Electromagnetic theory

设平面光波沿着 \(x\) 方向传播, \(\vec{E}\) 不依赖 \(y, z\) \[\nabla\cdot \vec{E} = 0\Rightarrow \frac{\partial E_x}{\partial x} = 0\] 故

• 电场只有 \(y, z\) 分量, 振动方向垂直于传播方向 (横波)
• 电场振动方向可以用两个正交的分量叠加-两个线偏振光组合.

设电场为沿 y 方向振动的线偏振光 \(\vec{E} = \hat{\jmath}E_y(x, t)\) \[\nabla\times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \Rightarrow \frac{\partial E_y}{\partial x} = -\frac{\partial B_z}{\partial t}, \;\frac{\partial B_x}{\partial t} = \frac{\partial B_y}{\partial t} = 0\] \[\Rightarrow \vec{B} = \frac{1}{\omega}\vec{k}\times \vec{E} \Rightarrow E_y = vB_z\] 在 均匀、线性、各向同性 的电介质传播的电磁波是电场与磁场传播方向互为正交的横波.

注意: 平面波并非麦克斯韦方程的唯一解.

任何稳定平衡点附近的振动可以近似为简谐振动.

分子或介质置于外电场中, 发生极化, 诱导出偶极 \(q \vec{x}\), 方向为 \(-q\) 指向 \(+q\).

在光场(高频交变电磁场)驱动下, 分子中偏移电荷在平衡位置附近做受迫运动:(忽略磁场力) \[m \frac{\mathrm{d}^2x}{\mathrm{d}t^2} + m\omega_0^2 x = qE_0\cos(\omega t)\] \[\Rightarrow x(t) = \frac{q/m}{\left(\omega_0^2 - \omega^2\right)^2} E_0\cos(\omega t) = \frac{q/m}{\left(\omega 0^2 - \omega^2\right)^2} E(t)\] 设单位体积中有 \(N\) 个光电场诱导的偶极子,则电极化强度为: \[P = qxN = \frac{q^2NE/m}{\left(\omega 0^2 - \omega^2\right)^2} = \varepsilon_0\chi E\] \[\Rightarrow\chi = \frac{q^2N/m}{\left(\omega 0^2 - \omega^2\right)^2\varepsilon_0}\] \[\therefore n^2(\omega) = 1 + \frac{Nq^2}{\varepsilon_0m} \left( \frac{1}{\omega_0^2 - \omega^2} \right)\]

当光频率 \(\omega \ll \omega_0\), 折射率可近似表示为:

\begin{aligned} n^2(\omega) &= 1+ \frac{Nq^2}{\varepsilon_0m\omega_0^2} \left( 1+ \frac{\omega^2}{\omega_0^2} \right)\\ &= 1 + \frac{Nq^2}{\varepsilon_0m\omega_0^2} + \frac{Nq^24\pi^2c^2}{\varepsilon_0m\omega_0^4} \frac{1}{\lambda^2} \end{aligned}

\[\therefore \left( n^2 - 1 \right)^{-1} = -C\lambda^{-2} + C\lambda_0^{-2}\]

• \(\lambda_0 = \frac{2\pi c}{\omega_0}\)
• \(C = \frac{4\pi^2c^2\varepsilon_0 m}{Nq^2}\)
• \(\lambda_0^2\ll\lambda^2\) 时上式成立.

如果分子有很多个简谐模式, 易得: (\(f_j\) 为跃迁几率) \[n^2(\omega) = 1+ \frac{Nq^2}{\varepsilon_0 m}\sum_j \frac{f_j}{\omega_{0j}^2 - \omega^2}\] 因为忽略阻尼, \(\omega\rightarrow\omega_0\), \(n\) 发散, 故引入阻尼, 得到复吸收率: \[n^2(\omega) = 1 + \frac{Nq^2}{\varepsilon_0 m} \sum_j \frac{f_j}{\omega_{0j}^2 - \omega^2 + \mathrm{i}\gamma_j\omega}\] \[n = \eta + \mathrm{i}\kappa \Rightarrow k = \frac{\omega}{v} = \frac{\omega}{c/n} = (\eta + \mathrm{i}\kappa)\frac{\omega}{c}\]

\begin{aligned} E &= E_0 \mathrm{e}^{\mathrm{i}(kz-\omega t)}\\ &= E_0\exp \left[ \mathrm{i}(\eta+\mathrm{i}\kappa)\frac{\omega}{c}z -\omega t \right]\\ &= E_0\exp \left[ \underbrace{-\mathrm{i}\omega \left( t-\frac{\eta z}{c} \right)}_{\text{transmission phase}} \;\underbrace{-\frac{\kappa\omega}{c}z}_{\text{damping}} \right] \end{aligned}

transmission phase 为传输相位, damping 为衰减项.

\[n^2(\omega) = 1 - \frac{Nq^2}{\varepsilon_0m\omega^2} = 1 - \frac{\omega_p^2}{\omega^2}, \;\omega_p = q \sqrt{\frac{N}{\varepsilon_0 m}}\]

• \(\omega > \omega_p\) 折射率为小于 1 的实数, 透明
• \(\omega < \omega_p\) 折射率为虚数, 强吸收(强反射).

Internal field (Lorentz local field) is different form the applied field due to polarization of the dielectric. \[E_{\text{tot}} = E + E_{\text{int}} = E + \frac{1}{3\varepsilon_0}P\] 可得

Clausis-Mossotti relation

\[\frac{n^2 - 1}{n^2 + 2} \frac{Nq_e^2}{\varepsilon_0 m_e} \sum_j \frac{f_j}{\omega_{0j}^2 - \omega^2 + \mathrm{i}\gamma_j\omega}\]

• Normal dispersion: \(n\) increases with frequency.
• Anomalous dispersion: \(n\) decreases with frequency.

The radiation pressure equals the energy density of the electromagnetic wave. \[P = u = \frac{\varepsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2 = \frac{S(t)}{c}\] totally absorbed.

Average radiation pressure

\[\left\langle P(t) \right\rangle = \frac{S(t)}{c} = \frac{I}{c}\]

电磁波的动量 \[p_V = \frac{u}{c} = \frac{S}{c^2}\]

\[p = \frac{\epsilon}{c} = \frac{h}{\lambda}\Rightarrow \vec{p} = \hbar \vec{k}\] 其中约化普朗克常量 \(\hbar = \frac{h}{2\pi}\)