Electromagnetic Waves in Media

Step 1: Equation of motion for a bound electron

$$m_e\ddot{x} + m_e\gamma\dot{x} + m_e\omega_0^2 x = -eE_0 e^{-i\omega t}$$

The restoring force gives $\omega_0$; the damping $\gamma$ accounts for radiation and collisional losses.

Step 2: Steady-state displacement

$$x_0 = \frac{-eE_0/m_e}{\omega_0^2 - \omega^2 - i\gamma\omega}$$

Step 3: Polarization

The polarization (dipole moment per volume) for $N$ electrons per unit volume: $P = -Nex_0 = Ne^2 E_0/[m_e(\omega_0^2 - \omega^2 - i\gamma\omega)]$.

Step 4: Dielectric function

Since $\mathbf{D} = \epsilon_0\mathbf{E} + \mathbf{P} = \epsilon_0\epsilon_r\mathbf{E}$:

$$\epsilon_r(\omega) = 1 + \frac{Ne^2}{\epsilon_0 m_e}\frac{1}{\omega_0^2 - \omega^2 - i\gamma\omega}$$

Step 5: Multiple resonances and sum rules

Real atoms have multiple resonances. Introducing oscillator strengths $f_j$: $$\boxed{\epsilon_r(\omega) = 1 + \frac{Ne^2}{\epsilon_0 m_e}\sum_j\frac{f_j}{\omega_j^2 - \omega^2 - i\gamma_j\omega}}$$

The Thomas-Reiche-Kuhn sum rule $\sum_j f_j = Z$ guarantees that at very high frequencies, $\epsilon \to 1 - \omega_p^2/\omega^2$ where $\omega_p = \sqrt{NZe^2/(\epsilon_0 m_e)}$.

Step 1: Causality in the time domain

The response function $G(t) = \chi(t)$ (the inverse Fourier transform of the susceptibility) must vanish for $t < 0$: the polarization at time $t$ depends only on the field at earlier times.

Step 2: Analyticity in the upper half-plane

Since $G(t) = 0$ for $t < 0$, the Fourier transform $\chi(\omega) = \int_0^\infty G(t)e^{i\omega t}dt$ converges for $\text{Im}(\omega) > 0$. Therefore $\chi(\omega)$ is analytic in the upper half-plane.

Step 3: Apply Cauchy's integral formula

For $\omega$ on the real axis, close the contour in the upper half-plane:

$$\chi(\omega) = \frac{1}{i\pi}\mathcal{P}\!\int_{-\infty}^{\infty}\frac{\chi(\omega')}{\omega'-\omega}\,d\omega'$$

The $1/(i\pi)$ comes from the half-residue at the pole on the real axis.

Step 4: Separate real and imaginary parts

Taking real and imaginary parts of the Cauchy relation gives the two Kramers-Kronig dispersion relations.

Step 5: Physical significance

The real part (dispersion) and imaginary part (absorption) of $\chi$ are not independent: measuring the absorption spectrum over all frequencies determines the refractive index at every frequency, and vice versa. This is a profound consequence of causality alone.

Step 1: Equation of motion for free electrons

With no restoring force ($\omega_0 = 0$) and negligible collisions ($\gamma \to 0$): $m_e\ddot{x} = -eE$. For harmonic fields: $x_0 = eE_0/(m_e\omega^2)$.

Step 2: Dielectric function

$\epsilon(\omega) = 1 - \omega_p^2/\omega^2$ where $\omega_p = \sqrt{Ne^2/(\epsilon_0 m_e)}$.

Step 3: Dispersion relation

The wave equation gives $k^2 c^2 = \omega^2\epsilon = \omega^2 - \omega_p^2$, i.e.:

$$\omega^2 = \omega_p^2 + c^2 k^2$$

Step 4: Cutoff frequency

For $\omega < \omega_p$: $k^2 < 0$, the wave is evanescent. The plasma reflects waves below $\omega_p$. For the ionosphere: $f_p \sim 10$ MHz, reflecting AM radio signals around the Earth. For metals: $\omega_p \sim 10$ eV, making them reflective in the visible spectrum.

Step 5: Phase and group velocity

$$v_p = \frac{c}{\sqrt{1-\omega_p^2/\omega^2}} > c, \qquad v_g = c\sqrt{1-\omega_p^2/\omega^2} < c$$

The product $v_p v_g = c^2$. Information and energy travel at the group velocity, always less than $c$.

Step 1: Huygens construction

The particle at position $z = vt$ emits spherical wavelets expanding at speed $c/n$. When $v > c/n$, the envelope forms a Mach cone with half-angle $\theta_c$.

Step 2: Cone angle

From the geometry: the wavelet travels $(c/n)\Delta t$ while the particle travels $v\Delta t$. The angle satisfies $\cos\theta_c = (c/n)/(v) = 1/(n\beta)$.

Step 3: Frank-Tamm formula

The energy radiated per unit path length per unit frequency interval is:

$$\frac{d^2W}{dx\,d\omega} = \frac{\mu_0 q^2\omega}{4\pi}\left(1-\frac{1}{n^2(\omega)\beta^2}\right)$$

valid for frequencies where $n(\omega)\beta > 1$.

Step 4: Number of photons

Converting to photon number using $dW = \hbar\omega\,dN$:

$$\frac{d^2N}{dx\,d\omega} = \frac{\alpha}{\hbar c}\left(1-\frac{1}{n^2\beta^2}\right) \approx \frac{\alpha}{c}\sin^2\theta_c$$

where $\alpha \approx 1/137$ is the fine structure constant.

Step 5: Spectral character

Since $dN/d\omega \propto 1$ (constant per unit frequency), $dN/d\lambda \propto 1/\lambda^2$. More blue-UV photons are emitted than red, giving Cherenkov radiation its characteristic blue glow. The total number of visible photons is about 300 per cm for a relativistic particle in water.

Step 1: High-frequency limit

For $\omega \gg \omega_j$ for all $j$: $\epsilon(\omega) \to 1 - \omega_p^2/\omega^2$ where $\omega_p^2 = Ne^2\sum_j f_j/(\epsilon_0 m_e)$.

Step 2: Physical requirement at high frequencies

At $\omega \gg \omega_j$, all electrons respond as free particles regardless of binding. Thus $\omega_p^2 = NZe^2/(\epsilon_0 m_e)$ where $Z$ is the total number of electrons per atom.

Step 3: Thomas-Reiche-Kuhn sum rule

$$\boxed{\sum_j f_j = Z}$$

Step 4: Integral form (f-sum rule)

Equivalently: $\int_0^\infty\omega\,\text{Im}[\epsilon(\omega)]\,d\omega = \frac{\pi}{2}\omega_p^2$. This constrains the total integrated absorption.

Step 5: Connection to Kramers-Kronig

The sum rule is a consequence of the Kramers-Kronig relation evaluated at $\omega = 0$ combined with the known high-frequency behavior. It connects the microscopic quantum structure (oscillator strengths, related to transition matrix elements) to macroscopic optical properties.