The index that depends on which way the light is going

At its defaults it draws the index that depends on which way the light is going. The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.

birefringence 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.

The two refractive indices of 3 uniaxial crystals, against the angle between the wave normal and the crystal's optic axis. The flat lines are the ordinary index, which is the same in every direction because the ordinary wave's field is always perpendicular to the axis. The curves are the extraordinary index, which runs from the ordinary value along the axis — where the two waves are identical and the crystal behaves like glass — to its extreme value at right angles to it. calcite (CaCO₃) has n_o = 1.6584 and n_e = 1.4864, so n_e − n_o = -0.1720; quartz (SiO₂) has n_o = 1.5443 and n_e = 1.5534, so n_e − n_o = 0.0091; lithium niobate has n_o = 2.3005 and n_e = 2.2075, so n_e − n_o = -0.0930. The sign of that difference is what makes a crystal positive or negative, and it decides which of the two images in a double-refracting crystal is the one that moves.

Two media with the same modulation depth, 0.2, computed the same way. Left: permittivity repeating in space, a stack of layers. For each frequency the wave equation is integrated across one spatial period, and the Bloch wavenumber drawn against frequency; between frequencies 0.478 and 0.526 (in units of c over the period) there is no real wavenumber, so light of those frequencies cannot travel and is reflected. Right: permittivity repeating in time. For each wavenumber the equation is integrated across one period, and the frequency drawn against wavenumber; between wavenumbers 0.474 and 0.518 there is no real frequency. The axes of the two panels are swapped, and so is everything else: the spatial gap is a band of frequencies that decays in space, the temporal gap is a band of wavenumbers that grows in time.

A uniform medium whose permittivity is modulated as 1 + 0.2 cos Ωt. Across: the wavenumber of a wave in it, in units of the modulation frequency divided by the speed of light. Up: the frequency the wave oscillates at, in units of the modulation frequency, folded into the range the modulation cannot distinguish. Each point comes from integrating the wave equation for that wavenumber over one period of the modulation. For most wavenumbers there is a real frequency. In one band of wavenumber there is none — from 0.475 to 0.521 — and a wave there grows instead, by up to 0.159 in its logarithm per modulation period; that band is shaded. The first gap is centred where the wave's own frequency is half the modulation's.

The growth rate of waves across the first wavenumber gap, in the logarithm of their amplitude per period of the modulation, for modulation depths of 0.1, 0.2, 0.4. At depth 0.1 the gap spans 4.8 per cent of its central wavenumber and the fastest growth is 0.079 per period; At depth 0.2 the gap spans 10.0 per cent of its central wavenumber and the fastest growth is 0.159 per period; At depth 0.4 the gap spans 20.3 per cent of its central wavenumber and the fastest growth is 0.333 per period. Both the width and the peak are close to proportional to the depth, and the small-depth estimate for the peak, an eighth of the depth times 2π, was checked against the integration. A wave at the centre of the gap at depth 0.4 grows by a factor of e in 3.0 periods of the modulation.

The amplitude of a single wave in the medium modulated to depth 0.2, on a logarithmic scale, over 30 periods of the modulation, integrated directly. At wavenumber 0.462 (in units of the modulation frequency over c), outside the gap, it wobbles with the modulation and returns to its starting size; at wavenumber 0.500 (in units of the modulation frequency over c), inside the gap, it grows by a factor of 73, matching the growth rate the one-period calculation predicts. The energy of the growing wave is supplied by whatever drives the modulation, and nothing in the equation limits it.

The frequency at which a medium's permittivity must be modulated to open a gap for waves of each kind, which is twice the waves' own frequency, on a logarithmic scale. Microwaves, 10 GHz: 2·10¹⁰ hertz; terahertz, 1 THz: 2·10¹² hertz; mid-infrared, 10 µm: 6·10¹³ hertz; telecom light, 1550 nm: 3.9·10¹⁴ hertz; green light, 530 nm: 1.1·10¹⁵ hertz. A gap for microwaves needs a modulation at tens of gigahertz, which electronic switches supply. A gap for visible light needs the index of a material to swing by a useful fraction of itself about a thousand million million times a second, faster than the response of any known optical material.

physicscheck asserts something about birefringence 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.