Three-dimensional polarization algebra for all polarization sensitive optical systems

Yahong Li; Yuegang Fu; Zhiying Liu; Jianhong Zhou; P. J. Bryanston-Cross; Yan Li; Wenjun He

doi:10.1364/OE.26.014109

1. Introduction

Polarization should be considered in 3D for these cases, e.g., the tightly focused fields of high NA lithography and microscopy objectives [1–5], non-linear optical scattering [6] and near-field optics [7, 8]. Theoretical and experimental studies [9–11] show that polarization aberration will affect the image resolution and accuracy of optical systems, especially for those with a high NA. When a polarization sensitive optical system interacts with an arbitrary 3D polarized light, accurately calculating the polarization properties of this optical system is extremely important to optimize and calibrate the performance of an optical system [12–17].

3D polarization has attracted more and more attention on the tightly focused fields [1–5], non-linear optical scattering [6] and near-field optics [7, 8]. In recent years, some mathematical representations of 3D polarization have been presented, such as Jones vector [18–20], coherency matrix [21–23], Stokes vector [24, 25]. As Jones vector can only be used to describe totally polarized light, the corresponding 3D Jones calculus cannot be available for the depolarizing optical systems. 3D coherency matrix is a 3 × 3 positive semidefinite Hermitian matrix, which can be applied to characterize all polarization states, including total polarization, partial polarization and un-polarization. 3D Stokes vector is defined as a 9 × 1 real vector, which cannot only describe total polarization but also partial polarization and un-polarization. In addition, using 3 × 3 coherency matrix and 9 × 1 Stokes vector, the physical concept of 3D degree of polarization (DoP) has been derived, and it is a useful tool to assess partial polarization of non-planar electromagnetic fields such as optical near fields [7,8].

When an arbitrary 3D polarized field incident on an optical system, it is of significance to quantify the polarization transformation effects of this optical system. Therefore, numerous scholars and experts have already started to explore and seek algebraic methods to accurately calculate the polarization transformation effects in the past several years. G. Yun, R. A. Chipman et al have developed a 3D polarization ray-tracing calculus, essentially, it is a 3D generalization of the traditional Jones calculus [18–20], which can only be used to calculate the diattenuation and retardance aberrations, not for the depolarization effect of the optical system [26,27]. Using Chandrasekhar phase matrices instead of Gell–Mann matrices [28], Colin J.R. Sheppard et al have proposed a 3D polarization algebra, and the resultant 3D Mueller matrix (9 × 9) representing a depolarizing or non-depolarizing optical system with a simpler form [25]. Based on 3 × 3 coherency matrix, a 3D polarization ray tracing calculus (3 × 3) for partially polarized light has been introduced by H. Zhang et al [21]. In this paper, using 9 × 1 coherency vector, we propose a new 3D polarization algebra (9 × 9) to preciously calculate the polarization transformation effects of all polarization sensitive optical system, including depolarizing, non-depolarizing and non-linearly scattering optical systems.

The rest of this paper is organized as follows: Section 2 presents the mathematical model of the proposed 3D polarization algebra, which contains two aspects: 1) definitions of local Cartesian coordinate system for each optical interface and global Cartesian coordinate system for the investigated optical system, 2) transforming the calculated results of 9 × 1 coherency vector into the global Cartesian coordinate system. In Section 3, the proposed 3D polarization algebra is applied to calculate the polarization properties of a high NA microscopy objective (NA = 1.25 immersed in liquid oil). In Section 4, in order to verify the validity of this 3D polarization algebra, the polarization simulations of this high NA optical system are undertaken by VirtualLab Fusion (VLF) software [29,30], which is compared with polarization calculations in Section 3. The comparative results demonstrate that a perfect matching relationship is obtained. Finally, we summarize the main conclusions of the work in Section 5.

2. A new 3D polarization algebra

Using 9 × 1 coherency vector, the new mathematical model of 3D polarization algebra (9 × 9) is proposed in this section.

In a 3D space, for a random electromagnetic field with arbitrary point $r$ and time $t$ , the 3 × 3 coherency matrix [13] is defined as,

Φ_{3 \times 3} = 〈 E (r, t) \otimes E^{†} (r, t) 〉 = (\begin{matrix} 〈 E_{x} (r, t) \cdot E_{x}^{*} (r, t) 〉 & 〈 E_{x} (r, t) \cdot E_{y}^{*} (r, t) 〉 & 〈 E_{x} (r, t) \cdot E_{z}^{*} (r, t) 〉 \\ 〈 E_{y} (r, t) \cdot E_{x}^{*} (r, t) 〉 & 〈 E_{y} (r, t) \cdot E_{y}^{*} (r, t) 〉 & 〈 E_{y} (r, t) \cdot E_{z}^{*} (r, t) 〉 \\ 〈 E_{z} (r, t) \cdot E_{x}^{*} (r, t) 〉 & 〈 E_{z} (r, t) \cdot E_{y}^{*} (r, t) 〉 & 〈 E_{z} (r, t) \cdot E_{z}^{*} (r, t) 〉 \end{matrix}),

where the subscripts

x, y, z

are three orthogonal coordinate axis of a space Cartesian coordinate system

{x, y, z}

,

\otimes

denotes the Kronecker product,

†

represents the complex conjugate transpose,

*

indicates the complex conjugate, and

〈 〉

denotes the average over the measurement time.

Then, using a particular 3 × 3 matrix basis [28] composed of eight linearly independent 3 × 3 Gell-Mann matrices $σ_{i} (i = 1, 2, ..., 8)$ plus the 3 × 3 identity matrix $σ_{0}$ ,

\begin{matrix} σ_{0} = \sqrt{\frac{2}{3}} \cdot (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}), σ_{1} = (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), σ_{2} = (\begin{matrix} 0 & - i & 0 \\ i & 0 & 0 \\ 0 & 0 & 0 \end{matrix}), σ_{3} = (\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & 0 \end{matrix}), \\ σ_{4} = (\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}), σ_{5} = (\begin{matrix} 0 & 0 & - i \\ 0 & 0 & 0 \\ i & 0 & 0 \end{matrix}), σ_{6} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}), σ_{7} = (\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & - i \\ 0 & i & 0 \end{matrix}), σ_{8} = \frac{1}{\sqrt{3}} (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 2 \end{matrix}) . \end{matrix}

Equation (1) can be written as,

Φ_{3 \times 3} = \frac{1}{2} \sum_{i = 0}^{8} s_{i} \cdot σ_{i},

where the nine real coefficients in Eq. (3) are defined as 3D Stokes vector (9 × 1),

S_{9 \times 1} = {(s_{0}, s_{1}, s_{2}, s_{3}, s_{4}, s_{5}, s_{6}, s_{7}, s_{8})}^{T} .

Correspondingly, 3 × 3 coherency matrix in Eqs. (1) and (3) can also be rewritten as 9 × 1 coherency vector,

Φ_{9 \times 1} = {(ϕ_{x x}, ϕ_{x y}, ϕ_{x z}, ϕ_{y x}, ϕ_{y y}, ϕ_{y z}, ϕ_{z x}, ϕ_{z y}, ϕ_{z z})}^{T} .

where

ϕ_{i j}

(

i, j = x, y, z

) are the elements of 3 × 3 coherency matrix.

Equations (4) and (5) can all be used to represent all polarization states, so there exists an inherent transformation relationship matrix,

S_{9 \times 1} = Q_{9 \times 9} \times Φ_{9 \times 1},

where

Q_{9 \times 9}

is a 9 × 9 constant matrix, the detail expression is provided in Appendix A.

According to Eqs. (1) and (5), it is known that the expression of 9 × 1 coherency vector is related to the choice of Cartesian coordinate system ${x, y, z}$ . For the sake of simplicity, we introduce two definitions of local and global Cartesian coordinate systems in the next section, which is necessary to calculate the polarization properties of the investigated optical system.

2.1 Local Cartesian coordinate system

The local and global Cartesian coordinate systems are defined for the investigated optical system, and we derive the general expression of 3D coherency transformation matrix (9 × 9) in a local Cartesian coordinate system, using 9 × 1 coherency vector shown in Eq. (5).

For a given optical system, a local Cartesian coordinate system is defined on each optical interface by which propagation vectors $k_{i}$ of each incident ray are aligned with the coordinate axis $z_{i}$ , and a global Cartesian coordinate system is defined by which the optical axis is along the coordinate axis $z$ , as shown in Fig. 1.

Fig. 1 Local Cartesian coordinate system of the propagation light.

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In a local Cartesian coordinate system, the polarization transformation effect of the investigated optical system can be expressed in terms of 9 × 1 coherency vector,

Φ^{'}_{9 \times 1, l} = N_{9 \times 9, l} \cdot Φ_{9 \times 1, l},

where

Φ_{9 \times 1, l}

and

Φ^{'}_{9 \times 1, l}

are 9 × 1 coherency vectors of incident and exiting lights.

N_{9 \times 9, l}

is 3D coherency transformation matrix (9 × 9) representing the investigated optical system. The subscript

l

means that the labeled operation is performed in the local Cartesian coordinate system.

When the polarization of incident field is given, as long as the 9 × 9 coherency transformation matrix of the investigated optical system can be calculated, the polarization of the exiting light can be obtained by combining Eq. (5) with Eq. (7). Therefore, in the next section, we mainly discuss the calculation algebra of 3D coherency transformation matrix (9 × 9) in a local Cartesian coordinate system.

The electric field is a transverse wave, so the field component $E_{z}$ is always equal to 0 in a local Cartesian coordinate system ( $k_{i} \equiv z_{i}$ ). Thus, the polarization transformation matrix representing the investigated optical interface can be written as,

S_{4 \times 1}^{'} = A_{4 \times 4} \cdot (J_{2 \times 2} \otimes J_{2 \times 2}^{*}) \cdot A_{4 \times 4}^{- 1} \cdot S_{4 \times 1},

S_{4 \times 1} = (\begin{matrix} < E_{x}^{2} > + < E_{y}^{2} > \\ < E_{x}^{2} > - < E_{y}^{2} > \\ < E_{x} E_{y}^{*} > + < E_{x}^{*} E_{y} > \\ i (< E_{x} E_{y}^{*} > - < E_{x}^{*} E_{y} >) \end{matrix}), A_{4 \times 4} = (\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 0 & 0 & - 1 \\ 0 & 1 & 1 & 0 \\ 0 & i & - i & 0 \end{matrix}),

where

J_{2 \times 2}

is the polarization transformation effect on the field components

E_{x}

and

E_{y}

.

Using the physical meaning of 3D Stokes vector include in Eq. (4), the expression of 3D Stokes vector in a local Cartesian coordinate system is derived,

S_{9 \times 1, l} = (\begin{matrix} < E_{x}^{2} > + < E_{y}^{2} > \\ < E_{x} E_{y}^{*} > + < E_{x}^{*} E_{y} > \\ i (< E_{x} E_{y}^{*} > - < E_{x}^{*} E_{y} >) \\ < E_{x}^{2} > - < E_{y}^{2} > \\ 0 \\ 0 \\ 0 \\ 0 \\ < E_{x}^{2} > + < E_{y}^{2} > \end{matrix}) .

Then, combining Eqs. (6)–(10) with Appendix A, we derive the general expression of 3D coherency transformation matrix (9 × 9) in a local Cartesian coordinate system,

N_{9 \times 9, l} = \frac{1}{2} (\begin{matrix} N_{11} & N_{12} & 0 & N_{14} & N_{15} & 0 & 0 & 0 & 0 \\ N_{21} & N_{22} & 0 & N_{24} & N_{25} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ N_{41} & N_{42} & 0 & N_{44} & N_{45} & 0 & 0 & 0 & 0 \\ N_{51} & N_{52} & 0 & N_{54} & N_{55} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}),

where the elements of

N_{9 \times 9, l}

are provided in Appendix B.

2.2 Global Cartesian coordinate system

In this section, we need to transform the calculation results shown in Eq. (11) into the defined global Cartesian coordinate system of the investigated optical system.

For an arbitrary optical system with a certain focal length, the propagation vectors of each incident rays are usually different, as shown in Fig. 1. Assuming that the propagation vector is $k_{i n} = {(a, b, c)}^{T}$ in the global Cartesian coordinate system ${x, y, z}$ , a transformation matrix $T_{3 \times 3}$ is introduced to realize the rotation transformation between the local Cartesian coordinate system and global Cartesian coordinate system, as shown in Fig. 2.

Fig. 2 Rotation transformation of Cartesian coordination systems.

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(\begin{matrix} E_{x} \\ E_{y} \\ E_{z} \end{matrix}) = T_{3 \times 3} \cdot (\begin{matrix} E_{x 1} \\ E_{y 1} \\ E_{z 1} \end{matrix}),

T_{3 \times 3} = (\begin{matrix} \cos α & 0 & - \sin α \\ 0 & 1 & 0 \\ \sin α & 0 & \cos α \end{matrix}) \cdot (\begin{matrix} 1 & 0 & 0 \\ 0 & \cos β & - \sin β \\ 0 & \sin β & \cos β \end{matrix}) = T (k_{i n}),

α = \arctan (b / c), β = \arcsin (a / \sqrt{a^{2} + b^{2} + c^{2}}) .

Combining Eqs. (1) with (12), the expressions of 3 × 3 coherency matrix in the global Cartesian coordinate system ${x, y, z}$ and local Cartesian coordinate system ${x_{n}, y_{n}, z_{n}}$ are derived,

Φ_{3 \times 3} = 〈 E \otimes E^{†} 〉 = 〈 (T (k_{i n}) \cdot E_{l}) \otimes {(T (k_{i n}) \cdot E_{l})}^{†} 〉, Φ_{3 \times 3, l} = 〈 E_{l} \otimes E_{l}^{†} 〉 .

Similarly, Eq. (15) can also be rewritten as two 9 × 1 vectors. Then, a 9 × 9 rotation transformation matrix between the global Cartesian coordinate system ${x, y, z}$ and local Cartesian coordinate system ${x_{n}, y_{n}, z_{n}}$ is derived,

Φ_{9 \times 1} = R_{9 \times 9} \cdot Φ_{9 \times 1}_{, l},

R_{9 \times 9} = T (k_{i n}) \otimes T {(k_{i n})}^{*},

where

R_{9 \times 9}

is the 9 × 9 rotation transformation matrix, which is only related to the introduced transformation matrix

T_{3 \times 3}

shown in Eq. (13).

Combining Eqs. (16)–(17) and (7), the general expression of 3D coherency transformation matrix (9 × 9) in the global Cartesian coordinate system is derived,

N_{9 \times 9} = (T (k_{o u t}) \otimes T {(k_{o u t})}^{*}) \cdot N_{9 \times 9, l} \cdot {(T (k_{i n}) \otimes T {(k_{i n})}^{*})}^{- 1} .

When the investigated optical system has m optical interfaces, the $q^{t h}$ optical interface is labeled by the subscript $q$ ( $q = 1, 2, ..., m$ ). Thus, the total 3D coherency transformation matrix (9 × 9) of the investigated optical system is calculated by,

N_{T o t a l} = \prod_{q = m, - 1}^{1} N_{q} = N_{m} \cdot ... \cdot N_{q} \cdot ... \cdot N_{1} .

Finally, when the polarization of incident light $Φ_{9 \times 1}$ is given, combining with Eq. (5), the 9 × 1 coherency vector of exiting light from the investigated optical system is calculated by,

Φ_{9 \times 1}^{'} = N_{T o t a l} \cdot Φ_{9 \times 1} .

In summary, Eq. (19) can be calculated by the proposed 3D polarization algebra, and it can completely characterize all polarization properties of the investigated optical system, including the retardance, diattenuation and depolarization. It is noteworthy that this algebra is applicable for all polarization sensitive optical systems, especially for the depolarizing optical systems with scatting effect. Combining the calculated Eq. (19) with incident light, we can easily obtain the polarization information of exiting light from the investigated optical system, such as 3D DoP, ellipticity, azimuth and degree of coherence. Moreover, this algebra can be used to handle the incident field with arbitrary polarization state, including totally polarized, partially polarized and unpolarized, whether in 2D or in 3D.

3. Application to a high NA microscopy objective

When a typical polarization sensitive optical system interacts with 3D totally polarized light and 3D partially polarized light, we apply the proposed 3D polarization algebra to calculate the polarization properties in the central and marginal field of view (FoV), respectively.

A high NA microscopy objective is investigated in this paper, and it is a rotationally symmetric optical system, as shown in Fig. 3. To achieve an optimal imaging quality and high resolution, two kinds of aberrations need to be strictly controlled: 1) correcting the field curvature for flattening the imaging field, 2) controlling chromatic aberration for realizing apochromatism. In addition, the polarization influence on imaging quality cannot commonly be ignored. The parameters of the investigated optical system are: NA of 1.25 immersed in oil, object height of 0.11 mm, paraxial magnification of 100, wavelength of 589 nm. The global Cartesian coordinate system is defined at the entrance pupil position, and $z$ axis is aligned with the optical axis.

Fig. 3 A high NA microscope objective (NA = 1.25 immersed in liquid oil).

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Table 1 lists the 9 × 1 coherency vectors of incident 3D totally and partially polarized lights, and the values of 3D DoP [7,8] respectively are: $P_{1} = 1$ and $P_{2} = 1 /2$ . Here, we need to interpret why the 3D polarized light with $P_{2} = 1 /2$ is unpolarized in 2D. From the perspective of statistic, at different times, the vibration directions of 3D polarized light with $P_{2} = 1 /2$ always lie on a 2D plane that is not fixed in a 3D space all the time.

Table 1. 3D coherency vectors (9 × 1) of incident lights.

View Table

In a 3D space, we choose three mutually orthogonal planes as the receiving planes: exit pupil plane, sagittal plane and tangential plane. The calculation results of polarization properties of the investigated high NA optical system are given in terms of three parameters: 3D DoP, ellipticity and azimuth of exiting light. When the incident 3D polarized lights have different polarization states, the polarization properties of the optical system in the central FoV (object height of 0 mm) and marginal FoV (object height of 0.11 mm) are calculated respectively. The theoretical calculation results are shown in Figs. 4–7, in which the horizontal and vertical axes denote the sampling points 256 × 256 of the receiving plane, and the color-bars correspond to the values of 3D DoP, ellipticity and azimuth.

Fig. 4 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D totally polarized light in the central FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.

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Fig. 5 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D totally polarized light in the marginal FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.

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Fig. 6 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light in the central FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.

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Fig. 7 Polarization properties of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light in the marginal FoV. (a) 3D DoP distribution. (b), (c) ellipticity and azimuth distributions on exit pupil plane. (d), (e) ellipticity and azimuth distributions on sagittal plane. (f), (g) ellipticity and azimuth distributions on tangential plane.

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In Figs. 4 and 5, sub-graphs (a) show that the distributions of 3D DoP are always equal to 1, which means that the exiting light is still 3D totally polarized, both in the central FoV and marginal FoV. Therefore, it can be inferred that the investigated high NA optical system may be non-depolarizing. Observing the sub-graphs (b) in Figs. 4 and 5, the uniform distributions of ellipticity are all indicated on the exit pupil, and the values of ellipticity are always equal to 1 (circularly polarized). On the exit pupil, the azimuth distribution is rotationally symmetric in the central FoV, but not in the marginal FoV, which can be clearly seen from sub-graphs (c) in Figs. 4 and 5. Comparing sub-graphs (d) with (f) in Fig. 4, an interesting phenomenon is found that the distribution of ellipticity on sagittal plane is orthogonal to that on tangential plane completely. Furthermore, a same phenomenon is seen for the distributions of azimuth on sagittal and tangential planes by observing sub-graphs (e) and (g) in Fig. 4. However, these orthogonal phenomena on sagittal and tangential planes are not appeared in the marginal FoV by comparing sub-graphs (d)–(g) in Fig. 4 with Fig. 5.

When the incident light is 3D totally polarized, the differences in the distributions of 3D DoP, ellipticity and azimuth are not obvious both in the central FoV and marginal FoV. These differences meanwhile also verified an important conclusion that the polarization transformation effect is proportional to the incident angle. For this high NA microscopic objectives, the change in the FoV does not cause dramatic changes in the incident angle. For example, when the object height is increased from 0 to 0.11mm, the maximum of incident angle has only changed from 49.77° to 52.41°. In addition, as the investigated high NA microscope objective does not include depolarizing or scattering elements, the distributions of 3D DoP are exactly the same in the central FoV and marginal FoV. Moreover, regardless of the central FoV or the marginal FoV, the ellipticity of the exit light is always equal to that of the incident light on the pupil plane. It is not difficult to explain this interesting phenomenon. When the sampled ray passes through an optical interface without coating or scattering, the amplitude transmittance coefficient is always real, that is, the polarization effects do not include retardance. Meanwhile, the optical path length of each sampled ray passing through a given optical system is only related to the incident angle. These distributions on exit pupil plane, sagittal and tangential planes are all (value reverse) symmetrical in the central FoV, but not in the marginal FoV. Similarly, when the 3D totally polarized light shown is interacted with this optical system without coating, scattering or depolarizing elements, the distributions of ellipticity and azimuth on sagittal and tangential planes are symmetric and orthogonal in the central FoV strictly. However, in the marginal FoV, these symmetrical and orthogonal distributions are no longer appeared on sagittal and tangential planes.

When a 3D partially polarized (2D unpolarized) incidents on the investigated high NA optical system, sub-graphs (a) in Figs. 6 and 7 indicate that the distributions of 3D DoP are uniform in the central and marginal FoV, and the values of 1/2 is the same to that of incident light. It is concluded that the investigated high NA optical system shown in Fig. 3 is a non-depolarizing optical system. In addition, a remarkable feature is appeared that, from the sub-graphs (b), (d) and (f) in Figs. 6 and 7, the distributions of ellipticity on the three orthogonally receiving planes are all equal to 0 (linearly polarized). This is due to the fact that the incident polarized light (a 2D un-polarized light in x-y plane) passes through this optical system without coating, scattering and depolarizing elements, therefore the exit light includes only x- and y-components. Moreover, the optical path length in this optical system is constant for each ray. Finally, it is concluded that the projected polarization states of exit light are always linearly polarized on exit pupil, sagittal and tangential planes. Similarly, sub-graphs(c) and (g) in Fig. 6 also show a completely same orthogonal relationship on sagittal and tangential planes. However, from sub-graphs(c), (e) and (g) in Fig. 7, the distribution of azimuth on exit pupil plane is no longer rotationally symmetric in the marginal FoV, and the distributions of azimuth on sagittal and tangential planes are not satisfied with the orthogonal relationship.

4. Simulation and discussion

In order to verify the validity of the proposed 3D polarization algebra, we perform the polarization simulation of the investigated high NA optical system by using the commercial software VLF. The comparative analyses of theoretical calculation and polarization simulation are made in this section.

In the VLF, the polarization simulations of the investigated high NA optical system are given in terms of polarization ellipse on the three mutually orthogonal receiving planes: x-y (exit pupil plane), x-z (sagittal plane) and y-z (tangential plane). The shape of polarization ellipse is determined by ellipticity and azimuth, so there is a one-to-two match relationship between polarization ellipse (one) and elliptic plus azimuthal (two). Figures 8–11 exhibit the polarization simulation results of exiting light from the investigated high NA optical system interacting with different 3D polarized lights shown in Table 1.

Fig. 8 Polarization simulation of exiting light from the investigated high NA optical system interacting with a 3D totally polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the central FoV.

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Fig. 9 Polarization simulation of exiting light from the investigated high NA optical system objective interacting with a 3D totally polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the marginal FoV.

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Fig. 10 Polarization simulation of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the central FoV.

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Fig. 11 Polarization simulation of exiting light from the investigated high NA optical system interacting with a 3D partially polarized light on the: (a) exit pupil plane (b) sagittal plane (c) tangential plane in the marginal FoV.

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When the high NA microscopy objective interacts with a 3D totally polarized light, the polarization simulation of exiting light in the central FoV are shown in Fig. 8. Now, the results shown in Fig. 8 and Fig. 4 can be compared. First, on exit pupil plane, the polarization ellipse of each sampled ray is circularly polarized, that is, the value of ellipticity is 1, which can be verified by sub-graph Fig. 4(b). Moreover, the azimuth of circular polarization along the horizontal axis is always equal to 0, as shown in Fig. 8(a), which is completely consistent with the results shown in sub-graph Fig. 4(c). On sagittal plane, linear polarization appears in the region near the horizontal axis, as shown in Fig. 8(b). It can be indicated that the values of ellipticity in this region are very close to 0. The distribution of ellipticity shown in sub-graph Fig. 4(d) just exhibits a light blue region along the horizontal axis, which is in good agreement with the results shown in Fig. 8(b). On tangential plane, Fig. 8(c) indicates that linear polarizations appear near the vertical axis, which is consistent with the vertically light blue area shown in Fig. 4(f). In addition, observing the Fig. 8 and sub-graphs(e), (f), (g) in Fig. 4, a very good match relationship among the distributions of azimuth can also be verified. In summary, these distribution phenomena are essentially caused by polarization aberration (PA) of the investigated high NA microscopy objective, mainly including skew aberration (SA), diattenuation aberration (DA) and retardance aberration (RA). For this typically high NA optical system without coatings, the distribution characteristics of SA, DA and RA are all symmetrical in the central FoV, but not for the results in the marginal FoV [31,32].

Figure 9 shows the exiting polarization ellipses of the investigated high NA optical system interacting with a 3D totally polarized light in the marginal FoV. Obviously, there is an obvious difference between the polarization ellipses in Figs. 8 and 9. Moreover, the difference caused by varying FoV can be distinguished in Fig. 9(c). In addition, the difference can be quantificationally described by the theoretical calculation results, as shown in the color-bars on the right of the sub-graphs(c), (e) and (f) included in Fig. 5. Observing Fig. 9(a), the circular polarization ellipses are still appeared on exit pupil plane, but the distribution of azimuth is no longer rotationally symmetric. On sagittal plane, linear polarizations still appear near the horizontal axis in Fig. 9(b). Correspondingly, a narrow light blue area can be clearly seen in sub-graph Fig. 5(d). The azimuth of polarization ellipse shown in Fig. 9(b) is value-inversion symmetry about the horizontal axis, which can be verified by the different color areas on either side of horizontal axis shown in Fig. 5(e). On tangential plane, compared with the results shown in Fig. 8(c), it is impossible to appear linear polarization for the exiting light in the marginal FoV, as shown in Fig. 9(c). This can be also proved by the fact that the light blue area is not appeared in Fig. 5 (f).

When the investigated high NA optical system interacts with a 3D partially polarized light, in the central FoV and marginal FoV, the distributions of exiting polarization ellipse are shown in Figs. 10 and 11, respectively. A comparative analysis of polarization simulation and theoretical calculation yields a number of observations. Obviously, Figs. 10 and 11 perfectly match with the results shown in Figs. 6(a) and 7(a). Namely, when the incident light is 3D partially polarized, the exiting light of the investigated high NA optical system is always linearly polarized, whether in the central FoV or in the marginal FoV. All linear polarizations coincide with ellipticity of 0 shown in sub-graphs(b), (d) and (f) in Figs. 6 and 7 very well. On the exit pupil plane, the distribution of polarization ellipse in the central FoV shown in Fig. 10(a) has a special distribution, which is consistent with sub-graph (c) in Fig. 6. Simultaneously, the distribution of polarization ellipse in the marginal FoV shown in Fig. 11(a) has a complete symmetry about the horizontal axis, which is in good agreement with the polarization calculation results in Fig. 7(c). On sagittal plane, from sub-graph(b) in Fig. 10, the distributions of exiting polarization ellipse in the central FoV have opposite azimuths about the vertical axis, as shown in Fig. 6(e). Similarly, on tangential plane, the distributions of exiting polarization ellipse in the central FoV have opposite azimuths about the horizontal axis, as shown in Fig. 6(f). However, the distributions of exiting polarization ellipse in the marginal FoV, the symmetric axes of mutual orthogonality (horizontal axis and vertical axis) are no longer appeared on sagittal and tangential planes, as shown in sub-graphs(b) and (c) included in Fig. 11.

The above comparative analyses of polarization simulations and theoretical calculations demonstrate that a perfect agreement between the two is obtained, and it is concluded that the proposed 3D polarization algebra is validity. Moreover, this new 3D polarization algebra has a wider range of application than the present 3D Jones calculus, especially in the case of partially polarized light passing through depolarizing, non-depolarizing and non-linearly scattering optical systems.

5. Conclusion

Using a 3D coherency vector (9 × 1), a new 3D polarization algebra is presented in this paper, which can be used to calculate the polarization properties of all polarization sensitive optical systems, including depolarizing and non-depolarizing optical systems. Namely, this 3D polarization algebra can be also applied to characterize 3D partially polarized light and 3D un-polarized light than the present 3D Jones calculus. In order to verify the validity of the proposed 3D polarization algebra, the polarization properties of a high NA microscope objective (NA = 1.25 immersed in oil) is calculated theoretically, and the polarization simulations of this optical system are performed by using the commercial software VLF. After a comparative analysis of theoretical calculations and polarization simulations, a perfect matching relation is obtained, which demonstrates the validity of this 3D polarization algebra. In addition, this 3D polarization algebra provides a new design idea of better optimizing optical capability, especially for several complex optical systems with folding mirror or bionic moth-eye microstructure.

Appendix A: Transformation relationship matrix $Q_{9 \times 9}$

Q_{9 \times 9} = (\begin{matrix} \frac{\sqrt{6}}{3} & 0 & 0 & 1 & 0 & 0 & 0 & 0 & \frac{\sqrt{3}}{3} \\ 0 & 1 & - i & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & - i & 0 & 0 & 0 \\ 0 & 1 & i & 0 & 0 & 0 & 0 & 0 & 0 \\ \frac{\sqrt{6}}{3} & 0 & 0 & - 1 & 0 & 0 & 0 & 0 & \frac{\sqrt{3}}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & - i & 0 \\ 0 & 0 & 0 & 0 & 1 & i & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & i & 0 \\ \frac{\sqrt{6}}{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - \frac{2 \sqrt{3}}{3} \end{matrix}) .

Appendix B: 3D coherency transformation matrix $N_{9 \times 9, l}$

\begin{matrix} N_{11} = m_{11} + m_{21} + m_{12} + m_{22}, N_{12} = m_{13} + m_{23} - i \cdot (m_{14} + m_{24}), \\ N_{14} = - m_{13} - m_{23} + i \cdot (m_{14} + m_{24}), N_{15} = m_{11} + m_{21} - m_{12} - m_{22}, \\ N_{21} = m_{31} - i \cdot m_{41} + m_{32} - i \cdot m_{42}, n_{22} = m_{33} - i \cdot m_{43} - i \cdot (m_{34} - i \cdot m_{44}), \\ N_{24} = - m_{33} + + m_{44} + i \cdot (m_{43} + m_{34}), N_{25} = m_{31} - m_{32} - i \cdot (m_{41} - m_{42}), \\ N_{41} = m_{31} + m_{32} + i \cdot (m_{41} + m_{42}), N_{42} = m_{33} + m_{44} - i \cdot (m_{34} - m_{43}), \\ N_{44} = - m_{33} - m_{44} - i \cdot (m_{43} + m_{34}), N_{45} = m_{31} - m_{32} + i \cdot (m_{41} - m_{42}), \\ N_{51} = m_{11} - m_{21} + m_{12} - m_{22}, N_{52} = m_{13} - m_{23} - i \cdot (m_{14} - m_{24}), \\ N_{54} = - m_{13} + m_{23} + i \cdot (m_{14} - m_{24})], N_{55} = m_{11} - m_{21} - m_{12} + m_{22}, \\ N_{i j} = 0 (i = 3, 6, ..., 9, j = 1, 2, ..., 9) . \end{matrix}

where $m_{i j} (i, j = 1, 2, 3, 4)$ are the elements of $M$ ,

M = A_{4 \times 4} \cdot (J_{2 \times 2} \otimes J_{2 \times 2}^{*}) \cdot A_{4 \times 4}^{- 1} .

Funding

National Natural Science Foundation of China (NSFC) (11474037,11474041). “111” Project of China (D17017); Jilin Province Science and Technology Development Project of China (20160520015JH).

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Three-dimensional polarization algebra for all polarization sensitive optical systems

Abstract

1. Introduction

2. A new 3D polarization algebra

2.1 Local Cartesian coordinate system

2.2 Global Cartesian coordinate system

3. Application to a high NA microscopy objective

4. Simulation and discussion

5. Conclusion

Appendix A: Transformation relationship matrix $Q_{9 \times 9}$

Appendix B: 3D coherency transformation matrix $N_{9 \times 9, l}$

Funding

References and links

Cited By

Figures (11)

Tables (1)

Equations (23)

Optics Express

3D DoP	polarization state	coherency vector
$P_{1} = 1$	3D totally polarization	$Φ_{1} =(\begin{matrix} 1 & - i & - i & i & 1 & - i & i & i & 1 \end{matrix})^{T}$
$P_{2} = 1 /2$	3D partially polarization (2D un-polarization)	$Φ_{2} =(\begin{matrix} 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \end{matrix})^{T}$

Abstract

1. Introduction

2. A new 3D polarization algebra

2.1 Local Cartesian coordinate system

2.2 Global Cartesian coordinate system

3. Application to a high NA microscopy objective

4. Simulation and discussion

5. Conclusion

Appendix A: Transformation relationship matrix Q9×9

Appendix B: 3D coherency transformation matrix N9×9,l

Funding

References and links

Cited By

Figures (11)

Tables (1)

Equations (23)

Optics Express

Appendix A: Transformation relationship matrix $Q_{9 \times 9}$

Appendix B: 3D coherency transformation matrix $N_{9 \times 9, l}$