4. Reflection and Refraction at Interfaces

Plane waves in infinite homogeneous media are an idealization. In real systems, waves hit boundaries, and what happens at those boundaries determines almost everything: how much signal a phone antenna captures, how an optical fiber traps light, how a Faraday cage shields, and how a PCB stackup couples or isolates layers.

4.1 Normal incidence

A plane wave hits a flat boundary head-on, going from medium 1 ($\eta_1$) to medium 2 ($\eta_2$). Tangential $\vec{E}$ and $\vec{H}$ must be continuous across the boundary (Section 2.9). Solving these conditions, you get the reflection coefficient

$\Gamma = \frac{\eta_2 - \eta_1}{\eta_2 + \eta_1}$

and transmission coefficient

$\tau = 1 + \Gamma = \frac{2\eta_2}{\eta_2 + \eta_1}$

Special cases:

• $\eta_1 = \eta_2$: $\Gamma = 0$, no reflection. The boundary is invisible to the wave. Matched.
• $\eta_2 = \infty$ (perfect dielectric to perfect insulator with no field allowed beyond): $\Gamma = +1$.
• $\eta_2 = 0$ (going into a perfect electric conductor): $\Gamma = -1$, full reflection with phase inversion. The tangential $\vec{E}$ at the metal surface must be zero, and a phase-inverted reflected wave is what makes that happen.

Wave-on-rope analogy. A pulse traveling on a rope tied to a heavier rope (higher $\eta$) reflects partially, same phase. Tie it to an immovable wall (infinite mechanical impedance): full reflection with phase inversion, as if a mirror image pulse came back. Tie it to nothing (zero mechanical impedance, free end): full reflection, same phase. Identical math to a transmission line.

4.2 Oblique incidence

When a wave hits a boundary at an angle, you must consider both polarizations. They behave differently.

• Parallel polarization (TM, transverse magnetic): $\vec{E}$ lies in the plane of incidence. Has a special angle (Brewster's, below) where reflection drops to zero.
• Perpendicular polarization (TE, transverse electric): $\vec{E}$ is perpendicular to the plane of incidence. No Brewster angle.

For both, Snell's law governs the angle of refraction:

$n_1\sin\theta_1 = n_2\sin\theta_2, \quad n = \sqrt{\varepsilon_r\mu_r}$

The Fresnel equations give the reflection coefficients for each polarization:

$\Gamma_{\parallel} = \frac{\eta_2\cos\theta_2 - \eta_1\cos\theta_1}{\eta_2\cos\theta_2 + \eta_1\cos\theta_1}$

$\Gamma_{\perp} = \frac{\eta_2\cos\theta_1 - \eta_1\cos\theta_2}{\eta_2\cos\theta_1 + \eta_1\cos\theta_2}$

These are the Fresnel equations. They tell you what fraction of an incident wave's amplitude reflects, as a function of angle and polarization. Glasses, lens coatings, RF radomes, and stealth-aircraft skins all rely on the Fresnel equations.

4.3 Brewster's angle

For parallel polarization, $\Gamma_{\parallel}$ goes to zero at the Brewster angle:

$\tan\theta_B = n_2/n_1$

At Brewster's angle, the parallel-polarized component is fully transmitted; only the perpendicular component reflects. This is why polarized sunglasses work. Light reflecting off horizontal surfaces (roads, water) tends to be horizontally polarized (the perpendicular component for that geometry). Vertically aligned polarizers in your sunglasses block it, killing the glare while letting most of the directly transmitted light through.

Brewster windows in laser cavities use the same principle in reverse: orient a window at Brewster's angle so that one polarization passes losslessly, building up only that polarization in the cavity.

4.4 Critical angle and total internal reflection

When going from a denser to a less dense medium ($n_1 > n_2$), Snell's law forces $\sin\theta_2 = (n_1/n_2)\sin\theta_1$. Beyond the critical angle

$\sin\theta_c = n_2/n_1$

the right side exceeds 1, so $\theta_2$ has no real solution. The wave cannot escape; it is totally reflected. This is the basis of:

• Optical fiber. Light bounces along the core by total internal reflection, traveling kilometers with very little loss. The entire global internet runs on this physics. We will go deeper in Chapter 20.
• Diamond brilliance. Diamond has $n \approx 2.4$, giving a low critical angle of about 24°. Light entering a faceted diamond gets trapped, bouncing through several total internal reflections before exiting at a sparkle-producing angle.
• Endoscopes and fiberscopes.
• Prisms in binoculars and SLR cameras.

4.5 The Poynting vector: where does the energy go?

The Poynting vector

$\vec{S} = \vec{E}\times\vec{H}$

has units of W/m² and points in the direction of energy flow. For a plane wave in free space, the time-averaged Poynting vector is

$\langle S\rangle = \frac{E_0^2}{2\eta_0}$

This tells you the power per unit area carried by the wave. A 1 W/m² plane wave (roughly the intensity of sunlight at Earth) has $E_0 \approx 27$ V/m. A typical Wi-Fi signal at the receiver is around $10^{-7}$ W/m², which corresponds to $E_0 \approx 10$ mV/m, just barely above noise.

The Poynting vector also explains an unsettling fact: in a coaxial cable, the energy flows through the dielectric, not the metal. The current in the conductors guides the field, but the actual energy is carried by the EM wave in the gap between inner and outer conductor. The wires are just the rails.