Illustrated Physics
physics, drawn
Concept

Critical angle — where it appears

The angle of incidence past which light cannot leave a denser medium at all, because the refracted direction would have to lie beyond the surface. Its sine is the ratio of the two indices, and past it the field on the far side does not vanish but decays exponentially over a fraction of a wavelength.

Named by 12 essays across 2 fields — each of them below, with the objects they name alongside it.

The critical angle for n = 1.5 into n = 1. Refracted angle against incident angle. It rises faster than the incident angle and reaches 90° at 41.8°, beyond which no refracted ray exists at all.

The angle past which light cannot leave

Snell's law asks for the sine of an angle greater than one, and no such angle exists. What happens instead is a perfect mirror made out of nothing but a change of speed.

optics · Total internal reflection The two reflectances, and the angle one of them loses. Reflectance against angle of incidence for light going from n = 1 into n = 1.5. The upper curve is light polarised with its electric field along the surface, which reflects more and more strongly until at grazing incidence everything reflects. The lower curve is light polarised in the plane of incidence, and it does something the other cannot: it falls to exactly zero at 56.31°, where tan θ = 1.5000, and then rises again. At normal incidence the two are equal at 4.00% because there is no plane of incidence to tell them apart. The dashed curve is the transmittance, computed from the transmission coefficients and the two media's projected impedances rather than as one minus the reflectance; it agrees with one minus the reflectance to 4.4e-16 across the whole range, which is where the energy accounting can be seen to close.

The angle at which reflection picks a side

At one angle of incidence, a water surface reflects no light at all of one polarisation. The Fresnel algebra says so, and says nothing about why. The reason is that the reflected ray would have to leave along the axis of the charges radiating it — and a shaking charge sends nothing along the direction it shakes in.

optics · Polarisation The reflected beam does not leave from where it arrived. The lateral displacement of a totally reflected beam along the surface, for the two polarisations, at 550 nm from n = 1.5 into n = 1. Geometrical optics puts the outgoing ray at the point where the incoming one struck. It is not there: it is displaced forward along the surface by a distance comparable with a wavelength, which was measured by Goos and Hänchen in 1947 by reflecting a beam many times and looking at the accumulated offset. The displacement is different for the two polarisations — at 42°, 1361 nm for s and 2991 nm for p, at 45°, 350 nm for s and 560 nm for p, at 50°, 237 nm for s and 261 nm for p, at 60°, 183 nm for s and 127 nm for p, at 75°, 161 nm for s and 79 nm for p — which is why an unpolarised beam comes back slightly split. A displacement is only possible if the light spent time on the far side of a boundary it never crossed, and it is the most direct evidence there is that the evanescent field is a real field rather than a bookkeeping term.

The reflection that happens where the glass is not

Total internal reflection sends back every photon, which is why it is called total. It does not send them back from where they arrived — the beam re-emerges displaced along the surface, by a fraction of a wavelength, and a displacement is only possible if the light spent time on the far side of a boundary it never crossed.

optics · Total internal reflection 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 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 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 A sheet of light a tenth of a micrometre thick. The intensity of the evanescent field in water beyond a glass coverslip (index 1.518 against 1.33), against distance from the glass, for 488 nm light totally reflected at 62°, 65°, 70°, 75°, s-polarised, relative to the light arriving in the glass. The critical angle is 61.2°. At 62° the field at the surface is 3.79 times the incident intensity and falls by e in 234 nm; at 65° the field at the surface is 3.07 times the incident intensity and falls by e in 110 nm; at 70° the field at the surface is 2.01 times the incident intensity and falls by e in 75 nm; at 75° the field at the surface is 1.15 times the incident intensity and falls by e in 63 nm. Ordinary illumination (dashed) lights the whole depth of a cell equally; totally reflected light lights only the layer against the glass, which is where a cell's membrane and the proteins in it are.

The mirror that lights a tenth of a micrometre

Total internal reflection sends back every photon, and it leaves behind a field in the medium beyond that dies away within a fraction of a wavelength. Put a living cell on a coverslip and reflect a laser inside the glass, and that field lights only the hundred nanometres of the cell pressed against the glass: the membrane, and the proteins moving in it. Nothing else glows. The same field works in reverse. A molecule close to the glass sends part of its own light into angles no ray from the water could reach, and how much it sends there says how close it is.

optics · Total internal reflection A mirror for X-rays that works only at a glance. The fraction of 8 keV X-rays reflected by a smooth surface of gold and of silicon, against the grazing angle between the ray and the surface, in milliradians, with absorption included (solid) and without it (dashed). Below the critical angle √(2δ) — 10.02 mrad (0.57°) for gold, 3.88 mrad (0.22°) for silicon — the X-rays are totally reflected from outside the material, because its index is below the vacuum's; above it almost nothing is. Absorption rounds the edge and keeps the plateau a little below one. A mirror for X-rays is therefore a surface met at less than a degree, and it reflects well only while it is met that way.

The index that falls below one

Glass bends light because light is slower in it. X-rays are not slower in glass, or in anything: every material's refractive index for them is slightly less than one, by a few parts in a million. Everything about X-ray optics follows from that small number with the wrong sign. A lens that focuses them must be thinner in the middle than at the edges, and so weak that dozens are stacked in a row. A mirror reflects them only if they arrive at a glancing angle of less than a degree — totally, from the outside, the reverse of the way a diamond traps light — which is why the mirrors of an X-ray telescope are long, nested barrels rather than dishes.

optics · Refraction Three arrivals, and the one that overtakes. Arrival time against distance from the shot for a 35 km crust at 6 km/s over a mantle at 8 km/s: the direct wave, a line through the origin with slope 1/v₁; the reflection from the base of the crust, a hyperbola; and the head wave, a straight line of slope 1/v₂ that begins at the critical distance, 79 km, where it touches the reflection. Close to the shot the direct wave arrives first. The head wave starts later but travels most of its path at the faster speed, and beyond the crossover distance, 185 km, it arrives first and stays first. At 300 km it beats the direct wave by 4.8 s, though it has travelled 26 km further.

The wave that arrives first by going the long way

A shot fired at the ground sends a wave straight along the surface layer to a distant listener, and another down to a faster rock beneath, along the boundary and back up. The second path is longer, and beyond a certain distance it arrives first, because most of it is run at the faster speed. Huygens's construction shows what the second wave is: a front drawn by wavelets that the disturbance racing along the faster floor sends up into the slower layer, the Mach cone of a source moving faster than the layer can carry sound. Reading where it overtakes the direct wave gave Andrija Mohorovičić the thickness of the Earth's crust in 1909, and gives every refraction survey since the depth to whatever lies beneath.

waves · Huygens 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 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 reflectionRefractive indexSnell's lawEvanescent waveFresnel equationsBoundary conditionsEscape coneFrustrated reflectionOptical fibrePolarisationRetroreflectorSolid angle

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