贡献者: addis
对非磁介质
\begin{equation}
\boldsymbol{\nabla}\boldsymbol{\times} ( \boldsymbol{\nabla}\boldsymbol{\times} \boldsymbol{\mathbf{E}} ) = -\mu_0 \frac{\partial^{2}{ \boldsymbol{\mathbf{D}} }}{\partial{t}^{2}} ~.
\end{equation}
平面波时
\begin{equation}
\boldsymbol{\nabla}^2 \boldsymbol{\mathbf{E}} - \mu_0 \frac{\partial^{2}{ \boldsymbol{\mathbf{D}} }}{\partial{t}^{2}} = 0~,
\end{equation}
\begin{equation}
\boldsymbol{\mathbf{D}} = \epsilon_0 \boldsymbol{\mathbf{E}} + \boldsymbol{\mathbf{P}} = \epsilon_0(1 + \tilde\chi) \boldsymbol{\mathbf{E}} + \boldsymbol{\mathbf{P}} ^{NL} = \tilde \epsilon \boldsymbol{\mathbf{E}} + \boldsymbol{\mathbf{P}} ^{NL}~,
\end{equation}
\begin{equation}
\boldsymbol{\nabla}^2 \boldsymbol{\mathbf{E}} - \mu_0\tilde\epsilon \frac{\partial^{2}{ \boldsymbol{\mathbf{E}} }}{\partial{t}^{2}} = \mu_0 \frac{\partial^{2}{ \boldsymbol{\mathbf{P}} ^{NL}}}{\partial{t}^{2}} ~.
\end{equation}
波浪号表示复数,$\tilde\epsilon$ 和 $\tilde\chi$ 都只是 $\omega$ 而不是场强的函数。先来看线性的情况($ \boldsymbol{\mathbf{P}} ^{NL} = \boldsymbol{\mathbf{0}} $),例如洛伦兹模型。
\begin{equation}
\boldsymbol{\nabla}^2 \boldsymbol{\mathbf{E}} - \mu_0 \tilde\epsilon \frac{\partial^{2}{ \boldsymbol{\mathbf{E}} }}{\partial{t}^{2}} = \boldsymbol{\mathbf{0}} ~.
\end{equation}
(见齐次波动方程,先做时间傅里叶变换,再解齐次亥姆霍兹方程,通解是所有平面波)然而这里的 $\tilde k^2 = \mu_0 \tilde \epsilon \omega$ 是复数,平面波变为指数衰减的单频单向波。以 $z$ 方向传播 $x$ 方向极化为例,令 $\tilde k = k + \kappa$
\begin{equation}
E_x(z,t) = \tilde E_{0x} \mathrm{e} ^{ \mathrm{i} (\tilde kz - \omega t)} = \tilde E_{0x} \mathrm{e} ^{-\kappa z} \mathrm{e} ^{ \mathrm{i} (kz - \omega t)}~,
\end{equation}
\begin{equation}
\tilde k = \omega \sqrt{\mu_0\tilde\epsilon} = \frac{\omega }{c}\sqrt {1 + \chi ^{(1)}}~,
\end{equation}
通解仍然为所有可能的单频单向波的线性组合。实折射率和吸收系数定义为
\begin{equation}
n = \frac{ck}{\omega } = \operatorname{Re} \left[\sqrt{1 + \chi ^{(1)}} \right] \approx 1 + \frac12 \operatorname{Re} \left[\chi^{(1)} \right] ~,
\end{equation}
\begin{equation}
\alpha = 2\kappa = \frac{2\omega}{c} \operatorname{Im} \left[\sqrt{1+\chi^{(1)}} \right] \approx \frac{\omega}{c} \operatorname{Im} \left[\chi ^{(1)} \right] ~,
\end{equation}
\begin{equation}
\tilde \epsilon = \epsilon_0 (n + \mathrm{i} \alpha c/2\omega )^2~.
\end{equation}
注意区分 $\epsilon$(permittivity),$\epsilon_r$(dielectric constant)和 $\chi$(susceptivility)。不同的书符号可能不一样,以名称和语境为准。