Illustrated Physics
physics, drawn
Concept

Ray tracing — where it appears

Following light through an optical system as straight lines that bend at each surface by Snell's law and reflect by the law of reflection. It ignores diffraction, and it is how lenses, prisms and cut gems are designed and how their aberrations are found.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Abbe's ratio, measured on the traced rays. The quantity h divided by the sine of the angle at which the ray converges on the focus, in units of the paraxial focal length, against how far up the aperture the ray entered. Abbe's sine condition says that a system already free of spherical aberration images a small region round the axis faithfully only if this ratio is the same for every ray. A horizontal line means the condition is met. The parabola departs by 12.96 per cent across the aperture; The sphere departs by 7.18 per cent across the aperture. The paraboloid is the interesting case, because it is exactly stigmatic on axis — every ray from infinity crosses at one point, which is the definition of the shape — and it still fails this test. Perfection at one point buys nothing at the next one along. What the departure predicts is coma, a blur that grows linearly with the distance off axis and quadratically with the aperture, and the offaxis figure measures exactly that blur on the same surfaces. The condition is not a design rule invented for telescopes: it follows from requiring that the same optical path length join object and image for every route, and any instrument that images a field rather than a point has to meet it.

The condition a lens must meet

A paraboloid brings every parallel ray to exactly one point. Move the source a fifth of a degree off axis and the image is a fan rather than a point, and the reason is a condition Abbe wrote down that has nothing to do with the axis — perfection at one point buys nothing at the next one along.

optics · Imaging Three conditions and the window they leave. For a ray entering a diamond's table straight down, the angle from the normal at which it meets the first pavilion face, the second pavilion face and the top again, against the pavilion angle; diamond's critical angle, 24.4°, is the dashed line. The first two must lie above it, so that the ray is totally reflected; the third must lie below it, so that the ray gets out. The first holds for every pavilion angle drawn. The second fails above 51.9°, where the ray would leak out of the far side of the pavilion. The third fails outside 38.9° to 51.1°, and it is the one that decides. Together they leave a window from 38.9° to 51.1°, and Tolkowsky's 40.75° sits inside it, towards the lower edge.

The window a diamond is cut inside

A well-cut diamond sends back out of its top most of the light that falls on it, and it does so without a single silvered surface. Light entering the top is totally reflected at one sloping face of the pavilion below, again at the opposite face, and then has to leave through the top — where it must not be totally reflected. Each of those three is a condition on one angle, and together they leave a window a dozen degrees wide inside which a stone looks bright and outside which it looks glassy. The window is wide for diamond and nearly closed for glass, which is why a cutter's skill cannot make glass sparkle.

optics · Total internal reflection The shadow a thunderclap's sound cannot enter. Sound rays from a point 3 km up a lightning channel, on an afternoon with 25 °C at the ground and the air cooling 6.5 K per kilometre upwards, so that sound travels slower the higher it goes; the vertical scale is stretched 3.5 times. Every ray curves upwards. Rays aimed below the horizontal reach the ground, reflect and climb away; the last ray to touch the ground grazes it 23.1 km from the channel. Beyond that, shaded, no ray arrives at all: a listener's ears 2 m up are inside the shadow from 23.7 km. The flash is visible there and the thunder is not, though the air between is perfectly clear and absorbs the low notes of thunder hardly at all.

The thunder that passes over the listener's head

Lightning twenty-five kilometres away is plainly visible and usually silent, though the air between is clear and absorbs the low notes of thunder hardly at all. The thunder has not faded; it has gone over the listener's head. Daytime air cools upwards, sound slows as it climbs, and Huygens's wavelets, smaller at the top of every front than at the bottom, tilt each front towards the sky. Beyond the ray that just grazes the ground there is a shadow the sound cannot enter — and a hundred and fifty kilometres further out, the stratosphere can bend it back down.

waves · Huygens

Named alongside it

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

AberrationAcoustic shadowAplanaticAtmosphereComaCritical angleDispersionField of viewGemstoneHuygens principleLapse rateOptical path length

All concepts