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或随机偏振光。
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 非偏振光或随机偏振光
线偏振光
任意\( \alpha\)的线偏光,相对相位差为0或π,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
- Pedrotti, F. L., Pedrotti, L. M. & Pedrotti, L. S. Introduction to optics. (Pearson/Prentice Hall, 2007).
- Collett, E. Field guide to polarization. (SPIE Press, 2005).
- https://spie.org/publications/fg05_p57-61_jones_matrix_calculus
- https://en.wikipedia.org/wiki/Jones_calculus
- http://electron9.phys.utk.edu/optics421/modules/m7/Jones.htm