Concept

Fresnel equations — where it appears

The formulas giving how much of a light wave is reflected and transmitted at a flat boundary between two media, for each polarisation and angle of incidence. They follow from the continuity of the fields at the surface and give Brewster's angle, total internal reflection and the phase shifts on reflection.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

The angle at which a total reflection stops being total. Reflectance of light at 633 nm inside a BK7 prism whose face carries a 50 nm gold film with water beyond it, against the angle of incidence, for the two polarisations, from the three-layer Fresnel sum in complex arithmetic. Past the critical angle for glass and water, 61.62°, s-polarised light is reflected almost completely, as total internal reflection requires, apart from what the thin gold absorbs. p-polarised light is reflected too — except near one angle, where the reflectance falls to 0.014 at 71.52°. The dispersion relation of a surface plasmon on gold against water puts its resonance at 71.67°; the difference is the film's finite thickness and the prism on its other side pulling the resonance slightly.

The angle at which total reflection stops being total

Coat the back face of a glass prism with fifty nanometres of gold and total internal reflection still returns almost all the light — except at one angle, where the reflection of one polarisation falls nearly to zero. The light has gone into an electron wave bound to the far surface of the gold, and because that wave's angle depends on whatever touches the gold, the missing reflection has become one of the most sensitive balances in biochemistry.

optics · Total internal reflection

Six ways into a corner, each leaving from the opposite side. A corner cube seen along its axis: its hexagonal face, divided into six sectors by the three edges where its mirror faces meet and by their extensions. A ray entering along the axis meets the three faces in an order fixed by the sector it enters: y → x → z, y → z → x, z → y → x, z → x → y, x → z → y, x → y → z. Traced reflection by reflection, every ray leaves travelling exactly back along its arrival direction, to rounding error, and it leaves from the point diametrically opposite where it entered — in the sector whose order is the reverse of its own.

The corner that sends light home

Three mirrors at right angles send light straight back the way it came, from any direction they can see, because each one reverses one component of its direction. Made from solid glass, a corner reflects by total internal reflection and needs no silver — which is why the reflectors left on the Moon are bare glass — but each of its six sectors then returns a different polarisation, and the returned spot loses three-quarters of its peak.

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

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

The surface that cancels whatever skims it

Glass reflects four per cent of light falling straight on it, and water at radio frequencies two-thirds of the wave's amplitude. Tilt either towards grazing and the difference disappears: every surface, whatever it is made of, reflects a wave that only skims it completely, and turns it upside down. A wave travelling along a surface therefore meets its own reflection reversed and cancels on the surface itself. That one limit explains a dark fringe Humphrey Lloyd saw with a single mirror in 1834, why a mobile signal over flat ground fades with the fourth power of distance rather than the second, why a radar above the sea is blind at the horizon, and how a cliff in Sydney became the first radio interferometer.

optics · Refraction $Rays inside a drop never reach the critical angle. Rays of light entering a drop of water (index 1.333) from the left at four heights — 0.3, 0.6, 0.85, 0.97 of the radius — followed through six internal reflections, each chord drawn with a width proportional to the light it carries. Every chord inside a sphere meets the surface at the same angle at both ends, so each ray strikes the inside of the drop again and again at its refraction angle: 13.0°, 26.8°, 39.6°, 46.7°, all below the critical angle of 48.6°. None is ever totally reflected; at each contact most of the light leaves, and what is left inside dwindles — the light that, leaving after one and two reflections, makes the primary and secondary rainbows.$

The drop that cannot hold the light it lets in

Light inside water meeting the surface at more than 48.6° is totally reflected, and every raindrop is made of water. Yet no ray of sunlight that enters a raindrop is ever totally reflected inside it. A sphere's geometry forbids it: a ray refracted on the way in meets the inside of the surface at its refraction angle, and keeps meeting it at that same angle every time it bounces, and that angle is always smaller than the critical one. The light leaks out a little at every bounce, which is why there are rainbows at all. Light born inside the drop is another matter, and a sphere can keep it for ever.

optics · Total internal reflection $A point of light split by an uncoated roof. The image of a distant point across the direction of the roof edge, behind an Amici roof in glass of index 1.517, for unpolarised light, as a fraction of the peak an undivided pupil would give, against position in units of the diffraction width λ/D: with a phase-coated roof, and with bare glass. The coated roof gives the ordinary diffraction pattern of the full aperture. Bare glass leaves the two halves of the pupil only partly able to interfere, and the image's peak falls to 0.70 of its proper height, its central lobe widening across the edge into a shoulder on each side — a slight doubling of every point, which an observer sees as a loss of crispness and contrast.$

The roof that splits a binocular's image

A slim, straight binocular turns its image upright with a roof — two glass faces meeting at a right angle, which reflect every ray twice by total internal reflection and lose no light at all. They lose sharpness instead. Total reflection shifts the phases of the two polarisations of light by different amounts, and the half of the beam on one side of the roof's edge meets the two faces in the opposite order from the other half. The halves leave in different polarisations, interfere only partly at the focus, and every star becomes a slightly doubled smudge — unless a coating a fraction of a micrometre thick makes the glass behave like a mirror.

optics · Total internal reflection $The angle at which water lets the light out. The fraction of light reflected inside a windscreen of index 1.52 at its outer surface, unpolarised, against the angle of incidence, when the outer surface is dry (glass to air) and when it is wet (glass to water). Beyond 41.2° all the light is reflected from a dry surface; under water total reflection does not begin until 61.3°. Between the two angles, shaded, the surface is a perfect mirror where it is dry and a window where it is wet. A sensor working at 45° sits in that band: at a wet spot only 1.2% of its light is reflected and the rest leaves the glass into the water. The difference between a mirror and a window is the refractive index of what is touching the glass — a property of contact, not of what can be seen through it.$

The windscreen that feels the rain

Behind the rear-view mirror of most modern cars a small box decides when the wipers should run, and it does so without looking at the rain. It shines infrared light into the windscreen at forty-five degrees, steep enough to be trapped inside the glass by total internal reflection, and counts how much comes back out of the far end of the glass a few centimetres away. A raindrop on the outside changes the medium the glass is touching from air to water, and water's critical angle against glass is sixty-one degrees, not forty-one: at forty-five the light is no longer trapped, and nearly all of it leaves into the drop. The sensor measures the area the rain is touching, and the rain rate from how fast that area grows.

optics · Total internal reflection

Named alongside it

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

Total internal reflection Critical angle Evanescent wave Polarisation Biosensor Brewster's angle Circular polarisation Critical coupling Diffraction Diffraction limit Duality symmetry Escape cone

All concepts