折射反射、菲涅尔方程
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反射与折射定律
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记$n_r=\frac{n_t}{n_i}$，$\theta_i$，$\theta_r$和$\theta_t$分别为入射角，反射角和透射角
$$单光子动量\\
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\boldsymbol{p}=\hbar \boldsymbol{k}=\frac{\hbar\omega}{\boldsymbol{v}}=\frac{n\hbar\omega}{\boldsymbol{c}}\propto n\\
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水平方向动量守恒\\
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\left\lbrace\begin{array}&\sin\theta_i&=&n_r\sin\theta_t\\
\sin\theta_i&=&\sin\theta_r
\end{array}\right.
$$
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Fresnel方程
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$$n_r=\sqrt{\varepsilon_r\mu_r}\\
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反射折射定律\\
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\theta_i=\theta_r\\
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\frac{\sin\theta_i}{\sin\theta_t}=n_r\\
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p光边界条件为平行界面E连续,垂直界面D连续\\
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E_i \cos \theta_i =E_t \cos \theta_t +E_r \cos \theta_r\\
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D_i \sin \theta_i =D_t \sin \theta_t -D_r \sin \theta_r\\
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s光边界条件为平行界面H连续,垂直界面B连续\\
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H_i \cos \theta_i =H_t \cos \theta_t +H_r \cos \theta_r\\
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B_i \sin \theta_i =B_t \sin \theta_t -B_r \sin \theta_r\\
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可得线性方程组\\
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\left\lbrace\begin{array} &\cos{\theta_t}\; t_p&+\cos{\theta_i}\; r_p& & &=&\cos{\theta_i}\\
\frac{n_r^2}{\mu_r}\sin\theta_t\; t_p&-\sin\theta_i\; r_p& & &=&\sin{\theta_i}\\
& &\frac{n_r}{\mu_r}\cos{\theta_t}\; t_s&+\cos{\theta_i}\; r_s&=&\cos{\theta_i}\\
& &n_r\sin{\theta_t}\; t_s&-\sin{\theta_i}\; r_s&=&\sin{\theta_i}\end{array}\right.\\
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解得\\
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\left\lbrace\begin{array}&r_s&=&-\frac{\frac{n_r}{\mu_r}\sin \theta_i \cos\theta_t-n_r\sin \theta_t \cos\theta_i}{\frac{n_r}{\mu_r}\sin \theta_i \cos\theta_t+n_r\sin \theta_t \cos\theta_i}\\
r_p&=&-\frac{\sin \theta_i \cos\theta_t-\frac{n_r^2}{\mu_r}\sin \theta_t \cos\theta_i}{\sin \theta_i \cos\theta_t+\frac{n_r^2}{\mu_r}\sin \theta_t \cos\theta_i}\\
t_s&=&+\frac{2\sin \theta_i \cos\theta_i}{\frac{n_r}{\mu_r}\sin \theta_i \cos\theta_t+n_r\sin \theta_t \cos\theta_i}\\
t_p&=&+\frac{2\sin \theta_i \cos\theta_i}{\sin \theta_i \cos\theta_t+\frac{n_r^2}{\mu_r}\sin \theta_t \cos\theta_i}\\
\end{array}\right.\\
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若\mu_r=1\\
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\left\lbrace\begin{array}&r_s&=&-\frac{\sin(\theta_i-\theta_t)}{\sin(\theta_i+\theta_t)}\\
r_p&=&+\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)}\\
t_s&=&+\frac{2\sin \theta_t \cos\theta_i}{\sin(\theta_i+\theta_t)}\\
t_p&=&+\frac{2\sin \theta_t \cos\theta_i}{\sin(\theta_i+\theta_t)\cos(\theta_i-\theta_t)}\\
\end{array}\right.\\
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若\varepsilon_r=1,即\mu_r=n_r^2\\
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\left\lbrace\begin{array}&r_s&=&+\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)}\\
r_p&=&-\frac{\sin(\theta_i-\theta_t)}{\sin(\theta_i+\theta_t)}\\
t_s&=&+\frac{2\sin \theta_i \cos\theta_i}{\sin(\theta_i+\theta_t)\cos(\theta_i-\theta_t)}\\
t_p&=&+\frac{2\sin \theta_i \cos\theta_i}{\sin(\theta_i+\theta_t)}\\
\end{array}\right.\\
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$$
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全内反射与隐失波
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由折射定律$\frac{\sin\theta_i}{\sin\theta_t}=n_r$有，当光线从光密介质射向光疏介质且入射角大于临界角，即$\sin\theta_i>n_r$时，$\theta_t=\frac{\pi}{2}+i\gamma$，$
k_t=n_r k_i$，$\boldsymbol{k_t}=k_{t}\sin\theta_t \hat{x}+k_{t}\cos\theta_t\hat{y}$，光矢垂直界面方向分量为纯虚数，非平面波，即出现所谓隐失波
$$\begin{aligned}E_t&=E_{i0}\times[t_p \;or\;t_s]e^{i(\boldsymbol{k_t\cdot r}-\omega t)}\\
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&=E_{t0}e^{i(k_t\sin\theta_t x+k_t\cos\theta_t y-\omega t)}\\
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&=E_{t0}e^{-\beta y} e^{i(k_i\sin\theta_i x-\omega t)}\end{aligned}\\
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其中\beta=k_i\sqrt{\sin^2\theta_i-n_r^2}$$
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参考与引用来源
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赫克特光学第五版