Two glasses that cancel a derivative

A simple lens has a different focal length for every colour. Point one at a bright star and there is no setting of the eyepiece that brings it to a point: at best focus the star is a small white disc surrounded by a purple halo, and racking the focus in or out turns the halo blue on one side and yellow on the other. For a century and a half after the telescope was invented this was considered incurable, and the response was to build refractors so long that the fault was diluted — Hevelius’s of 1673 was forty-six metres, slung from a mast, and unusable in any wind.

The cure, found by Chester Moore Hall in 1729 and patented by John Dollond in 1758, is to put a second lens of a different glass behind the first. What it cures, and what it does not, is a question about derivatives.

Why the focal length has a colour at all #

A lens bends light because the ray changes direction at each surface by the amount Snell’s law requires, and the amount depends on the index.

Everything about a lens is one refraction happening twice, at two curved surfaces. The focal length that results is set by the curvatures and by the index minus one — the whole of the lensmaker’s equation is those two factors multiplied — so any change in the index with wavelength changes the focal length in proportion. A glass whose index runs from 1.5225 in the blue to 1.5146 in the red has a focal length that runs by 1.5 per cent across the visible. Why the index depends on colour at all is a question about the electrons in the glass, and the answer is a resonance a long way into the ultraviolet.

For a thin lens the power is $\varphi = 1/f = (n-1)K$, where $K$ collects the two curvatures and is a fact about the glass’s shape, fixed once the lens is ground. Everything wavelength-dependent is in $n$.

What that costs is read off the imaging construction: an object, a lens of one focal length, and the image where the rays cross. Change the focal length by half a per cent and the image plane moves by half a per cent of the image distance — which for a telescope is a good deal more than the depth of focus, so the blue image and the red image cannot both be sharp on one sensor.

So the question becomes: by how much does $n$ change across the visible? The answer is measured, not derived, and every optical catalogue publishes it as a Sellmeier fit.

The ratio of a glass’s index-minus-one to the amount that index changes between two standard wavelengths is its Abbe number,

$$V = \frac{n_d - 1}{n_F - n_C},$$

with the three wavelengths being Fraunhofer’s helium and hydrogen lines at 587.6, 486.1 and 656.3 nm. It says how much bending is bought per unit of colour spread. Crown glass gives about 64, flint about 36, and the higher the number the less colour per unit of power.

The condition, in one line #

Two thin lenses in contact have powers that add: $\varphi = \varphi_1 + \varphi_2$. The chromatic spread of each is its own power divided by its own Abbe number, so requiring the change in total power between the F and C lines to vanish gives

$$\frac{\varphi_1}{V_1} + \frac{\varphi_2}{V_2} = 0.$$

Two equations — that one, and the requirement that the powers add to what is wanted — fix both elements. With $V_1 > V_2$ the first element must be positive and the second negative, and

$$\varphi_1 = \varphi\frac{V_1}{V_1 - V_2}, \qquad \varphi_2 = -\varphi\frac{V_2}{V_1 - V_2}.$$

For the pair drawn here that means a crown element of 4.62 dioptres and a flint element of −2.62, which combine to the 2 dioptres wanted. The elements are stronger than the pair they make, by a factor of $V_1/(V_1 - V_2) \approx 2.3$, and that is the price: every aberration that is not chromatic is worse in a doublet than in a singlet of the same focal length, because each surface is more strongly curved.

The derivation is worth doing slowly, because the step that matters is easy to skip. Each element’s power is $(n_i - 1)K_i$ with $K_i$ fixed by its shape. Differentiating with respect to wavelength gives $\mathrm{d}\varphi_i/\mathrm{d}\lambda = K_i\,\mathrm{d}n_i/\mathrm{d}\lambda$, and dividing by the element’s own power turns that into $\varphi_i \times (\mathrm{d}n_i/\mathrm{d}\lambda)/(n_i - 1)$ — a power times a property of the glass alone. That property, integrated across the F-to-C interval, is exactly $1/V_i$. So the achromatic condition is not a clever construction; it is what “the total power does not change” says once each element’s contribution has been split into a part belonging to the shape and a part belonging to the material.

What the condition does not do #

Setting a derivative to zero makes a function stationary. It does not make it constant, and the difference is the entire subject of the rest of this essay.

The corrected pair has the same focal length at the F and C lines, by construction. Between them the focus is shorter than at those two wavelengths, and outside them it is longer, so the curve of focal length against wavelength is a shallow parabola-like shape touching a horizontal line at two points. That leftover is the secondary spectrum.

The size of the residual is set by how the two glasses’ dispersions differ in shape rather than in magnitude, a quantity called the relative partial dispersion. For ordinary crowns and flints those shapes are nearly the same — the partial dispersions plot along a straight line, which is called the normal line — and a pair chosen from it always leaves a residual of about $f/2200$ across the visible. Escaping it requires a glass off the normal line, which is what fluorite and the special short-flints are, and a third element to use it: an apochromat, corrected at three wavelengths, at three or four times the price.

Why long telescopes worked, and why the fix is an aperture #

The comparison in that last figure is the practical heart of the matter, and it explains two centuries of instrument design in one relation. The depth of focus is $\pm 2\lambda N^2$ where $N$ is the focal ratio, and the secondary spectrum is proportional to the focal length. So the ratio of what is tolerable to what is left over goes as $N^2/f$, which for a fixed aperture $D = f/N$ is $N^3/D \cdot 1/N$ — in plain terms, slow lenses hide their colour error and fast ones show it.

That is why the great refractors were long. It is also why the same design stopped down becomes colourless, why a photographic lens at f/16 shows no colour fringing that is obvious at f/2, and why the pinhole camera — infinitely slow — is perfectly achromatic and always was.

The relation an imaging system obeys can be drawn once and for all: image distance against object distance, in units of the focal length. Changing the focal length with wavelength slides the lens along that curve, so a colour error at the glass becomes a displacement of the image plane — and the displacement is largest where the curve is steepest, which is why chromatic aberration is most obvious on close subjects and least on stars.

Who found it, and the argument about who did #

The history here earns its place because it is a case of a correct result being suppressed by a plausible argument from a great authority. Newton had measured the dispersion of one glass, found it proportional to the refraction, and concluded — in the Opticks, in as many words — that any combination of refractions strong enough to bend light would spread it in the same proportion, so that colour and bending could not be separated. The conclusion is false, and it is false for an empirical reason rather than a logical one: the ratio of dispersion to refraction differs from glass to glass, which is the whole content of the Abbe number having a range at all. Newton appears to have tested one type of glass and generalised.

Chester Moore Hall, a barrister with an interest in optics, worked out in 1729 that two glasses of different dispersion could be combined and had doublets made — reportedly by two different opticians, so that neither would learn the whole design. John Dollond arrived at the same idea independently around 1758, patented it, and his son enforced the patent vigorously. The courts upheld it on the grounds that Hall had kept his invention private: the doctrine, memorably, was that the public owed its knowledge to Dollond rather than to whoever had thought of it first. What the episode illustrates is not a moral about patents but the shape of the underlying claim. Newton’s error was to treat a measured ratio as a necessary one, and every later worker who checked more than one glass found the ratio varying — which is exactly the kind of assumption that a figure drawing two real dispersion curves, from two real catalogue entries, makes impossible to hold.

The one instrument that never had the problem #

Nothing in the argument applies to a mirror. A reflection obeys a law with no index in it, so every wavelength reflects at the same angle and a mirror has exactly one focal length — nor to a lens made of rings rather than glass, whose chromatic behaviour is worse rather than better and for an entirely different reason.

A mirror has its own failing and it is worth being exact about which. A spherical mirror does not bring parallel rays to a point, because the geometry is wrong away from the axis; that is spherical aberration, it is cured by grinding a paraboloid, and it is entirely independent of colour. A mirror has aberrations. It does not have a chromatic one, because the law of reflection contains no index at all.

Newton, who had convinced himself that dispersion and refraction were inseparable, drew exactly this conclusion and built a reflector in 1668. His premise was wrong — the ratio of dispersion to refraction does vary between glasses, which is the whole of Hall’s discovery — and his conclusion was nonetheless the one large telescopes eventually adopted, for the different reason that a mirror can be supported from behind and a lens can only be held at its rim.

Where the same coefficients turn up doing something else #

The Sellmeier fit used above is not a lens-maker’s convenience. It is the same function of wavelength that decides the rainbow.

The same coefficients turn up doing something nobody designed. A rainbow’s angular width exists because water’s index has a slope — compute the bow angle at two wavelengths and they differ by about two degrees — and that same slope in glass is what puts a lens’s blue focus in front of its red one. A rainbow is chromatic aberration on the scale of the sky, with the drop as the lens and no possibility of a second glass behind it.

The connection is worth taking seriously rather than treating as a pun. In both cases a quantity depends on wavelength only through $n(\lambda)$; in both the derivative $\mathrm{d}n/\mathrm{d}\lambda$ is what produces the spread; and in both the useful measure is a ratio of the effect to the derivative rather than either alone.

The eye that never had it corrected #

There is one imaging instrument in constant use that has no achromatic doublet in it and is not noticeably troubled by the fault, and working out why is more interesting than it sounds.

The human eye has about two dioptres of longitudinal chromatic aberration between 400 and 700 nanometres. That is an enormous amount — no camera lens with a tenth of it would be sold — and it is not corrected anywhere in the optics: the cornea and the lens are made of water-like material with ordinary dispersion, and nothing in the eye plays the part of the flint element.

Four things conspire to make it tolerable, and only the first is obvious.

The luminance signal is concentrated in the green. The eye’s spectral sensitivity peaks near 555 nanometres and falls steeply either side, so most of the information about fine detail arrives in a band much narrower than the visible, over which the focus shift is small.

The retina does not sample blue finely. There are essentially no short-wavelength cones in the central third of a degree of the fovea, and elsewhere they are sparse — so the part of the spectrum that is most badly out of focus is also the part that is least able to resolve anything. The optical defect and the sampling limitation are matched, which is unlikely to be an accident.

The macular pigment absorbs short wavelengths in front of the fovea, further reducing what the worst-focused part of the spectrum contributes.

And the aberration is used. Whether a blurred image has more red or more blue in its blur says which side of focus the eye is on, which a monochromatic blur cannot say. That signal drives accommodation: in monochromatic light the accommodation system loses the cue and hunts, focusing far less reliably than it does in white light. The eye’s largest optical fault is one of its focusing sensors.

The clinching evidence is what happens when it is removed. Achromatising lenses have been built for laboratory use, cancelling the eye’s chromatic aberration with an external doublet, and visual acuity improves only marginally. What limits ordinary vision is the sampling of the retina and the eye’s other aberrations, not this one — so the two centuries of effort that went into correcting it in telescopes would have bought nothing at all if it had been possible to correct in the eye.

Where the glasses came from #

The Abbe diagram above shows two families with a gap between them, and the essay called that gap a fact about what can be melted. It is worth saying where the two families came from, because for a century and a half the whole subject had exactly two materials to work with.

Crown glass is soda-lime — window glass, essentially, and as old as glass. Flint glass is lead glass, developed in England in the 1670s, and the lead is what gives it both its high index and its high dispersion. Those two, and nothing else, are what Hall and Dollond had; and since the achromatic condition uses only the two Abbe numbers, every achromat made between 1758 and about 1880 was the same pairing with the same residual secondary spectrum, differing only in size.

Making the flint element at all was the hard part. Lead glass melts unevenly and cools with striae — schlieren of varying index — that ruin an image, and large disc-shaped blanks were nearly impossible to obtain. The Swiss cabinetmaker Pierre-Louis Guinand solved it around the turn of the nineteenth century by stirring the melt with a fireclay rod, which sounds trivial and was the enabling invention: it made homogeneous blanks of a size nobody had managed before.

Guinand took the technique to Bavaria, where the young Joseph Fraunhofer learned it, and the consequences run in two directions. Fraunhofer’s objectives were the best in the world for a generation. And while measuring the dispersion of his glasses he needed a light source of known, sharply defined wavelength — so he dispersed sunlight through a prism to find one, and found instead the dark lines that now carry his name. The lines whose wavelengths define the Abbe number were discovered in the course of characterising glass for lenses.

The gap in the diagram was closed deliberately a lifetime later. Ernst Abbe at Zeiss wanted glasses with properties chosen rather than found, and with the chemist Otto Schott ran a systematic search over compositions in Jena through the 1880s — adding barium, boron and phosphates, producing dozens of new glasses that sat where no natural crown or flint did. That is the moment lens design stopped being a matter of selecting from what existed and became a matter of specifying what was wanted, and it is why a modern catalogue lists hundreds of glasses rather than two.

Where the model stops #

Three approximations run underneath everything above. The lenses are thin and in contact, so their powers simply add — a separated pair has a different condition, with a term in the separation, and the Huygens eyepiece exploits exactly that to achromatise using two lenses of the same glass. The elements are paraxial, so nothing here says anything about rays far from the axis, where spherical aberration and coma live and where the doublet’s stronger curvatures make things worse. And the design corrects focus and not magnification: a lens can be achromatised for focal length and still have a different image size for each colour, which is lateral colour, and needs a separated pair to fix.

There is also a floor beneath all of it that has nothing to do with glass.

What remains when every aberration has been removed is diffraction, and knowing where that floor sits decides how far correction is worth taking. A lens corrected far below the diffraction limit is corrected below the point at which the correction can be seen — and the limit falls as the aperture grows, so a large lens is harder to make well for two reasons at once: its tolerance shrinks while its residual aberration grows.

What the pictures cannot show #

The focus curves show where light of each colour comes to a point and say nothing about what an out-of-focus colour looks like. A point source imaged through this doublet is not a set of separated points but a disc with a coloured edge whose hue depends on which side of best focus the sensor sits — and no plot of focal length against wavelength contains that.

Nor is any of this an image-quality figure. Two lenses with the same focus curve can differ enormously in how much energy each puts inside the central spot, which is what actually decides whether a faint companion star can be seen next to a bright one.

And the Abbe diagram shows glasses as points with two coordinates, which is the designer’s summary and not the truth: two glasses can sit at the same point and have measurably different partial dispersions, which is precisely the difference an apochromat is built to exploit.

What the same condition looks like elsewhere #

The structure of the achromatic condition — combine two elements with opposite signs of a quantity, weighted so a derivative cancels — recurs often enough to be worth naming as a pattern.

A temperature-compensated pendulum does exactly this with lengths: a gridiron of steel and zinc rods, with the zinc pushing the bob up as fast as the steel lets it down, so the effective length’s derivative with respect to temperature vanishes. It cancels one derivative, leaves a residual second-order term, and the residual is the limit on such a clock in the same way the secondary spectrum limits a doublet.

A quartz oscillator cut at a particular angle has a frequency whose first temperature derivative vanishes at a chosen temperature, leaving a cubic residual — the AT cut, and the shape of that residual curve is what every quartz watch’s accuracy specification is a statement about.

In each case the same three facts hold: the condition sets a derivative rather than a value; agreement is exact at one or two points and approximate between; and improving it further requires a third component with a genuinely different dependence, not more of the same two. Cancelling the first derivative is cheap and cancelling the second is a different order of problem, and knowing which one a design has achieved is usually the whole of what separates a good instrument from an excellent one.

Where the ladder goes next #

Dispersion has now appeared twice: once as the reason a rainbow has an angular width, and here as the fault that made refracting telescopes nearly useless and the two-glass trick that repaired it. It appears a third time in a place with no lens in it at all: the wavelength at which a fibre stops smearing a pulse is the same Sellmeier arithmetic asked for a stationary group delay rather than a stationary focus. The next rungs are the ones this essay stopped at. A third glass off the normal line cancels the second derivative as well and gives an apochromat; a separated pair achromatises magnification rather than focus; and the same idea run in reverse — deliberately maximising rather than cancelling the derivative — is a prism spectrometer.

Beyond optics, the pattern is the general one. A single condition on a function of one variable buys a stationary point and nothing more; making a quantity genuinely independent of a variable takes as many conditions as the expansion has terms. The number of glasses in a lens is a count of how many derivatives have been cancelled, and the same arithmetic governs how many temperature-compensating elements a precision oscillator needs and how many terms a numerical scheme has to match to reach a given order.