复数形式表示电场：如何使用欧拉公式简化电场方程

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将'$\cos$'和'$\sin$'用欧拉公式表示：

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

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定义复数电场振幅：

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

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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。

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则电场可以写成复数形式：

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

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其中，$\tilde{E}^{*}$为复共轭。

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