Illustrated Physics
physics, drawn
Concept

Solid angle — where it appears

The area a shape covers on a unit sphere, measured in steradians, with a full sphere covering 4π. It is what a flux is divided by to get an intensity, and in the geometric phase it is the quantity a closed circuit of states encloses — with the phase acquired being exactly half of it.

Named by 5 essays across 2 fields — each of them below, with the objects they name alongside it.

The most any concentrator is allowed. The greatest concentration a receiver can be given, against the half-angle it accepts, both logarithmically, for a receiver in a medium of index 1. The upper curve is a three-dimensional concentrator, n²/sin²θ; the lower is a trough, which concentrates in one direction only, n/sin θ. The Sun's angular radius is 0.267°, so sunlight can be concentrated by at most 46165 times in a dish and 215 times in a trough — marked. Nothing about glass, mirrors, wavelength or aperture appears in either expression. The bound is thermodynamic: a receiver that accepts light out to θ also radiates out to θ, and the concentration at which what it emits balances what it absorbs is exactly where these curves are.

The brightness no lens can increase

A lens can make an image smaller and therefore hotter, and there is a temperature at which it stops — the temperature of the source. Every arrangement of glass and mirrors ever built obeys a bound that contains no wavelength, no aperture and no material — only the angle the receiver is allowed to accept — and the bound comes from thermodynamics rather than from optics.

optics · Etendue 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 A cone of 14.3°, from two indices and nothing else. On the left, the acceptance cone of a step-index fibre with a core index of 1.4677 and a cladding of 1.4624. A ray entering steeper than the cone reaches the wall inside the critical angle and is refracted out at the first bounce; one inside it is trapped. The sine of the half-angle is √(n₁² − n₂²) = 0.125, which is 7.2° in air. On the right, that number against the fractional index difference between core and cladding. Nothing about the core's diameter appears: a fibre a hundred times thicker accepts exactly the same cone, and takes a hundred times the area's worth of light through it.

The cone a fibre will accept

A fibre takes light from a cone whose half-angle depends on two refractive indices and nothing else — not on how thick it is, not on how long, not on what is shining into it. That single number, squared and multiplied by the core's area, is all the light it will ever carry.

optics · Etendue A hemisphere outside is a cone of 16.6° inside. Rays leaving a flat face from a material of index 3.5 into air. Every direction in the hemisphere above the face is reached by a ray from inside a cone of half-angle 16.6°, arcsin(1/n), because refraction at the face fans the cone out; rays steeper than that are reflected back, drawn in the warning colour. The squeeze is exact in the way the invariant requires. The projected solid angle of the inside cone is π sin²θc, which is π/12.25, and multiplied by n² it equals π, the projected solid angle of the whole hemisphere outside — checked on the drawn geometry. So the light that does get out has not gained anything: the radiance inside is n² times higher and the directions available are n² times fewer, and the étendue passing the face is the same on both sides.

The cone light has to find to get out

Inside a dense material the whole hemisphere of directions outside a flat face shrinks to a narrow cone, and light made inside can leave only if it happens to be travelling within it. For gallium arsenide that is two per cent of the light. Turned round, the same cone keeps light in: a slab of silicon with a rough surface holds the light it admits for fifty-one passes. Both numbers are the n² the optical invariant carries, and neither one breaks it.

optics · Etendue A potential that is lower every time round. The magnetic scalar potential along a path circling a wire carrying 10 amps, against the angle turned through, for 2 complete circuits. Away from the wire the magnetic field has no circulation round any small loop, so it is the gradient of something — and it is, except that the something does not come back to its own value. Each circuit lowers it by exactly the current, 10 amps, and a second circuit lowers it by 10 again. The potential is perfectly good locally and has no single value globally, and the amount by which it fails to close is the current threaded. So nothing has been lost in going from a circulation to a potential: Ampère's law has been rewritten as a statement about the shape of the region the potential lives in.

A potential that does not come back to itself

Where no current flows, the magnetic field has no circulation round any small loop, so it is the gradient of something and a magnetic problem becomes an electrostatic one. The catch is not that the potential fails to exist. It is that walking once round a wire lowers it by the current, and walking round again lowers it by the current again — so Ampère's law survives the translation as a statement about what the path encircles rather than about where it went.

electromagnetism · Ampere law

Named alongside it

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

Critical angleEtendueRadianceRefractive indexTotal internal reflectionAberrationAdiabatic theoremAmpere lawBlackbodyBoundary conditionsCirculationConcentration

All concepts