把$$E=E_y mcosleftomega t-k x+phirightoverrightarrowa_y+E_x msinleftomega t-k x+phirightoverrightarrowa_x$$改写为复数形式

$$E=E_{y m}\cos\left(\omega t-k x+\phi\right)\overrightarrow{a}{y}+E{x m}\sin\left(\omega t-k x+\phi\right)\overrightarrow{a}_{x}$$

将$\cos$和$\sin$用欧拉公式表示：

$$\begin{aligned} E&=E_{y m}\frac{e^{i(\omega t-kx+\phi)}+e^{-i(\omega t-kx+\phi)}}{2}\overrightarrow{a}{y}+E{x m}\frac{e^{i(\omega t-kx+\phi)}-e^{-i(\omega t-kx+\phi)}}{2i}\overrightarrow{a}{x} \ &=\frac{E{y m}}{2}\left(e^{i(\omega t-kx+\phi)}+e^{-i(\omega t-kx+\phi)}\right)\overrightarrow{a}{y}+\frac{E{x m}}{2i}\left(e^{i(\omega t-kx+\phi)}-e^{-i(\omega t-kx+\phi)}\right)\overrightarrow{a}{x} \ &=\frac{E{y m}}{2}e^{i(\omega t-kx+\phi)}\overrightarrow{a}{y}+\frac{E{y m}}{2}e^{-i(\omega t-kx+\phi)}\overrightarrow{a}{y}+\frac{E{x m}}{2i}e^{i(\omega t-kx+\phi)}\overrightarrow{a}{x}-\frac{E{x m}}{2i}e^{-i(\omega t-kx+\phi)}\overrightarrow{a}{x} \ &=\left(\frac{E{y m}}{2}e^{i\phi}\overrightarrow{a}{y}+\frac{E{x m}}{2i}e^{i\phi}\overrightarrow{a}{x}\right)e^{i(\omega t-kx)}+\left(\frac{E{y m}}{2}e^{-i\phi}\overrightarrow{a}{y}-\frac{E{x m}}{2i}e^{-i\phi}\overrightarrow{a}_{x}\right)e^{-i(\omega t-kx)} \end{aligned}$$

定义复数电场振幅：

$$\begin{aligned} \tilde{E}&=\frac{E_{y m}}{2}e^{i\phi}\overrightarrow{a}{y}+\frac{E{x m}}{2i}e^{i\phi}\overrightarrow{a}{x} \ &=\frac{E{y m}}{2}\cos\phi+i\frac{E_{y m}}{2}\sin\phi\overrightarrow{a}{y}+\frac{E{x m}}{2}\sin\phi-\frac{E_{x m}}{2}i\cos\phi\overrightarrow{a}{x} \ &=\left(E{y m}\cos\phi+E_{x m}\sin\phi\right)e^{i\phi}\frac{\overrightarrow{a}{y}}{2}+\left(E{y m}\sin\phi-E_{x m}\cos\phi\right)e^{i\phi}\frac{\overrightarrow{a}{x}}{2i} \ &=\left(E{y m}\cos\phi+E_{x m}\sin\phi\right)e^{i\phi}\frac{\overrightarrow{a}{y}}{2}+\left(E{y m}\sin\phi-E_{x m}\cos\phi\right)e^{i\phi}\frac{\overrightarrow{a}{x}}{2i} \ &=\left(E{y m}\cos\phi+E_{x m}\sin\phi\right)e^{i\phi}\frac{\overrightarrow{a}{y}}{2}+\left(E{y m}\sin\phi-E_{x m}\cos\phi\right)e^{i\phi}\left(-\frac{\overrightarrow{a}{y}\times\overrightarrow{a}{x}}{2}\right) \ &=\left(E_{y m}\cos\phi+E_{x m}\sin\phi\right)e^{i\phi}\frac{\overrightarrow{a}{y}}{2}-\left(E{y m}\sin\phi-E_{x m}\cos\phi\right)e^{i\phi}\frac{\overrightarrow{a}_{z}}{2} \end{aligned}$$

其中，$\overrightarrow{a}{z}=\overrightarrow{a}{y}\times\overrightarrow{a}_{x}$为垂直于平面波传播方向和偏振方向的单位矢量。

则电场可以写成复数形式：

$$E=\tilde{E}e^{i(\omega t-kx)}+\tilde{E}^{*}e^{-i(\omega t-kx)}$$

其中，$\tilde{E}^{*}$为复共轭