User:Zhennan/1DsommerfeldRadiationCondition

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1D Sommerfeld radiation condition is

\begin{equation} P^{up}(z_g',z_s,\omega)=P(z,z_s,\omega)\frac{d}{dz}G_0^+(z,z_g',\omega)-G_0^+(z,z_g',\omega)\frac{d}{dz}P(z,z_s,\omega) \end{equation}

is zero at .

Prove

From Lippman Schwinger equation, we know \begin{equation} \frac{d}{dz}G_0^+(z,z_g',\omega)|_{z=-\infty}=-ikG_0^+(z,z_g',\omega)|_{z=-\infty} \end{equation} and \begin{align} \frac{d}{dz}P(z,z_s,\omega)|_{z=-\infty}&=[\frac{d}{dz}\int_{-\infty}^{+\infty}\rho(z_s)G_0^+(z,z_s,\omega)dz_s]|_{z=-\infty} \\ \nonumber &=\int_{-\infty}^{+\infty}\rho(z_s)[\frac{d}{dz}G_0^+(z,z_s,\omega)]|_{z=-\infty}dz_s \\ \nonumber &=-ik\int_{-\infty}^{+\infty}\rho(z_s)G_0^+(z,z_s,\omega)|_{z=-\infty}dz_s \\ \nonumber &=-ikP(z,z_s,\omega)|_{z=-\infty} \\ \nonumber \end{align} Therefore at , \begin{align} P^{up}(z_g',z_s,\omega)&=P(z,z_s,\omega)\frac{d}{dz}G_0^+(z,z_g',\omega)-G_0^+(z,z_g',\omega)\frac{d}{dz}P(z,z_s,\omega) \\ \nonumber &=P(z,z_s,\omega)[-ikG_0^+(z,z_g',\omega)]-G_0^+(z,z_g',\omega)[-ikP(z,z_s,\omega)]\\ \nonumber &=0 \end{align}