The surface that images one point exactly

Fermat’s principle says light takes a path whose optical length is stationary. For a surface that brings every ray from a point $A$ to a point $B$, that has a stronger consequence: not merely stationary, but equal. If two rays from $A$ arrive at $B$ by different optical path lengths, they arrive out of step and there is no image; if the surface is to gather all of them, every path must take the same time.

Written as an equation that is one line:

$$n_1\,|PA| + n_2\,|PB| = \text{constant}.$$

The set of points $P$ satisfying it is a curve Descartes studied in the 1630s, and it is the exact answer to the problem every lens surface is an approximation to.

The figure is generated by the equal-path condition and then checked, at points along the curve, to refract according to Snell’s law — with the surface normal taken from the drawn points rather than from a formula. That check is Fermat’s principle being tested rather than assumed: a surface built to make paths equal turns out to bend rays by the angles Snell’s law requires, which is the content of the principle and is not obvious from the construction.

What the exact surface costs #

The oval is not a sphere and in general is not a conic either. Two special cases are, and they are the ones with names.

If one of the two points goes to infinity — a collimated beam on one side — the oval degenerates into a conic: an ellipse for one index ratio, a parabola for reflection, a hyperbola for another. That is why a parabolic mirror focuses a parallel beam with no spherical aberration and a spherical one does not: the parabola is the exact solution for the infinite-conjugate case, and the sphere is its second-order approximation.

If both points are at finite distances, as they are in most instruments, the oval is a quartic curve with no simpler description. It can be computed to any precision, it can be tested numerically, and until recently it could not be made. A grinding machine produces spheres for a geometric reason — two surfaces rubbed together with grit between them approach sphericity whatever they started as — and every other shape has to be made by a process that knows where it is, which is a modern capability. So the history of optics is largely the history of using the wrong surface and correcting it with more of them.

Comparing the two ovals says something the first alone does not. There is nothing special about the shape drawn; what is special is the condition, and each pair of conjugate points and each index ratio has its own surface. The family is two-parameter — the ratio of the distances and the ratio of the indices — and every member is exact for its own case and useless for the others.

That is worth holding onto because it is the reason the result is less useful than it sounds. A surface that is exact for one arrangement is not nearly exact for a neighbouring one; the whole curve moves. The exactness is not a property that degrades gracefully.

The two points a sphere gets right #

A sphere is the worst surface for almost every pair of points and a perfect one for a particular pair, and the exception is the most useful fact in this essay.

The two points are at $R/n$ and $nR$ from the centre — the first inside the glass, the second outside — and the check is direct: trace rays at every angle up to fifty degrees, extend each refracted ray backwards, and they all cross at one point to the accuracy of double-precision arithmetic. There is no small-angle assumption anywhere in the trace.

The reason is a similar-triangles argument that is worth having. For a point at $R/n$, every ray meets the sphere with an angle of incidence whose sine is proportional to its distance from the axis in just the way that makes the refracted ray point away from $nR$. The two triangles — centre, surface point, object; and centre, surface point, image — are similar with ratio $n$, and similarity is exactly what makes the construction independent of angle.

This is what the front element of an oil-immersion microscope objective does. The specimen sits at the aplanatic point of a hemisphere of glass with oil filling the gap, so the first surface increases the numerical aperture by the index of the glass and contributes no aberration at all. Every later element then works on a beam that has already been enlarged, and the condition a lens must meet to image an area rather than a point is satisfied at that first surface exactly rather than approximately. Without the aplanatic points, an aperture above one would cost more aberration than the rest of the design could remove.

And the point next door #

Here is what the exactness costs. A refracting surface has one degree of freedom: its shape. Imaging one pair of points perfectly uses all of it, and there is nothing left over for any other pair.

The blur grows almost linearly with the displacement, which identifies it: linear growth with field angle is coma, the aberration in which an off-axis point images as a comet-shaped smear rather than a disc. It is the aberration that ruins a single-surface design fastest, because it grows at the first power where spherical aberration grows at the third power of the aperture and astigmatism at the second power of the field.

That is the honest general statement about lens design. Surfaces are degrees of freedom and aberrations are constraints, and a design is a count. A single surface can kill one aberration for one point. Two surfaces can do rather more, and an objective with eight elements has sixteen surfaces plus the thicknesses and the glasses between them — perhaps forty free parameters — used to hold five or six aberrations small over a field and a wavelength range rather than to eliminate any of them anywhere.

Which is why nobody makes the exact surface. A Cartesian oval is perfect for one conjugate pair at one wavelength for one point in the field, and every one of those conditions is violated in every real instrument: objects have extent, light has bandwidth, and instruments are used at more than one distance. An exact surface is a solution to a problem nobody has.

Counting, before any ray is traced #

The degrees-of-freedom argument deserves stating as arithmetic, because it decides most questions about optical designs faster than tracing rays does.

A rotationally symmetric system’s departures from perfect imaging, expanded to third order in the aperture and the field, are five: spherical aberration, coma, astigmatism, field curvature and distortion. Add two chromatic terms — one for the focus and one for the magnification — and there are seven quantities to be made small.

Each surface contributes a curvature; each gap contributes a thickness; each element contributes a choice of glass, which is two numbers, the index and the dispersion. A cemented doublet therefore has three curvatures, one thickness that matters and two glasses: roughly seven or eight parameters against seven conditions, and that is exactly why a doublet can be made good on axis over a narrow field and no better. A four-element design has about twenty, and can hold the seven small over a real field. A modern photographic lens has ten or more elements and a hundred parameters, most of them spent not on the seven but on the higher-order terms the third-order expansion leaves out.

The count also says what cannot be bought. Distortion is a field-dependent magnification and is corrected by symmetry rather than by parameters — which is why a lens symmetric about its stop has almost none, and why an asymmetric one always does. And chromatic terms need different glasses rather than more surfaces, because a single material has one dispersion and no arrangement of it cancels its own.

Where it is used anyway #

There is one situation where the conditions genuinely hold, and it is worth naming because it is where the oval reappears.

A single-mode optical fibre emits from a point, at one wavelength, into a fixed cone, and it needs to be coupled to another such point. That is exactly one pair of conjugates, one wavelength and no field — so the exact surface is the right surface, and moulded aspheric lenses designed for the job are Cartesian ovals to within their manufacturing tolerance. The same is true of the lens that focuses a laser diode’s output onto a disc, and of the collimator on a fibre laser.

The other place is the reverse case. A parabolic mirror is the infinite-conjugate oval and is used precisely where the object is at infinity and the field is small: a telescope. Its aberrations off-axis are severe — coma again, growing linearly — and the whole business of telescope design past a certain field is about adding surfaces to cancel it.

So the exact surface is not a curiosity that turns out to be useless. It is the right answer to a narrow question, and the reason it is rare is that the question is narrow rather than that the answer is wrong.

What Descartes was doing with it #

The ovals are older than the principle they now illustrate, which is a common relationship between a piece of geometry and the physics that eventually needs it.

Descartes published them in 1637, in the appendix to the Discourse on Method that also introduced coordinate geometry, and his interest was in lens design: he wanted the shape that would focus light without the blur every real lens showed, and he found it by construction rather than by any variational argument. Fermat’s principle came twenty years later, and it recasts the ovals as the answer to a question about time rather than about geometry.

The two accounts explain different things. Descartes’ construction says what shape to grind. Fermat’s principle says why that shape and no other, and — more usefully — says what happens when the shape is wrong: paths of unequal length, arrivals out of step, and a spot rather than a point. It also connects the subject to everything else stationary principles cover, so that the surfaces here and the trajectory of a particle in a potential turn out to be the same kind of object.

The historical irony is worth recording. Descartes solved the design problem exactly and it made no difference to any instrument for three hundred years, because nobody could make the surface; and the approximate spherical surfaces everybody could make were improved by adding more of them, which is a completely different line of development. The exact solution and the useful practice diverged immediately and only met again when machines learned to cut a stated shape.

Where the model stops #

The whole treatment is ray optics and the surface is exact only in that sense. A perfect surface produces a diffraction-limited spot, not a point, and once the aberrations are below about a quarter of a wavelength there is nothing further to gain — a lens is already as good as its aperture allows. So “exact” here means exact in a description that is itself an approximation, and the practical target is not zero aberration but aberration below the diffraction blur.

One wavelength is assumed throughout. The index appears in the defining equation, so the oval for one colour is not the oval for another, and a surface exact at 550 nanometres is wrong at 450 by an amount set by the dispersion of the glass. That is chromatic aberration and no shaping of a single surface removes it, which is why two glasses are needed to cancel a derivative.

The surface is assumed to be reached by every ray drawn. At large angles a real oval curves back on itself and rays begin to be totally internally reflected rather than transmitted; the figures stay well inside that, and the limit is real.

And the aplanatic argument assumes the object is immersed in the glass. In practice it is immersed in oil chosen to match, and the match is never exact — a difference in the third decimal place of the index reintroduces spherical aberration in proportion to the depth below the coverslip, which is why an oil objective has a correction collar and why imaging deep into a watery specimen with an oil lens is a known and unsolved nuisance.

The wavefront view, and why designers use it #

There is a way of stating everything above that replaces surfaces and rays with a single object, and it is what an optical designer actually looks at.

A point source emits a spherical wavefront. A perfect imaging system converts it into a spherical wavefront converging on the image. Every aberration is a departure of the emerging wavefront from that sphere, measured in wavelengths across the pupil — and the aberrations named in the previous section are the leading terms of an expansion of that departure in the pupil coordinates and the field.

Two things become obvious in that language. Fermat’s principle is the statement that a wavefront is a surface of constant optical path, so a surface built to make paths equal is a surface that emits a spherical wavefront; the oval and the sphere are the same statement in two languages. And the tolerance is now a single number — Rayleigh’s quarter wave, which is where a focus stops being diffraction-limited — against which every design is judged, whatever combination of aberrations produces it.

That last point is why the wavefront view took over. Ray aberrations are several quantities with different units and different field dependences; the wavefront error is one number in wavelengths, it can be measured directly with an interferometer, and it is the quantity that decides how the image looks. A design is finished when the departure is under a quarter of a wave over the whole field, and how the remaining error divides among the named aberrations is a matter of taste.

What the pictures cannot show #

The oval figure draws a curve in a plane and the surface is that curve rotated about the axis. Nothing about the picture indicates that the rotational symmetry is doing work: the surface images the axial point exactly because every meridional plane is identical, and an object point off the axis breaks that symmetry, which is why the third figure exists.

The aplanatic figure shows rays extrapolated backwards to a virtual image and cannot show that the image is virtual — the light does not actually pass through the second point, and a detector placed there would find nothing. What follows in a real objective is another element that turns the virtual image into a real one, and the aplanatic surface’s contribution is entirely to have enlarged the cone without spoiling it.

Neither figure shows the wavefront, which is where the argument really lives. An exact surface converts a diverging sphere centred on the object into a converging sphere centred on the image, and the aberrations are departures of the second from a sphere. Drawn that way the whole subject is one picture, and it is a picture of a surface that is nearly spherical to within a wavelength, which is hard to draw and easy to compute.

Where the ladder goes next #

The Fermat ladder began with the path that does not change, went through the ray that bends without a surface and the path that takes the longest time. This rung asks what surface makes every path equal. The rungs after it: the eikonal, which is the wave statement the ray statement approximates and from which the surfaces here fall out as level sets; freeform surfaces, where the rotational symmetry is given up and the count of degrees of freedom changes entirely; and the aplanatic condition generalised, where a design is required to be exact for a point and correct to first order for its neighbours, which is what the sine condition asks.

The habit worth carrying away is to count what a shape can do. A surface is one function and each aberration killed uses part of it, so a design that promises to remove several with one surface has either found a coincidence or is quoting a limit. Asking how many free functions a system has and how many conditions are being imposed settles most claims about optical performance before any ray is traced.