The critical angle for n = 1.5 into n = 1

At its defaults it draws the critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.

critical-angle is one function in lib/figures/optics.js — rays, lenses, mirrors and what light does to a surface. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.

Reflectance of light at 633 nm inside a BK7 prism whose face carries a 50 nm gold film with water beyond it, against the angle of incidence, for the two polarisations, from the three-layer Fresnel sum in complex arithmetic. Past the critical angle for glass and water, 61.62°, s-polarised light is reflected almost completely, as total internal reflection requires, apart from what the thin gold absorbs. p-polarised light is reflected too — except near one angle, where the reflectance falls to 0.014 at 71.52°. The dispersion relation of a surface plasmon on gold against water puts its resonance at 71.67°; the difference is the film's finite thickness and the prism on its other side pulling the resonance slightly.

The p-polarised reflectance of the glass | gold | water arrangement against angle for gold films of 30, 40, 50, 60, 70 nm. At 30 nm the dip reaches 0.367 at 71.05°; at 40 nm the dip reaches 0.075 at 71.28°; at 50 nm the dip reaches 0.014 at 71.52°; at 60 nm the dip reaches 0.209 at 71.70°; at 70 nm the dip reaches 0.472 at 71.80°. The deepest dip, found by scanning the thickness in half-nanometre steps, is at 47.0 nm, where it reaches zero to four decimal places: there the energy the plasmon loses by leaking back through the film into the prism equals what it loses by heating the gold, and the reflected wave cancels. Thinner films leak too much and give a broad, shallow dip; thicker ones let too little light reach the plasmon at all.

The p reflectance of the glass | 50 nm gold | sample arrangement for samples of index 1.333 and 1.343, and the angle of the dip against the sample's index from 1.330 to 1.340. The dip moves from 71.52° to 72.99° for an index change of 0.010, and over the whole range at 144° per unit of index. A dip whose position is read to a thousandth of a degree therefore registers a change of index of 7.0 × 10⁻⁶, averaged over the thin layer next to the gold that the plasmon's field reaches — the scale on which molecules binding to the gold are detected.

The p reflectance against angle for films of silver, gold and aluminium between a BK7 prism and water at 633 nm, each at the thickness that gives it its deepest dip, with permittivities from tabulated values: silver, ε = −18.30 + 0.48i, 55 nm, dip at 67.82° and 0.79° wide at half depth; gold, ε = −12.45 + 1.29i, 47 nm, dip at 71.46° and 5.54° wide at half depth; aluminium, ε = −55.90 + 20.80i, 12 nm, dip at 64.78° and 18.38° wide at half depth. The width is set by how fast the plasmon loses energy into the metal, which grows with the imaginary part of the permittivity relative to the real: silver's resonance is the sharpest, aluminium's so broad it is barely a resonance. Gold is used anyway, because silver tarnishes and gold does not.

The intensity of the magnetic field just beyond the gold, in the water, relative to the incident light's, against the angle of incidence, for the same 50 nm film. At the resonance, 71.03°, it is 12.6 times the incident intensity, although almost no light is reflected and none is transmitted into the water as a propagating wave: the energy is held in the plasmon and dissipated in the metal. Into the water the intensity falls by a factor of e in 96 nm — set, as for any evanescent wave, by the angle alone — so the resonance senses a layer of the order of a hundred nanometres thick and nothing beyond it.

physicscheck asserts something about critical-angle that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.