Illustrated Physics
physics, drawn
Concept

Circular polarisation — where it appears

A wave whose field vector rotates at a fixed rate rather than oscillating along a line, in one of two senses. Which sense a medium favours is what makes a magnetised plasma transparent below its own plasma frequency to one and reflecting to the other, and it is what a whistler travels as.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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.

The crystal that answers twice

Lay a piece of calcite on a printed page and the print appears twice. One image sits still when the crystal is turned and the other goes round it. Nothing has been done to the light except pass it through a material whose response to a field is not a number.

optics · Polarisation A click that arrives sorted by pitch. Arrival time against frequency for a broadband pulse travelling 40 megametres along a magnetic field line through a plasma of 1e+8 electrons per cubic metre in a field of 300 nanotesla — the magnetosphere, roughly. The gyrofrequency is 8.4 kilohertz and the plasma frequency 90 kilohertz, so this is the regime where a right-hand circularly polarised wave travels happily below both of them. It does not travel at one speed. The group velocity is 2c√ω(ω_c − ω)^(3/2)/(ω_p ω_c), which is zero at zero frequency and zero again at the gyrofrequency and peaks in between — at 2.10 kilohertz here, which is a quarter of the gyrofrequency, found by searching the drawn function rather than quoted. Everything either side of that peak is slower, so the curve has a nose: 3.21 s at 500 Hz, 2.20 s at 2000 Hz, 3.59 s at 5000 Hz. The branch below the nose is the classic whistler — a lightning stroke in one hemisphere reaching a receiver in the other as a note gliding downward over about a second, which is what gave the phenomenon its name — and the branch above it arrives as a rising tone at the same time. Both are observed, and a recording that shows the two joined at the nose is how the gyrofrequency along the path is read off directly. Read the other way it is an instrument: the product of the delay and the square root of the frequency is 79 s·Hz^½ here, nearly constant across the low end of the band, and it measures the electron content of a path through the magnetosphere that nothing else could reach.

The whistle that arrives sorted

A lightning stroke in one hemisphere reaches a receiver in the other as a note gliding downward over about a second. Nothing dispersed the sound; there was no sound. A radio pulse travelled forty megametres along a magnetic field line through a plasma whose group velocity depends on frequency, and arrived with its frequencies separated by up to three seconds.

astrophysics · Plasma oscillation Past the critical angle, all that is left of a reflection is its phase. The phase each polarisation acquires on total internal reflection at an index ratio of 1.518, against angle of incidence, together with the difference between them. Below the critical angle of 41.19° there is a transmitted beam and the reflection coefficients are real; above it the transmitted wavenumber is imaginary, the coefficients have modulus exactly one — every photon comes back — and the only thing that distinguishes one angle from another is the phase. The two polarisations acquire different phases, and their difference peaks at 46.533° at an incidence of 51.05°, which the closed form puts at the same place. That difference is a retardation: a wave plate made out of an angle, with no birefringent material anywhere in it, and — because the expression contains only the index ratio — one that barely changes with colour.

The retarder with no crystal in it

A wave plate turns linear polarisation into circular by making one component travel a little further than the other, which requires a birefringent crystal cut to a thickness and works properly at one wavelength. Total internal reflection does the same job with a phase that comes from the geometry instead — and because a refractive index barely changes across the visible where a wavelength changes by a factor of two, the same block of ordinary glass is a quarter-wave plate for every colour at once.

optics · Polarisation One measurement that separates unpolarised from polarised. What a rotating linear polariser passes, against its angle, for beams of the same total intensity and degrees of polarisation 0, 0.35, 0.7, 1. Every curve has the same average — a polariser passes half of any beam over a whole turn, whatever its state — and they differ only in how deeply they modulate. The depth of the modulation is the degree of polarisation, exactly: a fully polarised beam goes to zero at one angle and a beam with no preferred direction gives a flat line at a half. That is the whole measurement, and it is why "unpolarised" is a statement about a modulation depth rather than about what a wave is doing.

The light with no direction of shaking

Unpolarised light is not a state of a wave; it is the absence of one, and no description of a single wave can represent it. What can is a set of four numbers, all of them powers a detector reads — and they describe every beam there is, including the ones that are neither polarised nor not.

optics · Polarisation The path the tip of the field traces. The tip of the total field over one cycle, for four arrangements of coils, with each coil's axis as a faint line and the field at one moment as an arrow. One coil: forward-turning part 0.50, backward-turning part 0.50; Two coils, 90° and a quarter cycle apart: forward-turning part 1.00, backward-turning part 0.00; Three balanced coils: forward-turning part 1.50, backward-turning part 0.00; Three coils, the third at 70 per cent current: forward-turning part 1.35, backward-turning part 0.15. One coil traces a line: equal forward and backward parts, a field that pulses and does not turn. Two coils at right angles a quarter cycle apart trace a circle of radius 1, three balanced coils a circle of 1.5, and a weakened third coil an ellipse, the sum of a large forward circle and a small backward one.

The field that turns with nothing turning

A coil fed with alternating current makes a magnetic field that grows, shrinks and reverses along the coil's axis and never leaves it. Put three such coils at 120° to one another and feed them currents a third of a cycle apart, and their fields add to one of constant strength that turns steadily round the centre — a rotating magnet made of nothing that moves. The trick is superposition read backwards: a field pulsing along a line is already two fields turning opposite ways, and the three coils are arranged so that one pair cancels and the other adds. Everything that goes wrong afterwards is a field turning the wrong way.

waves · Superposition How much reflected light keeps its handedness. Circularly polarised light reflected from a flat surface: the fraction of the reflected power that keeps its handedness, against the angle of incidence, for a perfect conductor, for glass, n = 1.5, for a medium with ε = μ = 1.5 (the same index, and the same impedance as empty space), and for one in between. A metal mirror reverses the handedness of every photon at every angle. Glass reverses it at normal incidence, returns an even split at Brewster's angle (56.3°, where only one linear polarisation reflects), and keeps 60 per cent at 60°. The medium with ε = μ keeps it at every angle: it respects the duality symmetry, and so conserves what the symmetry conserves.

The handedness a mirror cannot keep

Maxwell's equations in empty space do not change if the electric field is turned partly into the magnetic field and the magnetic partly into the electric. A symmetry that exact must hand over a conserved quantity, and this one hands over the handedness of light — how many more photons spin one way than the other. Every mirror breaks it: circularly polarised light comes back from a mirror with its handedness reversed. Only a material that responds to the magnetic field exactly as strongly as to the electric one keeps it, and such a material, it turns out, sends nothing straight back.

electromagnetism · Maxwell equations Light whose crests wind round its axis. Cross-sections of three beams carrying orbital angular momentum, with phase exp(iℓφ) round the axis, for ℓ = 1, 2 and 3, at one instant: solid lines where the field is at a crest, dashed where it is at a trough. Along the beam the crests form ℓ interleaved helices, and at a fixed plane the pattern turns about the axis once every ℓ optical periods. Looked at from a frame turning about the axis at a rate Ω, every crest arrives ℓΩ/2π times a second more or less often: the field's frequency is shifted by ℓΩ. That is a Doppler shift with no approach and no recession — the angular counterpart of k·v, with the winding number ℓ standing in for the wavenumber and the rotation rate for the speed. Each photon of such a beam carries ℓħ of orbital angular momentum about the axis.

The Doppler shift with nothing coming closer

Every Doppler shift so far has needed something to approach or recede. Light can be shifted in frequency by something that does neither: a surface spinning in place, or a polarising plate turning in a beam that passes straight through it. A beam whose wavefronts wind round its axis, ℓ times per wavelength, comes back from a surface spinning at Ω shifted by exactly ℓΩ, at every distance from the axis. A circularly polarised beam through a spinning half-wave plate comes out shifted by exactly 2Ω, because the plate takes two units of angular momentum from each photon and does work on it. The shift is tiny and perfectly measurable, and satellite navigation has to correct for it.

waves · Doppler

Named alongside it

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

PolarisationSuperpositionWave plateBirefringencePhaseRefractive indexAchromaticAngular momentumAnisotropyCoherenceComplex amplitudeCrystal

All concepts