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Simplified analysis and suppression of polarization aberration in planar symmetric optical systems

Kaikai Wang, Qiang Fu, Boshi Wang, Haodong Shi, Jianan Liu, Qi Wang, and Chao Wang

Open Access Open Access

Abstract

Polarized remote sensing imaging has attracted more attention in recent years due to its wider detection information dimension compared to traditional imaging methods. However, the inherent instrument errors in optical systems can lead to errors in the polarization state of the incident and outgoing light, which is the polarization aberration of the optical system, resulting in a decrease in polarization detection accuracy. We propose a polarization aberration simplification calculation method for planar symmetric optical systems, by what only three ray samples are needed to obtain the distribution of polarization aberrations within the pupil. This method has a calculation accuracy close to traditional methods, and the sampling rate is 0.003 times that of traditional methods. Based on this, we designed a merit function that optimizes both wavefront and polarization aberrations simultaneously. It is found that diattenuation and retardance of the optical system are 62% and 58% of the original, and the polarization crosstalk term is reduced by 37% when the polarization weight factor takes an appropriate value. And at the same time, the wavefront aberration has also been well optimized.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Polarization aberration is defined as the changes in amplitude, phase, and polarization of the optical wavefront at the exit pupil of the optical system, and these changes will occur with changes in wavelength and object coordinates. The main causes of polarization aberrations are Fresnel effects, thin film coating effects, birefringence of the medium, and birefringence caused by stress, moreover, the geometry of the lens can also alter coordinate systems [1]. In addition, polarization aberration may also occur due to imperfect polarization optical components or incorrect alignment of polarization components in the optical system [2].

Polarization aberrations can significantly compromise the image quality of certain high-resolution optical systems. This impact becomes particularly pronounced in exoplanet detection configurations, where PSF ghost images caused by polarization aberrations extend to twice the radius of Airy diffraction patterns, leading to interference with accurate exoplanet measurements [3]. In the polarization remote sensing detection, the effectiveness of the system may be compromised by the presence of system polarization aberrations which can reduce the accuracy of polarimetry [4]. In addition, the polarization aberration of high NA projection lenses can cause various undesirable effects, such as image placement error, best focus shift, critical dimension errors, and so on, leading to the degradation of image quality and process window [5,6].

Currently, there are two main methods for solving polarization aberration: Three-dimensional polarization ray-tracing calculus and polarization aberration function method [7–10]. The former technique is capable of accurately calculating the polarization aberration of the optical system, but it requires a large number of ray samples to effectively analyze the polarization characteristics. Meanwhile, the latter method is an approximation technique that can provide a closer description of the polarization physics characteristics of the system, which represents each polarization aberration coefficient by a Taylor series expansion of the Fresnel coefficients and incident angle. It is important to note that the value of polarization aberration obtained using the Taylor series expansion method may deviate significantly from the actual value when the incident angle is large. To effectively evaluate the impact of polarization aberrations in off-axis multi-reflectors super-resolution optical systems, we propose a new analysis method that offers both high accuracy and speed. Optical designers have a variety of methods to change the polarization aberrations of any particular optical system. Lam pointed out that the method of using cross folded mirrors can appropriately compensate for the polarization aberration generated by each reflector, but there will be significant difficulties in aligning the optical path [11,12]. Jia designed a low polarization film layer to reduce the polarization aberration of large numerical aperture lithography objectives, but cannot zero out diattenuation or retardance for substantial angle and wavelength ranges [13]. Yao proposed the method of increasing the lens group to optimize the polarization aberration of the Cassegrain system which was based on three-dimensional polarization ray-tracing calculus and needed large mount of ray sampling [14]. This may lead to a slow speed and high difficulty in optimizing optical systems, and ultimately lead to optimization falling into a local minima. Therefore, the present study provides a new merit function based on polarization aberration simplified method, which can ensure simultaneous optimization of both wavefront and polarization characteristics of the system, which can suppress the polarization aberration quickly without having to add any optical components.

The remainder of this paper is structured as follows. In Section 2, we present an introduction to existing methods for analyzing polarization aberrations, along with a simplified algorithm for polarization aberrations that is applicable to planar symmetric optical systems. The feasibility of the simplified method is verified in Section 3. In Section 4, we propose a collaborative optimization method for both polarization and wavefront aberrations based on the simplified theory, and outlines the design of a planar symmetric optical system that suppresses polarization aberrations with near distortion free wavefront. Finally, in Section 5, we summarize our conclusions and future directions.

2. Principles and methods

In this section, we propose a new method based of three-dimensional polarization ray-tracing calculus and polarization aberration function for quickly solving polarization aberrations. This method is applicable for planar symmetric optical system composed of lenses and mirrors which are weakly polarized elements.

2.1 Polarization aberration function method

Polarization aberrations in non-scattering, non-depolarizing systems can be readily modeled using a spatially varying Jones matrix. A series of Pauli spin matrices can be decomposed into its Jones matrices, defined as [15]:

$${{\boldsymbol{\mathrm{\sigma}} }_0} = \left[ {\begin{array}{{cc}} 1&0\\ 0&1 \end{array}} \right],{{\boldsymbol{\mathrm{\sigma}}}_1} = \left[ {\begin{array}{{cc}} 1&0\\ 0&{ - 1} \end{array}} \right],{{\boldsymbol{\mathrm{\sigma}}}_2} = \left[ {\begin{array}{{cc}} 0&1\\ 1&0 \end{array}} \right],{{\boldsymbol{\mathrm{\sigma}}}_3} = \left[ {\begin{array}{{cc}} 0&{ - i}\\ i&0 \end{array}} \right], $$
then any Jones matrix can be written as:
$${\mathbf J} = \sum\limits_{i = 0}^3 {{a_i}{{\boldsymbol{\mathrm{\sigma}}}_i}}, $$
with complex coefficients, ai. The Identity Matrix and Pauli Spin Matrices and their eigenstates as listed in Table 1.

Tables Icon

Table 1. Identity Matrix and Pauli Spin Matrices and their eigenstates

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Optical systems with isotropic surfaces have no circular diattenuation and circular retardence, so the σ3 is usually ignored. This matrix, denoted as the polarization aberration function J(ρ,ϕ), is a function of the polar pupil coordinates ρ, the normalized radial distance, and ϕ, the azimuth measured from the x-axis. By decomposing the polarization aberration function using the Pauli matrices, one can derive the diattenuation and retardance pupils of the polarization aberration function [16]:

$${{\mathbf J}_{dia}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + {d_0}{{\boldsymbol{\mathrm{\sigma}}}_1} + {d_1}\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ {d_2}{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } ),$$
$${{\mathbf J}_{ret}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + i{\Delta _0}{{\boldsymbol{\mathrm{\sigma}}}_1} + i{\Delta _1}\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ i{\Delta _2}{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } ).$$
where d0 is coefficient of diattenuation piston, d1 is coefficient of diattenuation tilt, d2 is coefficient of diattenuation defocus; Δ0 is coefficient of retardance piston, Δ1 is coefficient of retardance tilt, Δ2 is coefficient of retardance defocus.

2.2 Simplified calculation of polarization aberration

When the elements in an optical system rotate around a certain axis or off axis, the degree of symmetry is reduced, it becomes a plane symmetric optical system that is symmetric about one plane [17,18]. As shown in Fig. 1, the surface rotates around the x-axis at an angle of I to the optical axis before tilted, the system becomes a plane symmetric optical system that is symmetric about y-z plane. After tilt, the optical surface curvature center of is point C.C. The coordinate of filed is h and the coordinate of pupil is ρ.

 figure: Fig. 1.

Fig. 1. Plane symmetric system with one tilted surface.

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In Fig. 1, OAR refers to optical axis ray. Due to the plane symmetric character of system, the angle of incidence i will also exhibit plane symmetry characteristics. The incidence angle i is between the ray propagation direction vector R and the normal N:

$$i = \arcsin |{{\mathbf R} \times {\mathbf N}} |. $$

In terms of rays vectors R and surface normals N lay on a symmetric plane can be represented by pupil coordinates ρ :

$${\mathbf N} = {\left( {\begin{array}{{ccc}} 0&{\textrm{sinI} + \frac{\rho }{r}}&{\cos \left[ {\arcsin \left( {\textrm{sinI} + \frac{\rho }{r}} \right)} \right]} \end{array}} \right)^\textrm{T}}, $$
$${\mathbf R} = {\left( {\begin{array}{{ccc}} 0&{\sin \left[ {\arctan \frac{{({h + \rho } )}}{L}} \right]}&{\cos \left[ {\arctan \frac{{({h + \rho } )}}{L}} \right]} \end{array}} \right)^\textrm{T}}, $$
where r represents the radius of the surface, and L represents the distance from an object to the intersection of light and surface. Therefore, according to Eq. (5), the incidence angles lay on y-z plane can be expressed by:
$$i(\rho )\approx \sin \left( {\arctan \frac{{({h + \rho } )}}{L}} \right)\cos \left( {\arcsin \left( {\sin I + \frac{\rho }{r}} \right)} \right) - \cos \left( {\arctan \frac{{({h + \rho } )}}{L}} \right)\left( {\textrm{sinI} + \frac{\rho }{r}} \right).$$

The diattenuation of reflector defined as the difference of reflectance between p-polarized light and s-polarized light, and retardance is the difference of phases:

$$D = \left|{\frac{{\alpha_s^2(i )- \alpha_p^2(i )}}{{\alpha_s^2(i )+ \alpha_p^2(i )}}} \right|, $$
$$\delta = |{{\varphi_s}(i )- {\varphi_p}(i )} |, $$
where α is the reflective coefficient of the mirror surface or the refractive coefficient of the lens surface, φ is phase reflective coefficient or refractive coefficient, and i is incidence angle. The diattenuation and retardance of a bare aluminum reflector at different incident angles are shown in Fig. 2.

 figure: Fig. 2.

Fig. 2. The diattenuation and retardance curves of uncoated aluminum mirror.

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The solid line represents the accurate value, and the dashed line represents the quadratic approximate value. The diattenuation D(i) and retardance δ(i) of interfaces are well approximated for the isotropic coatings and uncoated interfaces at small angles of incidence i by simple quadratic equations:

$$D(i )\approx {D_2}{i^2},\delta (i )\approx {\delta _2}{i^2}. $$

Combining the Eqs. (8) and (11), we can obtain expressions for the variation of diattenuation and retardance with pupil coordinates distributed on the y-z plane,

$$D(\rho )\approx {d_2}{^{\prime}}{\rho ^2} + {d_1}{^{\prime}}\rho + {d_0}{^{\prime}}, $$
$$\delta (\rho )\approx {\Delta _2}{^{\prime}}{\rho ^2} + {\Delta _1}{^{\prime}}\rho + {\Delta _0}{^{\prime}}. $$

According to Eqs. (12) and (13), it can be seen that the relationship between the polarization aberration values on the symmetry plane and the pupil coordinates approximated as a quadratic trinomial. Where the pupil coordinate quadratic coefficients d2’ and Δ2’ are the defocus coefficients; the pupil coordinate linear coefficients d1’ and Δ1’ are the tilt coefficients; the constant coefficients d0’ and Δ0’ are the piston coefficients.

From Reference 7, the three-dimensional polarization ray-tracing calculus can calculate the diattenuation and retardance of an optical systems accurately. We can calculate the diattenuation and retardance values when the marginal (ρ=±1) and central (ρ=0) light of the symmetric plane incident, and express them as follows:

$$\begin{array}{l} {D^ + } = D({\rho = 1} )= {d_2}{^{\prime} + }{d_1}{^{\prime} + }{d_0}{^{\prime} ,}\begin{array}{{cc}} {}&{} \end{array}{\delta ^ + } = \delta ({\rho = 1} )= {\Delta _2}{^{\prime} + }{\Delta _1}{^{\prime} + }{\Delta _0}{^{\prime} ,}\\ {D^o} = D({\rho = 0} )= {d_0}{^{\prime} ,}\begin{array}{{cccc}} {\begin{array}{{cc}} {}&{} \end{array}}&{}&{}&{{\delta ^o} = \delta ({\rho = 0} )= {\Delta _0}{^{\prime},}} \end{array}\\ {D^ - } = D({\rho ={-} 1} )= {d_2}{^{\prime} - }{d_1}{^{\prime} + }{d_0}{^{\prime} ,}\begin{array}{{cc}} {}&{{\delta ^ - } = \delta ({\rho ={-} 1} )= {\Delta _2}{^{\prime} - }{\Delta _1}{^{\prime} + }{\Delta _0}{^{\prime} }{.}} \end{array} \end{array}$$

After simplifying Eq (14), each polarization aberration coefficient can be represented by the three specific polarization aberration which are calculated by three-dimensional polarization ray-tracing calculus,

$$\begin{array}{l} {d_2}{^{\prime}} = ({{D^ + } + {D^ - } - 2{D^o}} )/2\begin{array}{{cc}} ,&{} \end{array}{\Delta _2}{^{\prime}} = ({{\delta^ + } + {\delta^ - } - 2{\delta^o}} )/2,\\ {d_1}{^{\prime}} = ({{D^ + } - {D^ - }} )/2\begin{array}{{cc}} {\begin{array}{{cc}} ,&{} \end{array}}&{} \end{array}\begin{array}{{cc}} {}&{{\Delta _1}{^{\prime}} = ({{\delta^ + } - {\delta^ - }} )/2,} \end{array}\\ {d_0}{^{\prime}} = {D^o}\begin{array}{{cccc}} {\begin{array}{{cc}} {\begin{array}{{cc}} {\begin{array}{{cc}} ,&{} \end{array}}&{} \end{array}}&{} \end{array}}&{}&{}&{} \end{array}{\Delta _0}{^{\prime}} = {\delta ^o}. \end{array}$$

Substitute the above polarization aberration coefficients into polarization aberration function Eqs. (3) and (4) to obtain the diattenuation and retardance of the entire pupil of the system:

$$\scalebox{0.87}{$\displaystyle{{\mathbf J}_{dia}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + {D^o}{{\boldsymbol{\mathrm{\sigma}}}_1} + [{({{D^ + } - {D^ - }} )/2} ]\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ [{({{D^ + } + {D^ - } - 2{D^o}} )/2} ]{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } )$}$$
$$\scalebox{0.89}{$\displaystyle{{\mathbf J}_{ret}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + i{\delta ^o}{{\boldsymbol{\mathrm{\sigma}}}_1} + i[{({{\delta^ + } - {\delta^ - }} )/2} ]\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ i[{({{\delta^ + } + {\delta^ - } - 2{\delta^o}} )/2} ]{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } )$}$$

3. Simulation verification of plane symmetric optical systems

In this study, two different polarization aberrations results of a plane symmetric optical systems calculated by the simplified method and three-dimensional polarization ray-tracing calculus are given.

3.1 Polarization aberrations calculated by the simplified method

In this section, we calculate the diattenuation and the retardance pupils using the simplified method. This system is a plane symmetric, and the symmetry plane is y-z plane, as shown in Fig. 3.

 figure: Fig. 3.

Fig. 3. Plane symmetric system structure.

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The diattenuation and retardance values when the marginal (ρ=±1) and central (ρ=0) light of the symmetric plane incident calculated by three-dimensional polarization ray-tracing calculus and the coefficients solved by Eq. (15) are listed in Table 2.

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Table 2. Polarization aberration values and aberration coefficients

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Using the aberration coefficients listed in Table 2 and combining the Eqs. (16) and (17), the approximate diattenuation and retardance pupils calculated by the method proposed in section 2 are shown in Fig. 4. The accurate diattenuation and retardance pupils calculated by three-dimensional polarization ray-tracing calculus are shown in Fig. 5. Where (a) and (b) describe the diattenuation and the retardance of this optical system respectively, the length of each line represents the magnitude, and the line's orientation corresponds to the orientation of the polarization aberration. Due to the eccentricity of the field of view in this system, the angle of incidence of each mirror gradually increases from the negative direction of the pupil to the positive direction. Therefore, amplitude of the polarization aberration gradually increases from the negative direction of the pupil to the positive direction. Besides, due to the system is plane symmetric, the aberrations are symmetric about symmetry plane y-z. Significantly, comparing Figs. 4 and 5 calculated by this two methods respectively, the polarization aberrations distributions are highly similar, and the trend of change is consistent.

 figure: Fig. 4.

Fig. 4. Diattenuation and Retardance pupils calculated by simplified method.

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 figure: Fig. 5.

Fig. 5. Diattenuation and Retardance pupils calculated by three-dimensional polarization ray-tracing calculus.

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Moreover, the difference in the average polarization aberration values obtained by these two methods under different fields of view is also very small, as shown in Fig. 6.

 figure: Fig. 6.

Fig. 6. Average values of diattenuation and retardance obtained from three-dimensional polarized ray tracing and simplified algorithm for different field of view.

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3.2 Verify the simplified calculation

In this section, we provide evaluation criteria which can validate the practicality of the simplified polarization aberration calculation. It is known that the simplified calculation is an approximate solution method, and the three-dimensional polarization ray-tracing calculus is an accurate solution method. So we proposed that using the root mean square error (RMSE) of the difference between the two results as the evaluation criterion,

$${D_{RMSE}} = \sqrt {\frac{{{{\sum\limits_{i = 1}^n {({{D_{3Di}} - {D_i}} )} }^2}}}{n}}, $$
$${\delta _{RMSE}} = \sqrt {\frac{{{{\sum\limits_{i = 1}^n {({{\delta_{3Di}} - {\delta_i}} )} }^2}}}{n}}, $$
where n represents the number of pupil sampling, i is the sort of pupil coordinates, D3D and δ3D are the accurate values, the Di and δi are the approximate values.

According to the Eqs. (18) and (19), the RMSE of diattenuation and retardance of different field of view are shown in Table 3.

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Table 3. RMSE values of different field of view

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The RMSEs of diattenuation and retardance are close to zero and the difference between these two methods is about 10−5 times the accurate value. It means that the error value can be acceptable for most optical systems. Therefore, the simplified polarization aberration calculation is practical. Last but not least, the sampling number of the simplified polarization aberration calculation is three, and the minimum sampling number for polarized ray tracing pupil applied in commercial optical software Zemax is 32 × 32. So that the sampling rate of the method proposed in this article is 3/(32 × 32) times that of the three-dimensional polarization ray-tracing calculus. This simplified method not only has high accuracy in the calculation results, but also accelerates the calculation speed.

4. Low polarization optical system based on simplification method

Many optical systems are desired to be non-polarizing ideally, the diattenuation and retardance would be zero everywhere. However, it is quite complex project to design a non-polarizing optical system, the researchers only tried to reduce the polarization aberration as soon as possible to meet the requirements of non-polarized systems. In this section, we will optimize an optical system based on the simplified polarization aberration calculation method proposed in Section 3 quickly. Due to the fact that the polarization aberration simplification algorithm mentioned in the previous section has a light sampling rate of 0.003 times that of traditional methods, the calculation speed of polarization aberration is 145 times faster than traditional methods. Resulting in significantly faster optimization speed of the system.

4.1 Wavefront and polarization quality

The ideal wavefront exiting an imaging system is a spherical wavefront with a uniform amplitude and a uniform polarization state. One metric for wavefront quality is the RMS wavefront aberration, the square root of the wavefront aberration integrated over the aperture and normalized by the area,

$$\Delta {W_{RMS}}\frac{{\sqrt {\int\!\!\!\int\limits_{pupil} {{W^2}({\rho ,\phi } )d\rho d\phi } } }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}, $$
where the W represents the wavefront aberration coefficients.

According to Eqs. (16) and (17), we can describe the RMS diattenuation and retardance of weak polarization properties as:

$$\scalebox{0.83}{$\displaystyle{D_{RMS}} = \frac{{\sqrt {\int\!\!\!\int\limits_{pupil} {\left[ {{{\left( {{D^o} - \frac{{{D^ + } - {D^ - }}}{2}\rho \sin \phi + \frac{{{D^ + } + {D^ - } - 2{D^o}}}{2}{\rho^2}\cos 2\phi } \right)}^2} + {{\left( {\frac{{{D^ + } - {D^ - }}}{2}\rho \cos \phi + \frac{{{D^ + } + {D^ - } - 2{D^o}}}{2}{\rho^2}\sin 2\phi } \right)}^2}} \right]d\rho d\phi } } }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}$}$$
$$\scalebox{0.85}{$\displaystyle{\delta _{RMS}} = \frac{{\sqrt {\int\!\!\!\int\limits_{pupil} {\left[ {{{\left( {{\delta^o} - \frac{{{\delta^ + } - {\delta^ - }}}{2}\rho \sin \phi + \frac{{{\delta^ + } + {\delta^ - } - 2{\delta^o}}}{2}{\rho^2}\cos 2\phi } \right)}^2} + {{\left( {\frac{{{\delta^ + } - {\delta^ - }}}{2}\rho \cos \phi + \frac{{{\delta^ + } + {\delta^ - } - 2{\delta^o}}}{2}{\rho^2}\sin 2\phi } \right)}^2}} \right]d\rho d\phi } } }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}$}$$

Thus the RMS polarization aberration, PolRMS, indicates how far the pupil is from non-polarizing,

$$Po{l_{RMS}} = \frac{{\sqrt {D_{RMS}^2 + \delta _{RMS}^2} }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}. $$

To design an optical system, we design the merit function can optimize the wavefront aberration and polarization aberrations simultaneously,

$${\Phi ^2} = \frac{{{w_1}\Delta W_{RMS}^2 + {w_2}Pol_{RMS}^2}}{{{w_1} + {w_2}}}, $$
where w represents the weight. We write the Eq. (24) in Zemax macro and the design process is illustrated in Fig. 7. It is known that in the newly established merit function, only the diattenuation and retardance values of three pupil points are used in the polarization aberration section, which can greatly improve the optimization speed of optical system aberrations.

 figure: Fig. 7.

Fig. 7. Optimization design process of optical systems.

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4.2 Optical system design results

In this section, three different design results are obtained by changing the wavefront aberration and polarization aberration weight factors w1 and w2. The system consists of three even degree aspherical mirrors, with a focal length of 1000 mm and a F-number of 1.9. The wavelength range of the system is 8-12 µm. We analyze the polarization image difference generated by the optical system at wavelength of 8µm. We assume that the coating of the reflector is all metallic aluminum with a complex refractive index of n = 7.08 + 40.83i and the maximum field of the system is 1°. The main optical parameters of the planar symmetric optical system with three off axis mirrors are listed in Table 4.

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Table 4. The main optical parameters of three different design results

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The optical structure diagrams of the three design results are shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. Optical system structure diagrams for three design results: (a) Design1; (b) Design2; (c) Design3.

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4.3 Polarization aberration optimization results

We analyzed the polarization aberration of the three different design results. To demonstrate a physically intuitive interpretation of the Jones matrix, the singular value decomposition (SVD) theorem is used to decompose the Jones matrix into two parts,

$${\mathbf J = \mathbf{WD}}{{\mathbf W}^{\dagger}} \cdot {\mathbf{WV}}, $$
where † is the conjugate transpose, WDW† is Hermitian components of the polar decomposition of J associated with diattenuation aberration; WV is the pure retarder associated with retardance aberration.

The diattenuation and retardace pupils are shown in Figs. 9 and 10 when the polarization aberration weight factors w2 are 0.5, 1, and 2. It can be seen that the diattenuation and retardance exhibits planar symmetry, this phenomenon is mainly due to the fact that the optical system is also planar symmetric. When the polarization aberration weight factor is 0.5, the diattenuation of the optical system is shown in Fig. 9 (a). The Fig. 9 (b) represents the diattenuation for the weight factor is 1, and Fig. 9 (c) is the diattenuation for the weight factor is 2. It can be seen that the diattenuation of the optical system significantly decreases with the increase of the polarization aberration weight factor. The average values of diattenuation of three design results are show in Fig. 9 (d). The average diattenuation value of Design 2 is 75% of Design 1, and Design 3 is 62% of Design 1.

 figure: Fig. 9.

Fig. 9. The Diattenuation pupils of three design results and the average values

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 figure: Fig. 10.

Fig. 10. The Retardance pupils of three design results and the average values.

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Similarly, when the polarization aberration weight factor is 0.5, the retardance of the optical system is shown in Fig. 10 (a). The Fig. 10 (b) presents the retardance for the weight factor is 1, and Fig. 10 (c) is the retardance for the weight factor is 2. The average values of retardance of three design results are show in Fig. 10 (d). The average retardance value of Design 2 is 71% of Design 1, and Design 3 is 58% of Design 1. It can be seen that the retardance of the optical system significantly decreases with the increase of the polarization aberration weight factor. Therefore, the method of simultaneously optimizing polarization aberration and wave aberration proposed in section 4.1 is effective.

4.4 Jones pupils

Each incident of light transmitted through the optical system corresponds to a Jones matrix [19]. Jones pupils are the set of Jones matrices of specific object points at different exit pupil positions, composed of a 2 × 2 Jones matrix [20]:

$$\begin{aligned} {\mathbf{JP}}({x,y} )&= \left( {\begin{array}{{cc}} {{J_{\textrm{XX}}}({x,y} )}&{{J_{\textrm{X}\textrm{Y}}}({x,y} )}\\ {{J_{\textrm{YX}}}({x,y} )}&{{J_{\textrm{YY}}}({x,y} )} \end{array}} \right)\\& = \left( {\begin{array}{{cc}} {{{A}_{{XX}}}({x,y} ){e^{i{\phi_{{XX}}}({x,y} )}}}&{{{A}_{{XY}}}({x,y} ){e^{i{\phi_{{XY}}}({x,y} )}}}\\ {{{A}_{{YX}}}({x,y} ){e^{i{\phi_{{YX}}}({x,y} )}}}&{{{A}_{{YY}}}({x,y} ){e^{i{\phi_{{YY}}}({x,y} )}}} \end{array}} \right) \end{aligned}.$$

In Eq. (26), (x, y) represents the ray coordinates at the exit pupil. Due to the existence of polarization aberrations, there is a difference between Jones and identity matrix. Because of the reflection loss caused by the inherent reflectivity of each reflective surface in the system, the AXX and AYY are smaller than 1. The off-diagonal elements are polarization coupled energy, AXY represents the energy of light coupled to the y direction after incident polarized light in the X direction into the system and AYX represents the energy of light coupled to the x direction after incident polarized light in the y direction. The term ϕXY indicates the X-polarized phase shift from the entrance pupil to the exit pupil. Similarly, ϕYX is the phase shift from the X-polarized light coupled the Y-polarized light at the exit pupil. The off-diagonal elements ϕXY and ϕYX change discontinuously because the phase of a complex number changes by π when amplitude passes through zero due to the Fresnel's law.

In Fig. 11, the Jones pupil diagrams for three different design results are presented with different polarization aberration weight factors of 0.5, 1, and 2, respectively. They are Design 1, Design 2, and Design 3. Obviously, with the increase of polarization aberration weight factor, the optimized design results show a trend that AXX is closer to AYY, and ϕXX is closer to ϕYY, making the Jones pupil of the system closer to the identity matrix. The crosstalk terms of non-diagonal elements have also been significantly reduced, close to zero, indicating that the optimization method can not only reduce the polarization aberration of the system, but also reduce the polarization cross talk caused by polarization aberration. The polarization crosstalk term Axy and Ayx decrease by 37% when the weight factor of polarization aberration is suitable.

 figure: Fig. 11.

Fig. 11. The Jones pupils of three design results.

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4.5 Comprehensive evaluation indicators for optical systems

This section provides comprehensive evaluation indicators for polarization aberration and wave aberration. For incoherent light, a point spread function (PSF) describes the response of the system to a point source object in intensity space [21]. Similarly, a Mueller point spread matrix (MPSM) can relate the Stokes parameters of a point object to the Stokes parameter distribution of its image. MPSM can be obtained by Jones pupils’ Fourier and mathematical transformation,

$${\mathbf{MPSM}} = {\mathbf U} \cdot \{{{{({\Im [{{\mathbf{JP}}} ]} )}^\ast } \otimes ({\Im [{{\mathbf{JP}}} ]} )} \}\cdot {{\mathbf{U}}^{ - 1}},$$
where JP represents Jones pupils (see Eq. (26)), the $\otimes$ represents tensor product, and U is an the unitary matrix:
$${\mathbf U} = \frac{1}{2}\left( {\begin{array}{{cccc}} 1&0&0&1\\ 1&0&0&{ - 1}\\ 0&1&1&0\\ 0&i&{ - i}&0 \end{array}} \right). $$

According to the analysis in Section 4.4, it has been determined that the polarization crosstalk in the optical system gradually decreases as the weight factor of polarization aberration increases. Therefore, we compared the values of MPSM with the maximum and minimum polarization aberration weight factors, the Design1 and Design 3, as shown in Fig. 12. Figure 13 provides the polarization crosstalk terms in log10 scale of Design 1 and Design 3. It can be observed that the values of the non-diagonal crosstalk elements of MPSM, with a large polarization aberration weight factor (2, Design3), are 0.65-0.8 times those with a small polarization aberration weight factor (0.5, Design1). This indicates that the optimization method effectively minimizes polarization crosstalk in the optical system. When the system is used for polarization detection, the interference between Stokes images will be reduced, and the polarization detection accuracy of the system can be improved.

 figure: Fig. 12.

Fig. 12. The MPSM of Design1 and Design3.

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 figure: Fig. 13.

Fig. 13. Comparison of peak value of polarization crosstalk terms of MPSM in Design 1 and Design 3.

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In traditional methods of optimizing polarization aberrations, it often leads to wavefront image quality deterioration in optical systems. From Fig. 14, it can be seen that the optical system maintains good wavefront image quality in all three design scenarios, and the OTF is close to the diffraction limit. In Design 1, we give the merit function a polarization aberration factor 0.5, the wavefront image quality is good, and in Design 2 and Design 3, we increase the polarization aberration factor to 1 and 2, the wavefront imaging quality remains in good condition. At the same time, the polarization aberration of designs 2 and 3 is gradually decreasing. It shows that the method of simultaneous optimization of wavefront aberration and polarization aberration proposed in the paper can consider two kinds of aberrations at the same time, and finally achieve the effect of polarization aberration reduction, good wavefront image quality, and improving the imaging performance of the system while ensuring the polarimetry accuracy of the system. Therefore, Design 3 with good wavefront image quality and minimal polarization aberration can be selected as the final design result.

 figure: Fig. 14.

Fig. 14. The OTF of Design1, Design2 and Design3.

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5. Conclusion

Polarization aberration is a common issue in both optical systems and polarization detection equipment, and is considered an inherent instrument error. The polarization aberrations analyze and optimize typically need a lot of three-dimensional polarization ray tracing for previous non rotational symmetric optical systems. In this work, we propose a simplified polarization aberration analysis method that is suitable for planar symmetric systems. The polarization aberration coefficient is approximated by the polarization aberration of chief ray and marginal ray on the symmetric plane which can extend to the entire pupil surface to calculate the polarization aberration at any coordinate within the pupil. The simplified method combined the advantages of three-dimensional polarization ray tracing and polarization aberration function method. The final calculation result was accurate meanwhile avoided the complexity of the calculation process.

For planar symmetric optical systems in this article, the polarization aberration pupils were achieved by using the simplified method only need three ray sampling. The sampling rate is 0.003 times that of traditional methods, while still maintaining accuracy close to the three-dimensional polarization ray-tracing calculus. Furthermore, based on this simplified method and combined with the wavefront aberration function, a merit function which can improve the optimization speed for optimizing polarization aberrations and wavefront aberration greatly was established. Besides it can also reduce the polarization aberrations and polarization crosstalk of the system effectively. It was found that the diattenuation decreases to 62% of the original value, the retardance decreases to 58%, and the polarization crosstalk term decreases by 37% when the weight factor of polarization aberration is suitable. Not only that, the optical transfer function of the system can still approach the diffraction limit and the wavefront aberration is also well optimized.

Funding

National Natural Science Foundation of China (62375027, 61805027, 61705019, 62127813); General Program of Chongqing Natural Science Foundation (CSTB2023NSCQ-MSX0504); Natural Science Foundation of Jilin Province (222621JC010498735, YDZJ202201ZYTS41); Education Department of Jilin Province (JJKH20220742KJ).

Disclosures

The authors declare that they have no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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7. G. Yun, K. Crabtree, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus I: definition and diattenuation,” Appl. Opt. 50(18), 2855–2865 (2011). [CrossRef]  

8. G. Yun, S. C. McClain, and R. A. Chipman, “Three-dimensional polarization ray-tracing calculus II: retardance,” Appl. Opt. 50(18), 2866–2874 (2011). [CrossRef]  

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11. W. S. Tiffany Lam and R. Chipman, “Balancing polarization aberrations in crossed fold mirrors,” Appl. Opt. 54(11), 3236–3245 (2015). [CrossRef]  

12. S. Banerjee, R. Chipman, and Y. Otani, “Simultaneous balancing of geometric transformation and linear polarizations using six-fold-mirror geometry over the visible region,” Opt. Lett. 45(9), 2510–2513 (2020). [CrossRef]  

13. W. Jia, W. He, Y. Li, Y. Fu, and L. Zhang, “optimization method of multilayer coatings to correct polarization aberration of optical systems,” Opt. Express 30(17), 30611–30622 (2022). [CrossRef]  

14. C. Jiang, D. Yao, L. Meng, C. Yan, and H. Shen, “Suppressing the polarization aberrations by combining reflection and refraction optical groups,” Opt. Express 30(23), 41847–41861 (2022). [CrossRef]  

15. R. A. Chipman, “Polarization aberrations (thin films)”, Ph.D. thesis, University of Arizona (1987).

16. R. A. Chipman, W. S. T. Lam, and G. Young, Polarized light and optical systems (CRC press, 2018).

17. J. M. Sasian, “Review of methods for the design of unsymmetrical optical systems,” in Applications of Optical Engineering: Proceedings of OE/Midwest'90, (SPIE, 1991), 453–466.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (14)
Fig. 1.
Fig. 1. Plane symmetric system with one tilted surface.

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Fig. 2.
Fig. 2. The diattenuation and retardance curves of uncoated aluminum mirror.

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Fig. 3.
Fig. 3. Plane symmetric system structure.

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Fig. 4.
Fig. 4. Diattenuation and Retardance pupils calculated by simplified method.

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Fig. 5.
Fig. 5. Diattenuation and Retardance pupils calculated by three-dimensional polarization ray-tracing calculus.

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Fig. 6.
Fig. 6. Average values of diattenuation and retardance obtained from three-dimensional polarized ray tracing and simplified algorithm for different field of view.

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Fig. 7.
Fig. 7. Optimization design process of optical systems.

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Fig. 8.
Fig. 8. Optical system structure diagrams for three design results: (a) Design1; (b) Design2; (c) Design3.

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Fig. 9.
Fig. 9. The Diattenuation pupils of three design results and the average values

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Fig. 10.
Fig. 10. The Retardance pupils of three design results and the average values.

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Fig. 11.
Fig. 11. The Jones pupils of three design results.

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Fig. 12.
Fig. 12. The MPSM of Design1 and Design3.

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Fig. 13.
Fig. 13. Comparison of peak value of polarization crosstalk terms of MPSM in Design 1 and Design 3.

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Fig. 14.
Fig. 14. The OTF of Design1, Design2 and Design3.

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Tables (4)

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Table 1. Identity Matrix and Pauli Spin Matrices and their eigenstates

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Table 2. Polarization aberration values and aberration coefficients

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Table 3. RMSE values of different field of view

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Table 4. The main optical parameters of three different design results

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Equations (28)

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$${{\boldsymbol{\mathrm{\sigma}} }_0} = \left[ {\begin{array}{{cc}} 1&0\\ 0&1 \end{array}} \right],{{\boldsymbol{\mathrm{\sigma}}}_1} = \left[ {\begin{array}{{cc}} 1&0\\ 0&{ - 1} \end{array}} \right],{{\boldsymbol{\mathrm{\sigma}}}_2} = \left[ {\begin{array}{{cc}} 0&1\\ 1&0 \end{array}} \right],{{\boldsymbol{\mathrm{\sigma}}}_3} = \left[ {\begin{array}{{cc}} 0&{ - i}\\ i&0 \end{array}} \right], $$
$${\mathbf J} = \sum\limits_{i = 0}^3 {{a_i}{{\boldsymbol{\mathrm{\sigma}}}_i}}, $$
$${{\mathbf J}_{dia}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + {d_0}{{\boldsymbol{\mathrm{\sigma}}}_1} + {d_1}\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ {d_2}{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } ),$$
$${{\mathbf J}_{ret}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + i{\Delta _0}{{\boldsymbol{\mathrm{\sigma}}}_1} + i{\Delta _1}\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ i{\Delta _2}{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } ).$$
$$i = \arcsin |{{\mathbf R} \times {\mathbf N}} |. $$
$${\mathbf N} = {\left( {\begin{array}{{ccc}} 0&{\textrm{sinI} + \frac{\rho }{r}}&{\cos \left[ {\arcsin \left( {\textrm{sinI} + \frac{\rho }{r}} \right)} \right]} \end{array}} \right)^\textrm{T}}, $$
$${\mathbf R} = {\left( {\begin{array}{{ccc}} 0&{\sin \left[ {\arctan \frac{{({h + \rho } )}}{L}} \right]}&{\cos \left[ {\arctan \frac{{({h + \rho } )}}{L}} \right]} \end{array}} \right)^\textrm{T}}, $$
$$i(\rho )\approx \sin \left( {\arctan \frac{{({h + \rho } )}}{L}} \right)\cos \left( {\arcsin \left( {\sin I + \frac{\rho }{r}} \right)} \right) - \cos \left( {\arctan \frac{{({h + \rho } )}}{L}} \right)\left( {\textrm{sinI} + \frac{\rho }{r}} \right).$$
$$D = \left|{\frac{{\alpha_s^2(i )- \alpha_p^2(i )}}{{\alpha_s^2(i )+ \alpha_p^2(i )}}} \right|, $$
$$\delta = |{{\varphi_s}(i )- {\varphi_p}(i )} |, $$
$$D(i )\approx {D_2}{i^2},\delta (i )\approx {\delta _2}{i^2}. $$
$$D(\rho )\approx {d_2}{^{\prime}}{\rho ^2} + {d_1}{^{\prime}}\rho + {d_0}{^{\prime}}, $$
$$\delta (\rho )\approx {\Delta _2}{^{\prime}}{\rho ^2} + {\Delta _1}{^{\prime}}\rho + {\Delta _0}{^{\prime}}. $$
$$\begin{array}{l} {D^ + } = D({\rho = 1} )= {d_2}{^{\prime} + }{d_1}{^{\prime} + }{d_0}{^{\prime} ,}\begin{array}{{cc}} {}&{} \end{array}{\delta ^ + } = \delta ({\rho = 1} )= {\Delta _2}{^{\prime} + }{\Delta _1}{^{\prime} + }{\Delta _0}{^{\prime} ,}\\ {D^o} = D({\rho = 0} )= {d_0}{^{\prime} ,}\begin{array}{{cccc}} {\begin{array}{{cc}} {}&{} \end{array}}&{}&{}&{{\delta ^o} = \delta ({\rho = 0} )= {\Delta _0}{^{\prime},}} \end{array}\\ {D^ - } = D({\rho ={-} 1} )= {d_2}{^{\prime} - }{d_1}{^{\prime} + }{d_0}{^{\prime} ,}\begin{array}{{cc}} {}&{{\delta ^ - } = \delta ({\rho ={-} 1} )= {\Delta _2}{^{\prime} - }{\Delta _1}{^{\prime} + }{\Delta _0}{^{\prime} }{.}} \end{array} \end{array}$$
$$\begin{array}{l} {d_2}{^{\prime}} = ({{D^ + } + {D^ - } - 2{D^o}} )/2\begin{array}{{cc}} ,&{} \end{array}{\Delta _2}{^{\prime}} = ({{\delta^ + } + {\delta^ - } - 2{\delta^o}} )/2,\\ {d_1}{^{\prime}} = ({{D^ + } - {D^ - }} )/2\begin{array}{{cc}} {\begin{array}{{cc}} ,&{} \end{array}}&{} \end{array}\begin{array}{{cc}} {}&{{\Delta _1}{^{\prime}} = ({{\delta^ + } - {\delta^ - }} )/2,} \end{array}\\ {d_0}{^{\prime}} = {D^o}\begin{array}{{cccc}} {\begin{array}{{cc}} {\begin{array}{{cc}} {\begin{array}{{cc}} ,&{} \end{array}}&{} \end{array}}&{} \end{array}}&{}&{}&{} \end{array}{\Delta _0}{^{\prime}} = {\delta ^o}. \end{array}$$
$$\scalebox{0.87}{$\displaystyle{{\mathbf J}_{dia}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + {D^o}{{\boldsymbol{\mathrm{\sigma}}}_1} + [{({{D^ + } - {D^ - }} )/2} ]\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ [{({{D^ + } + {D^ - } - 2{D^o}} )/2} ]{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } )$}$$
$$\scalebox{0.89}{$\displaystyle{{\mathbf J}_{ret}} = {{\boldsymbol{\mathrm{\sigma}}}_0} + i{\delta ^o}{{\boldsymbol{\mathrm{\sigma}}}_1} + i[{({{\delta^ + } - {\delta^ - }} )/2} ]\rho ({ - {{\boldsymbol{\mathrm{\sigma}}}_1}\sin \phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\cos \phi } )+ i[{({{\delta^ + } + {\delta^ - } - 2{\delta^o}} )/2} ]{\rho ^2}({{{\boldsymbol{\mathrm{\sigma}}}_1}\cos 2\phi + {{\boldsymbol{\mathrm{\sigma}}}_2}\sin 2\phi } )$}$$
$${D_{RMSE}} = \sqrt {\frac{{{{\sum\limits_{i = 1}^n {({{D_{3Di}} - {D_i}} )} }^2}}}{n}}, $$
$${\delta _{RMSE}} = \sqrt {\frac{{{{\sum\limits_{i = 1}^n {({{\delta_{3Di}} - {\delta_i}} )} }^2}}}{n}}, $$
$$\Delta {W_{RMS}}\frac{{\sqrt {\int\!\!\!\int\limits_{pupil} {{W^2}({\rho ,\phi } )d\rho d\phi } } }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}, $$
$$\scalebox{0.83}{$\displaystyle{D_{RMS}} = \frac{{\sqrt {\int\!\!\!\int\limits_{pupil} {\left[ {{{\left( {{D^o} - \frac{{{D^ + } - {D^ - }}}{2}\rho \sin \phi + \frac{{{D^ + } + {D^ - } - 2{D^o}}}{2}{\rho^2}\cos 2\phi } \right)}^2} + {{\left( {\frac{{{D^ + } - {D^ - }}}{2}\rho \cos \phi + \frac{{{D^ + } + {D^ - } - 2{D^o}}}{2}{\rho^2}\sin 2\phi } \right)}^2}} \right]d\rho d\phi } } }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}$}$$
$$\scalebox{0.85}{$\displaystyle{\delta _{RMS}} = \frac{{\sqrt {\int\!\!\!\int\limits_{pupil} {\left[ {{{\left( {{\delta^o} - \frac{{{\delta^ + } - {\delta^ - }}}{2}\rho \sin \phi + \frac{{{\delta^ + } + {\delta^ - } - 2{\delta^o}}}{2}{\rho^2}\cos 2\phi } \right)}^2} + {{\left( {\frac{{{\delta^ + } - {\delta^ - }}}{2}\rho \cos \phi + \frac{{{\delta^ + } + {\delta^ - } - 2{\delta^o}}}{2}{\rho^2}\sin 2\phi } \right)}^2}} \right]d\rho d\phi } } }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}$}$$
$$Po{l_{RMS}} = \frac{{\sqrt {D_{RMS}^2 + \delta _{RMS}^2} }}{{\int\!\!\!\int\limits_{pupil} {d\rho d\phi } }}. $$
$${\Phi ^2} = \frac{{{w_1}\Delta W_{RMS}^2 + {w_2}Pol_{RMS}^2}}{{{w_1} + {w_2}}}, $$
$${\mathbf J = \mathbf{WD}}{{\mathbf W}^{\dagger}} \cdot {\mathbf{WV}}, $$
$$\begin{aligned} {\mathbf{JP}}({x,y} )&= \left( {\begin{array}{{cc}} {{J_{\textrm{XX}}}({x,y} )}&{{J_{\textrm{X}\textrm{Y}}}({x,y} )}\\ {{J_{\textrm{YX}}}({x,y} )}&{{J_{\textrm{YY}}}({x,y} )} \end{array}} \right)\\& = \left( {\begin{array}{{cc}} {{{A}_{{XX}}}({x,y} ){e^{i{\phi_{{XX}}}({x,y} )}}}&{{{A}_{{XY}}}({x,y} ){e^{i{\phi_{{XY}}}({x,y} )}}}\\ {{{A}_{{YX}}}({x,y} ){e^{i{\phi_{{YX}}}({x,y} )}}}&{{{A}_{{YY}}}({x,y} ){e^{i{\phi_{{YY}}}({x,y} )}}} \end{array}} \right) \end{aligned}.$$
$${\mathbf{MPSM}} = {\mathbf U} \cdot \{{{{({\Im [{{\mathbf{JP}}} ]} )}^\ast } \otimes ({\Im [{{\mathbf{JP}}} ]} )} \}\cdot {{\mathbf{U}}^{ - 1}},$$
$${\mathbf U} = \frac{1}{2}\left( {\begin{array}{{cccc}} 1&0&0&1\\ 1&0&0&{ - 1}\\ 0&1&1&0\\ 0&i&{ - i}&0 \end{array}} \right). $$
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