The corner that sends light home

The angle past which light cannot leave lists, among the uses of total internal reflection, the prisms of binoculars and the retroreflectors left on the Moon. The second deserves an essay of its own, because it combines two things that seem unrelated: a geometrical fact about three perpendicular mirrors, and the phase shifts that total internal reflection imposes on the two polarisations of light. The geometry makes a device that returns light home from any direction. The phases decide what the returning light looks like when it gets there.

The geometrical fact takes one line. A mirror whose face is perpendicular to the $x$ axis reverses the $x$ component of a ray’s direction and leaves the other two alone. Three mirrors perpendicular to $x$, $y$ and $z$ reverse all three components, whatever order the ray meets them in. A direction with all three components reversed is the opposite direction. A corner of three perpendicular mirrors sends every ray it catches straight back the way it came.

Six ways into a corner #

A corner cube is usually made as a solid piece of glass cut from the corner of a cube, with the corner at the back and a flat face in front perpendicular to the corner’s diagonal. Seen along that diagonal, the three edges where the faces meet divide the front into sectors, and a ray entering meets the three faces in an order fixed by which sector it enters.

There are six orders, one per sector. A ray entering near the top right meets one face, then a second, then the third, and leaves from the point directly opposite through the centre — in the sector whose order is its own reversed. The returning beam is therefore turned inside out: it has the same cross-section as the arriving beam, rotated by half a turn about the axis. That inversion is harmless for returning light to its source, and it is the reason a corner cube is not a mirror in the ordinary sense. It does not form an image; it sends every ray home.

From any direction it can see #

A flat mirror returns light to its source only when it faces the source squarely. Tilted by a degree, it sends the light back two degrees away, which at the distance of the Moon is thirteen thousand kilometres. A corner cube has no such requirement. Every ray traced here, arriving up to 45 degrees off the cube’s axis and from ninety different azimuths, leaves travelling back along its arrival direction to the precision of the arithmetic. The only limit is geometric — a ray arriving too far off the axis misses one of the three faces and leaks away — and it shows up as a shrinking of the part of the aperture that works, not as an error in direction.

The property is what makes corner cubes the standard retroreflector: on road signs, bicycles and high-visibility clothing, on survey instruments, on satellites for laser ranging, and in the falling mirror of an absolute gravimeter, which the length nobody has to measure describes replacing the pendulum. The gravimeter drops a corner cube in a vacuum and counts interference fringes as it falls; a corner cube is used because the fringe count does not change if the cube tumbles as it drops, which a flat mirror’s would.

There is a second way to make a retroreflector, and it is worth setting beside this one. The lens with no axis builds one from a sphere of graded refractive index, which focuses any parallel beam onto its far surface, where a mirror sends it back; it works from a whole hemisphere of directions. The corner cube does the same job with three flat faces and no focusing at all.

A mirror that needs no silver #

A corner cube can be made of three silvered mirrors. Made of solid glass it can dispense with the silver, because a ray travelling along the axis strikes each internal face at 54.74 degrees — the angle whose cosine is $1/\sqrt3$ — which is past the critical angle of any ordinary glass against air, 43.2 degrees for fused silica. Every reflection is total. No metal is needed, none of the light is absorbed by a coating, and a bare cube reflects better than a silvered one could.

The price is a limited field of view. Tilt the arriving light and the angle at which it meets each face changes; for one of the faces it becomes steeper, and past a certain tilt that face’s reflection stops being total and the light leaks out through the back of the cube.

The range is lopsided. Tilting the light towards one of the edges makes the face opposite that edge meet the light more steeply and the other two less, and the steep face reaches its critical angle first; tilting the other way makes all three faces shallower and nothing fails until the light misses a face altogether. For fused silica, total reflection holds from 16 degrees one way to beyond 45 the other. A denser glass pushes the critical angle down and the range out: at an index of 1.8, from 40 degrees to beyond 45. The critical angle of the angle past which light cannot leave is doing all of this, one face at a time.

The reflectors on the Moon #

Apollo 11 left an array of a hundred corner cubes on the Moon in July 1969, Apollo 14 another hundred, Apollo 15 an array of three hundred. Each cube is 3.8 centimetres across and made of bare fused silica, reflecting by total internal reflection. The binding energy that has to fall too describes what the arrays have measured since: the distance to the Moon to millimetres, the Moon’s slow recession, and the equivalence principle for the Earth’s own gravitational binding energy.

The choice of bare glass was deliberate. A metal coating absorbs a few per cent of the sunlight that falls on it, and a cube heated unevenly by the Sun develops gradients of refractive index that spoil its return beam. Bare glass absorbs almost nothing, and the cubes were set back in recesses in their mounting so that the Sun rarely strikes their faces directly — which also keeps the incoming laser light within the range over which every reflection stays total. Earth-based stations see the arrays within a few degrees of their axes, well inside the fused-silica range.

What total reflection does to the returned beam #

Total internal reflection returns all the light, but it does not return it unchanged. The angle past which light cannot leave and the reflection that happens where the glass is not show that the two polarisations come back with different phase shifts, and at 54.74 degrees in fused silica the difference is tens of degrees per reflection. A corner cube applies three such reflections, and the planes of incidence of the three are at different orientations, so the combined effect on the light’s polarisation depends on the order in which the faces were met — which is to say, on the sector.

With silvered faces, which reflect both polarisations with the same phase, every sector returns the arriving polarisation. With bare glass, each sector returns a different ellipse: some nearly circular, some elongated, each tilted differently. The polarisation of the returned beam is no longer a single state but a patchwork of six, arranged around the axis.

The spot that comes back #

A patchwork of polarisations across an aperture changes how the returned light spreads. The far-field pattern of a beam — the spot it makes at a great distance — is the Fourier transform of the field across the aperture it leaves, and six sectors carrying different fields interfere with each other there.

Silvered, the cube returns a single compact spot, the diffraction pattern of a hexagonal aperture. Bare, the peak at the centre falls to 0.266 of the silvered cube’s, and the light that is missing from the centre is spread into six lobes around it: the same total light, spread over roughly four times the area. For a reflector whose job is to put as much light as possible back into a telescope a long way away, that is a factor of about four in signal lost to polarisation alone.

Why about a quarter is simple arithmetic once the sectors are seen as six separate sources. At the exact centre of the far field, light from every part of the aperture arrives in step as far as path length goes, so the field there is the sum of the six sectors’ fields, each weighted by its area. Silvered, the six fields are identical and add to six times one of them. Bare, they are six different polarisation states, each a vector with its own direction and phase, and six such vectors partly cancel: the sum comes to about half of six, and the intensity, its square, to about a quarter. The energy is not lost. What adding does to the energy shows that wherever waves add to less than their sum, the missing energy appears somewhere else in the pattern, and here it is in the six lobes. The direction of the shaking is the other ingredient: two fields at right angles to each other cannot cancel at all, and only the components that share a direction interfere, which is why a patchwork of polarisations spreads light rather than extinguishing it.

For the lunar arrays the loss is tolerated, and it is partly useful. The Moon moves relative to the Earth, so light returning from it arrives at the Earth displaced from where it was sent by a few microradians — the velocity aberration — and a return spot somewhat broader than the ideal is more likely to cover the telescope that sent the pulse. A 3.8 centimetre cube’s diffraction spread is itself a few times that, so the arrays were already designed around a spot wide enough to catch it.

Two faces instead of three #

Take away one of the three faces and what remains is a roof: two faces at right angles, which reverse the ray’s direction in one plane and leave it unchanged in the other. A roof is half a corner cube, and it is how binoculars turn an image the right way up. The prism in a pair of straight-through binoculars has a roof edge along which the light is reflected twice by total internal reflection, and the two halves of the beam — one striking the left face first, the other the right — play the part of the corner cube’s sectors.

They cause the same trouble. The two halves leave with different polarisations, because they met the two faces in opposite orders, and where they recombine at the focus they interfere imperfectly. The image of a point is widened across the roof edge, and the resolution of a roof-prism binocular is measurably worse than the aperture alone allows. Since the late 1980s the roof faces of good binoculars have carried a thin multilayer coating — a phase coating — that equalises the phase shifts of the two polarisations at the angle of the reflections, and with it the two halves recombine as one. It is the corner cube’s polarisation problem solved for two sectors, and it is the reason a phase coating appears in the specification of a binocular at all.

The same geometry works at every wavelength where the faces are flat and reflective. A trihedral of three metal plates on the mast of a small boat is a corner cube for radar: it sends the radar’s pulse straight back to the ship that sent it, and a corner half a metre across returns as much signal as a far larger object, because it concentrates its return into a narrow beam aimed at the source. Everything a scatterer removes defines the cross-section a target presents; a corner reflector’s is its area squared divided by the wavelength squared, times a few, which at radar wavelengths makes a small corner look to a radar like a ship.

Where the traced corner stops #

Perfect geometry. Every face is exactly flat and every angle exactly a right angle. Real corner cubes are made with angle errors of a fraction of an arcsecond, deliberately so in some designs: a small error in one angle splits the returned beam into slightly separated spots, which can be used to compensate the velocity aberration of a moving target. The ideal cube returns everything exactly back, and a slightly imperfect one can be better for a target that moves.

Ideal materials. The silvered cube is drawn with a perfect conductor; real metal coatings reflect 90 to 97 per cent and impose small phase differences of their own. The bare cube is drawn without absorption in the glass or at its surfaces, and without the front face’s own four per cent reflection.

Plane waves on axis. The far field and the polarisation figures are for light arriving exactly along the axis. Off axis the three reflection angles differ from each other, the phase shifts change, and the pattern changes with them.

Solid, not hollow. Everything here is a solid glass cube. A hollow corner of three silvered mirrors has no front face to refract at, no critical angle to limit it and no total-reflection phases to scramble its polarisation, and it is light for its size; it is also harder to keep square, because three separate mirrors must be bonded at right angles and held there as the temperature changes. Satellites and instruments use both kinds, chosen by exactly the trade the figures set out.

One wavelength. The phase shifts of total internal reflection depend only weakly on wavelength, through the refractive index, so the pattern is nearly the same across the visible; its scale, in angle, is proportional to the wavelength over the aperture.

The image a corner turns upside down #

The sector figure and the reversal figure are ray drawings and cannot show that a corner cube also returns a wave whose phase has been reversed across its face — the inversion of the beam through the axis. For a laser beam with a smooth wavefront that makes no difference; for a beam carrying an image or a wavefront distortion, the corner cube returns the distortion rotated by half a turn, which is why a corner cube cannot be used to undo the atmosphere’s blurring the way a phase-conjugating mirror can. And the far-field figure is drawn at one polarisation of the arriving light. For circularly polarised input the pattern of an uncoated cube is different again, which is why polarisation is chosen with care by stations that range to uncoated reflectors.

Still open: how much of a lunar return is lost on the Moon #

The lunar reflectors return far less light than their design predicts — by a factor of ten or more, and worse around full Moon, when the arrays are in direct sunlight. The favoured explanation is dust: lunar dust, lifted by electrostatic charging or by nearby impacts, settling on the front faces and absorbing sunlight, which heats the cubes unevenly and spoils their return. How the dust gets there, how much of the loss it accounts for, and whether new reflectors of other designs — single large corner cubes, or hollow cubes of three mirrors — would suffer less, are being studied, and new reflectors have been sent to the Moon on recent landers partly to find out.

The habit worth carrying away is to separate what a device does to a direction from what it does to a wave. A corner cube reverses every ray exactly, which is a statement about geometry that no material can spoil; what it returns is also a field, and a field carries phases that each reflection can change. Total internal reflection makes a perfect mirror for the direction and an imperfect one for the wave, and whether that matters depends on whether the light is being sent home or being looked at when it arrives.