Open AccessArticle

Four-Channel Polarimetric-Spectral Intensity Modulation Imager

Jian Bo

^1,*,

Xueping Ju

¹ and

Changxiang Yan

^1,2

Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, China

Center of Materials Science and Optoelectrics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11759; https://doi.org/10.3390/app142411759

Submission received: 7 November 2024 / Revised: 4 December 2024 / Accepted: 6 December 2024 / Published: 17 December 2024

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Abstract

To solve the problems of channel crosstalk and edge jitter caused by the Fourier transform demodulation of polarimetric-spectral intensity modulation in polarization spectral data, this paper proposes a Four-Channel Polarimetric Spectrometer (FCPS) with two groups of polarimetric-spectral intensity modulation (PSIM). FCPS can demodulate the full Stokes spectra information by system matrix calibration in the spatial domain. The traditional channel filtering method and the FCPS data demodulation method are simulated, and their results are compared. The simulated results show that the FCPS does not have the problem of the edge jitter, and the demodulation accuracy is higher. It is confirmed that the angle error of phase retarders has little influence on the data reconstruction, and the maximum allowable angle error of the calibration light linear polarizer cannot exceed 0.4°.

Keywords:

polarization modulation; spectropolarimeter instrument; Stokes

1. Introduction

With the increasing demand for optical remote sensing information acquisition, many researchers are deeply studying polarization spectral imaging technology and the technique is widely used in many fields, including atmospheric aerosol characterization [1,2], remote sensing [3,4,5,6], and material characterization [7,8,9,10,11,12]. These fields require an imaging spectropolarimeter to simultaneously measure the spatial, spectral, and polarization information of a scene.

Polarimetric-spectral intensity modulation (PSIM) imaging is one of these polarization spectral imaging technologies. It measures the polarization information of the target by inserting an intensity modulation module into the traditional spectrometer system. PSIM was first proposed by Nordsieck and OKA [13,14]. The system has a simple structure, and there are no moving parts in it. In order to detect polarization spectra, Iannarilli combined PSIM with a grating or dispersion prism [15] which can acquire the spectrometer and polarimeter of a target [15]. Zhang et al. combined the Fourier transform spectrometer with PSIM to obtain a Fourier transform spectral imaging polarizer [7]. Computed tomography imaging spectropolarimeter was proposed by Vandervlugt et al., which can simultaneously obtain information from a Stokes spec, polarimeter, and image [16]. Ren et al. designed a PSIM polarization measurement method of coded aperture based on intensity-modulated, coded aperture measurement [17], which comprises spectral polarization intensity modulation and a compressive sensing spectral imaging system. Mu et al. analyzed the angle error of a channeled spectropolarimeter [18]. Zhang et al. proposed tempo-spatially modulated imaging spectropolarimetry based on a polarization modulation array in order to realize the demodulation of target polarization spectral information [19].

The polarization spectrum instrument based on PSIM needs to recover the polarization spectrum data by Fourier transform channel filtering. It is necessary to calibrate the phase delay of the instrument for high precision polarization spectral data demodulation. Mu used 22.5° and 45° linearly polarized beams to calibrate this. Yang used auxiliary waveplates to calibrate this instead [20,21]. However, PSIM applies the channel filtering method to Stokes data demodulation, which results in a limited Stokes spectral resolution and a crosstalk between channels [22,23,24].

Li et al. combined the PSIM with two orthogonal optical axes using a Wollaston prism, and obtained two orthogonal interferograms on the detector. By processing the positive interferograms, the space range between each channel was enlarged, channel crosstalk was avoided, and spectral resolution was improved. However, the data demodulation still uses Fourier transform, which has the problem of edge jitter [25]. Li et al. combined four PSIMs with the Savart polarizer. The four PSIMs have different thicknesses, resulting in different phase delays. Four groups of interferogram are obtained on the detector, and the data demodulation is realized by the spatial processing of four interferograms. However, it has high requirements for PSIM thickness [26]. Zhao proposes an efficient calibration method for the spectral modulation transfer function via channel shifting in a channeled spectropolarimeter using PSIM [27]. Huang proposes a reconstruction method, incorporating the complete physical model into a traditional deep neural network for a channeled spectropolarimeter [28].

In this paper, a Four-Channel Polarimetric Spectrometer (FCPS) is proposed for the simultaneous measurement of the target spectrum and polarization states based on PSIM. To measure the modulated spectra, this paper proposes a spatial data demodulation method that can demodulate Stokes vector information using a calibrated system matrix. Spatial data demodulation can avoid the influence of channel crosstalk.

This paper is organized as follows: Section 2 shows the optical layout and theoretical model of the system, Section 3 carries out a system matrix calibration method and a mathematical simulation to verify the effectiveness of the scheme. Section 4 discusses the angle error range of calibrating light. Our conclusions are set out in Section 5.

2. Optical Layout and Principle

The optical layout of FCPS is depicted in Figure 1. L1, S, and L2 denote the objective lens, field stop, and collimating lens, respectively. R1, R2, R3, and R4 denote the phase retarders used to modulate incident light, and they are placed behind the collimating lens, where the thickness ratio of R1 and R2 is 1:2 and that of R3 and R4 is 2:1. The fast axes of R1 and R3 are 0°, and that of R2 and R4 are 45°. Linear polarizers P1, P2, P3, and P4 are placed behind the phase retarder so that the fast axes of P1, P2, P3, and P4 are 0°, 90°, 45°, and 0°, respectively. Following the linear polarizer is a dispersion prism for light splitting. L denotes an imaging lens, and the CCD is placed on the back focal plane of L.

The light from a scene is collected and collimated by the front telescopic system (L1, S, and L2), and then it passes through phase retarders R1, R2, R3, and R4. Next, the light is modulated and then passed through P1, P2, P3, and P4, respectively. In this way, four beams of modulated light can be obtained, and they are imaged on the CCD after being dispersed through the dispersion prism. Finally, four modulation curves can be acquired on the CCD.

According to the definition of the Stokes parameters as expressed Equation (1), six polarimetric intensities of a scene should be measured for the determination of full Stokes parameters.

S (λ) = [\begin{matrix} S_{0} (λ) \\ S_{1} (λ) \\ S_{2} (λ) \\ S_{3} (λ) \end{matrix}] = [\begin{matrix} I_{0} (λ) + I_{90} (λ) \\ I_{0} (λ) - I_{90} (λ) \\ I_{45} (λ) - I_{135} (λ) \\ I_{R} (λ) - I_{L} (λ) \end{matrix}]

(1)

where

S_{0}

and

S_{1}

present the total power of the light and the preference for the linear 0° over 90° polarization,

S_{2}

indicates the difference between the linear 45° and 135° polarization, while

S_{3}

for the right circular over left circular polarization states. The Muller matrix of phase retarder and linear polarizer is shown in Equations (2) and (3).

M_{R} (φ (λ), α) = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos^{2} (2 α) + \cos (φ (λ)) \sin^{2} (2 α) & (1 - \cos (φ (λ))) \sin (2 α) \cos (2 α) & - \sin (φ (λ)) \sin (2 α) \\ 0 & (1 - \cos (φ (λ))) \sin (2 α) \cos (2 α) & \sin^{2} (2 α) + \cos (φ (λ)) \cos^{2} (2 α) & \sin (φ (λ)) \cos (2 α) \\ 0 & \sin (φ (λ)) \sin (2 α) & - \sin (φ (λ)) \cos (2 α) & \cos (φ (λ)) \end{matrix}]

(2)

M_{P} (β) = [\begin{matrix} 1 & \cos (2 β) & \sin (2 β) & 0 \\ \cos (2 β) & \cos^{2} (2 β) & \sin (2 β) \cos (2 β) & 0 \\ \sin (2 β) & \sin (2 β) \cos (2 β) & \sin^{2} (2 β) & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

(3)

In Equation (2),

α

is the fast axis direction of the phase retarder and

φ (λ)

is the phase delay generated by the phase retarder. The

φ (λ)

can be expressed as

φ = \frac{2 π}{λ} Δ n d

where

Δ n

and

d

is the birefringence and thickness of the phase retarder. In Equation (3),

β

is the direction of the linear polarizer.

In FCPS, we defined that d is the thickness of R2 and R3 and 2d is the thickness of R1 and R4. Therefore, the phase delay generated by R2 and R3 is

φ_{1} (λ) = \frac{2 π}{λ} Δ n d

and the phase delay generated by R1 and R4 is

φ_{2} (λ) = \frac{4 π}{λ} Δ n d

. We let

φ (λ) = φ_{1} (λ) = \frac{1}{2} φ_{2} (λ)

. Through R1, R2, R3, and R4 and the polarizers P1, P2, P3, and P4, the output Stokes vector can be expressed as Equations (4) to (7):

I_{2 : 1}^{0} (λ) = M_{P 4} M_{R 2} M_{R 1} S_{i n} (λ)

(4)

I_{1 : 2}^{90} (λ) = M_{P 3} M_{R 4} M_{R 3} S_{i n} (λ)

(5)

I_{1 : 2}^{45} (λ) = M_{P 2} M_{R 4} M_{R 3} S_{i n} (λ)

(6)

I_{1 : 2}^{0} (λ) = M_{P 1} M_{R 4} M_{R 3} S_{i n} (λ)

(7)

where

S_{i n}

denotes the input Stokes.

M_{P i}

represents the Muller matrix of linear polarizers P1, P2, P3, and P4, and

M_{R i}

denotes the Muller matrix of phase retarders R1, R2, R3, and R4. Equations (3) and (2) into Equations (4) to (7) yields the following:

I_{2 : 1}^{0} (λ) = \frac{1}{2} S_{0} (λ) + \frac{1}{2} S_{1} (λ) \cos (φ (λ)) + \frac{1}{2} S_{2} (λ) \sin (φ (λ)) \sin (2 φ (λ)) - \frac{1}{2} S_{3} (λ) \sin (φ (λ)) \cos (2 φ (λ))

(8)

I_{1 : 2}^{90} (λ) = \frac{1}{2} S_{0} (λ) - \frac{1}{2} S_{1} (λ) \cos (2 φ (λ)) - \frac{1}{2} S_{2} (λ) \sin (2 φ (λ)) \sin (φ (λ)) + \frac{1}{2} S_{3} (λ) \sin (2 φ (λ)) \cos (φ (λ))

(9)

I_{1 : 2}^{45} (λ) = \frac{1}{2} S_{0} (λ) + \frac{1}{2} S_{2} (λ) \cos (φ (λ)) + \frac{1}{2} S_{3} (λ) \sin (φ (λ))

(10)

I_{1 : 2}^{0} (λ) = \frac{1}{2} S_{0} (λ) + \frac{1}{2} S_{1} (λ) \cos (2 φ (λ)) + \frac{1}{2} S_{2} (λ) \sin (2 φ (λ)) \sin (φ (λ)) - \frac{1}{2} S_{3} (λ) \sin (2 φ (λ)) \cos (φ (λ))

(11)

The upper right part of

I

represents the angle of the linear polarizer and the lower right part of

I

represents the thickness ratio of the phase retarders. Equations (4) to (7) can change to Equation (12):

I = P S_{i n}

(12)

P (λ) = [\begin{matrix} \frac{1}{2} & \frac{1}{2} \cos (2 φ (λ)) & \frac{1}{2} \sin (2 φ (λ)) \sin (φ (λ)) & - \frac{1}{2} \sin (2 φ (λ)) \cos (φ (λ)) \\ \frac{1}{2} & - \frac{1}{2} \cos (2 φ (λ)) & - \frac{1}{2} \sin (2 φ (λ)) \sin (φ (λ)) & \frac{1}{2} \sin (2 φ (λ)) \cos (φ (λ)) \\ \frac{1}{2} & 0 & \frac{1}{2} \cos (φ (λ)) & \frac{1}{2} \sin (φ (λ)) \\ \frac{1}{2} & \frac{1}{2} \cos (φ (λ)) & \frac{1}{2} \sin (φ (λ)) \sin (2 φ (λ)) & - \frac{1}{2} \sin (φ (λ)) \cos (2 φ (λ)) \end{matrix}]

(13)

P is referred to as the system matrix. Since P has a full rank, the full Stokes can be solved as follows:

P {(λ)}^{- 1} [\begin{matrix} S_{o u t}^{1} (λ) \\ S_{o u t}^{2} (λ) \\ S_{o u t}^{3} (λ) \\ S_{o u t}^{4} (λ) \end{matrix}] = [\begin{matrix} S_{0} (λ) \\ S_{1} (λ) \\ S_{2} (λ) \\ S_{3} (λ) \end{matrix}]

(14)

By Equation (14), Stokes in space can be demodulated while avoiding channel crosstalk and resolution reduction.

3. Effectiveness Validation for the FCPS Scheme

In this section, numerical simulation is conducted to demonstrate the feasibility of the FCPS system. The simulation is based on the theory proposed in Section 2. In this simulation, the related parameters are set as follows: the spectral measurement range of the system is 470 nm to 900 nm; the thickness of retarders are 1.8 mm and 3.6 mm, and the retarder is made of quartz; and the polarization spectral resolution is 0.2 nm.

3.1. System Matrix Calibration

Calibrating the system matrix P is the key to realizing data demodulation in the FCPS scheme. By the system parameter of the FCPS scheme, the system matrix P can be obtained, but there are installation angle errors which will cause the system matrix P to vary. Therefore, it is necessary to calibrate the demodulation matrix to obtain high-precision demodulation data.

In the calibration process, it is assumed that the angle of the phase retarder, the thickness of the phase retarder, and the angle of the line analyzer are not known. Therefore, when the polarized light passes through the phase retarder and line polarizer, the modulated beams can be expressed as follows:

\begin{matrix} I (λ) & = \frac{1}{2} S_{0} (λ) \\ + \{\begin{matrix} [\cos (2 β) [\cos^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \sin^{2} (2 α_{2})] + \sin (2 β) [(1 - \cos (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 α_{2})]] [\cos (2 α_{1}) + \cos (φ_{1} (λ)) \sin^{2} (2 α_{1})] \\ + [\cos 2 β (1 - \cos (φ_{2} (λ))) \sin (2 α_{2}) \cos (2 α_{2}) + \sin (2 β) (\sin^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \cos^{2} (2 α_{2})] [(1 - \cos φ_{1} (λ)) \sin (2 α_{1}) \cos (2 α_{1})] \\ + [- \sin (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 β) + \sin (2 β) \sin (φ_{2} (λ)) \cos (2 α_{2})] [\sin φ_{1} (λ) \sin (2 α_{1})] \end{matrix}\} S_{1} (λ) \\ + \{\begin{array}{l} [\cos (2 β) [\cos^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \sin^{2} (2 α_{2})] + \sin (2 β) [(1 - \cos (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 α_{2})]] [(1 - \cos φ_{1} (λ)) \sin (2 α_{1}) \cos (2 α_{1})] \\ + [\cos 2 β (1 - \cos (φ_{2} (λ))) \sin (2 α_{2}) \cos (2 α_{2}) + \sin (2 β) (\sin^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \cos^{2} (2 α_{2})] [\sin^{2} (2 α_{1}) + \cos (φ_{1} (λ)) \cos^{2} (2 α_{1})] \\ + [- \sin (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 β) + \sin (2 β) \sin (φ_{2} (λ)) \cos (2 α_{2})] [- \sin φ_{1} (λ) \sin (2 α_{1})] \end{array}\} S_{2} (λ) \\ + \{\begin{array}{l} [\cos (2 β) [\cos^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \sin^{2} (2 α_{2})] + \sin (2 β) [(1 - \cos (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 α_{2})]] [- \sin φ_{1} (λ) \sin (2 α_{1})] \\ + [\cos 2 β (1 - \cos (φ_{2} (λ))) \sin (2 α_{2}) \cos (2 α_{2}) + \sin (2 β) (\sin^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \cos^{2} (2 α_{2})] [\sin φ_{1} (λ) \cos (2 α_{1})] \\ + [- \sin (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 β) + \sin (2 β) \sin (φ_{2} (λ)) \cos (2 α_{2})] [\cos φ_{1} (λ)] \end{array}\} S_{3} (λ) \end{matrix}

(15)

where

β

is the direction of the linear polarizer,

α_{1}

and

α_{2}

are the fast axis direction of the phase retarders,

φ_{1}

and

φ_{2}

are the phase delay generated by the phase retarders. Equation (15) can be rewritten as

I = E (λ) S_{0} (λ) + B (λ) S_{1} (λ) + C (λ) S_{2} (λ) + D (λ) S_{3} (λ)

(16)

where

E (λ) = \frac{1}{2}

(17)

\begin{matrix} B (λ) & = [\cos (2 β) [\cos^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \sin^{2} (2 α_{2})] + \sin (2 β) [(1 - \cos (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 α_{2})]] [\cos (2 α_{1}) + \cos (φ_{1} (λ)) \sin^{2} (2 α_{1})] \\ + [\cos 2 β (1 - \cos (φ_{2} (λ))) \sin (2 α_{2}) \cos (2 α_{2}) + \sin (2 β) (\sin^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \cos^{2} (2 α_{2})] [(1 - \cos φ_{1} (λ)) \sin (2 α_{1}) \cos (2 α_{1})] \\ + [- \sin (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 β) + \sin (2 β) \sin (φ_{2} (λ)) \cos (2 α_{2})] [\sin φ_{1} (λ) \sin (2 α_{1})] \end{matrix}

(18)

\begin{matrix} C (λ) & = [\cos (2 β) [\cos^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \sin^{2} (2 α_{2})] + \sin (2 β) [(1 - \cos (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 α_{2})]] [(1 - \cos φ_{1} (λ)) \sin (2 α_{1}) \cos (2 α_{1})] \\ + [\cos 2 β (1 - \cos (φ_{2} (λ))) \sin (2 α_{2}) \cos (2 α_{2}) + \sin (2 β) (\sin^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \cos^{2} (2 α_{2})] [\sin^{2} (2 α_{1}) + \cos (φ_{1} (λ)) \cos^{2} (2 α_{1})] \\ + [- \sin (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 β) + \sin (2 β) \sin (φ_{2} (λ)) \cos (2 α_{2})] [- \sin φ_{1} (λ) \sin (2 α_{1})] \end{matrix}

(19)

\begin{matrix} D (λ) & = [\cos (2 β) [\cos^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \sin^{2} (2 α_{2})] + \sin (2 β) [(1 - \cos (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 α_{2})]] [- \sin φ_{1} (λ) \sin (2 α_{1})] \\ + [\cos 2 β (1 - \cos (φ_{2} (λ))) \sin (2 α_{2}) \cos (2 α_{2}) + \sin (2 β) (\sin^{2} (2 α_{2}) + \cos (φ_{2} (λ)) \cos^{2} (2 α_{2})] [\sin φ_{1} (λ) \cos (2 α_{1})] \\ + [- \sin (φ_{2} (λ)) \sin (2 α_{2}) \cos (2 β) + \sin (2 β) \sin (φ_{2} (λ)) \cos (2 α_{2})] [\cos φ_{1} (λ)] \end{matrix}

(20)

where

β

denotes the direction of the linear polarizer;

α_{1}

and

α_{2}

denote the fast axis direction of the phase retarders;

φ_{1}

and

φ_{2}

denote the phase delay generated by the phase retarders.

There are four channels in the FCPS scheme, so if B, C, and D of each channel are marked out, the system matrix can be obtained. Let 0° linearly polarized light, 45° linearly polarized light, and left-handed circularly polarized light pass through each channel, respectively. According to Equation (1), we have

S_{0}^{0} (λ) = S_{1}^{0} (λ), S_{2}^{0} (λ) = S_{3}^{0} (λ) = 0

(21)

S_{0}^{45} (λ) = S_{2}^{45} (λ), S_{1}^{45} (λ) = S_{3}^{45} (λ) = 0

(22)

S_{0}^{L} (λ) = - S_{3}^{L} (λ), S_{1}^{L} (λ) = S_{2}^{L} (λ) = 0

(23)

Then, Equation (16) becomes the following:

I_{i}^{0} (λ) = \frac{1}{2} S_{0}^{0} (λ) + B (λ) S_{1}^{0} (λ)

(24)

I_{i}^{45} (λ) = \frac{1}{2} S_{0}^{45} (λ) + C (λ) S_{2}^{45} (λ)

(25)

I_{i}^{L} (λ) = \frac{1}{2} S_{0}^{L} (λ) + D (λ) S_{3}^{L} (λ)

(26)

According to Equations (24) to (26), we have

B (λ) = (I_{i}^{0} (λ) - \frac{1}{2} S_{0}^{0} (λ)) / S_{1}^{0} (λ)

(27)

C (λ) = (I_{i}^{45} (λ) - \frac{1}{2} S_{0}^{45} (λ)) / S_{2}^{45} (λ)

(28)

D (λ) = (I_{i}^{L} (λ) + \frac{1}{2} S_{0}^{L} (λ)) / S_{3}^{L} (λ)

(29)

Through the above method, the values of B, C, and D of every channel can be determined, and based on this, the system matrix can be calibrated. The values of B, C, and D of different wavelengths and different channels are different. The system matrix of each wavelength is composed of the B, C, and D of different linear polaroid angles and the same wavelength, as shown in Equation (30):

A (λ) = [\begin{matrix} \frac{1}{2} & B_{1} (λ) & C_{1} (λ) & D_{1} (λ) \\ \frac{1}{2} & B_{2} (λ) & C_{2} (λ) & D_{2} (λ) \\ \frac{1}{2} & B_{3} (λ) & C_{3} (λ) & D_{3} (λ) \\ \frac{1}{2} & B_{4} (λ) & C_{4} (λ) & D_{4} (λ) \end{matrix}]

(30)

where the subscripts of B, C, and D represent different channels.

3.2. Matrix Rank Calculation

According to Section 2, we learned that the system matrix must be full of rank to solve Equation (13). Therefore, before system feasibility simulation verification, we first verify that the system matrix is always full rank. When there no errors in the FCPS, the rank of the system matrix is 4, as shown in Figure 2. In Figure 2, the x-coordinate represents the number of system matrix, and the y-coordinate represents the rank of system matrices. The number of system matrices is the same as the number of bands.

When there is a ± 3° angle error for the first phase retarder and there is no angle error for the second phase retarder, the rank of the system matrix is revealed, as in Figure 3. It can be seen from Figure 3 that the system matrix’s rank is still 4. When the second phase retarder has an angle error of ±3° and the first phase retarder has no angle error, the rank of the system matrix is revealed, as in Figure 4. The rank of the system matrix also stays the same at 4. When both phase retarders have angle errors of ±3°, the rank of the system matrix is shown in Figure 5. The system matrix’s value is still 4. It can be seen from Figure 3, Figure 4 and Figure 5 that when the phase retarder has angle errors, the rank of the system matrix remains unchanged and is always 4. Therefore, as long as we calibrate the system matrix after installing and adjusting FCPS, the demodulation of Stokes data can be completed.

3.3. The Effectiveness of the FCPS

In this section, the following simulations are carried out to prove the effectiveness of the FCPS. The data demodulation results of the FCPS are compared with those of the traditional channel filtering method. According to Section 3.1, the parameters of the system matrix calibrated for the FCPS are shown in Figure 6, where Figure 6a–d represent the matrix calibration parameters of the four channels, respectively.

The simulated input Stokes spectra for the simulation are shown in Figure 7. It is a 30° line polarized light.

D O L P = \frac{\sqrt{S_{1}^{2} + S_{2}^{3} + S_{3}^{2}}}{S_{0}^{2}} = 1

(31)

The simulated spectra are input, and four modulated simulation curves are obtained through Equation (1), as shown in Figure 8. It can be seen from Figure 8 that each channel is modulated well. Using calibrated system matrix, the reconstructed Stokes spectra are obtained and shown in Figure 9.

In Figure 9, the two lines overlap in Figure 9a–c, indicating that the reconstructed

S_{0}

S_{1}

, and

S_{2}

are well. In Figure 9d, although the two curves do not overlap perfectly, the maximum error is less than 5 × 10⁻¹³, and the average error is 1.5 × 10⁻¹⁶, which proves the accuracy of the reproduction.

To compare the accuracy of this method, the Stokes vector is demodulated using the traditional channel filter Fourier transform (CFFT), and the simulation conditions are the same as the first channel. Yang’s [18] calibration method is utilized to demodulate the data during demodulation. The simulated modulation curve is shown in Figure 10. The demodulated Stokes vector is shown in Figure 11.

As shown in Figure 11, the middle part of the spectra obtained by the channel filter data demodulation method coincides well with the input spectrum, while the edge part has jitter. Comparing Figure 9 and Figure 11, it can be seen that the effect of the FCPS is better. The edge data of demodulation and the simulation input data well coincide, indicating that this method has a better effect.

The root mean square error (RMSE) of the FCPS and CFFT is calculated to prove the accuracy of the method. Due to the serious edge jitter of the CFFT, the edge part is discarded when calculating the RMSE. The RMSE is listed in Table 1, where the PMSE of the FCPS is much smaller than that of the CFFT.

4. Simulation of Alignment Errors in FCPS

According to Section 3, the system matrix calibration requires 0° linearly polarized light, 45° linearly polarized light, and left-handed circularly polarized light. Left-handed circular light can be obtained by the combination of 45° linearly polarized light and a quarter-wave plate (QWP). A linear polarizer is placed behind the left-rotated circular light, and the detector receives the light intensity passing through the polarizer. If the received light intensity remains unchanged when the line polarizer is rotavated, it proves that the light produced is via left-handed circular polarization.

The FCPS can demodulate the Stokes vector well. However, this is based on the premise that the calibration of the system matrix is accurate. During the installation of the PSIM, the fast axis direction of the linear polarizer P is taken as the reference. During the installation process, there will be phase retarder angle error, as shown in Figure 12.

In order to verify the influence of phase retarder angle error, the Monte Carlo method was adopted for random selection. There are 440 random error combinations. The rank of the system matrix and the demodulated DOLP are shown in Figure 13 and Figure 14.

It can be seen from Figure 13 that the rank of the matrix remains at 4, which proves that all system matrices are of full rank. The maximum DOLP error of demodulation is much less than 0.1%. The RMS of 440 randomly error combinations’ DOLP is shown in Figure 15. The small value of RMS proves that DOLP inversion is effective. Through the above analysis, it can be concluded that the angle error of the phase retarders has little effect on data inversion.

The calibration light is accurate and there is an angle error between the fast axis direction of the linear polarizer and the calibrated light. The angle error of the linear polarizer is 0.2°, 0.3°, 0.4°, and 0.5°, respectively, and the data demodulation is shown in Figure 16.

In Figure 16b, the angle error of the linear polarizer is 0.3° and the spectral curve of the reconstructed

S_{3}

begins to separate from the simulation input curve. In Figure 16c, the linear polarizer angle error is 0.4°, the reconstructed

S_{2}

begins to separate from the input

S_{2}

, and the separation angle between the spectral curve of the reconstructed

S_{3}

and the simulation input curve gradually increases. In Figure 16d, the reconstructed

S_{2}

and

S_{3}

are separated from the input

S_{2}

and

S_{3}

at a greater angle, when the linear polarizer angle error is 0.5°. The maximum deviation between the reconstructed and the input is calculated in Table 2. The maximum deviation of reconstructed

S_{3}

increases by one order of magnitude when the linear polarizer angle error is 0.4, which shows that the inversion effect decreases. In summary, we obtain that the maximum calibration light angle error allowed by the linear polarizer cannot exceed 0.4°.

5. Discussion

Due to the complexity of FCPS system calibration, it is necessary to carry out four separate calibration groups. In order to improve the calibration efficiency, the calibration method of the system will be improved. At the same time, in the aspect of data demodulation, a wavelet change and optimization algorithm will be considered to realize spectral data demodulation.

6. Conclusions

In this paper, the four-channel polarimetric-spectral intensity modulation imager (FCPS) is proposed to realize the real-time and error-free acquisition of the incident Stokes vector spectra. Four Stokes parameters are modulated by four phase retarders, and polarization spectra of the incident light can be reconstructed from four modulation curves. The principle and the detection mode of the scheme have been demonstrated. Comparing the simulation results of the FTTS and FCPS data demodulation, the results show that FCPS Stokes demodulation is better. The influence of phase retarders angle error and the maximum calibration light allowable angle error for Stokes demodulation has been discussed. The phase retarders have little influence and the maximum allowable angle error of the linear polarizer cannot exceed 0.4°. FCPS does not have the problem of edge jitter and the FCPS-reconstructed Stokes is better; therefore, the FCPS can promote polarimetric-spectral intensity modulation engineering development.

Author Contributions

Conceptualization, J.B. and X.J.; methodology, J.B.; software, J.B.; validation, J.B. and C.Y.; formal analysis, J.B.; investigation, J.B.; resources, C.Y.; data curation, J.B.; writing—original draft preparation, J.B.; writing—review and editing, J.B.; funding acquisition, C.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China (NSFC) under Grant 61627819, Grant 61727818, Grant 6187030909, Grant 61875192; in part by the HY-multiangle polarizing spectral camera; in part by the K.C.Wong Education Foundation; in part by High-altitude large UAV airborne visible infrared spectral imager procurement project under Grant 21540; and in part by the National Science Fund for Distinguished Young Scholars under Grant 62105331.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

We declare no conflicts of interest.

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Figure 1. The optical layout of FCPS.

Figure 2. The rank of the system matrix.

Figure 3. The rank of the system matrix with first phase retarder angle error.

Figure 4. The rank of the system matrix with second phase retarder angle error.

Figure 5. The rank of the system matrix with both phase retarder angle error. (a) First phase retarder angle error is −3° and second phase retarder angle error ranges from −3° to 3°. (b) First phase retarder angle error is −2° and second phase retarder angle error ranges from −3° to 3°. (c) First phase retarder angle error is −1° and second phase retarder angle error ranges from −3° to 3°. (d) First phase retarder angle error is 1° and second phase retarder angle error ranges from −3° to 3°. (e) First phase retarder angle error is 2° and second phase retarder angle error ranges from −3° to 3°. (f) First phase retarder angle error is 3° and second phase retarder angle error ranges from −3° to 3°.

Figure 6. The calibrated system matrix. (a) The first channel’s system matrix calibration parameters. (b) The second channel’s system matrix calibration parameters. (c) The third channel’s system matrix calibration parameters. (d) The fourth channel’s system matrix calibration parameters.

Figure 7. The input Stokes spectra.

Figure 8. The simulated modulated spectra. (a) R3, R4, and P1 modulation spectra; (b–d) the modulation spectra through R1, R2, P2, P3, and P4, respectively.