Illustrated Physics
physics, drawn
Concept

Geometric phase — where it appears

A phase a wave or quantum state acquires from the path it is taken round in its space of states, independent of how long the journey took. In polarisation it is Pancharatnam's phase, half the solid angle enclosed on the Poincaré sphere; made to grow steadily, it shifts a frequency.

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

A circuit on the sphere of polarisations. The Poincaré sphere, on which every polarisation state is a point: linear states around the equator, circular at the poles, and orthogonal states at opposite ends of a diameter. The triangle is a closed circuit — linear at 0°, then linear at 23°, then left circular, and back — taken along geodesics, which is what a sequence of ideal polarisers does. Bringing a state round it returns it to exactly the state it started in, and multiplied by a phase: 0.3927 radians here, against minus half the enclosed solid angle of -0.7854 steradians. The two agree exactly, and neither calculation knows about the other — the phase is the argument of a product of three overlaps between Jones vectors, and the solid angle is spherical geometry. Nothing about the elements used, their thickness, or the wavelength enters. A phase that depends only on the shape of a path is the signature of a geometric phase, and this is the oldest known example of one.

The phase that is only a shape

Take a beam of polarised light through a sequence of elements that returns it to the polarisation it started with, and it comes back with a phase it did not have before. That phase is not an optical path length — it does not depend on the thickness of anything, or on the wavelength, or on how slowly the sequence was carried out. It is minus half the area the path enclosed on the sphere of polarisation states, and nothing else.

optics · Polarisation 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.

Adiabatic theoremAngular momentumCircular polarisationDoppler effectFrequency shiftInterferenceInterferometryJones calculusOrbital angular momentumPoincare spherePolarisationSatellite navigation

All concepts