Cours 2 傅里叶光学，电磁学基础，波导

傅里叶光学

散射

关于傅里叶变换到频率还是角频率：

\[ \frac{2\pi}{p}\sum\delta(k_x-m\frac{2\pi}{p}) = \frac{2\pi}{p}\sum\delta(2\pi \sigma_x-m\frac{2\pi}{p}) = \frac{2\pi}{|2\pi|p}\sum\delta( \sigma_x-m\frac{1}{p}) = \frac{1}{p}\sum\delta( \sigma_x-m\frac{1}{p}) \]

注意，傅里叶变换完之后的结果是\(K_x\)和\(K_y\)，我们要将其转换为坐标\(\xi,\eta\)转换公式：

\[ K_x = k\frac {\xi} z = \frac {2\pi\xi} {\lambda z}\\K_y = k\frac \eta z = \frac {2\pi\eta} {\lambda z} \]

电磁学基础

真空中

结构关系

\(\vec B = \frac{\vec k \land \vec E}{\omega}\)

色散关系relation de dispersion

\(k = \frac{\omega}{c}\)

能量关系

\(\frac{du}{dt} + div(\vec S) = 0\)
能量：\(u = \epsilon_0\frac{\vec E^2}{2}+\frac{\vec B}{2\mu_0}\)
Poynting矢量：\(\vec S = \frac{\vec E \land \vec B}{\mu_0}\)

介质中

色散关系

\(\vec k^2 = \frac{\omega^2}{c^2}\widetilde\epsilon\mu \Rightarrow k = \frac \omega c \widetilde n\)

吸收和色散

\(\vec E = \vec E_0 exp(-\frac{\omega \kappa}{c})exp(i\omega(t-\frac{n}{c}z)), n = Re(\widetilde n), \kappa = Im(\widetilde n)\)

穿透深度 Profondeur de pénétration

\(\delta = \frac c {\omega\kappa} = \frac{\lambda}{2\pi\kappa}\)

衰减效率 Coefficient d’atténuation

\(\alpha = 2\frac{\omega\kappa}c = \frac{4\pi\kappa}\lambda\)

波导

导向模式

模态

有\(AC-AB = 2dsin(\theta)\Rightarrow sin(\theta_m) = m\frac{\lambda}{2d}\)
m被称为模态modes
第m模态的频率：\(\nu_m = m\frac{c}{2d}\)

纵向守恒

对于同一模态，波矢的y方向分量为常数：\(k_y = k_y(m)\)

\[ k_{ym} = m\frac \pi d \]

\[ \beta_m^2 = k^2-\frac{m^2\pi^2}{d^2} \Leftrightarrow k^2 = \beta^2+k_y^{2} \]

模态数量

设M为最大可允许的模态数量，根据\(sin(\theta_m) = m\frac{\lambda}{2d}\)，有：

\[ M = E(\frac{2d}{\lambda}) = E(\nu\frac{2d}{c}) \]

\(\lambda\)的最大值(\(M= 0\))longueur d ’onde de coupure du guide：\(\lambda_{max} = 2d\)
\(\nu\)的最小值fréquence de coupure：\(\nu_{min} = \Delta\nu = \frac c{2d}\)
单模态条件： \(d\le\lambda\le2d\)