Reflection and Refraction at Boundaries

A uniform plane wave travelling through one medium eventually arrives at a surface where the material properties change. At that surface the single travelling wave can no longer satisfy Maxwell's equations by itself: the medium beyond it has a different intrinsic impedance, so the ratio of \(\vec{E}\) to \(\vec{H}\) it would demand is wrong. Nature resolves this by launching two new waves — a reflected wave that returns into the first medium and a transmitted wave that continues into the second. Their amplitudes are not free; they are precisely what is needed to keep the fields continuous across the surface.

The whole chapter rests on the two boundary conditions established back in electrostatics and magnetostatics, now read at a single instant for a time-varying field. With no surface currents on an ordinary dielectric interface, the tangential components of both \(\vec{E}\) and \(\vec{H}\) must be continuous:

Every result below — coefficients, Snell's law, Brewster's angle — is just these two statements written out for a particular geometry. We start with the simplest: the wave striking the surface head-on.

Let medium 1 (\(z<0\), intrinsic impedance \(\eta_1\)) carry a wave straight toward the boundary at \(z=0\), into medium 2 (\(z>0\), impedance \(\eta_2\)). The incident and reflected waves live in medium 1; the transmitted wave lives in medium 2:

The matching \(\vec{H}\) fields follow from each wave's own impedance, with the reflected wave's magnetic field reversed because it travels the other way: \(H_y=E_x/\eta\) for a forward wave and \(H_y=-E_x/\eta\) for the backward (reflected) wave. Imposing the two boundary conditions at \(z=0\) gives two equations in the two unknown amplitudes \(E_{r0}\) and \(E_{t0}\).

Continuity of \(E_x\) gives \(E_{i0}+E_{r0}=E_{t0}\). Continuity of \(H_y\) gives \(\dfrac{E_{i0}-E_{r0}}{\eta_1}=\dfrac{E_{t0}}{\eta_2}\). Solving the pair yields the two coefficients that define the interface:

A single identity ties them together and is worth memorising as a check on any calculation:

The sign of \(\Gamma\) carries the physics. If medium 2 is "denser" in the impedance sense (\(\eta_2<\eta_1\)), then \(\Gamma<0\): the reflected field flips by \(180^\circ\). If \(\eta_2>\eta_1\), the reflection is in phase. When the impedances match (\(\eta_2=\eta_1\)) there is no reflection at all — \(\Gamma=0\), \(\tau=1\) — the principle behind anti-reflection coatings and matched loads. Power obeys its own bookkeeping: the fraction reflected and the fraction transmitted must add to one for lossless media.

Push the second medium to the extreme of a perfect conductor: \(\eta_2\to 0\), so \(\Gamma=-1\) and \(\tau=0\). The wave is thrown back entirely, and nothing penetrates. The incident and reflected waves of equal amplitude, travelling in opposite directions, superpose into a pattern that no longer moves — a standing wave:

The field magnitude is fixed in space: zero (a node) at the conductor and at every half wavelength back from it, maximum (an antinode) a quarter wavelength away. For a general interface the reflection is partial, so the pattern does not reach zero. How deep the dips run is captured by the standing-wave ratio, the ratio of the largest to the smallest field magnitude:

A matched interface gives \(\Gamma=0\) and \(s=1\) (no standing wave, a flat envelope); total reflection gives \(|\Gamma|=1\) and \(s\to\infty\). This same number returns in Part 6 as the voltage standing-wave ratio on a mismatched transmission line — the analogy is exact, and not a coincidence.

Tilt the incoming wave so it strikes the surface at an angle. The plane that holds both the incident ray and the surface normal is the plane of incidence, and all three rays — incident, reflected, transmitted — lie in it. Three angles, measured from the normal, describe the geometry:

Requiring the fields to match along the whole interface — not just at one point — forces the phase of all three waves to advance together across the surface. That single demand produces the two laws of geometric optics, here derived from electromagnetics rather than assumed:

The reflected angle equals the incident angle, and the bend on transmission is set by the ratio of refractive indices (or equivalently of phase constants, or of \(\sqrt{\varepsilon_r}\) for non-magnetic media). A wave entering a denser medium (\(n_2>n_1\)) bends toward the normal; leaving into a rarer one, it bends away. How much energy goes each way, though, depends on something Snell's law is silent about: the orientation of the electric field.

At oblique incidence the answer splits in two, according to how the electric field sits relative to the plane of incidence. Any wave is a combination of these two independent cases:

• Perpendicular polarization (also called \(s\)- or TE-polarization): \(\vec{E}\) points out of the plane of incidence, parallel to the surface.
• Parallel polarization (also called \(p\)- or TM-polarization): \(\vec{E}\) lies within the plane of incidence.

Each case has its own pair of Fresnel coefficients, obtained — as always — by matching the tangential fields, now with the extra factors of \(\cos\theta\) that the tilt introduces:

Set \(\theta_i=\theta_t=0\) and both pairs collapse back to the normal-incidence formulas of Section 22-3 — a useful sanity check. The two polarizations reflect by different amounts at the same angle, and that difference is exactly what makes reflection a tool for controlling polarization.

Two special angles emerge from the Fresnel formulas. The first is the Brewster angle \(\theta_B\), the incidence angle at which the parallel-polarized reflection vanishes (\(\Gamma_\parallel=0\)). At this angle the reflected and refracted rays are exactly \(90^\circ\) apart, and any reflected light is purely perpendicular-polarized — reflection has acted as a polarizer:

The second appears only when the wave moves from a denser into a rarer medium (\(n_1>n_2\)). As \(\theta_i\) grows, the refracted ray bends ever closer to the surface until, at the critical angle \(\theta_c\), it grazes along it (\(\theta_t=90^\circ\)). Beyond \(\theta_c\) Snell's law has no real solution: nothing is transmitted and \(|\Gamma|=1\) for both polarizations — total internal reflection:

Total internal reflection is the mechanism that traps light inside an optical fibre: a glass core of slightly higher index is sheathed in a lower-index cladding, and any ray steep enough bounces along the core for kilometres with almost no loss. The Brewster angle, meanwhile, is why glare off water and roads is strongly polarized, and why polarized sunglasses — and camera filters — can cut it.

For each item, decide first whether the incidence is normal or oblique, and which medium is denser — those two facts steer the choice of formula. Difficulty rises down the list.