偏振态、偏振光学元件
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椭圆偏振光
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两束偏振方向分别在$x$、$y$轴上的同频同速沿$z$轴传播的线偏振光叠加后可得到椭圆偏振光
$$
\varphi_x=E_x\cos(kx-\omega t-\varepsilon/2)\\
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i\varphi_y=iE_y\cos(kx-\omega t+\varepsilon/2)\\
\quad\\
\varphi=\varphi_x+i\varphi_y\\
\quad\\
\small\begin{aligned}\left(\frac{\varphi_x}{E_x}\right)^2+\left(\frac{\varphi_y}{E_y}\right)^2
&=\cos^2(kx-\omega t-\varepsilon/2)+\cos^2(kx-\omega t+\varepsilon/2)\\
\quad\\
&=2\cos^2(kx-\omega t)\cos^2(\frac{\varepsilon}{2})+2\sin^2(kx-\omega t)\sin^2(\frac{\varepsilon}{2})\\
\quad\\
&=2\cos^2(kx-\omega t)(\cos^2(\frac{\varepsilon}{2})-\sin^2(\frac{\varepsilon}{2}))+2\sin^2(\frac{\varepsilon}{2})\\
\quad\\
&=2\cos^2(kx-\omega t)\cos\varepsilon+2\sin^2(\frac{\varepsilon}{2})\\
\quad\\
&=[1+\cos(2kx-2\omega t)]\cos\varepsilon+2\sin^2(\frac{\varepsilon}{2})\\
\quad\\
&=\cos(2kx-2\omega t)\cos\varepsilon+1\\
\quad\\
&=[2\cos(kx-\omega t-\varepsilon/2)\cos(kx-\omega t+\varepsilon/2)-\cos\varepsilon]\\
\quad\\
&\qquad\cos\varepsilon+1\\
\quad\\
&=2\left(\frac{\varphi_x}{E_x}\right)\left(\frac{\varphi_y}{E_y}\right)\cos\varepsilon+\sin^2\varepsilon\\
\end{aligned}\\
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\small即\\
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\left(\frac{\varphi_x}{E_x}\right)^2+\left(\frac{\varphi_y}{E_y}\right)^2
-2\frac{\varphi_x}{E_x}\frac{\varphi_y}{E_y}\cos\varepsilon
=\sin^2\varepsilon\\
\quad\\
\small二次型相伴矩阵为\left(\begin{array}&\frac{1}{E_x^2}&-\frac{\cos\varepsilon}{E_x E_y}&0\\
-\frac{\cos\varepsilon}{E_x E_y}&\frac{1}{E_y^2}&0\\
0&0&-\sin^2\varepsilon
\end{array}\right)\\
\quad\\
I_1=\frac{1}{E_x^2}+\frac{1}{E_y^2}\qquad I_2=\frac{\sin^2\varepsilon}{E_x^2 E_y^2}\qquad I_3=-\frac{\sin^4\varepsilon}{E_x^2 E_y^2}\\
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\small特征方程为\lambda^2-(\frac{1}{E_x^2}+\frac{1}{E_y^2})\lambda+\frac{\sin^2\varepsilon}{E_x^2 E_y^2}=0\\
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\small解得特征根\lambda=\frac{\frac{1}{E_x^2}+\frac{1}{E_y^2}\pm\sqrt{\frac{1}{E_x^4}+\frac{2\cos2\varepsilon}{E_x^2E_y^2}+\frac{1}{E_y^4}}}{2}\\
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\small曲线简化方程为椭圆方程\\
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\lambda_负X^2+\lambda_正Y^2-\sin^2\varepsilon=0\\
\quad\\
\small原坐标系中椭圆长轴方向与x轴夹角正切值为\\
\quad\\
\tan\alpha=\frac{\lambda_负-\frac{1}{E_x^2}}{-\frac{\cos\varepsilon}{E_xE_y}}=\frac{\frac{\cos\varepsilon}{E_xE_y}}{\lambda_正-\frac{1}{E_x^2}}=\sqrt{-\frac{\lambda_负-\frac{1}{E_x^2}}{\lambda_正-\frac{1}{E_x^2}}}\\
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\begin{aligned}\tan2\alpha&=\frac{2\sqrt{-(\lambda_负-\frac{1}{E_x^2})(\lambda_正-\frac{1}{E_x^2})}}{\lambda_负+\lambda_正-\frac{2}{E_x^2}}\\
\quad\\
&=\frac{2E_xE_y\cos\varepsilon}{E_x^2-E_y^2}\\
\quad\\
\end{aligned}
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\small椭圆长短轴之比为\\
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\sqrt{\frac{\lambda_正}{\lambda_负}}=\frac{\frac{E_x}{E_y}+\frac{E_y}{E_x}+\sqrt{\frac{E_x^2}{E_y^2}+
2\cos2\varepsilon+\frac{E_y^2}{E_x^2}}}{2|\sin\varepsilon|}
$$
从$\varphi$的表达式可以看出，若$\pi>\varepsilon>0$，即$\varphi_y$落后于$\varphi_x$，迎着波动传播的方向看去，振动方向随着时间增大逆时针旋转，若$-\pi<\varepsilon<0$，即$\varphi_y$超前于$\varphi_x$，振动方向随着时间增大顺时针旋转，$\varepsilon=0$或$\pi$时，振动方向不随时间而改变。
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偏振态
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对于满足特殊条件的椭圆偏振光[$\mathscr{E}$态]，可将其分为左旋圆偏振光[$\mathscr{L}$态]，右旋圆偏振光[$\mathscr{R}$态]和线偏振光[$\mathscr{P}$态]
$$\pi>\varepsilon>0,E_x=E_y\qquad\mathscr{L}\\
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-\pi<\varepsilon<0,E_x=E_y\qquad\mathscr{R}\\
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\varepsilon=0或\pi\qquad\mathscr{P}$$
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旋转偏振光的角动量
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由于光子的角动量永远为$\hbar$或$-\hbar$，故
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偏振器
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对于平行于透振方向透射率为$T_0$，垂直于透振方向透射率为$T_{90}$的线偏振片，振动方向与偏振片透振方向成$\theta$角的线偏振光垂直入射的透射率为
$$T=T_0\cos^2\theta+T_{90}\sin^2\theta$$
两个平行的透振方向成$\theta$角的线偏振片对垂直入射的自然光的透射率为
$$T=\frac{1}{2}(T_0T_0+T_{90}T_{90})\cos^2\theta+T_0T_{90}\sin^2\theta$$