光的偏振

偏振是横波的一种性质,电磁波的偏振态通常是指电场矢量的方向。

实际生活中的光源,大多是非相干光,他们叠加后成为非偏振光随机偏振光

这里我们只讨论简单的模型,即单色平面波。

偏振的表示

假设电磁波沿z方向传播,它的数学表达式可以写作

$$ \vec{\mathbf{E}}= E_{x}\vec{\mathbf{x}} +E_{y}\vec{\mathbf{y}} $$

复数域表示为

$$ \widetilde{\mathbf{E}}= \widetilde{E_{x}}\vec{\mathbf{x}} +\widetilde{E_{y}}\vec{\mathbf{y}} $$

其中

$$ \widetilde{E_{x}} = E_{0x}e^{i(kz-\omega t+\varphi _{x})} $$

$$ \widetilde{E_{y}} = E_{0y}e^{i(kz-\omega t+\varphi _{y})} $$

所以电场在复数域的表示可以写为

$$ \widetilde{\mathbf{E}} = \left [ E_{0x}e^{i\varphi _{x}} \vec{\mathbf{x}} + E_{0y}e^{i\varphi _{y}} \vec{\mathbf{y}} \right ]e^{i(kz-\omega t)} = \widetilde{ \mathbf {E_{0}}} e^{i(kz-\omega t)} $$

因为偏振态完全由电场分量的相对强度和相位决定,根据 \( \widetilde{ \mathbf {E_{0}}} \)就可以完全确定光的偏振态,其矩阵形式称为琼斯矢量(Jones vector)

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} E_{0x}e^{i\varphi_{x}} \\ E_{0y}e^{i\varphi _{y}} \end{bmatrix} \end{equation}

琼斯矢量无法表示 非偏振光随机偏振光

线偏振光

任意\( \alpha\)的线偏光,相对相位差为0π,可以让\( \varphi_{x} =\varphi_{y} = 0\)

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} Acos\left ( \alpha \right ) \\ Asin\left ( \alpha \right ) \end{bmatrix} \end{equation}

为了使 \( \widetilde{ \mathbf {E_{0}}} \) 归一化,此处 A = 1

假设 \( \varphi_{x} = 0 \),\( \varphi_{y} = \varepsilon \) ,定义相位差 \( \Delta \varphi = \varphi_{y} – \varphi_{x} = \varepsilon \)

当相位差为\( +\pi /2 \)时,为左旋光(left-circularly polarized,LCP),电场矢量的箭头逆时针旋转

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} Acos\left ( \alpha \right ) \\ Asin\left ( \alpha \right ) \end{bmatrix} =\begin{bmatrix} A \\ A e^{i\pi /2} \end{bmatrix} =A\begin{bmatrix} 1 \\ i \end{bmatrix} \end{equation}

此处\( A =\frac{1}{\sqrt{2}} \)

当相位差为\( -\pi /2 \)时,为右旋光(right-circularly polarized,RCP) ,电场矢量的箭头顺时针旋转

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ -i \end{bmatrix} \end{equation}

圆偏振光

更加一般的形式

顺时针

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} A \\iB \end{bmatrix} \end{equation}

逆时针

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} A \\-iB \end{bmatrix} \end{equation}

椭圆偏振光

顺时针

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} A \\ B+iC \end{bmatrix} \end{equation}

逆时针

\begin{equation} \widetilde{ \mathbf {E_{0}}} = \begin{bmatrix} A \\ B-iC \end{bmatrix} \end{equation}

更加一般的形式的描述,椭圆方程

\begin{equation} \frac{ \widetilde{ E_{x}}^{2}} {E_{0x}^{2}} + \frac{ \widetilde{ E_{y}}^{2}} {E_{0y}^{2}} – \frac{ 2\widetilde{ E_{x}}\widetilde{ E_{y}}} {E_{0x}E_{0y}} cos\varepsilon = sin^{2}\varepsilon \end{equation}

椭圆的参数可由方向角(orientation angle)\( \psi\left ( 0 \leq \psi \leq \pi \right ) \)和椭圆度角表示(ellipticity angle) \( \chi \left ( -\pi/4 \leq \chi \leq \pi/4 \right ) \)

引入辅助角(auxiliary angle)

$$ tan\alpha = \frac{E_{0y}}{E_{0x}}, 0 \leq \alpha \leq \pi/2 $$

所以

\begin{equation} tan2\psi = \left ( tan2\alpha \right )cos\varepsilon \\ sin2\chi = \left ( sin2\alpha \right )sin\varepsilon \end{equation}

引入庞加莱球(Poincare sphere)

\begin{equation} \begin{aligned} x &= cos(2\chi)cos(2\psi ) , 0 \leq \psi \leq \pi \\ y &= cos(2\chi)sin(2\psi) , -\pi/4 \leq \chi \leq \pi/4 \\ z &= sin2\chi \\ 1 &= x^{2}+y^{2}+z^{2} \end{aligned} \end{equation}

琼斯矩阵

光学元件可以改变透射光的偏振态,用琼斯矩阵表示光学元件对透射光偏振态的改变。

偏振器

只透过某一线偏振态的光,下面的矩阵表示,只透过\(\vec{x}\)方向的分量

\begin{equation} \mathbf{M} = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \end{equation}

相位延迟器

使电场分量的相位分别延迟\( \varepsilon _{x}\) 和 \( \varepsilon _{y}\)

\begin{equation} E_{0x}e^{i\varphi_{x}} \rightarrow E_{0x}e^{i(\varphi_{x}+\varepsilon _{x})} \\ E_{0y}e^{i\varphi _{y}} \rightarrow E_{0y}e^{i(\varphi_{y}+\varepsilon _{y})} \end{equation}

写成矩阵形式

\begin{equation} \begin{bmatrix} e^{i\varepsilon_{x}} & 0 \\ 0 & e^{i\varepsilon_{y}} \end{bmatrix} \begin{bmatrix} E_{0x}e^{i\varphi_{x}} \\ E_{0y}e^{i\varphi_{y}} \end{bmatrix} = \begin{bmatrix} E_{0x}e^{i(\varphi_{x}+\varepsilon_{x})} \\ E_{0y}e^{i(\varphi_{y}+\varepsilon_{y})} \end{bmatrix} \end{equation}

所以普适的相位延迟器的琼斯矩阵为

\begin{equation} \mathbf{M} = \begin{bmatrix} e^{i\varepsilon_{x}} & 0 \\ 0 & e^{i\varepsilon_{y}} \end{bmatrix} \end{equation}

四分之一波片

quarter-wave plate (QWP) 满足\( \left | \varepsilon_{x} – \varepsilon_{y} \right | = \pi/2 \)

慢轴垂直时 \( \varepsilon_{x} – \varepsilon_{y} = -\pi/2 \)

\begin{equation} \mathbf{M} = \begin{bmatrix} e^{-i\pi/4} & 0 \\ 0 & e^{i\pi/4} \end{bmatrix} = e^{-i\pi/4}\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \end{equation}

慢轴水平时\( \varepsilon_{x} – \varepsilon_{y} = \pi/2\)

\begin{equation} \mathbf{M} = \begin{bmatrix} e^{i\pi/4} & 0 \\ 0 & e^{-i\pi/4} \end{bmatrix} = e^{i\pi/4}\begin{bmatrix} 1 & 0 \\ 0 & -i \end{bmatrix} \end{equation}

二分之一波片

half-wave plate (HWP) 满足\( \left | \varepsilon_{x} – \varepsilon_{y} \right | = \pi \)

同理

慢轴垂直时 \( \varepsilon_{x} – \varepsilon_{y} = -\pi \)

\begin{equation} \mathbf{M} = \begin{bmatrix} e^{-i\pi/2} & 0 \\ 0 & e^{i\pi/2} \end{bmatrix} = e^{-i\pi/2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{equation}

慢轴水平时\( \varepsilon_{x} – \varepsilon_{y} = \pi \)

\begin{equation} \mathbf{M} = \begin{bmatrix} e^{i\pi/2} & 0 \\ 0 & e^{-i\pi/2} \end{bmatrix} = e^{i\pi/2}\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \end{equation}

旋转

将某一角度 \(\theta\) 的线偏振光旋转 \( \beta \) 角度

\( \theta \rightarrow \left ( \theta+ \beta \right ) \)

\begin{equation} \mathbf{R}(\beta) = \begin{bmatrix} cos\beta & -sin\beta \\ sin\beta & cos\beta \end{bmatrix} \end{equation}

旋转矩阵的性质\( \mathbf{R}\mathbf{R}^{-1} = \mathbf{R}\mathbf{R}^{T} = \mathbf{I} = \mathbf{R}(\beta)\mathbf{R}(-\beta) \)

若将光学元件旋转 \( \beta \) 角度

$$ \mathbf{M}(\beta) = \mathbf{R}(\beta) \mathbf{M} \mathbf{R}(-\beta) $$

斯托克斯参数

$$ S_{0}^{2} = S_{1}^{2} +S_{2}^{2} +S_{3}^{2} $$

\begin{equation} S = \begin{pmatrix} S_{0}\\ S_{1}\\ S_{2}\\ S_{3} \end{pmatrix} = \begin{pmatrix} E_{0x}^{2} + E_{0y}^{2}\\ E_{0x}^{2} – E_{0y}^{2}\\ 2E_{0x}E_{0y}cos\varepsilon \\ 2E_{0x}E_{0y}sin\varepsilon \end{pmatrix} \end{equation}

其中\( \varphi_{y} – \varphi_{x} = \varepsilon \)

考虑到一般椭圆偏振的情况, 实用椭圆参数,也可以写为

\begin{equation} S = \begin{pmatrix} S_{0}\\ S_{1}\\ S_{2}\\ S_{3} \end{pmatrix} = S_{0}\begin{pmatrix} 1\\ cos(2\chi)cos(2\psi )\\ cos(2\chi)sin(2\psi )\\ sin(2\chi ) \end{pmatrix} \end{equation}

斯托克斯分量可以表示偏振光,也可以表示部分偏振光或随机偏振光

定义偏振度P(degree of polarization ,DOP)

\begin{equation} S = \begin{pmatrix} S_{0}\\ S_{1}\\ S_{2}\\ S_{3} \end{pmatrix} = \left ( 1- \mathbf{\mathit{P}} \right ) \begin{pmatrix} S_{0}\\ 0\\ 0\\ 0 \end{pmatrix} + \mathbf{\mathit{P}} \begin{pmatrix} S_{0}\\ S_{1}\\ S_{2}\\ S_{3} \end{pmatrix}, 0 \leq \mathbf{\mathit{P}} \leq 1 \end{equation}

偏振度P可由下列公式计算

$$ \mathbf{\mathit{P}} = \frac{I_{pol}}{I_{tot}} = \frac{\sqrt{S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}} $$

参考文献

  1. Pedrotti, F. L., Pedrotti, L. M. & Pedrotti, L. S. Introduction to optics. (Pearson/Prentice Hall, 2007).
  2. Collett, E. Field guide to polarization. (SPIE Press, 2005).
  3. https://spie.org/publications/fg05_p57-61_jones_matrix_calculus
  4. https://en.wikipedia.org/wiki/Jones_calculus
  5. http://electron9.phys.utk.edu/optics421/modules/m7/Jones.htm

发表评论

您的电子邮箱地址不会被公开。