The medium that images every point

Every lens is a compromise, and the reason is counting. A single refracting surface can be shaped to image one point perfectly — Descartes’ oval does it — but the shape is used up in doing so, and the point next to the one it was built for comes out blurred. More surfaces buy more corrected aberrations, one condition per degree of freedom, and a camera lens with a dozen elements is an elaborate negotiation over which errors to leave behind. It is natural to conclude that perfect imaging of more than one point is impossible. In 1854, the year he graduated, James Clerk Maxwell published a problem in a Cambridge mathematical journal that showed it is not. The medium he described images every point in space onto another point, perfectly, with every ray.

A medium with no surfaces #

The trick is to give up surfaces. Instead of glass with sharp boundaries, the fish-eye is a medium whose index varies continuously, highest at a centre and falling with distance:

$$n(r) = \frac{n_0}{1 + r^2/R^2}.$$

In such a medium a ray does not refract at boundaries; it curves continuously, bending towards higher index at a rate set by the gradient, the way light bends over a hot road. The path is found by the same rule that governs every ray — the optical path length is stationary — applied to a medium where the index is a function of position, which turns Snell’s law at a boundary into a differential equation for the curvature of the ray. For the fish-eye’s particular profile, the solutions are circles. Every ray is a circle or an arc of one, and the family of circles has a remarkable property: all the circles through any one point $P$ also pass through a second point $P'$, the point on the far side of the centre at distance $R^2/|P|$. That second point is the image.

The first figure traces that claim rather than assuming it. Each ray is integrated through the graded medium with a step a five-hundredth of the radius, with nothing about circles built in, and every one of them passes through the predicted image point to within a thousandth of the radius — which is the accuracy of the integration, not a limit of the medium. Rays leaving the source directly towards the image go straight across. Rays leaving at an angle curve round the centre and come back. Rays leaving directly away from the image travel a great arc out into the weak-index region far from the centre and return from the other side. They take wildly different routes, and they arrive together.

What stays constant along a curving ray #

A graded medium whose index depends only on the distance from a centre has a conserved quantity that makes every ray in it tractable, and it is worth having because it is the same quantity that governs an orbit. Along any ray, the product $n\,r\sin\psi$ stays fixed, where $\psi$ is the angle between the ray and the line from the centre. It is Snell’s law applied to a nest of thin spherical shells — a ray crossing from one shell to the next conserves $n\sin\psi$ at the local boundary, and the geometry of nested spheres converts that into conservation of $n r \sin\psi$ — and it is the optical twin of the angular momentum of a particle moving in a central force. A ray in a spherically graded medium is an orbit, with the index playing the part of the speed and $n r\sin\psi$ the part of the angular momentum.

That analogy says at once why most graded spheres do not image perfectly and why a few do. Of all central forces, only two close every orbit, and of all spherical index profiles, only special ones return every ray from a point to a single point. The fish-eye’s profile belongs with the closing cases: its rays close, and the symmetry that closes them is the same hidden rotation in four dimensions that closes Kepler’s orbits — Fock found it in the hydrogen atom by exactly the stereographic projection that turns a sphere into this medium. It is as rare among index profiles as the inverse square is among force laws. A graded index chosen for convenience — a parabolic fall-off, say, which is easy to manufacture — gives rays that wander like the orbits of an arbitrary force law and come back near the axis without ever meeting at a point.

The conserved quantity also fixes where the image lies. A ray that leaves the source at distance $r$ from the centre, moving at right angles to the line from the centre, carries $n r$ as its constant, and when it reaches its image, moving at right angles again, it must carry the same. With the fish-eye’s index, $n r$ takes the same value at exactly one other radius, $R^2/r$, since $r/(1 + r^2/R^2)$ is unchanged when $r$ is replaced by $R^2/r$; the tracing confirms that every other ray, whatever its starting direction, arrives at the point on the far side at that radius. A source moved outwards has an image moved inwards, and the product of their distances from the centre stays $R^2$.

The two traced cases say the same thing from different ends. Near the centre the image is far away, because the index is high and flat there and the rays need a long circuit to turn round; near the unit radius the image approaches the source’s mirror point, and a source on the unit sphere images onto the diametrically opposite point of the same sphere. Nothing in the tracing knew about the product rule, and both figures obey it to the accuracy of the step.

It also says why the rays are circles rather than any other closed curve. Under the fish-eye’s index the ray equation becomes the equation of a circle in the inverted coordinates, and inversion maps circles to circles. The fish-eye’s whole behaviour is the geometry of inversion made physical: the index is exactly the factor by which inversion rescales lengths, so a medium with that index cannot tell a point from its inverse.

Every point at once #

A Cartesian oval does this for one pair of points. The fish-eye does it for every point in space simultaneously.

An instrument that images every point of a region sharply is called an absolute instrument, and the fish-eye is the first one ever described. The map from object to image is inversion through the centre combined with a reflection through it, $P \mapsto -R^2P/|P|^2$, and it is its own inverse: the image of the image is the source again, which is how it must be, since a ray run backwards is still a ray.

What the map is not is a copy. Distances near the centre are stretched and distances far out are compressed, so a small object near the centre forms a large image far away, and a straight line of sources that does not pass through the centre images onto an arc of a circle. The image is sharp at every point and distorted everywhere. That distinction — between the sharpness of each point’s image and the fidelity of the whole picture — is easy to run together, because in an ordinary camera both fail together and both are called aberration. The fish-eye separates them completely, and the separation is not a defect that better design could remove. There is a theorem about it.

Why lenses cannot do this and a graded medium can #

Maxwell asked the obvious next question himself, and the answer, completed by Carathéodory, is a strong negative. An absolute instrument that forms images of one uniform medium in another uniform medium must reproduce every length exactly — magnification one in every direction — so that the only perfect instrument between uniform media is, in effect, a flat mirror. A telescope, a microscope or a camera, all of which exist to change sizes, cannot be perfect. Every one of them must have aberrations somewhere, and optical design is the business of putting them where they matter least.

The argument behind the theorem is a statement about optical length. If every ray from a point reaches its image, every path between them must be stationary, and a family of stationary paths between two fixed points must all have the same optical length. For an instrument between uniform spaces, applying that to pairs of neighbouring points forces the optical distance between any two object points to equal the optical distance between their images, which in a uniform medium means equal geometric distances — magnification one. The same accounting sits under the principle that no passive optical system can brighten an image beyond its source: optics is a transformation that preserves a measure of phase space, and perfection would have to preserve distances as well.

The fish-eye escapes because its objects and images are inside the graded medium, where optical distance is not geometric distance. An object near the centre, where the index is high, occupies a large optical extent in a small geometric space; its image far out, where the index is low, occupies the same optical extent in a much larger geometric space. The theorem still holds — optical lengths are preserved exactly — and the magnification that looks wildly non-uniform in geometric terms is exactly uniform in optical ones. Perfect imaging with apparent magnification is possible precisely because the medium itself does the rescaling.

The same time along every path #

The equality of optical lengths is not just the mechanism of the theorem. It is visible directly in the traced rays.

A ray that goes the long way round travels more than twenty times as far as the one that goes straight across, and it does so through the region far from the centre where the index is tiny. Distance and index compensate exactly, and every route takes the same time. That is what it means, in Fermat’s terms, for a point to have a perfect image: light takes the stationary path, a point imaged by every ray has infinitely many stationary paths to its image, and all of them must be equally stationary.

The common value is $\pi n_0 R$ whatever the position of the source, and the $\pi$ is a clue to what the fish-eye really is. Its index profile is exactly the one that makes the optical geometry of the plane equal to the geometry of the surface of a sphere, mapped onto the plane by stereographic projection. On a sphere the straightest paths are great circles, every great circle through a point passes through the diametrically opposite point, and the distance to the opposite point along any of them is half a great circle — $\pi$ times the radius. Project that sphere onto a plane and its great circles become the fish-eye’s circular rays, its antipodal points become each point’s image, and half the circumference becomes $\pi n_0 R$. Maxwell’s medium is a sphere, flattened out, and imaging on a sphere is perfect because every straight line through a point meets every other at the antipode.

A sphere that images its own surface #

The full fish-eye extends to infinity, with an index falling towards zero far from the centre, which no material provides. Cut it off at the radius where the index has fallen to the index of the surroundings and it becomes a real sphere.

With $n_0 = 2$ the index at the rim is exactly one, the sphere joins the surrounding air with no jump and no reflection, and a point on its surface is imaged onto the antipodal point on its surface. Light from a source on one side of the sphere is collected over the whole hemisphere of directions and delivered, every ray, to a single point on the other side. A device of this kind was proposed in the early twenty-first century as a route to imaging finer than the wavelength — the argument being that a perfect instrument for rays might also be one for waves — and it provoked a long argument about whether the image of a point source can be sharper than diffraction allows if a suitable absorber sits at the image point. The consensus that emerged is that the resolution in a real fish-eye is limited by the wavelength in the ordinary way; the perfect focusing of rays does not transfer to the sub-wavelength detail that rays do not describe.

How far this is from what a lens does #

It is worth measuring the fish-eye against the ordinary lens, because the comparison is what makes it remarkable. A lens deflects each ray in proportion to its height and forms an image of a single plane, and only approximately: a spherical surface does not bring parallel light to one point, and even a surface shaped to focus one point exactly fails for the point beside it unless a further condition on the angles is also met. Each condition consumes a surface or a glass, and the list of conditions never ends, because an image of an extended object requires the imaging to be correct not for one point but for a continuum of them.

The fish-eye meets the whole continuum at once, with no surfaces at all, by giving every point of space its own index. It pays for that in two ways that no designer of cameras would accept: its images are inverted through a centre rather than projected onto a plane, so they are warped out of all resemblance, and the object has to be inside the medium. Both costs are forced by the theorem; they are what perfection between points costs when it is not allowed to be perfection between uniform spaces.

Where rays stop being enough #

The fish-eye is perfect for rays. Everything in the figures is geometrical optics, which treats light as rays of zero width. A real wave converging on the image point forms a spot about half a wavelength across, set by the angles the rays arrive from, exactly as in any other instrument. Perfect ray imaging removes every aberration; it does not remove the diffraction that limits every image.

The index must be what the formula says. The full fish-eye requires an index falling towards zero, which no transparent material has; the truncated sphere requires a factor of two between centre and rim, which is large for a graded glass and is achieved in practice for microwaves with artificial materials. Any departure from the profile brings back aberration in proportion to the departure.

The profile is exact at one colour. A real graded material has an index that depends on wavelength, differently at different radii, so the profile is correct at one colour and slightly wrong at others. The fish-eye is free of monochromatic aberrations and has chromatic ones like any other glass.

The source and image are inside. The perfect imaging is between points in the medium. Light from an object outside it enters through the surface, and the truncated sphere images only its own surface perfectly. Using it to image an external scene gives up the property that made it interesting.

Circles in every plane through the source #

The figures are cross-sections. In three dimensions the fish-eye’s rays are circles in every plane containing the source and the centre, the images form in three dimensions, and the inversion maps spheres to spheres. A two-dimensional drawing of a three-dimensional absolute instrument shows one slice of the family and hides the rotational symmetry that fills it out.

Nor do the drawings show how much light each ray carries. A ray fan drawn at equal angular spacing suggests equal amounts of light in each ray, and a point source radiating equally in all directions does send equal power into equal angles at the source. The rays arrive at the image with a different angular spacing, crowded in some directions and sparse in others, so the image point receives light unevenly from the different directions it arrives from. A sharp image with an uneven distribution of arriving directions is still a sharp image, but the drawing’s even fan says nothing about it.

Still open: whether perfect ray imaging can be made to matter for waves #

The debate the truncated fish-eye provoked has settled on the conventional answer for ordinary use, and it left a narrower question open. In a medium where every ray from a point returns to a point, the waves in the medium have a highly degenerate set of modes — the fish-eye’s wave modes are the spherical harmonics of the sphere it is a projection of — and a detector or absorber placed at the image interacts with all of those modes at once. Whether that degeneracy can be exploited, for instance to couple light between two distant emitters placed at a source and its image more strongly than any conventional arrangement allows, has been studied in theory and in microwave experiments, with results that depend on how the absorber and the emitter are modelled. How far the perfection of the ray picture survives in the wave picture, and what it can be used for, is not fully resolved.

The next question is the one a designer would ask: whether a graded sphere can be made perfect not for points inside it but for parallel light arriving from outside, and from every direction at once. The habit worth carrying from here is to ask where a limit applies before accepting it. The theorem that forbids perfect imaging is a theorem about instruments between uniform media, and the moment the medium itself is allowed to vary, the counting that forced every lens to compromise no longer applies.