Multi-Parameter Model-Based Polarimetric Calibration for Dual-Coded Spectral Polarization Imaging System

Abstract

A polarization analysis method based on a multi-parameter model is proposed to address the polarization effect analysis and calibration requirements of a dual-coded snapshot spectral polarization imaging system. A full-link polarization effect model for a spectral polarization imaging system is established that includes a digital micromirror array (DMD), prism grating prism (PGP), micro-polarizer array detector (MPA), and multi-film. The influence of parameters such as the refractive index, incident angle, grating refractive index, constant, prism refractive index, vertex angle, multi-layer film complex refractive index, and film thickness on the optical transmittance of the system are analyzed. Using a dynamic data exchange mechanism to perform full-link, full-FOV, and full-pupil ray tracing on the optical system, the polarization effect distribution of the system under different degrees of polarization (DOP) and wavelengths is obtained. A calibration experiment for the controllable incident wavelength and DOP using narrowband filters and glass stacks is established. The experimental results show that in the 420 nm, 532 nm, and 635 nm wavelength bands, the MSEs of the calibrated values are 1.3924 × 10⁻⁴, 1.6852 × 10⁻⁴, and 1.6735 × 10⁻⁴, respectively. It is proven that the calibration method based on a multi-parameter model is feasible. Finally, the spectral polarization image at 532 nm is calibrated. The contrast ratio of metallic aluminum is calibrated from 7.13 to 15.33. This study provides a theoretical basis for the analysis and calibration of polarization effects in a dual-coded snapshot spectral polarization imaging system.

1. Introduction

Spectral polarization imaging (SPI) is a technique capable of acquiring data cubes of targets in terms of spectral, polarization, and spatial attributes. The combination of SPI can obtain the spatial light intensity distribution characteristics of the target and invert the target. The polarization characteristics can be used to obtain its surface texture, refractive index, roughness, and other physical properties, to reduce the interference of complex environments [1]. SPI technology has been gradually applied to astronomical observation, meteorological detection, environmental monitoring, geological exploration, and other fields [2,3]. The difference in target and background polarization is the key to highlighting the target in SPI. However, the polarization state of the target undergoes refraction or reflection in the optical system to produce diattenuation or retardence [4], which is collectively referred to as the polarization effect of the optical system. Polarization effects can seriously affect the accuracy of polarization detection in SPI, causing bias in the degree of polarization and leading to the ineffective highlighting of the target [5]. The polarization effect of the calibration and alignment system is the focus of SPI.

In 2017, Yang introduced a field-usable polarization calibration and reconstruction method for a channel dispersive imaging spectropolarimeter, and a theoretical model for polarization calibration was derived [6]. In 2020, Pamba proposed a new polarization radiation calibration model that decoupled the radiation calibration coefficients and polarization characteristics of the optical system. The alignment error of the polarization module and the variation in the delay under different fields of view were considered and calibrated independently [7]. In 2021, Gogler proposed an integrated method that combined radiometric calibration, non-uniformity correction, and polarization calibration. It improved the reconstruction accuracy of the linear Stokes vector by a factor of 2.4, compared to the 0.83% error of the conventional technique [8]. In 2021, Xing established a polarization radiation transmission model for channel dispersive polarization spectral imaging systems. They gave the Mueller matrix in the global coordinate systems of devices such as phase delayers to improve the recovery accuracy of the polarization information of the target light [9]. In 2022, Miller S. proposed a method for Mueller spectrum reconstruction and Mueller CSP calibration. The method improves the practicability of Mueller CSP and has the potential to become a general reconstruction and calibration method for both imaging and non-imaging Stokes Mueller CS. The method also takes into account beam drift and cold reflection effects and enables simple and direct polarization measurements [10]. In 2022, Tian studied the effect of the grating structure on polarization characteristics at the 1.1–1.6 µm wavelength, offering a guideline to further improve the polarization performance of gratings [11]. Although the above methods have analyzed and calibrated the polarization effects of different optical systems, there is still a lack of effective polarization effect optimization and calibration methods for computational spectral polarization imaging systems.

Previously, we designed a snapshot dual-coded spectral polarization imaging system based on a digital micromirror device (DMD) with a micro-polarizer array (MPA) detector to acquire hyperspectral polarization data of moving targets in a single-shot. The DMD is used to encode spatial data and the MPA is used to encode polarization data [12]. The snapshot dual-coded spectral polarization imaging system consists of a DMD, PGP, MPA detector, objective, relay, and imaging lens. Firstly, the 4D data cube is imaged on the main image plane through the objective lens, and the DMD is placed on the main image plane of the objective lens to encode and modulate the data cube. Secondly, the light passes through the relay lens to achieve PGP. Dispersed rays converge to the MPA detector through an imaging lens. Adjacent spectra are evenly distributed on the focal plane at regular distances. Thirdly, encoding and dispersion data cubes containing degrees of polarization (DOP) are obtained through an MPA detector. Finally, spectral polarization imaging is achieved through unmixing and reconstruction. However, the traditional modeling method does not consider the transmittance of the film, micro-polarizer array, DMD reflection, and prism grating prism (PGP) dispersion. The systematic polarization effect model has a high error. The polarization effects of partially incident polarized light with different polarization degrees are different, resulting in a large and difficult-to-calibrate polarization effect of the snapshot dual-coded spectral polarization imaging system.

Therefore, a polarization effect analysis and calibration method for a dual-coded spectral polarization imaging system with multi-parameter model authorization is proposed. A polarization effect model for the DMD, MPA, PGP, and multilayer film lenses is established. A full-link, full-field-of-view (FOV), and full-pupil ray tracing of the system-based dynamic data exchange (DDE) is realized. The polarization effect of the system under different DOP and wavelengths of incident light is analyzed. A polarimetric calibration experiment with controlled DOP and wavelengths of incident light is built. The polarization light source consists of a visible light source, narrowband filters, and glass stacks. By comparing the results of the dual-coded spectral polarization imaging system with a commercial polarization state measuring instrument, the accuracy of the polarization effect is verified. This study provides a theoretical basis for the analysis and calibration of polarization effects in dual-coded snapshot spectral polarization imaging.

2. Principle

2.1. Stokes Vector

In 1852, Stokes proposed a four-dimensional mathematical vector derived from the intensity of light to characterize the different polarization states of light. Many representations of polarized light currently use this vector method because of its simple expression. The Stokes vector method uses four covariates $I$, $M$, $C$, and $S$, with each parameter.

$$\mathrm{Stokes}=\left[\begin{array}{c}I\\ M\\ C\\ S\end{array}\right]=\left[\begin{array}{c}{I}_0+I_{90}\\ I_0-I_{90}\\ I_{45}-I_{135}\\ I_R-I_L\end{array}\right],$$

(1)

where $I$ is the total irradiance of the beam. $M$ is the horizontal ($I_0$) polarized flux component minus the vertical ($I_{90}$) flux component. $C$ is the 45° flux ($I_{45}$) minus the 135° flux ($I_{135}$). Finally, $S$ measures the difference in the right ($I_R$) minus left ($I_L$) circularly polarized flux.

After reflection and refraction, the incident light and the emergent light vectors can be characterized by a matrix that represents the relationship between the incident light and the emergent light. The Mueller matrix can represent the polarization characteristics of the optical element alone.

$$\left[\begin{array}{c}{I}_2\\ M_2\\ C_2\\ S_2\end{array}\right]=\left[\begin{array}{c}{M}_{11}\hspace{1em}{M}_{12}\hspace{1em}{M}_{13}\hspace{1em}{M}_{14}\\ M_{21}\hspace{1em}{M}_{22}\hspace{1em}{M}_{23}\hspace{1em}{M}_{24}\\ M_{31}\hspace{1em}{M}_{32}\hspace{1em}{M}_{33}\hspace{1em}{M}_{34}\\ M_{41}\hspace{1em}{M}_{42}\hspace{1em}{M}_{43}\hspace{1em}{M}_{44}\end{array}\right]\left[\begin{array}{c}{I}_1\\ M_1\\ C_1\\ S_1\end{array}\right],$$

(2)

The Mueller matrix of the light incident optical element can be expressed as $M_i(i=1,2,\dots n)$. After $n$ times the light refracted or reflected by the optical element, the outgoing light Stokes vector is a leftward superposition of the form

$$\left[\begin{array}{c}I\prime \\ M\prime \\ C\prime \\ S\prime \end{array}\right]=M_n{M}_{n-1}\dots M_2{M}_1\left[\begin{array}{c}I\\ M\\ C\\ S\end{array}\right],$$

(3)

The DOP of light represents the percentage of polarized light in the total light intensity, and the Stokes vector method defines polarization as

$$DOP=\frac{\sqrt{M^2+C^2+S^2}}{I}(0\le DOP\le 1),$$

(4)

2.2. Mueller Matrix of DMD

A DMD is an array device composed of controllable reflective aluminum micromirrors. According to Fresnel’s Diffraction Laws, the amplitude ratio of the reflected wave to the incident wave is

$$\left\{\begin{array}{c}{r}_{\perp }=-\frac{\sin \left({\theta }_i-{\theta }_t\right)}{\sin \left({\theta }_i+{\theta }_t\right)}=\frac{n_1\cos {\theta }_i-n_2\cos {\theta }_i}{n_1\cos {\theta }_i+n_2\cos {\theta }_t}\\ r_{//}=\frac{\tan \left({\theta }_i-{\theta }_t\right)}{\tan \left({\theta }_i+{\theta }_t\right)}=\frac{n_2\cos {\theta }_i-n_1\cos {\theta }_t}{n_2\cos {\theta }_i+n_1\cos {\theta }_t}\end{array}\right.,$$

(5)

where $r_{\perp }$ is the reflected wave, $r_{//}$ is the incident wave, ${\theta }_i$ is the angle of incidence, ${\theta }_t$ is the angle of refraction, $n_1$ is the incident light medium, and $n_2$ is the reflected light medium.

Since ${\theta }_t$ and $n_2$ are complex numbers, both $r_{//}$ and $r_{\perp }$ are complex numbers, and taking $n_2=n_t\cos {\theta }_t=n-i\chi$ and $n_i\sin {\theta }_i=n_t\sin {\theta }_t$ into account,

$$n_t\cos {\theta }_t=\sqrt{n_t^2-n_t^2{\sin }^2{\theta }_2}=\sqrt{n_t^2-n_i^2{\sin }^2{\theta }_i}=\sqrt{{(n-i\chi )}^2-{\tilde{n}}_i^2{\sin }^2{\theta }_i}=N-i{\chi }^{\prime },$$

(6)

Then, it has

$$\left\{\begin{array}{l}N=\sqrt{\frac{1}{2}\{{(n_{DMD}-i\chi )}^2-n_{air}^2{\sin }^2{\theta }_{DMD}+\sqrt{{[{(n_{DMD}-i\chi )}^2-n_{air}^2{\sin }^2{\theta }_{DMD}]}^2+4n_{DMD}^2{\chi }^2}\}}\hfill \\ {\chi }^{\prime }=\sqrt{-\frac{1}{2}\{{(n_{DMD}-i\chi )}^2-n_{air}^2{\sin }^2{\theta }_{DMD}-\sqrt{{[{(n_{DMD}-i\chi )}^2-n_{air}^2{\sin }^2{\theta }_{DMD}]}^2+4n_{DMD}^2{\chi }^2}\}}\hfill \end{array}\right.,$$

(7)

According to the polarization bidirectional reflectance distribution function, the Mueller matrix of the mirror is [13]

$$M_{DMD}=\left[\begin{array}{cccc}{r}_p{r}_p^{*}+r_s{r}_s^{*}& r_p{r}_p^{*}-r_s{r}_s^{*}& 0& 0\\ r_p{r}_p^{*}-r_s{r}_s^{*}& r_p{r}_p^{*}+r_s{r}_s^{*}& 0& 0\\ 0& 0& r_p{r}_s^{*}+r_s{r}_p^{*}& i(r_p{r}_s^{*}-r_s{r}_p^{*})\\ 0& 0& i(r_s{r}_p^{*}-r_p{r}_s^{*})& r_s{r}_p^{*}+r_p{r}_s^{*}\end{array}\right],$$

(8)

where $r_p$ and $r_s$ are the reflection coefficients of the $p$ and $s$ components of the incident light from the metal surface. They are defined, respectively, as

$$\left\{\begin{array}{l}{r}_p=\frac{{(n_{DMD}^2+{\chi }^2)}^2{\cos }^2{\theta }_{DMD}-n_{air}^2(N^2+{{\chi }^{\prime }}^2)+2i{n}_{air}\cos {\theta }_{DMD}[(n_{DMD}^2-{\chi }^2){\chi }^{\prime }-2N{n}_{DMD}\chi ]}{{[(n_{DMD}^2-{\chi }^2)\cos {\theta }_{DMD}+n_{air}N]}^2+{(2n_{DMD}\chi \cos {\theta }_{DMD}+n_{air}{\chi }^{\prime })}^2}\\ r_s=\frac{(n_{air}^2{\cos }^2{\theta }_{DMD}-N^2)-{{\chi }^{\prime }}^2+2i{\chi }^{\prime }{n}_{air}\cos {\theta }_{DMD}}{{(n_{air}\cos {\theta }_{DMD}+N)}^2+{{\chi }^{\prime }}^2}\end{array}\right.,$$

(9)

where $n-i\chi$ is the complex refractive index of the metal. $i$ is an imaginary unit. $\chi$ is the attenuation of the light in the absorbing medium. $n_{DMD}$ and $n_{air}$ are the refractive indices of the DMD and incident medium, respectively. ${\theta }_{DMD}$ is the angle of incidence of the DMD metal surface.

2.3. Mueller Matrix of Coated Lenses

SPI is focused on using polarization to enhance image contrast. Thus, the diattenuation of the optical system is described. Refraction and reflection at an isotropic interface establish a coordinate system with the direction of vibration of the s-state as the x-axis and the direction of vibration of the p-state as the y-axis. Then, the Mueller matrix of the coated lens interface is

$$M=\frac{1}{2}\left[\begin{array}{cccc}{t}_s+t_p& t_s-t_p& 0& 0\\ t_s-t_p& t_s+t_p& 0& 0\\ 0& 0& 2\sqrt{t_s{t}_p}& 0\\ 0& 0& 0& 2\sqrt{t_s{t}_p}\end{array}\right],$$

(10)

where $t_s$ and $t_p$ are the transmittance of s-light and p-light, respectively.

For the coated lens interface, the distribution is as shown in Figure 1.

When rays are reflexed at the junctions of different media, from Snell’s law, it is known that

$$n_0\sin {\theta }_0=n_1\sin {\theta }_1=n_2\sin {\theta }_2=\dots =n_c\sin {\theta }_c,$$

(11)

According to the constant relation of Equation (13), the refractive angle of any film can be inferred from the incident angle:

$$\cos {\theta }_i=\sqrt{1-\frac{n_0^2{\sin }^2{\theta }_0}{n_i^2}},$$

(12)

Let the environment, film 1, film 2, and the substrate be homogeneous and optically isotropic.

According to Fresnel’s formula, the incident ray is partially reflected in the medium and partially refracted in the membrane, and the refracted ray inside the membrane is subsequently reflected internally several times at the boundary interface. The reflectance and transmittance at each interface are denoted by $r_{0-1}$, $r_{1-2}$, $r_{2-3}\dots$ and $t_{0-1}$, $t_{1-2}$, $t_{2-3}\dots$, respectively. The total transmission amplitude of the refraction is given by the infinite geometric series [14]

$$\begin{array}{l}T=t_{0,1}{t}_{1,2}\dots t_{m_c-1,m_c}{\mathrm{e}}^{-i\beta }+t_{0,1}{t}_{1,2}\dots t_{m_c-1,m_c}\times r_{0,1}{r}_{1,2}\dots r_{m_c-1,m_c}{\mathrm{e}}^{-i3\beta }\\ \hspace{1em}+ t_{0,1}{t}_{1,2}\dots t_{m_c-1,m_c}\times r_{1,0}^2{r}_{1,2}^2\dots r_{m_c-1,m_c}^2{\mathrm{e}}^{-i5\beta }+\cdots \\ \hspace{1em}=\frac{t_{0,1}{t}_{1,2}\dots t_{m_c-1,m_c}{\mathrm{e}}^{-i\beta }}{1+r_{0,1}{r}_{1,2}\dots r_{m_c-1,m_c}{\mathrm{e}}^{-i2\beta }}\end{array},$$

(13)

where

$$\beta ={\displaystyle \sum _{j=1}^{m_c}2\pi \left(\frac{d_j}{{\lambda }_j}\right)}{N}_j\cos {\theta }_j,$$

(14)

where ${\lambda }_j$ is the corresponding wavelength, $d_j$ is the thickness of the corresponding film layer, and $N_j$ is the complex refractive index of the corresponding film layer.

These equations are valid when the incident wave is linearly polarized parallel (_p) or perpendicular (_s) to the plane of incidence [15]. Therefore, the permeability of the multilayer membrane is

$$\left\{\begin{array}{c}{t}_p=\frac{{\displaystyle \sum t_{p(m_c-1,m_c)}}{\mathrm{e}}^{-i\beta }}{1+{\displaystyle \sum r_{p(m_c-1,m_c)}}{\mathrm{e}}^{-i/\beta }}\\ t_s=\frac{{\displaystyle \sum t_{s(m_c-1,m_c)}}{\mathrm{e}}^{-i\beta }}{1+{\displaystyle \sum r_{s(m_c-1,m_c)}}{\mathrm{e}}^{-i2\beta }}\end{array}\right.,$$

(15)

$$\left\{\begin{array}{c}{t}_{p(m_c-1,m_c)}=\frac{2N_{m_c-1}\cos {\theta }_{m_c-1}}{N_{m_c}\cos {\theta }_{m_c-1}+N_{m_c-1}\cos {\theta }_{m_c}}\\ r_{p(m_c-1,m_c)}=\frac{N_{m_c}\cos {\theta }_{m_c-1}-N_{m_c-1}\cos {\theta }_{m_c}}{N_{m_c}\cos {\theta }_{m_c-1}+N_{m_c-1}\cos {\theta }_{m_c}}\\ t_{s(m_c-1,m_c)}=\frac{2N_{m_c-1}\cos {\theta }_{m_c-1}}{N_{m_c-1}\cos {\theta }_{m_c-1}+N_{m_c}\cos {\theta }_{m_c}}\\ r_{s(m_c-1,m_c)}=\frac{N_{m_c-1}\cos {\theta }_{m_c-1}-N_{m_c}\cos {\theta }_{m_c}}{N_{m_c-1}\cos {\theta }_{m_c-1}+N_{m_c}\cos {\theta }_{m_c}}\end{array},\right.$$

(16)

Finally, the Muller matrix of the coated lens is represented by Equation (10), where $t_p$ and $t_s$ are chosen from Equation (15).

2.4. Mueller Matrix of PGP

For the Mueller matrix of the PGP, the superposition properties of the polarization elements from Equation (3) are

$$M_{PGP}=M_{P1}{M}_G{M}_{P2},$$

(17)

where $M_{PGP}$ represents the Mueller matrix of the PGP, and $M_{P1}$ and $M_{P2}$ represent the Mueller matrices of prism 1 and prism 2, respectively. $M_G$ represents the Mueller matrix of the grating.

The incident photoelectric vector perpendicular to the grating will pass through, while the polarized light parallel to the line grating is reflected by the grating. Therefore, the light transmittance of the grating is [16]

$$\left\{\begin{array}{c}{t}_S^G=\frac{4n_G^2{A}^2}{1+n_G^2{A}^2}\\ t_P^G=\frac{4n_G^2{B}^2}{1+n_G^2{B}^2}\end{array}\right.,$$

(18)

where $n_G^{}$ is the refractive index of the grating substrate material; $A$ and $B$ are expressed as follows:

$$\left\{\begin{array}{c}A=\frac{\lambda }{4d}{\left[\ln \left(\sqrt{2}\right)+\frac{\frac{1}{4}\times \left(\frac{1}{{\left[1-{(d/\lambda )}^2\right]}^{\frac{1}{2}}}-1\right)}{1+\left(\frac{1}{{\left[1-{(d/\lambda )}^2\right]}^{\frac{1}{2}}}-1\right)}+\frac{1}{1024}{\left(\frac{d}{\lambda }\right)}^2\right]}^{-1}\\ B=\frac{d}{\lambda }\left[\ln \left(\sqrt{2}\right)+\frac{\frac{1}{4}\times \left(\frac{1}{{\left[1-{(d/\lambda )}^2\right]}^{\frac{1}{2}}}-1\right)}{1+\frac{1}{4}\times \left(\frac{1}{{\left[1-{(d/\lambda )}^2\right]}^{\frac{1}{2}}}-1\right)}+\frac{1}{1024}{\left(\frac{d}{\lambda }\right)}^2\right]\end{array}\right.,$$

(19)

For the Muller matrix of a prism, the form remains as in formula (10), but the $t_s$ and $t_p$ in it are replaced with [17]

$$\left\{\begin{array}{c}{t}_p^p=\frac{\sqrt{4n_P^{}\cos {\theta }_{\lambda }\left(1-n_p^2{\sin }^2{\theta }_{\lambda }\right)}}{\left(n_p^2+1\right)-\left(n_p^4+1\right){\sin }^2{\theta }_{\lambda }+\sqrt{2n_p\cos {\theta }_{\lambda }\left(1-n_p^2{\sin }^2{\theta }_{\lambda }\right)}}\\ t_s^p=\frac{\sqrt{4n_p\cos {\theta }_{\lambda }\left(1-n_p^2{\sin }^2{\theta }_{\lambda }\right)}}{1+n_p^2\cos 2{\theta }_{\lambda }+\sqrt{2n_p\cos {\theta }_{\lambda }\left(1-n_p^2{\sin }^2{\theta }_{\lambda }\right)}}\end{array}\right.,$$

(20)

where $n_p$ is the refractive index of the prism to the surrounding air, and ${\theta }_{\lambda }$ is the refractive angle corresponding to the wavelength in the prism.

The derivation process of $n_p$ is as follows

$$\left\{\begin{array}{c}{\theta }_{P1}+{\theta }_{P8}=\epsilon +{\alpha }_1\\ {\theta }_{P2}+{\theta }_{P3}={\alpha }_1\end{array}\right.,$$

(21)

where $\epsilon$ is the internal deviation angle of the prism.

According to Snell’s law of refraction, in prism 1, there is the following formula

$$\left\{\begin{array}{c}\sin {\theta }_{P1}=n_{P1}\sin {\theta }_{P2}\\ n_{P1}\sin {\theta }_{P3}=n_G\sin {\theta }_{P4}\end{array}\right.,$$

(22)

where ${\theta }_{P1}$ is the angle of incidence of the first face of prism 1, $n_{P1}$ is the refractive index of the first face of the prism, ${\theta }_{P2}$ is the angle of refraction of the first face of prism 1, ${\theta }_{P3}$ is the angle of incidence of the second face of prism 1, and ${\theta }_{P4}$ is the angle of refraction of the second face of prism 1.

From Equations (21) and (22), it can be concluded that

$$\begin{array}{c}\frac{d{\theta }_{P2}}{d{\theta }_{P1}}=-\frac{d{\theta }_{P8}}{d{\theta }_{P1}}\\ \cos {\theta }_{P1}=n_P\cos {\theta }_{P2}\frac{d{\theta }_{P2}}{d{\theta }_{P1}}\\ \cos {\theta }_{P8}\frac{d{\theta }_{P8}}{d{\theta }_{P1}}=n_P\cos {\theta }_{P3}\frac{d{\theta }_{P3}}{d{\theta }_{P1}}\end{array},$$

(23)

Thus, the $n_P$ of the prism is

$$n_P=-\frac{\sin \left[\frac{1}{2}\left(\epsilon +\alpha \right)\right]}{\sin \left(\frac{1}{2}\alpha \right)},$$

(24)

Because the gratings and prisms are in the transmission form, we choose the Mueller matrix in the transmission form to obtain the Mueller matrix of the PGP.

2.5. Mueller Matrix of MPA

The MPA detector integrates pixel-level micro-polarizers on the focal plane of the sensor, and the Muller matrix of the micro-polarizer can be represented as

$$M_{MPA}=t_d^2\left[\begin{array}{cccc}1+{\epsilon }^2& 1-{\epsilon }^2& 0& 0\\ 1-{\epsilon }^2& 1+{\epsilon }^2& 0& 0\\ 0& 0& 2\epsilon & 0\\ 0& 0& 0& 2\epsilon \end{array}\right],$$

(25)

where $\epsilon$ represents the extinction ratio in the corresponding direction, and $t_d$ represents the maximum transmittance in the direction of the micro-polarizer. The commonly used micro-polarizer angles are 0°, 45°, 90°, and 135°.

For spectrometers, tilt detectors are used to correct longitudinal chromatic aberration, so the Muller matrix of the tilted MPA detector is

$$\begin{array}{cc}M{\prime }_{MPA}& =R(\varphi )\cdot M_{MPA}\cdot R(-\varphi )\\ & =t_d^2\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos 2\varphi & -\sin 2\varphi & 0\\ 0& \sin 2\varphi & \cos 2\varphi & 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1+{\epsilon }^2& 1-{\epsilon }^2& 0& 0\\ 1-{\epsilon }^2& 1+{\epsilon }^2& 0& 0\\ 0& 0& 2\epsilon & 0\\ 0& 0& 0& 2\epsilon \end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& -\cos 2\varphi & \sin 2\varphi & 0\\ 0& -\sin 2\varphi & -\cos 2\varphi & 0\\ 0& 0& 0& 1\end{array}\right],\end{array}$$

(26)

where $\varphi$ is the angle between the micro-polarizer and the optical axis.

Polarization-sensitive interfaces of optical systems can be determined by a full-link polarization effect model of the dual-coded snapshot spectral polarization imaging system. By controlling the incident angle and refractive index of the lens, the refractive index and grating constant of the grating, the refractive index and top angle of the prism, the compound refractive index and thickness of the film, and the tilt angle of the MPA detector, we can suppress the polarization effect at the interface.

3. Simulation Analysis

3.1. Model of Dual-Coded Snapshot Spectral Polarization Imaging System

A plane figure is shown in Figure 2; the dual-coded snapshot spectral polarization imaging system operates at wavelengths ranging from 400 nm to 650 nm, and the FOV is 2.86° (the figure shows a half FOV of 0–2.86°). The DMD is on the primary image plane of the objective with a resolution of 1920 × 1080, a pixel size of 7.65 μm, and a rotation angle of 45 degrees. The PGP disperses the rays passing through the relay lens and converges on the MPA detector through the imaging lens, which has a pixel size of 3.45 μm. The MPA detector tilt is 2.24°.

3.2. Dynamic Data Exchange-Based Full-Link, Full-Pupil, and Full-FOV Ray Tracing

In the analysis of the polarization effect of the SPI system, the polarization effect of the system is mostly achieved by means of ray tracing. However, for dual-coded snapshot spectral polarization imaging, the polarization effect of single ray tracing is different for a different FOV and pupils when the incident light is incompletely polarized. The dispersion of the PGP causes the linear arrangement of the polarized rays in different bands, resulting in the distortion of the polarization effect model in traditional ray tracing methods. A schematic diagram of ray tracing with different FOVs, pupils, and wavelengths is shown in Figure 3 ($H_Y$ represents the FOV in the Y direction; $P_Y$ represents the pupil position in the Y direction). The polarization effects obtained by changing the wavelength, FOV, and pupil under different incident light polarization conditions are shown in

Table 1. Polarization effect of 0° FOV and 0% pupil.

	${\mathit{P}}_{\mathit{i}\mathit{n}}$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Wavelength
420 nm		0.5854	0.5401	0.4966	0.4476	0.3938	0.3359	0.2743	0.2096	0.1421	0.0722	0
532 nm		0.8734	0.7961	0.7145	0.6304	0.5442	0.4563	0.3671	0.2766	0.1752	0.093	0
635 nm		0.9432	0.8576	0.7653	0.6716	0.5776	0.4825	0.3868	0.2907	0.1941	0.0972	0

,

Table 2. Polarization effect of 0% pupil and 532 nm wavelength.

	${\mathit{P}}_{\mathit{i}\mathit{n}}$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
FOV (Hy)
0°		0.8693	0.7918	0.71	0.6275	0.5418	0.4545	0.3657	0.2756	0.1846	0.0927	0
1.4°		0.8712	0.7951	0.7137	0.6297	0.5436	0.4559	0.3667	0.2764	0.1851	0.0929	0
2.8°		0.8743	0.7961	0.7145	0.6304	0.5442	0.4563	0.3671	0.2766	0.1552	0.093	0

and

Table 3. Polarization effect of 0° FOV and 532 nm wavelength.

	${\mathit{P}}_{\mathit{i}\mathit{n}}$	0	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1
Pupil (Py)
0%		0.8743	0.7961	0.7145	0.6304	0.5442	0.4563	0.3671	0.2766	0.1779	0.093	0
50%		0.8712	0.792	0.71111	0.6277	0.5419	0.4545	0.3657	0.2757	0.1846	0.0927	0
100%		0.8531	0.7775	0.699	0.6177	0.5339	0.4482	0.3609	0.2722	0.1824	0.0916	0

.

According to Table 1, when the FOV is 0°, the pupil is 0%, the DOP of the incident light is 0.1, the relative error of the DOP at 420 nm is 0.5401, and the relative error of the DOP at 635 nm is 0.9432. When the FOV is 0°, the pupil is 0%, the DOP of the incident light is 0.9, the relative error of the DOP at 420 nm is 0.0722, and the relative error of the DOP at 635 nm is 0.0972. It can be seen that the polarization effect of the system is proportional to the wavelength after dispersion. The longer the wavelength, the greater the polarization effect of the system. According to Table 2, when the pupil is 0%, the wavelength is 532 nm, the DOP of the incident light is 0.1, the relative error of the DOP of the central FOV is 0.7918, and the relative error of the DOP of the edge FOV is 0.7961. When the FOV is 0°, the wavelength is 532 nm, the DOP of the incident light is 0.9, the relative error of the DOP of the 0% pupil is 0.0927, and the relative error of the DOP of the 100% pupil is 0.093. It can be seen that the polarization effect of the system is inversely proportional to the FOV. The larger the field of view, the greater the polarization effect of the system. According to Table 3, when the FOV is 0°, the wavelength is 532 nm, and the DOP of the incident light is 0.1, the relative error of the DOP of the 0% pupil is 0.7961, and the relative error of the DOP of the 100% pupil is 0.7775. When the FOV is 0°, the wavelength is 532 nm, the DOP of the incident light is 0.9, the relative error of the DOP of the 0% pupil is 0.093, and the relative error of the DOP of the 100% pupil is 0.0916. It can be seen that the polarization effect of the system is inversely proportional to the position of the pupil. The smaller the position of the pupil, the greater the polarization effect of the system. It can be seen that the polarization effect of the system is inversely proportional to the pupil position.

It can be seen that for the dual-coded snapshot spectral polarization imaging system, it is inappropriate to choose the DOP, FOV, and pupil of a single incident light to determine the polarization effect of the system. Therefore, a multi-parameter model polarization effect evaluation method based on DDE is proposed. DDE is a mechanism of computer data communication, and data can be exchanged and shared between programs. Through optimized sampling methods and iterative operations, the full-link, full-FOV, and full-pupil polarization effects can be obtained, as shown in Figure 4.

The specific content of the multi-parameter model polarization effect evaluation method based on DDE is as follows.

(1)

Establish MATLAB and ZEMAX communication; read the wavelength, FOV, pupil, material of refractive or reflective surfaces, and other basic information of the optical system in ZEMAX.

(2)

The wavelength, FOV, and pupil are allocated to sampling points. In this paper, we choose wavelengths of 420 nm, 532 nm, and 635 nm; the FOV sampling points include 50 sampling points from 0° to 2.86°; and the pupil sampling points include 20 sampling points from 0% to 100%.

(3)

Select a wavelength in any wavelength band and select any FOV within the range of sampling points in the FOV; select any pupil in this field of view.

(4)

Use MATLAB to read the ZEMAX refraction angle function RAID and calculate the full-link system Mueller matrix under this wavelength, FOV, and pupil.

(5)

Repeat steps (3)–(4) and iteratively calculate the polarization effect under each wavelength, FOV, and pupil.

3.3. Simulation and Analysis of Polarization Effect

We analyze the polarization effect of the system in 3.2 using the DDE-based full-link, full-pupil, and full-FOV ray tracing methods. In the polarization degree of the incident light, values from 0 to 1, 0.1, 0.4, and 0.7 are selected. We simulate the polarization effect of the system by choosing an incident light DOP of 0.1, 0.4, and 0.7 and incident light wavelengths of 420 nm, 532 nm, and 635 nm, as shown in Figure 5, Figure 6 and Figure 7.

The XY axis in the figure represents the X, Y FOV at the full pupil, while the Z axis represents the absolute error of the DOP of the emergent light of the system. The absolute error of the DOP of the emergent light is used as the evaluation standard for the polarization effect of the system. As shown in the figure, when the wavelength $\lambda$ = 420 nm, the mean absolute error of the system decreases from 0.2305 to −0.1456 with the increase in the DOP of the incident light ($P_{in}$). At 532 nm, with the increase in $P_{in}$, the mean absolute error of the system decreases from 0.2734 to −0.0966. At 635 nm, with the increase in $P_{in}$, the mean absolute error of the system decreases from 0.4035 to −0.0898. When the incident light wavelength $\lambda$ remains constant, the polarization effect of the system decreases as $P_{in}$ increases. When $P_{in}$ is low, the system will experience bias. When $P_{in}$ is high, the system will generate depolarization. When $P_{in}$ remains constant, the polarization effect of the system decreases with the increase in the incident light wavelength $\lambda$. Due to the presence of dispersion, the polarization effect is not an axisymmetric pattern but decreases in the direction of dispersion.

Compared to the single ray tracing method, the full-link, full-pupil, and full-FOV ray tracing maximally avoids the randomness caused by single ray tracing. It can be used to analyze the polarization effects of optical systems with an arbitrary DOP of incident light and assist in system polarimetric calibration.

4. Experiment and Discussion

Adjustable polarization can be achieved using glass stacks to verify the polarization calibration accuracy of the instrument, with a controllable DOP of the polarized light emitted from the glass stacks [18]. The working principle of a glass pile is shown in Figure 8.

When the incident light is incident on the surface at Brewster’s angle, the phenomena of reflection (producing reflected light) and refraction (producing transmitted light) occur. The reflected light only has s-shaped linearly polarized light, while the transmitted light has s-shaped linearly polarized light as well as p-shaped linearly polarized light. This beam of light will generate multiple external reflections between the glass slides. Finally, the degree of polarization of the light transmitted from the m-plate wave plate is

$$P=\frac{2^{2m-2}-\frac{2n^2}{n^4+1}{(1+\frac{2n^2}{n^4+1})}^{2m-2}}{2^{2m-2}+\frac{2n^2}{n^4+1}{(1+\frac{2n^2}{n^4+1})}^{2m-2}},$$

(27)

where $m$ is the number of glass stacks, and $n$ is the refractive index of the wave plate.

The entire experiment includes a polarized light source, the dual-coded spectral polarization imaging system, and a commercial polarization state measurement instrument. Firstly, the DOP of the glass stack is measured using the polarization state measurement instrument. Secondly, the dual-coded spectral polarization imaging system obtains the DOP of the glass stack. Thirdly, the DOP of the system is calibrated using the multi-parameter model polarization effect evaluation method. Fourthly, we compare the polarization data of the polarization state measurement instrument with the calibrated polarization data. The entire experimental schematic is shown in Figure 9. Polarization sources include light sources, filters, and polarizers. The polarizer consists of two symmetrically placed planar H-k9L glasses. The filter selection includes three simulated bands (Thorlabs FBH420-10, FLH532-10, FBH635-10). The coding aperture of the system is a digital micromirror device (DMD, Texas Instruments DLP6500). The detector is a micro-polarizer array detector (MPA, Flir Blackfly BFS-U3-123S6C-C). The working wavelength of the prism is 425–675 nm (Edmund #35-788). The central wavelength of the grating is 532 nm (WP-360/532-25.4). The working wavelength of the polarization state measuring instrument is 400–700 nm (Thorlabs PAX1000VIS (/M): 400–700 nm).

When increasing the angle of the glass stack, the polarization increases, and we choose 30 angles to be fitted to the curve. Finally, the standard value is measured by the polarization state measuring instrument. The corrected value after polarization correction, and the relationship between the measured value obtained by the system and the incident angle of the glass pile, are shown in Figure 10.

Mean squared error (MSE) refers to the expectation of the square of the difference between the estimated value of the parameter and the true value of the parameter. In this paper, MSE is used to reflect the difference between the standard value and the calibrated value. The true value here is the calibrated value. The standard value is measured using the polarization state measuring instrument. The smaller the MSE value, the smaller the difference between the standard value and the true value, and the better the curve similarity. The experimental results show that the polarization effect of the system is low at short waves, and the error of the measurement value decreases as the polarization degree of the incident light increases. At 420 nm, the MSE before calibration is 0.2224, and the calibrated MSE is 1.3924 × 10⁻⁴. At 532 nm, the MSE before calibration is 0.3266, and the calibrated MSE is 1.6852 × 10⁻⁴. At 635 nm, the MSE before calibration is 0.3839, and the calibrated MSE is 1.6735 × 10⁻⁴. After calibrating and using the MSE to express the differences in the polarization of different types of incident light, the accuracy in obtaining polarization in the system is found to be essentially the same as that of commercial instruments. It can be seen that the accurate polarization correction of spectral polarization systems can be achieved using the multi-parameter model polarization effect evaluation method.

By analyzing the polarization effect of the system, the two-dimensional spectral polarization images also can be corrected. We select the reconstructed 532 nm spectral polarization image, as shown in Figure 11. Figure 11a shows the target material, Figure 11b shows the ground truth values, Figure 11c shows the case before correction, and Figure 11d shows the case after correction.

In spectral polarization images, contrast is used to quantify the calibration effect. In Figure 11c,d, the contrast of the letter “c” is calibrated from 7.13 to 15.33. The contrast of the letter “U” is calibrated from 11.11 to 6.13. The contrast of the letter “S” is corrected from 10.39 to 5.86. The contrast of the letter “T” is corrected from 19.73 to 14.69. The DOP of aluminum is higher than that of other materials in the experiment, but the contrast ratio in the image is the lowest. After calibration, the contrast ratio of aluminum is significantly improved.

The system polarization effect model established based on the full-link, full-FOV, and full-pupil ray tracing of DDE has certain accuracy. However, in the polarization calibration accuracy verification experiment, limited by the basic working principle and size of the polarization light source, only a certain FOV can be measured in one experiment. To ensure accuracy, the coordinate system must be reunified after each device movement. Moreover, due to setup errors, there will be unpredictable random errors in the measurement and calibration values of the system during the experimental setup process. It was shown through research that controlling the incidence angle and refractive index of the lens is necessary. The refractive index and grating constant of the grating, the refractive index and vertex angle of a prism, and the complex refractive index and film thickness of multilayer films can be optimized in advance during the design process to optimize the polarization effect of the system. The polarization effect model can be used as the optimization criterion to optimize the system’s polarization effect, making the system a low polarization effect system.

5. Conclusions

In this article, a full-link polarization effect model for a spectral polarization imaging system based on a DMD, PGP, MPA, and multi-film layer system is established. A polarization analysis and calibration method based on a multiparametric polarization effect model is proposed. The effects of the refractive index, angle of incidence, grating refractive index and constant, prism refractive index and top angle, multi-layer film compound refractive index, film thickness, and other parameters on the analysis of the optical transmittance of the system are studied. The correspondence among the incident light wavelength, polarization degree, and system polarization effect is revealed.

In order to prevent the different wavelengths, FOVs, and pupils of a single ray tracing from resulting in the insufficient accuracy of the system polarization effect, a multi-parameter model polarization effect evaluation method based on DDE is proposed. By utilizing full-link, full-FOV, and full-pupil ray tracing, the distribution of polarization effects in the system under different DOPs and wavelengths is achieved. It can visually demonstrate the impact of polarization detection imaging systems on the DOP of incident light. When the incident light wavelength is kept constant, the polarization effect of the system decreases as the DOP of the incident light increases. When the DOP of the incident light is low, the system is polarized. When the DOP of the incident light is high, the system is depolarized. When the incident light is kept constant, the polarization effect of the system decreases with the increase in the incident light wavelength.

Subsequently, calibration experiments for a controllable incident wavelength and polarization are conducted using narrowband filters and glass stacks. The system is polarimetrically calibrated by full-link, full-FOV, and full-pupil polarization effects at different DOPs and wavelengths. At 420 nm, the MSE decreases from 0.2224 to 1.3924 × 10⁻⁴. At 532 nm, the MSE decreases from 0.3266 to 1.6852 × 10⁻⁴. At 635 nm, the MSE decreases from 0.3839 to 1.6735 × 10⁻⁴. Finally, each material in the 532 spectral polarization image is corrected, and the aluminum is calibrated from 7.13 to 15.33. The letter plastics are calibrated from 11.11 to 6.13. The cardboard is corrected from 10.39 to 5.86. The rubber is corrected from 19.73 to 14.69. It can be seen that the polarization calibration of full-link, full-FOV, and full-pupil polarization effect models under different DOP and wavelengths is meaningful. The method improves the efficiency of polarimetric calibration. It has theoretical and practical implications for the polarimetric calibration of dual-coded snapshot spectral polarization imaging systems.