The lens with no axis

A camera lens is built round an axis, and the axis is both what makes it work and what limits it. Light arriving along the axis can be focused very well; light arriving at an angle to it meets the glass asymmetrically and suffers a list of aberrations — coma, astigmatism, a focus that lies on a curved surface rather than a plane — each of which grows as the angle grows. Wide-angle lenses are expensive because they spend most of their glass fighting those penalties. There is an obvious way out, which is to use a lens with no axis at all. A sphere looks the same from every direction, so light arriving from any direction meets exactly the same lens, and nothing can depend on the angle. The difficulty is that a sphere of glass is a bad lens. In 1944 Rudolf Luneburg found the sphere that is a good one.

Why a glass ball will not do #

A uniform glass ball is the simplest lens there is, and it has the symmetry that the argument wants: a ball has no axis, so a beam from any direction meets the same lens and is focused the same way. What it does not have is enough freedom. A sphere does not bring parallel light to a point — the rays striking its edge are bent more strongly than the rays near its centre, out of proportion to their distance from the centre, and they cross the axis sooner. A ball has one index and one curvature to spend, and no choice of the two brings every ray to one place.

For a ball of index $n$ the rays close to the centre focus at $nR/2(n-1)$ from its centre — one and a half radii for ordinary glass, and exactly on the far surface for $n = 2$. The rays further out focus closer in. The spread between the two is the ball’s spherical aberration, and it is not small: for a glass bead it is about four-tenths of a radius between the central rays and those entering nine-tenths of the way out, a blur so large that a ball lens is used only where its compactness matters more than its image, in fibre couplers and in the cheapest of magnifiers. Raising the index shrinks the spread without removing it.

The ball does keep one promise. Its aberration is exactly the same for every direction the light comes from, because every direction is the same to it. A ball lens is, in a precise sense, equally bad everywhere. What Luneburg supplied was a way to be equally good everywhere, by giving up the uniform index.

A sphere graded to the right profile #

The idea is the one that Maxwell’s fish-eye used for points inside a medium: let the index vary continuously, so that each ray bends gradually along its whole path rather than twice at the surfaces, and choose the variation to make every path from a source to its focus take the same time. For parallel light arriving from outside and focusing on the far surface, the profile that does it is

$$n(r) = \sqrt{2 - r^2/R^2}.$$

At the centre the index is $\sqrt2$, about 1.41; at the rim it is exactly one. A ray along the diameter passes straight through and is not bent at all; a ray entering further out must be turned through a larger angle to reach the same point on the far rim, and it spends its path in the lower-index outer shells, where the gradient that bends it is steepest. The profile balances the two so that every ray of a parallel beam reaches the same point on the far rim, and the tracing in the first figure confirms it to within the step of the integration. The rays bend without any surface, continuously, curving towards the higher index at the centre and then out to the focus.

Two features of the profile are worth noticing. The index at the rim is exactly that of the surroundings, so there is no jump at the surface: light enters without refraction at the boundary, and without the few per cent reflected at every glass–air surface. And the focus lies on the surface itself, not in space behind the lens. A ball lens’s focus hangs in the air on the far side, where a detector can be placed; a Luneburg lens delivers its light onto its own skin.

Equal times, and a condition met for free #

Why this profile and no other is a statement about travel times. A parallel beam is a set of rays that all start on one flat wavefront, perpendicular to their direction, at the same moment. For all of them to arrive at one point together, each path from the wavefront to the focus must be stationary, and a family of stationary paths to one point must have one common optical length — the same requirement that made a single surface exact for one pair of points, now imposed on a medium instead of a surface. Luneburg’s contribution was to turn that requirement into an integral equation for the index profile of a sphere and to solve it; for a focus on the rim, with the sphere in air, the solution is the square root in the formula above, and the ray that crosses the whole diameter and the ray that skims past the edge arrive together because the first travels further through slower glass.

The profile also meets, without being asked, the condition that separates a lens that images a point from one that images a neighbourhood of it. A lens that is exact on its axis must also satisfy the sine condition to be free of coma: the height at which each ray enters must be proportional to the sine of the angle at which it arrives at the focus. In Luneburg’s sphere a ray entering at height $h$ arrives at the focus on the rim at exactly the angle whose sine is $h/R$ — the geometry of a sphere and its conserved quantity $n r\sin\psi$ guarantee it — so the sine condition holds identically, with the sphere’s radius as the focal length. For an ordinary lens the sine condition is one more requirement to be bought with another element. For the sphere it is a consequence of the symmetry, which is another way of saying that a lens with no axis has no off-axis to be wrong about.

The focus also collects light from the widest cone there is. Rays arrive at the rim point from every direction within the hemisphere facing the beam, up to grazing incidence along the surface, so the lens has the largest numerical aperture a lens in air can have. That is the limit set by étendue reached exactly: the brightness at the focus equals the brightness of the source, as it must, and the spot is as small as diffraction allows for a cone of that width, about half a wavelength across. A Luneburg lens is not only aberration-free; it is as fast as a lens in air can be.

Every direction is on axis #

What makes the design worth its difficulty is the symmetry it keeps.

For an ordinary lens, a beam arriving at an angle is an off-axis beam, and all the aberrations that grow with field angle — coma, astigmatism, distortion and the curvature of the image surface — apply to it. For a Luneburg sphere, every beam is on axis: the axis is simply the diameter pointing towards wherever the beam came from, and the sphere is identical about every diameter. The image of a distant scene is formed on the surface of the sphere itself, with every direction focused as sharply as every other.

The price of that symmetry is that the image surface is the sphere — curved, and facing inward. That is the extreme version of the curvature of field that every lens has: an ordinary lens images a flat scene onto a gently curved bowl and must be corrected to flatten it, while a Luneburg lens images the whole sky onto a sphere and makes no attempt to flatten anything. For a flat photographic sensor that is fatal. For an array of detectors or antennas arranged on a spherical surface, it is exactly right.

Where it is used, and the mirror on the back #

Luneburg lenses exist, but not for visible light. Grading the index of glass smoothly by a factor of $\sqrt2$ across a sphere is beyond practical manufacture, so the lens is made where the wavelength is long enough to build the grading from a stack of shells or from a foam whose density varies with radius: at microwave and radio frequencies. There it is an antenna. A feed placed at one point on the surface transmits a beam in the diametrically opposite direction; a ring of feeds round the surface transmits beams in many directions at once, all equally sharp; and moving a feed round the sphere steers its beam with no loss of focus and without moving the lens.

The most familiar use turns the lens back on itself.

Silver the back half of the sphere and a beam arriving from any direction within that hemisphere is focused onto the mirror, reflected, and — because the path back through a spherically symmetric medium is the mirror image of the path in — sent back as a parallel beam in exactly the reverse direction. The sphere becomes a retroreflector that works from a whole hemisphere of directions. Small boats and aircraft carry them to show up on radar; targets used to calibrate radars are built round them; and the returned beam is strong because all of the light collected over the sphere’s full cross-section is sent back along one direction instead of scattered. A flat mirror returns a beam only when it faces the radar squarely; a corner reflector returns it from a limited range of angles; a Luneburg reflector returns it from anywhere in front of it.

The graded sphere in a fish’s eye #

The problem Luneburg solved for radio waves had been solved long before, in water.

A land animal’s eye focuses mostly with its cornea, the curved front surface where light passes from air into the eye’s watery interior. Under water that surface does almost nothing, because the water outside has nearly the same index as the fluid inside, and an aquatic eye has to do all its focusing with its lens. Fish lenses are consequently spheres — the strongest lens shape there is — and a uniform sphere of the protein in a lens would have the spherical aberration of the second figure, far too much for a sharp image.

In 1877 Ludwig Matthiessen measured the focal length of fish lenses and found it consistently about two and a half times the lens’s radius, which is far shorter than a uniform sphere of the lens’s outer index could give. The explanation is that the lens is graded, densest at the centre and thinning towards the edge, and the grading does two things at once: it gives the sphere much more power than its surface index alone would, and it corrects the spherical aberration that a uniform sphere would have. The model in the figure, with a simple parabolic grading, brings the focus from four radii to two and a half and cuts the spread by a factor of five. Real fish lenses are graded more subtly than a parabola and do better still, and different species have profiles tuned to different wavelengths.

A fish’s lens is not a Luneburg lens — it sits in water, focuses behind itself rather than on its surface, and uses an index that never falls to that of its surroundings — but it is the same answer to the same problem. A sphere is the shape that works equally well in every direction, a uniform sphere cannot focus, and grading its index is how to make it focus without giving up the symmetry. The eye’s retina is curved to match the curved image surface, which a spherical lens also produces, so the fish has solved the field-curvature problem as well by making its detector the shape of its image.

Where the perfect profile stops being perfect #

It is perfect for rays. Every figure is geometrical optics. A real beam focused by a Luneburg lens forms a spot about half a wavelength across, and for a lens a few wavelengths in diameter — common at radio frequencies — the focus is correspondingly broad, and the lens is designed with the diffraction in view rather than the ray picture.

The profile is approximated in steps. A manufactured lens is a set of concentric shells of different index, not a continuous grading, and each step reflects a little and refracts a little. With enough shells the approximation is excellent; with few, it adds its own aberrations. The same applies to foams whose index is set by their density.

The rim index must match the surroundings. The profile is perfect only if the index outside is exactly the index at the rim. A Luneburg lens designed for air and immersed in water is no longer perfect; the fish’s lens, designed for water, would be a poor lens in air.

The focus is on the surface. A feed or detector of finite size at the rim samples a patch of the focal surface rather than a point, which is harmless when the feed is small compared with the wavelength and matters when it is not.

One slice of a lens that is the same in every plane #

The traced rays are drawn in one plane through the centre, and a Luneburg lens is three-dimensional: a beam fills a disc of cross-section, and every plane through the beam’s direction looks like the figures. That symmetry is what makes the lens useful and what the drawing, a single slice, only implies.

Nor can the rays show how much energy reaches the focus. A graded sphere made of a real material absorbs a little along each path, and the paths are of different lengths — longest for the central ray, which crosses the whole diameter, shortest for the rays that skim the edge. The central rays therefore arrive slightly weaker, and a lens made of a lossy foam has an efficiency that depends on its size and its material as well as on its profile. Equal travel time, which is what the figures demonstrate, is not equal attenuation.

Still open: a Luneburg lens for light #

The design is eighty years old at radio frequencies and still largely out of reach at optical ones. Grading the index of a solid by a factor of about 1.4 across a sphere is beyond any glass-making process, and the approaches under way build the grading from structures smaller than the wavelength — arrays of posts, holes or layers whose effective index depends on how densely they are packed — printed or etched on a chip or assembled in three dimensions. Flat, two-dimensional Luneburg lenses for light confined in thin films have been made and work; a full three-dimensional lens for visible light, with the performance of the microwave versions, has not been demonstrated at a useful size. Whether such structures can reach the required index range with low enough loss and scattering across a sphere many wavelengths across is the question that decides whether the lens with no axis ever becomes a camera lens.

The broader question the two graded spheres raise is what other perfect instruments exist. Maxwell’s and Luneburg’s are both members of a family of spherically symmetric profiles found by solving for the index that makes every path from a given source to a given image equal, and the family has been extended to lenses that focus at points inside, outside or on themselves, and to lenses that work between two different surrounding media. The habit worth carrying from here is to ask which symmetry a design is paying to keep. An axis costs off-axis aberrations; a sphere costs a curved image; and grading the index is how a sphere is made to keep its symmetry and focus at the same time.