Generator

The critical angle for n = 1.5 into n = 1

One function in the optics library, called 29 times across 5 essays. Below: what it draws at its defaults, what it draws at every branch an essay asks for, whether the site's own gate puts a claim to it, and everywhere it is called.

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.

At its defaults

Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.

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.

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.

The critical angle for n = 1.5 into n = 1

The options are the ones The angle past which light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

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.

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.

How far past the boundary the light that does not cross it reaches

The options are the ones The angle past which light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

How far past the boundary the light that does not cross it reaches. The depth at which the field beyond the boundary has fallen to 1/e of its value at the surface, for 550 nm light going from n = 1.5 to n = 1, against angle of incidence past the 41.81° critical angle. The usual sentence — "the evanescent field reaches about a wavelength" — is true in the middle of the range and nowhere near the critical angle, where the depth diverges: the exponent is the square root of sin²θ − (n₂/n₁)², which goes to zero there. The depth is at 42° it is 1017 nm, at 45° it is 248 nm, at 50° it is 155 nm, at 60° it is 106 nm, at 75° it is 83 nm. That is the whole of what makes total internal reflection total: not that the field is zero on the far side, which it is not, but that a purely decaying field carries no time-averaged energy away.

The depth at which the field beyond the boundary has fallen to 1/e of its value at the surface, for 550 nm light going from n = 1.5 to n = 1, against angle of incidence past the 41.81° critical angle. The usual sentence — "the evanescent field reaches about a wavelength" — is true in the middle of the range and nowhere near the critical angle, where the depth diverges: the exponent is the square root of sin²θ − (n₂/n₁)², which goes to zero there. The depth is at 42° it is 1017 nm, at 45° it is 248 nm, at 50° it is 155 nm, at 60° it is 106 nm, at 75° it is 83 nm. That is the whole of what makes total internal reflection *total*: not that the field is zero on the far side, which it is not, but that a purely decaying field carries no time-averaged energy away.

The critical angle for n = 1.33 into n = 1

The options are the ones The angle past which light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The critical angle for n = 1.33 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 48.8°, beyond which no refracted ray exists at all.

Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 48.8°, beyond which no refracted ray exists at all.

The critical angle for n = 2.42 into n = 1

The options are the ones The angle past which light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

The critical angle for n = 2.42 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 24.4°, beyond which no refracted ray exists at all.

Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 24.4°, beyond which no refracted ray exists at all.

Light crossing a gap it cannot enter

The options are the ones The angle past which light cannot leave passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.

Light crossing a gap it cannot enter. The fraction of a totally reflected beam that crosses a gap of air between two pieces of glass, against the width of the gap, at 550 nm and 5 angles past the critical angle. At zero gap the two pieces are one piece and everything crosses. As the gap opens the transmission falls off as the square of the evanescent field, which is an exponential in the gap — at 42° it is half at 156 nm, at 45° it is half at 141 nm, at 50° it is half at 122 nm, at 60° it is half at 93 nm, at 75° it is half at 51 nm. The formula is the quantum tunnelling formula: a barrier the wave decays inside, a transmission governed by sinh² of the decay constant times the width, and the same divergence of the decay length as the barrier is made shallow. This is the classical rehearsal of an effect that is usually taught as having no classical analogue, and it was demonstrated by Newton, with two prisms, more than two centuries before anyone needed it to explain a nucleus.

The fraction of a totally reflected beam that crosses a gap of air between two pieces of glass, against the width of the gap, at 550 nm and 5 angles past the critical angle. At zero gap the two pieces are one piece and everything crosses. As the gap opens the transmission falls off as the square of the evanescent field, which is an exponential in the gap — at 42° it is half at 156 nm, at 45° it is half at 141 nm, at 50° it is half at 122 nm, at 60° it is half at 93 nm, at 75° it is half at 51 nm. The formula is the quantum tunnelling formula: a barrier the wave decays inside, a transmission governed by sinh² of the decay constant times the width, and the same divergence of the decay length as the barrier is made shallow. This is the classical rehearsal of an effect that is usually taught as having no classical analogue, and it was demonstrated by Newton, with two prisms, more than two centuries before anyone needed it to explain a nucleus.

What checks 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.

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

The angle past which light cannot leave

Snell's law asks for the sine of an angle greater than one, and no such angle exists. What happens instead is a perfect mirror made out of nothing but a change of speed.

The angle that is two angles

Snell's law survives a complex index by giving a complex answer, and a complex angle is not an angle. What the phase-matching argument actually fixes is the tangential wavenumber, and when the medium absorbs, the surfaces of constant phase and the surfaces of constant amplitude stop being parallel. In silver at 550 nanometres the phase fronts run within four degrees of the surface while the amplitude decays straight into it.

The bend at the boundary, and what it is really about

Light changes direction when it changes speed. Snell's law is the geometry of that statement, and it can be derived without knowing anything about light at all.

The channel with no walls

A pipe will not carry a note below its cutoff, and no length of pipe helps. Replace the walls with nothing but a region where the wave travels slightly slower, and the cutoff disappears — however weak the contrast and however thin the channel, at least one mode is bound. The difference is not a matter of degree; it is the difference between a boundary condition and a potential well.

The reflection that happens where the glass is not

Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed.

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