$\hat{u}$ 방향으로 진행하는 평면파는 다음과 같이 기술된다. \begin{equation} e^{-i(\omega t-\kappa \hat{u} \cdot \vec{r})} \end{equation} 전파 벡터(propagation vector) $\vec{\kappa}$는 다음과 같이 정의된다. $$ \vec{\kappa}=\kappa \hat{u} $$ 그러면 다음과 같이 기술된다. $$ e^{-i(\omega t-\vec{\kappa} \cdot \vec{r})} $$ $$ \begin{aligned} & \nabla \cdot \vec{D} = 0, \\ & \nabla \cdot \vec{B} = 0, \\ & \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}, \\ & \nabla \times \vec{H} = \frac{\partial \vec{D}}{\partial t} \end{aligned} $$ $$ \vec{E}(\vec{r}, t)=\vec{E}_0 e^{-i(\omega t-\vec{\kappa} \cdot \vec{r})} $$ $$ \frac{\partial}{\partial t}=-i \omega $$ $$ \nabla=i \vec{\kappa} . $$ $$ \begin{aligned} \vec{\kappa} \cdot \vec{D}_0 & =0 \\ \vec{\kappa} \cdot \vec{B}_0 & =0 \\ \vec{\kappa} \times \vec{E}_0 & =\omega \vec{B}_0, \\ \vec{\kappa} \times \vec{H}_0 & =-\omega \vec{D}_0 . \end{aligned} $$ $$ \begin{aligned} & \vec{D}_0=\epsilon \vec{E}_0, \\ & \vec{H}_0=\frac{1}{\mu} \vec{B}_0 . \end{aligned} $$ $$ \begin{aligned} K \vec{\kappa} \cdot \vec{E}_0 & =0 \\ \vec{\kappa} \cdot \vec{B}_0 & =0 \\ \vec{\kappa} \times \vec{E}_0 & =\omega \vec{B}_0 \\ \vec{\kappa} \times \vec{B}_0 & =-\frac{\omega}{c^2} K \vec{E}_0 \end{aligned} $$ $$ \vec{\kappa} \times(\vec{\kappa} \times \vec{E}_0)=\omega \vec{\kappa} \times \vec{B}_0=-K(\omega / c)^2 \vec{E}_0 $$ $$ \vec{\kappa} \times(\vec{\kappa} \times \vec{E}_0)=(\vec{\kappa} \cdot \vec{E}_0) \vec{\kappa}-\kappa^2 \vec{E}_0, $$ $$ -K(\omega / c)^2 \vec{E}_0=-\kappa^2 \vec{E}_0, $$ $$ \kappa=\sqrt{K} \omega / c . $$ $$ \begin{aligned} & \vec{E}(\vec{r}, t)=\vec{E}_0 e^{-i(\omega t-\vec{\kappa} \cdot \vec{r}),} \\ & \vec{B}(\vec{r}, t)=\vec{B}_0 e^{-i(\omega t-\vec{\kappa} \cdot \vec{r})}, \end{aligned} $$ $$ \hat{u} \cdot \vec{E}_0=0 . $$ $$ \kappa=n \omega / c, $$ $$ n=\sqrt{K} . $$ $$ \vec{B}_0=\frac{n}{c} \hat{u} \times \vec{E}_0 . $$ $$ \vec{E}(\vec{r}, t)=\sum_i \vec{E}_0\left(\vec{\kappa}_i, \omega_i\right) \exp \left[-i\left(\omega_i t-\vec{\kappa}_i \cdot \vec{r}\right)\right] $$ $$ n=n(\kappa, \omega) . $$ $$ \vec{E}_0=\vec{E}_{\vec{r}}+i \vec{E}_i ; $$ $$ \vec{E}_0=\hat{E}_p \mathbf{p}+\hat{E}_s \mathbf{s}+\hat{E}_u \hat{u} . $$ $$ E_{p_r}=E_{r_p}, \quad E_{p_i}=E_{i_p,} \quad E_{s_r}=E_{r_s}, \text { etc. } $$ $$ \vec{E}_0=\hat{E}_p \mathbf{p}+\hat{E}_s \mathbf{s} . $$ $$ \hat{E}_p=E_p e^{i \phi_p}, \quad \hat{E}_s=E_s e^{i \phi_s} . $$ $$ \hat{E}_s e^{-i(\omega t-\kappa \cdot \vec{r})}=E_s e^{-i\left(\omega t-\vec{\kappa} \cdot \vec{r}-\phi_s\right)} ; $$ $$ \phi_p-\phi_s=\phi, \quad \phi_s=0, $$ $$ \begin{aligned} \vec{E}_0 & =E_p e^{i \phi} \mathbf{p}+E_s \mathbf{s}, \\ \vec{E}(\vec{r}, t) & =E_p \mathbf{p} e^{-i(\omega t-\kappa \cdot \vec{r}-\phi)}+E_s \mathbf{s} e^{-i(\omega t-\kappa \cdot \vec{r})}, \end{aligned} $$ $$ \vec{E}(\vec{r}, t)=E_p \mathbf{p} \cos (\omega t-\kappa \cdot \vec{r}-\phi)+E_s \mathbf{s} \cos (\omega t-\kappa \cdot \vec{r}) $$ $$ \vec{E}(0, t)=\left(E_p \mathbf{p}+E_s \mathbf{s}\right) \cos \omega t $$ $$ \vec{E}(0, t)=\left(-E_p \mathbf{p}+E_s \mathbf{s}\right) \cos \omega t $$ $$ \vec{E}(0, t)=E_p \mathbf{p} \sin \omega t+E_s \mathbf{s} \cos \omega t $$ $$ \vec{B}_0=\frac{n}{c} \hat{u} \times \vec{E}_0 . $$ $$ \vec{B}_0 \cdot \vec{E}_0=0 \text {. } $$ $$ \vec{E}=E_p \mathbf{p} \cos (\omega t-\phi)+E_s \mathbf{s} \cos \omega t $$ \begin{equation} \vec{B}=(n / c)\left[E_p s \cos (\omega t-\phi)-E_{s p} \cos \omega t\right] \end{equation} $$ \begin{aligned} & u=\frac{1}{2}(\vec{E} \cdot \vec{D}+\vec{B} \cdot \vec{H}) \\ & \mathbf{S}=\vec{E} \times \vec{H} \end{aligned} $$ $$ \begin{aligned} & E^2=E_p^2 \cos ^2(\omega t-\phi)+E_s^2 \cos ^2 \omega t \\ & B^2=(n / c)^2 E^2=\epsilon \mu_0 E^2 \end{aligned} $$ $$ \begin{aligned} \vec{B} \cdot \vec{H} & =\vec{D} \cdot \vec{E}, \\ u & =\epsilon E^2=\frac{1}{\mu_0}\left(\frac{n}{c}\right)^2 E^2 . \end{aligned} $$ $$ S=\frac{1}{\mu_0} \frac{n}{c} E^2 $$ $$ S=\frac{c}{\pi} u . $$ $$ \begin{aligned} v_p & =\frac{c}{n}, \\ \mathrm{~S} & =u \mathbf{v}_p . \end{aligned} $$ $$ \vec{J}=\rho \mathbf{v}, $$ $$ E^2=E_p^2 \sin ^2 \omega t+E_p^2 \cos ^2 \omega t=E_p^2 $$ $$ E^2=\left(E_p^2+E_s^2\right) \cos ^2 \omega t $$ $$ \overline{E^2}=\frac{1}{2}\left(E_p^2+E_s^2\right) . $$ $$ \overline{\operatorname{Re} f \operatorname{Re} g}=\frac{1}{2} \operatorname{Re} f^* g . $$ $$ \operatorname{Re} f^* g=u \xi+v \eta . $$ $$ \begin{aligned} \lim _{T \rightarrow \infty} \frac{1}{T} \int_0^T \sin ^2 \omega t d t & =\frac{1}{2}, \\ \lim _{T \rightarrow \infty} \frac{1}{T} \int_0^T \cos ^2 \omega t d t & =\frac{1}{2}, \\ \lim _{T \rightarrow \infty} \frac{1}{T} \int_0^T \sin \omega t \cos \omega t d t & =0 . \end{aligned} $$ $$ \overline{\operatorname{Re} f \operatorname{Re} g}=\frac{1}{2}(u \xi+v \eta) \text {. } $$ $$ \overline{E^2}=\frac{1}{2} \operatorname{Re}\left(\vec{E}^* \cdot \vec{E}\right) \text {. } $$ $$ \nabla \times \vec{H}=\frac{\partial \vec{D}}{\partial t}+g \vec{E} $$ \begin{equation} \boldsymbol{\kappa} \times \vec{H}_0=-\omega \vec{D}_0-i g \vec{E}_0 \end{equation} $$ \vec{\kappa} \times \vec{B}_0=-\frac{\omega}{c^2}\left(K+i \frac{g}{\epsilon_0 \omega}\right) \vec{E}_0 . $$ $$ \hat{K}=K+i \frac{g}{\epsilon_0 \omega}, $$ $$ \kappa \times \vec{B}_0=-\frac{\omega}{c^2} \hat{K} \vec{E}_0 . $$ $$ \kappa=\sqrt{\hat{K}} \omega / c=\hat{n} \omega / c, $$ $$ \hat{n}^2=\hat{K} \text {. } $$ $$ \hat{\boldsymbol{\kappa}}=\boldsymbol{k}_{\vec{r}}+i \boldsymbol{k}_i $$ $$ \begin{aligned} & \vec{E}(\vec{r}, t)=\left(\vec{E}_0 e^{-\boldsymbol{x}_i \cdot \vec{r}}\right) e^{-i\left(\omega t-\boldsymbol{k}_{\vec{r}} \cdot \vec{r}\right)} \\ & \vec{B}(\vec{r}, t)=\left(\vec{B}_0 e^{-\boldsymbol{k}_i \cdot \vec{r}}\right) e^{-i\left(\omega t-k_i \cdot \vec{r}\right)} \end{aligned} $$ \begin{equation} \hat{\kappa}=\sqrt{\hat{\kappa} \cdot \hat{\boldsymbol{\kappa}}}=\sqrt{\kappa_r^2-\kappa_i^2+2 i \kappa_r \cdot \kappa_i} \end{equation} $$ \hat{n}=n+i k $$ $$ \hat{\vec{\kappa}}=\left(\kappa_r+i \kappa_i\right) \hat{u}=\hat{\kappa} \hat{u}, $$ $$ \hat{u} \cdot \vec{E}_0=0=\hat{u} \cdot \vec{B}_0 $$ $$ \vec{B}_0=\frac{\hat{n}}{c} \hat{u} \times \vec{E}_0 . $$ $$ \hat{n}=n+i k, $$ $$ \kappa_r=n \omega / c, \quad \kappa_i=k \omega / c . $$ $$ \vec{E}(\vec{r}, t)=\left(\vec{E}_0 e^{-k \omega \xi / c}\right) e^{-i \omega(t-n \xi / c)} . $$ $$ \delta=c / k \omega $$ \begin{equation} \delta=\frac{n}{k} \frac{\lambda}{2 \pi} \end{equation} $$ \begin{aligned} & \hat{n}=n+i k \\ & \hat{K}=K+i \frac{g}{\epsilon_0 \omega}, \\ & \hat{K}=\hat{n}^2 . \end{aligned} $$ $$ \hat{K}=K_r+i K_i, $$ $$ K_r=K, \quad K_i=g / \epsilon_0 \omega . $$ $$ \begin{aligned} & K_r=n^2-k^2, \\ & K_i=2 n k . \end{aligned} $$ $$ \begin{aligned} & n=\sqrt{\frac{1}{2}\left[K_r+\sqrt{K_r^2+K_i^2}\right]}, \\ & k=\sqrt{\frac{1}{2}\left[-K_r+\sqrt{K_r^2+K_i^2}\right]} \end{aligned} $$ 1. $K_i \ll\left|K_r\right|, K_r>0\left(\omega \gg \frac{g}{\epsilon}\right)$ : $$ n \cong \sqrt{K_r}, \quad k=K_i / 2 n \ll n . $$ 2. $K_i \ll\left|K_r\right|, K_r?0\left(\omega \gg \frac{g}{-\epsilon}\right)$ : $$ $$ 3. $K_i \gg\left|K_r\right|,\left(\omega \ll \frac{g}{|\epsilon|}\right)$ : $$ n \cong k \cong \sqrt{K_i / 2} . $$ $$ \delta \cong \frac{c}{\omega} \sqrt{\frac{2}{K_i}}=\sqrt{\frac{2}{\mu_0 \omega g}} . $$ $$ g=3 \times 10^7 \mathrm{~S} / \mathrm{m} \dagger $$ $$ \delta=\sqrt{\frac{2}{\left(2 \pi \times 10^{10}\right)\left(3 \times 10^7\right)\left(4 \pi \times 10^{-7}\right)}}=9.2 \times 10^{-5} \mathrm{~cm} . $$ $$ \begin{gathered} \omega=\frac{2}{g \mu_0 \delta^2}=\frac{2}{4.3 \times 4 \pi \times 10^{-7} \delta^2} \mathrm{~s}^{-1}=\frac{3.70 \times 10^5}{\delta^2} \mathrm{~s}^{-1}, \\ f=58.6 \times 10^3 \mathrm{~Hz}, \end{gathered} $$ \begin{equation} \nabla^2 \vec{E}-\frac{1}{c^2} \frac{\partial^2 \cdot}{\partial t^2}=0 \end{equation} $$ \nabla^2 \vec{E}(\vec{r})+\left(\frac{\omega}{c}\right)^2 \vec{E}(\vec{r})=0 $$ $$ \nabla^2 \vec{E}=-\nabla \times \nabla \times \vec{E}+\nabla \nabla \cdot \vec{E} $$ $$ \nabla^2 \psi+\left(\frac{\omega}{c}\right)^2 \psi=0, $$ $$ -\nabla \times \nabla \times \vec{E}+\nabla \cdot \vec{E}+\left(\frac{\omega}{c}\right)^2 \vec{E}=0 $$ $$ \vec{E}=\vec{r} \times \nabla \psi=-\nabla \times(\vec{r} \psi), $$ $$ \nabla \times(\mathbf{F} \varphi)=\varphi \nabla \times \mathbf{F}-\mathbf{F} \times \nabla \varphi $$ $$ \nabla \times \vec{r}=0 \text {. } $$ \begin{equation} \nabla \times(\mathbf{F} \times \mathbf{G})=\mathbf{F} \nabla \cdot \mathbf{G}-\mathbf{G} \nabla \cdot \mathbf{F}+(\mathbf{G} \cdot \nabla) \mathbf{F}-(\mathbf{F} \cdot \nabla) \mathbf{G} \end{equation} $$ \nabla \times(\vec{r} \times \nabla \psi)=\vec{r} \nabla^2 \psi-\nabla \psi \nabla \cdot \vec{r}+(\nabla \psi \cdot \nabla) \vec{r}-(\vec{r} \cdot \nabla) \nabla \psi $$ $$ \begin{aligned} & \nabla(\mathbf{F} \cdot \mathbf{G})=(\mathbf{F} \cdot \nabla) \mathbf{G}+(\mathbf{G} \cdot \nabla) \mathbf{F}+\mathbf{F} \times \nabla \times \mathbf{G}+\mathbf{G} \times \nabla \times \mathbf{F}, \\ & =\vec{r} \text { and } \mathbf{G}=\nabla \psi \text {, gives } \\ & \nabla(\vec{r} \cdot \nabla \psi)=(\vec{r} \cdot \nabla) \nabla \psi+(\nabla \psi \cdot \nabla) \vec{r} . \\ & \end{aligned} $$ $$ \nabla \times(\vec{r} \times \nabla \psi)=-\left(\frac{\omega}{c}\right)^2 \vec{r} \psi-3 \nabla \psi+\nabla \psi-\nabla(\vec{r} \cdot \nabla \psi)+\nabla \psi $$ $$ \nabla \times \nabla \times(\vec{r} \times \nabla \psi)=-\left(\frac{\omega}{c}\right)^2 \nabla \times \vec{r} \psi=\left(\frac{\omega}{c}\right)^2 \vec{r} \times \nabla \psi, $$ $$ \vec{E}=\vec{r} \times \nabla \psi . $$ $$ \nabla \times \vec{E}=i \omega \vec{B}, $$ $$ \vec{B}=-i \frac{1}{\omega} \nabla \times(\mathrm{r} \times \nabla \psi) $$ $$ \vec{B}^{\prime}=\frac{1}{c} \vec{r} \times \nabla \psi $$ $$ \vec{E}^{\prime}=\frac{i c}{\omega} \nabla \times(\vec{r} \times \nabla \psi) . $$ $$ \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial \psi}{\partial r}\right)+\frac{1}{r^2 \sin \theta} \frac{\partial}{\partial \theta}\left(\sin \theta \frac{\partial \psi}{\partial \theta}\right)+\frac{1}{r^2 \sin ^2 \theta} \frac{\partial^2 \psi}{\partial \phi^2}+\kappa^2 \psi=0 $$ $$ \psi=R(r) \Theta(\theta) \Phi(\phi) . $$ $$ \frac{1}{R} \sin ^2 \theta \frac{d}{d r} r^2 \frac{d R}{d r}+\frac{1}{\Theta} \sin \theta \frac{d}{d \theta} \sin \theta \frac{d \Theta}{d \theta}+\frac{1}{\Phi} \frac{d^2 \Phi}{d \phi^2}+\kappa^2 r^2 \sin ^2 \theta=0 $$ $$ \frac{d^2 \Phi_m}{d \phi^2}+m^2 \Phi_m=0 $$ $$ \frac{1}{R} \frac{d}{d r} r^2 \frac{d R}{d r}+\kappa^2 r^2+\frac{1}{\Theta} \frac{1}{\sin \theta} \frac{d}{d \theta} \sin \theta \frac{d \Theta}{d \theta}-\frac{m^2}{\sin ^2 \theta}=0 $$ $$ \begin{gathered} \frac{1}{\sin \theta} \frac{d}{d \theta} \sin \theta \frac{d \Theta_{l m}}{d \theta}+\left[l(l+1)-\frac{m^2}{\sin ^2 \theta}\right] \Theta_{l m}=0 \\ \frac{d}{d r} r^2 \frac{d R_l}{d r}-\left[l(l+1)-\kappa^2 r^2\right] R_l=0 \end{gathered} $$ $$ \Phi_m=e^{\mp i m \phi} . $$ $$ P_l^m(u)=\left(1-u^2\right)^{m / 2} \frac{d^m}{d u^m} P_l(u), $$ $$ \frac{d}{d \xi} \xi^2 \frac{d}{d \xi} R_l-\left[l(l+1)-\xi^2\right] R_l=0 $$ \begin{equation} \begin{array}{lc} P_0(u) & 1 \\ P_1(u) & u=\cos \theta \\ P_1^1(u) & \left(1-u^2\right)^{1 / 2}=\sin \theta \\ P_2(u) & \frac{1}{2}\left(3 u^2-1\right)=\frac{1}{4}(3 \cos 2 \theta+1) \\ P_2^1(u) & 3 u\left(1-u^2\right)^{1 / 2}=\frac{3}{2} \sin 2 \theta \\ P_2^2(u) & 3\left(1-u^2\right)=\frac{3}{2}(1-\cos 2 \theta) \\ P_3(u) & \frac{1}{2}\left(5 u^3-3 u\right) \\ P_3^1(u) & \frac{3}{2}\left(1-u^2\right)^{1 / 2}\left(5 u^2-1\right) \\ P_3^2(u) & 15 u\left(1-u^2\right) \\ P_3^3(u) & 15\left(1-u^2\right)^{3 / 2} \end{array} \end{equation} $$ \xi^2 \frac{d^2 Z_l}{d \xi^2}+\xi \frac{d Z_l}{d \xi}-\left[\left(l+\frac{1}{2}\right)^2-\xi^2\right] Z_l=0 $$ $$ j_l(\kappa r)=\sqrt{\pi / 2 \kappa r} J_{l+1 / 2}(\kappa r), \quad n_l(\kappa r)=\sqrt{\pi / 2 \kappa r} N_{l+1 / 2}(\kappa r) \text {; } $$ $$ h_l^{(1)}(\kappa r)=j_l(\kappa r)+\operatorname{in}_l(\kappa r), \quad h_l^{(2)}=j_l(\kappa r)-\operatorname{in}_l(\kappa r) . $$ $$ \begin{aligned} & h_l^{(1)}(\kappa r) \underset{\kappa r \rightarrow \infty}{ } \frac{(-i)^{l+1} e^{i \kappa r}}{\kappa r}, \\ & h_l^{(2)}(\kappa r) \underset{\kappa r \rightarrow \infty}{\longrightarrow} \frac{i^{l+1} e^{-i \kappa r}}{\kappa r}, \end{aligned} $$ \begin{array}{lc} j_0(\rho) & (1 / \rho) \sin \rho \\ n_0(\rho) & -(1 / \rho) \cos \rho \\ h_0^{(1)}(\rho) & -(i / \rho) e^{i \rho} \\ h_0^{(2)}(\rho) & (i / \rho) e^{-i \rho} \\ j_1(\rho) & \left(1 / \rho^2\right) \sin \rho-(1 / \rho) \cos \rho \\ n_1(\rho) & -(1 / \rho) \sin \rho-\left(1 / \rho^2\right) \cos \rho \\ h_1^{(1)}(\rho) & -(1 / \rho) e^{i \rho}(1+i / \rho) \\ h_1^{(2)}(\rho) & -(1 / \rho) e^{-i \rho}(1-i / \rho) \\ j_2(\rho) & {\left[\frac{3}{\rho^3}-\frac{1}{\rho}\right] \sin \rho-\frac{3}{\rho^2} \cos \rho} \\ n_2(\rho) & -\frac{3}{\rho^2} \sin \rho-\left[\frac{3}{\rho^3}-\frac{1}{\rho}\right] \cos \rho \\ h_2^{(1)}(\rho) & (i / \rho) e^{i \rho}\left(1+\frac{3 i}{\rho}-\frac{3}{\rho^2}\right) \\ h_2^{(2)}(\rho) & -(i / \rho) e^{-i \rho}\left(1-\frac{3 i}{\rho}-\frac{3}{\rho^2}\right) \end{array} $$ \psi_{l m}=\sqrt{\pi / 2 \kappa r} Z_l(\kappa r) P_l^m(\cos \theta) e^{\mp i m \phi} . $$ $$ \psi_{10}=\frac{1}{\kappa r} e^{i \kappa r}\left[1+\frac{i}{\kappa r}\right] \cos \theta . $$ $$ \left.\nabla \psi_{10}=\mathbf{a}_r e^{i \kappa r}\left[\frac{i}{r}-\frac{2}{\kappa r^2}-\frac{2 i}{\kappa^2 r^3}\right]\right] \cos \theta-\mathbf{a}_\theta e^{i \kappa r}\left[\frac{1}{\kappa r^2}+\frac{i}{\kappa^2 r^3}\right] \sin \theta . $$ $$ \vec{E}=\vec{r} \times \nabla \psi_{10}=-\mathbf{a}_\phi E_0 e^{i \kappa r}\left[\frac{1}{\kappa r}+\frac{i}{\kappa^2 r^2}\right] \sin \theta, $$ $$ \begin{aligned} \vec{B}= & -i \frac{1}{\omega} \nabla \times \vec{E} \\ = & i \frac{1}{\omega} E_0 e^{i \kappa r}\left[\frac{1}{\kappa r^2}+\frac{i}{\kappa^2 r^3}\right] 2 \cos \theta \mathbf{a}_r \\ & -i \frac{1}{\omega} E_0 e^{i \kappa r}\left[\frac{i}{r}-\frac{1}{\kappa r^2}-\frac{i}{\kappa^2 r^3}\right] \sin \theta \mathbf{a}_\theta . \end{aligned} $$
12.3.1 미분
12.4.1 Gradient
12.5.1 Gradient
12.5.1 Gradient