# 光的偏振

## 偏振的表示

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

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

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

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

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

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

### 线偏振光

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

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

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

### 圆偏振光

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

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

### 椭圆偏振光

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

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

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

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

$$tan2\psi = \left ( tan2\alpha \right )cos\varepsilon \\ sin2\chi = \left ( sin2\alpha \right )sin\varepsilon$$

\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}

## 琼斯矩阵

### 偏振器

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

### 相位延迟器

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

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

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

#### 四分之一波片

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

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

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

#### 二分之一波片

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

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

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

### 旋转

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

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

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

## 斯托克斯参数

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

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

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

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

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

## 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