Illustrated Physics
physics, drawn
Concept

Floquet theory — where it appears

The theory of linear systems whose coefficients repeat periodically, from a pumped swing to a crystal or a laser cavity. Each period acts as a fixed matrix, and the motion stays bounded for ever exactly when half that matrix's trace lies between −1 and 1.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

A repeat in space gaps the frequencies; a repeat in time gaps the wavenumbers. 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.

The crystal made of moments

A stack of layers that repeats in space refuses a band of frequencies and reflects them. A medium that repeats in time — its refractive index swung up and down everywhere at once — refuses a band of wavenumbers instead, and a wave with a wavenumber in that band does not reflect. It grows, exponentially, drawing on whatever is swinging the index. The construction is exact, the gap is computable from one period of the modulation, and the obstacle to building one for light is how fast a material would have to change.

optics · Refraction The cavities that hold a beam. Every two-mirror cavity as a point (g1, g2), with g = 1 − L/R for each mirror. The shaded region, between the axes and the hyperbolas g1 g2 = 1, holds a ray for ever; everywhere else a ray is walked out. The commonest designs sit on the boundary: two flat mirrors (1, 1), the confocal cavity with the mirrors' foci together (0, 0), the concentric cavity with their centres together (−1, −1), and a flat mirror facing a curved one at its own radius (1, 0). A boundary cavity is stable in principle and marginal in practice, which is why the confocal one, sitting at the corner where the two shaded regions touch, is usually built a little inside it.

The two mirrors that hold a beam

A laser is two mirrors facing each other, and the light in it makes thousands of round trips before it leaves. Whether a ray stays between the mirrors for ever or is walked out after a few bounces depends on a single number, built from their spacing and their curvatures, and the answer has a sharp edge: two flat mirrors sit exactly on it, which is why the first lasers were so hard to align. The criterion is the one that decides whether a pumped swing grows, whether a wave crosses a crystal, and whether a particle beam stays in an accelerator.

optics · Imaging

Named alongside it

The objects these essays reach for when they reach for this one.

Band gapGaussian beamLaserMomentum conservationOptical cavityParametric amplificationPermittivityPhotonic time crystalRay transfer matrixStabilityTime-varying mediaUnstable resonator

All concepts