom part when $\delta {d_\beta }$ remains unchanged. If $\delta {d_\alpha }$ was a constant, the values of $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ were an even function as $\delta {d_\beta }$, and the small $\Delta {S_0}$ was mainly concentrated in a range of $\delta {d_\alpha } \in [{ - 0.0340\textrm{mm}, - 0.0207\textrm{mm}} ]$ and $\delta {d_\beta } \in [{0.0059\textrm{mm},0.0209\textrm{mm}} ]$, and its maximum was less than 0.0142. Similarly, the small $\Delta {S_2}$ and $\Delta {S_3}$ occurred at the positions $\delta {d_\alpha } \in [{ - 0.0600\textrm{mm}, - 0.0358\textrm{mm}} ]$ and $\delta {d_\beta } \in [{0.0123\textrm{mm},0.0276\textrm{mm}} ]$ together with $\delta {d_\alpha } \in [{0.0516\textrm{mm},0.0596\textrm{mm}} ]$ and $\delta {d_\beta } \in [{ - 0.0247\textrm{mm}, - 0.0071\textrm{mm}} ]$. That is to say the absolute errors of the Stokes parameters ${S_0}$, ${S_2}$ and ${S_3}$ were less affected by the thickness deviations when the values of $\delta {d_\alpha }$ and $\delta {d_\beta }$ were in the above ranges.
Fig. 14. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters as a function of the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$ under the condition of ${\varepsilon _1} = {\varepsilon _2} = 0^\circ$.
Fig. 15. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters as a function of the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$ under the condition of ${\varepsilon _1} = {\varepsilon _2} = 1^\circ$.
Fig. 16. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters as a function of the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$ under the condition of ${\varepsilon _1} = {\varepsilon _2} = \textrm{ - }1^\circ$.
According to Figs. 15, 16, the variations of $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ presented an odd-symmetry distributes with respect to $\delta {d_\alpha }$ while $\Delta {S_1}$ were even symmetrical. For one period of $\delta {d_\alpha } \in [{ - 0.03\textrm{mm},0.03\textrm{mm}} ]$, initially, the values of $\Delta {S_0}$ and $\Delta {S_2}$ decreased slowly, then increased gradually with the increasing $\delta {d_\alpha }$, and dropped again for the top part, while their variation behaviors just the opposite for the bottom part for the constant $\delta {d_\beta }$. However, if $\delta {d_\alpha }$ remained unchanged, the $\Delta {S_0}$ value at the top part increased with $|{\delta {d_\beta }} |$ increasing and its values at the bottom part decreased when $\delta {d_\beta } \in [{ - 0.02\textrm{mm},0.04\textrm{mm}} ]$. With ${\varepsilon _1} = {\varepsilon _2} = 1^\circ$, the $\Delta {S_2}$ value first decreased then increased with the increasing of $|{\delta {d_\beta }} |$ for the top part, together with it firstly rising then dropping for the bottom part if $\delta {d_\beta } \in [{ - 0.01\textrm{mm},0.05\textrm{mm}} ]$. With ${\varepsilon _1} = {\varepsilon _2} ={-} 1^\circ$, the $\Delta {S_2}$ value increased with the increasing of $|{\delta {d_\beta }} |$ for the top part and decreased for the bottom part.
The following analysis and discussion were concerned with the effect of the thickness deviation $\delta {d_\alpha }$ on the Stokes parameter errors, and two-dimensional plots of $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ with $\delta {d_\alpha }$ were depicted in Figs. 17 –20 for the cases of ${\varepsilon _1} \ne {\varepsilon _2}$. $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ as a function of $\delta {d_\alpha }$ showed similar behavior for different values of ${\varepsilon _1}$ and ${\varepsilon _2}$, and the $\Delta {S_0}$, $\Delta {S_2}$ and $\Delta {S_3}$ plots had one intersection for various $\delta {d_\beta }$, respectively, while there were two intersections for $\Delta {S_1}$. From Figs. 17, 18, it is easy to find that, with the same $\delta {d_\alpha }$ value, the $\Delta {S_0}$ and $\Delta {S_2}$ values reached a maximum for $\delta {d_\beta } ={-} 0.05\textrm{ mm}$ and a minimum for $\delta {d_\beta } = 0.05\textrm{ mm}$ on the left of the intersection, while $|{\Delta {S_0}} |$ had little changes for various $\delta {d_\beta }$ on the right of the intersection together with $\Delta {S_2}$ presenting the opposite characteristic. The $\Delta {S_3}$ value had a maximum at $\delta {d_\beta } ={-} 0.05\textrm{ mm}$ and $\delta {d_\beta } = 0.025\textrm{ mm}$ together with a minimum at $\delta {d_\beta } = 0.05\textrm{ mm}$ for the left of the intersection, while it had the maximum of 1.1993 at $\delta {d_\beta } = 0.05\textrm{ mm}$ and the minimum of -1.2675 at $\delta {d_\beta } ={-} 0.05\textrm{ mm}$ for the right of the intersection. In addition, the maximums of $\Delta {S_1}$ appeared in the positions $\left\{ \begin{array}{l} \delta {d_\alpha } ={-} 0.02\textrm{ mm}\\ \delta {d_\beta } = 0.05\textrm{ mm} \end{array} \right.$ and $\left\{ \begin{array}{l} \delta {d_\alpha } = 0.02\textrm{ mm}\\ \delta {d_\beta } ={-} 0.05\textrm{ mm} \end{array} \right.$, and they were equal to -1.1368 and -1.0424, while its minimums of -1.6093 and -1.6838 occurred at $\left\{ \begin{array}{l} \delta {d_\alpha } ={-} 0.02\textrm{ mm}\\ \delta {d_\beta } = 0.025\textrm{ mm} \end{array} \right.$ and $\left\{ \begin{array}{l} \delta {d_\alpha } = 0.02\textrm{ mm}\\ \delta {d_\beta } ={-} 0.025\textrm{ mm} \end{array} \right.$, respectively.
Fig. 17. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the thickness deviation $\delta {d_\alpha }$ when ${\varepsilon _1} = 0.5^\circ$ and ${\varepsilon _2} = 1^\circ$.
Fig. 18. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the thickness deviation $\delta {d_\alpha }$ when ${\varepsilon _1} = 1^\circ$ and ${\varepsilon _2} = 0.5^\circ$.
Fig. 19. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the thickness deviation $\delta {d_\alpha }$ when ${\varepsilon _1} = 0^\circ$ and ${\varepsilon _2} = 0.5^\circ$.
Fig. 20. Absolute errors $\Delta {S_0}$, $\Delta {S_1}$, $\Delta {S_2}$ and $\Delta {S_3}$ of the Stokes parameters changing with the thickness deviation $\delta {d_\alpha }$ when ${\varepsilon _1} = 0.5^\circ$ and ${\varepsilon _2} = 0^\circ$.
On basis of Figs. 19, 20, $\Delta {S_0}$ and $\Delta {S_2}$ had the maximums at $\delta {d_\beta } ={-} 0.05\textrm{ mm}$ and the minimums at $\delta {d_\beta } = 0.05\textrm{ mm}$ for the left of the intersection, while the $|{\Delta {S_0}} |$ values were hardly affected by $\delta {d_\beta }$ for the right of the intersection. With the condition of ${\varepsilon _1} = 0^\circ$ and ${\varepsilon _2} = 0.5^\circ$, the maximum and minimum of $\Delta {S_3}$ occurred at $\delta {d_\beta } = \left\{ \begin{array}{l} - 0.05\textrm{ mm}\\ 0.025\textrm{ mm} \end{array} \right.$ and $\delta {d_\beta } = 0.5\textrm{ mm}$ when $\delta {d_\alpha } \in [{ - 0.02\textrm{mm}, - 0.01\textrm{mm}} )$, but they appeared in $\delta {d_\beta } ={\pm} 0.05\textrm{ mm}$, respectively, for $\delta {d_\alpha } \in [{ - 0.01\textrm{mm},0.02\textrm{mm}} ]$. Moreover, $\Delta {S_1}$ arrived at the maximums of -1.1460 and -1.0512 at the positions $\left\{ \begin{array}{l} \delta {d_\alpha } ={-} 0.02\textrm{ mm}\\ \delta {d_\beta } = 0.05\textrm{ mm} \end{array} \right.$ and $\left\{ \begin{array}{l} \delta {d_\alpha } = 0.02\textrm{ mm}\\ \delta {d_\beta } ={-} 0.05\textrm{ mm} \end{array} \right.$, while it had the minimums of -1.6047 and -1.2671 at $\left\{ \begin{array}{l} \delta {d_\alpha } ={-} 0.02\textrm{ mm}\\ \delta {d_\beta } = 0.025\textrm{ mm} \end{array} \right.$ and $\left\{ \begin{array}{l} \delta {d_\alpha } = 0.02\textrm{ mm}\\ \delta {d_\beta } ={-} 0.025\textrm{ mm} \end{array} \right.$. With the condition of ${\varepsilon _1} = 0.5^\circ$ and ${\varepsilon _2} = 0^\circ$, the maximum and minimum for $\Delta {S_1}$ and $\Delta {S_3}$ occurred at the same positons as the above analysis, and two maximums were equal to -1.1372 and -1.0425 while two minimums equal to -1.6096 and -1.6843. To sum up, the alignment deviations had a relatively weak effect on the reconstructed Stokes vector spectra, and the condition of ${\varepsilon _1} \in [{ - 0.43^\circ , + 0.43^\circ } ]$ and ${\varepsilon _2} \in [{ - 0.22^\circ , + 0.22^\circ } ]$ should be satisfied to ensure the reconstruction accuracy of the incident Stokes vector spectra within 5%. However, the thickness deviations of the phase delay had an obvious influence on the reconstructed Stokes vector spectra, so the thickness deviations were established $\delta {d_\alpha } \in [{ - 0.03{\mathrm{\mu} \mathrm{m}}, + 0.03{\mathrm{\mu} \mathrm{m}}} ]$ and $\delta {d_\beta } \in [{ - 0.03{\mathrm{\mu} \mathrm{m}}, + 0.03{\mathrm{\mu} \mathrm{m}}} ]$ for the reconstructed Stokes vector spectra with high precision.
4. Conclusion
In this paper, the FQRAISP was proposed to realize the real time and no error acquisition of the incident Stokes vector spectra. The polarization demodulation for the FQRAISP did not require the channel filtering, and it could measure the incident spectra with a narrow-band, overcoming the problem of the spectral aliasing. The feasibility of the FQRAISP scheme was validated by theoretical deduction and experimental simulation, and the spectral resolution of the Stokes vector spectrum recovered by the FQRAISP was seven times higher than that by single-channel imaging spectropolarimeter. The exact expressions of the incident Stokes parameters changing with the alignment deviations ${\varepsilon _1}$ and ${\varepsilon _2}$ together with the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$ were given, and the Stokes parameter errors due to the alignment deviations of the FPR along with its thickness deviations were analyzed and discussed in detail. The range ability of the Stokes parameter errors with the increasing ${\varepsilon _1}$ were larger than those with various ${\varepsilon _2}$, and the thickness deviations $\delta {d_\alpha }$ and $\delta {d_\beta }$ had remarkable influences on the reconstructed Stokes vector spectra. Hence, the alignment deviations of the FPR should meet the conditions of ${\varepsilon _1} \in [{ - 0.43^\circ , + 0.43^\circ } ]$ and ${\varepsilon _2} \in [{ - 0.22^\circ ,\textrm{ + }0.22^\circ } ]$, along with its thickness deviations $\delta {d_\alpha } \in [{ - 0.03{\mathrm{\mu} \mathrm{m}},\textrm{ + }0.03{\mathrm{\mu} \mathrm{m}}} ]$ and $\delta {d_\beta } \in [{ - 0.03{\mathrm{\mu} \mathrm{m}},\textrm{ + }0.03{\mathrm{\mu} \mathrm{m}}} ]$ for the engineering purpose.
Funding
Natural Science Foundation of Shandong Province (ZR2022MF266); National Natural Science Foundation of China (11704213, 61605098).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data was generated in the presented research.
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