Reflectance is pure impedance mismatch — and total internal reflection is tunnelling, done with light
We have been talking about refraction, but something else always happens at a boundary: reflection. And there is exactly one satisfying principle here. Reflection happens not because matter is present, but because there is a mismatch. Match the indices exactly and the boundary reflects nothing at all, even though it is still there. Anti-reflection coatings and Brewster's angle are both corollaries of that single line. But the real business of this episode is total internal reflection — light is supposed to transmit nothing, and yet an \(e^{-\kappa z}\) wave is seeping beyond the boundary. Put a second piece of glass right next to it, and light gets through. The transmission is \(T\propto e^{-2\kappa d}\) — the same equation as Episode 1 of the sister series Tunneling That Clicks, with different letters. Tunnelling, done with light. Newton saw it in 1704.
Matching electric and magnetic fields across the boundary gives the amplitude reflection coefficient
$$r=\frac{n_1-n_2}{n_1+n_2},\qquad R=|r|^2=\left(\frac{n_1-n_2}{n_1+n_2}\right)^2$$Air (1) to glass (1.5) gives \(r=-0.2\) and \(R=4\%\). A pane of glass has two surfaces, so it loses about 8%.
If \(n_1=n_2\), then \(r=0\). The boundary is there, and it does not reflect. What causes reflection is not matter but a difference.
About the sign
For \(n_2>n_1\), \(r<0\): the reflected wave is shifted by \(\pi\). The same thing as shaking a rope tied to a wall — the returning pulse is inverted. Going dense to rare (glass to air), it is not.
Come in at an angle and polarization matters. For light polarized in the plane of incidence (p-polarization), there is an angle at which the reflection drops exactly to zero.
Why it vanishes — recall Episode 1. Both the reflected and the transmitted wave are light emitted by oscillating dipoles inside the material. At \(\theta_B\) the refracted and reflected rays are exactly 90° apart, so for p-polarization the dipole's oscillation direction coincides exactly with the reflected ray direction.
And a dipole does not radiate along its own axis (the intensity goes as \(\sin^2\theta\)). There is nobody to emit toward, so the reflected wave cannot exist.
The upshot is that light reflected near \(\theta_B\) is almost purely s-polarized. Glare off a road or a water surface is horizontally polarized for this reason. Polarizing sunglasses are a sheet that cuts exactly that direction.
Going from dense to rare (glass to air) at increasing angle, Snell's law \(n_1\sin\theta_1=n_2\sin\theta_2\) begins to demand \(\sin\theta_2>1\). No such angle exists. That is total internal reflection, with the boundary at
Beyond \(\theta_c\), the component of the wavevector normal to the boundary becomes imaginary:
$$k_z=\frac{\omega}{c}\sqrt{n_2^2-n_1^2\sin^2\theta}\ \longrightarrow\ i\kappa,\qquad \kappa=\frac{2\pi}{\lambda_0}\sqrt{n_1^2\sin^2\theta-n_2^2}$$So \(e^{ik_zz}\to e^{-\kappa z}\): not a travelling wave but an exponentially decaying one, called an evanescent wave.
Numbers: at \(\lambda_0=550\) nm, glass (1.5) to air, \(\theta=45^\circ\), the decay length \(1/\kappa\approx\) 248 nm. Less than a wavelength.
Bring a second piece of glass within a few hundred nanometres of a totally reflecting surface. Light gets through. Close the gap and more gets through; open it and the transmission falls exponentially.
| Electron tunnelling (Tunneling Ep. 1) | Total reflection (here) | |
|---|---|---|
| The wave | wavefunction \(\psi\) | electric field \(E\) |
| Why it is forbidden | \(E| \(n_1\sin\theta>n_2\) (too steep an angle) | |
| Decay constant | \(\kappa=\dfrac{\sqrt{2m(V-E)}}{\hbar}\) | \(\kappa=\dfrac{2\pi}{\lambda_0}\sqrt{n_1^2\sin^2\theta-n_2^2}\) |
| Transmission | \(T=\left[1+\dfrac{(k^2+\kappa^2)^2}{4k^2\kappa^2}\sinh^2\kappa d\right]^{-1}\ \xrightarrow{\ \kappa d\gg1\ }\ \propto e^{-2\kappa d}\) identical | |
| Decay length | ~0.2 nm (\(V-E=1\) eV) | ~248 nm (visible, 45°) |
These are not merely similar equations. They are the same equation. Both describe a region where the wave equation has \(k^2<0\). The difference is that the decay length is over a thousand times larger for light, which is why you can do the experiment with two slabs of glass held by hand.
Electron tunnelling feels strange mostly because of the phrasing "it gets over the wall without enough energy." But look at the table above — exactly the same thing happens with light, and nobody finds that strange.
Because light is a wave to begin with. That a wave goes as \(e^{-\kappa z}\) in a forbidden region is just the solution of a differential equation. There is nothing quantum in the exponential at all.
What is quantum is that the electron is a wave — that one point alone. Grant that, and the rest is the same classical wave physics Newton saw in 1704.
— In this collection's vocabulary, this is the second grade of "similar": different solutions of one framework. The same wave equation, in the region where \(k^2<0\).
Established: the normal-incidence Fresnel coefficient \(r=(n_1-n_2)/(n_1+n_2)\) and glass's 4% reflection; the anti-reflection condition \(n=\sqrt{n_1n_3}\) at \(\lambda/4\) thickness, and MgF₂ (1.38) taking 4% to about 1.4%; Brewster's angle \(\tan\theta_B=n_2/n_1\) and the dipole-axis explanation; the critical angle \(\sin\theta_c=n_2/n_1\); the evanescent wave \(e^{-\kappa z}\) with \(\kappa=(2\pi/\lambda_0)\sqrt{n_1^2\sin^2\theta-n_2^2}\); zero time-averaged normal energy flux in steady state; frustrated total internal reflection and the fact that the transmission through a symmetric gap has exactly the form of the quantum rectangular barrier; Newton's two-prism observation in Opticks (1704); TIRF, near-field microscopy and optical fingerprint sensors. All standard optics.
Caveats: (1) The transmission in the table and figure is for s-polarization (field perpendicular to the plane of incidence). For p-polarization the prefactor changes, but the \(e^{-2\kappa d}\) exponent is the same. (2) The figure idealises to non-absorbing, non-dispersive, perfectly flat surfaces and a monochromatic plane wave; real glass has surface roughness of a few nanometres, which dominates the near-contact behaviour. (3) "The same equation" means the same mathematical form. Photon tunnelling and electron tunnelling are not the same phenomenon; what is shared is the structure of crossing a region where the wave equation has \(k^2<0\) (the second grade, in the text's phrasing). (4) Brewster's angle gives zero reflection only for p-polarization; s-polarization does not vanish, and magnetic materials (\(\mu\neq1\)) change the condition. (5) "No energy crosses" for the evanescent wave is a statement about the steady-state time average; transiently, energy does flow both ways. (6) The 0.2 nm electron decay length is for \(V-E=1\) eV and varies strongly with barrier height.
Reflectance is fixed by index mismatch alone (\(r=(n_1-n_2)/(n_1+n_2)\); 4% for glass). Zero difference, zero reflection, boundary or not. Anti-reflection coatings split the difference in two and cancel out of phase — \(n=\sqrt{n_1n_3}\) is ideal, MgF₂ gets 4% down to 1.4%. At Brewster's angle (\(\tan\theta_B=n_2/n_1\), 56.3° for glass) p-polarization vanishes because a dipole does not radiate along its own axis — a direct consequence of Episode 1's "refraction is forward scattering."
Past the critical angle you get total internal reflection, but an evanescent wave \(e^{-\kappa z}\) still seeps beyond the boundary (decay length 248 nm for visible light at 45°). In steady state no energy crosses — until you put a second slab there. Then light does get through, with transmission
\(T=\left[1+\frac{(k^2+\kappa^2)^2}{4k^2\kappa^2}\sinh^2\kappa d\right]^{-1}\ \propto\ e^{-2\kappa d}\)
That is Episode 1 of Tunneling That Clicks with different letters. Newton saw it with two prisms in 1704 — 220 years before quantum mechanics. Which licenses the conclusion: the "quantumness" of tunnelling is not in the exponential. What is quantum is only that the electron is a wave.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, move the gap d and watch the marker travel along the log plot, landing on a straight line of slope −2κ. Bring the incidence angle toward the critical angle (41.8°) and κ shrinks, so the wave reaches much further. "Show answer" reveals the solutions.