A reflector built wrong so the light comes home

In 1728 James Bradley found that every star in the sky moves round a small ellipse once a year, twenty arcseconds across at most, and that the displacement points in the direction the Earth is moving. The other twenty arcseconds was the first direct proof that the Earth goes round the Sun: light arriving from a star appears to come from slightly ahead of where the star is, by the ratio of the Earth’s orbital speed to the speed of light, just as rain falling vertically appears to come from ahead to someone walking through it.

The same effect, in a form Bradley could not have imagined, decides how the reflectors on geodetic satellites have to be built. A laser pulse fired at a satellite and reflected back by a corner cube returns to the ground displaced from the telescope that fired it, by an angle proportional to the satellite’s speed across the line of sight. For a satellite in low orbit the displacement is larger than the beam a small perfect reflector sends back. The reflectors are therefore made deliberately imperfect — their angles cut slightly wrong, by about an arcsecond — so that the returning light spreads out to the angle at which the telescope will be.

That is a strange thing to have to do to a precision optical instrument, and the reason for it is aberration taken twice.

A mirror that returns light to where it came from #

A corner cube is three mutually perpendicular reflecting faces, like the inside corner of a box. Any ray entering it reflects once off each face and leaves travelling exactly antiparallel to the direction it arrived from, whatever the cube’s orientation. That property is what makes it the standard target for laser ranging: a satellite covered in corner cubes reflects a pulse back towards the station that sent it without needing to be pointed, and the round-trip time measures the range to millimetres.

The return is not a ray, though. Each cube has a finite aperture, a few centimetres across, and its reflected beam diffracts. A perfect cube of diameter $D$ returns an Airy pattern whose central bright disc has an angular radius of $1.22\lambda/D$ — seventeen microradians for a 3.8-centimetre cube at the green wavelength of a frequency-doubled laser. Most of the reflected light lies inside that disc, centred exactly on the direction back to the source.

Exactly back, that is, in the cube’s own frame of rest. On a moving satellite that is not the direction to the telescope.

Aberration, twice #

The satellite moves at several kilometres a second across the line of sight, and the station, in the frame of the satellite, moves at the same speed in the opposite direction. Light that leaves the station is received by the satellite from a direction aberrated by $v_\perp/c$, exactly as starlight is received by a moving Earth. The cube sends it back along that aberrated direction. And the station, receiving the return while moving relative to the satellite, sees it aberrated again by the same amount in the same sense. The total displacement of the return beam from the station is

$$\delta = \frac{2v_\perp}{c},$$

where $v_\perp$ is the component of the satellite’s velocity relative to the station perpendicular to the line of sight.

The same result can be had without aberration at all, by following the light. The pulse takes a time $2\rho/c$ to make the round trip of range $\rho$, and during that time the station and the satellite move relative to each other by $v_\perp \cdot 2\rho/c$ sideways. Seen from the satellite, the return is sent towards where the station was, and the station has moved on by an angle $2v_\perp/c$. Aberration and light travel time are two descriptions of one fact.

Fifty microradians is ten arcseconds — half of Bradley’s annual aberration, for a velocity a quarter of the Earth’s orbital speed, because the effect here is two-way. It is small by any ordinary standard and enormous by the standard of the beam a small cube returns. At the telescope, after a round trip to a satellite a thousand kilometres away, it is a displacement of fifty metres, and the bright core of the returned beam is only thirty-five metres across.

How it changes through a pass #

The displacement is not a fixed number for a given satellite, because the geometry changes as the satellite crosses the sky.

So a reflector cannot be designed for one offset. It must return useful light over a range of angles from the exact retro-direction — twenty-five to fifty microradians for a low satellite, a narrower band for a high one — and in every direction round that axis, because the offset points along the satellite’s apparent motion, which rotates as the satellite rises, culminates and sets.

The figures ignore the Earth’s rotation, which adds the station’s own speed of up to half a kilometre a second to the relative velocity and shifts the values by up to about ten per cent. They also assume an overhead pass. A pass that culminates lower has a smaller maximum, which only widens the range of angles the reflector has to serve.

A perfect cube is the wrong reflector #

Put the numbers side by side and the problem is plain. A 3.8-centimetre cube’s central disc is seventeen microradians in radius. A LAGEOS station sits thirty-two to thirty-eight microradians from the retro-direction. The station lies in the dark beyond the first ring of the cube’s diffraction pattern, catching only the faint second ring.

The remedy is to make the cube imperfect. If the angles between its faces are not exactly ninety degrees but differ by a small angle $\varepsilon$, the return is no longer a single beam. The face of the cube divides into six sectors, according to the order in which a ray meets the three reflecting faces, and each sector returns its light in a slightly different direction. For a solid cube of refractive index $n$, the six beams are deflected from the retro-direction by about

$$\theta \approx \frac{4\sqrt6}{3}\,n\,\varepsilon \approx 3.27\,n\,\varepsilon,$$

arranged symmetrically round the axis. With fused silica’s index of 1.46 and an offset of 1.25 arcseconds, that is six beams twenty-nine microradians out. Each beam comes from a sixth of the aperture and is correspondingly broader, about thirty-five microradians across, so that together they cover an annulus around the retro-direction — which is where the telescope will be.

The arithmetic of the trade is worth stating. Each of the six beams carries a sixth of the light from a sixth of the area, so its peak intensity is a thirty-sixth of what the perfect cube puts at its centre. That sounds like a heavy loss and is not, because the perfect cube’s light is in the wrong place: at the station’s angle, the spoiled cube returns about ten times as much.

An offset with an optimum #

The spoiling is a design parameter, and like most design parameters it has a best value.

Too little offset and the beams stay near the retro-direction, where the station is not. Too much and they overshoot it.

The two failures are not symmetric in what they cost the rest of the pass. An offset too small leaves light concentrated near the retro-direction, which is wasted at every point of the pass; an offset too large wastes the light low in the sky, where the aberration is smallest, but can still serve the part of a low satellite’s pass near the zenith, where it is largest. The best offset puts the beams’ centres a little inside the band of aberrations the station will see during a pass, so that their broad profiles cover the whole of it, and the optimum depends on the orbit. A low satellite with aberrations from twenty-five to fifty microradians needs a different cube from a high one where the range is twenty-four to twenty-six.

That is why the reflectors on different geodetic satellites are cut differently. LAGEOS, launched in 1976 as a dense sixty-centimetre sphere covered with 426 cubes and designed to be tracked for decades, carries cubes with dihedral offsets of 1.25 arcseconds — tuned to its own altitude. The arrays on navigation satellites, higher and with smaller aberrations, are spoiled less. A cube from one satellite on another would return a fraction of the light.

The far field, face-on #

The profile shows one direction; the station can be in any.

The six-fold pattern means a single cube’s return varies as the satellite’s apparent motion turns: brighter when the offset direction falls on a lobe, dimmer between. A satellite carries hundreds of cubes at different orientations, and their patterns add, so the variation averages out over the array. What does not average out is the requirement that every cube send its light to the right annulus, and that is set by the dihedral offset alone.

The photon budget makes this more than a refinement. A ranging pulse to LAGEOS leaves the station with something like 10¹⁷ photons and returns, at the telescope, with a handful — often fewer than one per pulse, so that the range is built up from single-photon detections over thousands of pulses. A factor of ten in returned light is the difference between a ranging session that produces a normal point every couple of minutes and one that produces nothing.

What the light is for #

The effort is worth describing because the measurements it buys are among the most precise in the solar system.

LAGEOS and its successors are passive spheres with nothing on board but reflectors, made dense so that drag and radiation pressure barely move them. Ranging to them from a few dozen stations round the world gives the positions of those stations relative to the Earth’s centre of mass to millimetres, and that network is what defines the scale and origin of the terrestrial reference frame every map and every navigation satellite ultimately refers to. The same ranges measure the slowest-changing parts of the Earth’s gravity field — its flattening, which changes as ice melts and the solid Earth rebounds — and they measured the dragging of the satellites’ orbital planes by the Earth’s rotation, the relativistic effect called frame dragging, to about ten per cent.

The lunar reflectors measure the Earth–Moon distance to millimetres, and the residuals of that measurement have become a laboratory of their own. They give the Moon’s recession, about 3.8 centimetres a year, which is the present rate of the tidal transfer that has lengthened the day. They give the Moon’s physical librations, the slight rocking of its face, and through them the size of its liquid core. And they test whether the Earth and the Moon fall towards the Sun at the same rate — the equivalence principle at the scale of whole bodies, which a violation would show as a Moon displaced towards or away from the Sun by an amount the ranging now bounds to a few millimetres.

None of that would be possible if the light did not come back, and for the satellites it comes back because of an arcsecond cut into each cube.

Why the range itself is unaffected #

The aberration displaces the returning beam sideways; it does not change the round-trip time in any way that matters. The time is measured between the departure of the pulse and the arrival of whatever photons return, and the photons that reach the telescope travelled a path that differs from the ideal by a negligible amount. What matters to the range is the same list of corrections that matters to any ranging measurement: the delay in the atmosphere, which grows from about two metres at the zenith to tens of metres near the horizon; the relativistic delay in the Sun’s and the Earth’s gravity; and the light time, the fact that where the target is and where it is seen differ by the distance it moved while the light was in flight.

There is one correction the spoiling does touch. A sphere covered in cubes returns light from many cubes at different depths, so the returned pulse is smeared out in time by a few centimetres’ worth, and the range has to be referred from the effective reflecting point to the satellite’s centre of mass. Which cubes contribute most depends on their orientations relative to the station and on how their spoiled patterns overlap the station’s annulus — so the centre-of-mass correction, a quarter of a metre for LAGEOS, depends slightly on the spoiling, the laser and the detector, and is computed for each combination.

The Moon does not need it #

The reflectors left on the Moon by Apollo astronauts and by Soviet rovers are corner cubes of the same size, and they were not spoiled. The reason is the same arithmetic run with a different velocity.

The Moon moves across the sky at about a kilometre a second relative to the Earth’s centre, and a station on the rotating Earth moves at up to half a kilometre a second. The relative transverse velocity is about a kilometre a second, and $2v_\perp/c$ is about seven microradians — well inside the seventeen-microradian central disc of a 3.8-centimetre cube. The return from the Moon lands on the telescope that sent it, aberration and all, and spoiling would only have spread the already scarce light.

Scarce it is. A lunar ranging pulse that leaves with 10¹⁷ photons returns at most a few photons per pulse at the largest telescope used for the purpose, and far fewer at most stations, because the beam spreads over kilometres on the way out and over kilometres again on the way back. The return beam from a lunar cube, diffraction-limited at seventeen microradians, illuminates a patch of the Earth whose bright core is some six and a half kilometres in radius, and the telescope collects the fraction of it that falls on a mirror a metre or two across.

The same angle on the way out #

Aberration also changes where the station must point. A telescope sees the satellite where it was when the light it is receiving left it, displaced by aberration from where it is now; a pulse fired now arrives after the satellite has moved on. To hit the satellite, the outgoing beam must lead the satellite’s apparent position by an angle that is again $2v_\perp/c$ in total — the point-ahead angle. For a low satellite and a ranging laser whose outgoing beam is spread over tens of microradians, the correction is comparable to the beam’s width and is applied by the tracking system.

For optical communication links to spacecraft, where beams are narrower still, point-ahead is a central part of the design: a laser terminal on a geostationary satellite talking to one in low orbit must aim at where its partner will be, not where it appears, and the two angles differ by several times the beam width. It is the oldest effect in astrometry, carried into an engineering specification.

What the figures leave out #

An idealised geometry. The passes are overhead and the Earth does not rotate. The station’s rotation adds up to about half a kilometre a second to the relative velocity, depending on latitude and the direction of the pass, and a pass that culminates at low elevation never reaches the zenith values.

A simplified lobe shape. Each of the six beams is drawn as the Airy pattern of a circular aperture with a sixth of the cube’s area. The true sector of a hexagonal-faced cube has a different shape and a far-field pattern with more structure, and the six beams interfere where they overlap. The design calculations for real arrays use the full diffraction integral; the figures show the scale and the trade, not the precise pattern.

Uncoated silica cubes. The deflection formula is for a solid cube with total internal reflection at the faces. Hollow cubes with metal-coated faces behave differently, and the polarisation of the return depends on the coating and on the incidence angle.

Thermal distortion. A cube heated unevenly by sunlight bends its faces and changes its far-field pattern, often by more than the deliberate offset. Much of the engineering of retroreflector arrays is about keeping the cubes isothermal, and some of the observed variation in return strength from satellites is attributed to it.

The surprising connection #

The dihedral offset on a LAGEOS cube is an arcsecond, cut into a piece of glass to compensate for the ratio of a satellite’s speed to the speed of light. Bradley measured twenty arcseconds of the same ratio, with a telescope pointing at the zenith, as a proof that the Earth moves. Two and a half centuries later the effect is specified to a fraction of an arcsecond in the machining tolerances of optical components, and getting it wrong would make the components fail at their job.

The two are the same physics at the same order — the effect that, applied to the Sun’s changing velocity, moves every quasar towards the Galactic centre, and that every astrometric catalogue removes alongside the bending of light by the Sun: a first-order correction in $v/c$, and the exact form of aberration has no higher terms either effect could see. What changed is that the observer started sending the light as well as receiving it, which doubled the effect, and that the target started being something built rather than a star, which made the effect something to design for.

Still open: one reflector for every orbit #

A spoiled cube is optimised for one band of aberrations, and that band depends on the orbit and on the station’s latitude. The satellites now carrying reflectors range from a few hundred kilometres to geostationary, and new ones include constellations of small satellites whose arrays are designed for one altitude and flown at another. Arrays with mixed dihedral offsets, cubes of different sizes, and hollow cubes whose offsets can be tuned are all in use; whether a single array design can serve a satellite throughout a mission whose orbit changes — or serve stations at very different latitudes equally — without losing the factor of ten the spoiling buys, is a question of optical design with an aberration at its centre.