Nothing is in focus at both ends

Everything else on this site treats the eye as three spectral sensitivities and stops there. That is the right abstraction for colorimetry, it is the abstraction the whole 1931 system is built on, and it hides an enormous fact about the physical organ.

The eye is made of water and protein. Water and protein disperse — their refractive index falls with wavelength, like glass, like everything. So the eye is a lens with chromatic aberration, and it has a great deal of it.

Where the curve crosses zero is a choice the eye makes, and moving it moves which end of the spectrum is blurred.

Sliding it the other way puts the same amount of blur on the other end, and there is no setting in between that escapes both.

Two more settings say that the trade is continuous in both of its arguments rather than a choice between three cases.

Two dioptres is not a residual #

A camera lens with two dioptres of longitudinal chromatic aberration would be a defective lens. Photographic optics are corrected to a small fraction of that, by combining glasses with different dispersions, and the residual is one of the things a lens review measures.

The eye has no such correction. Its span across the visible band is 2.10 dioptres, computed here from Thibos’s chromatic eye — a reduced-eye fit whose entire content is

$$R(\lambda) = 1.731 - \frac{633.46}{\lambda - 214.10}$$

in dioptres, with the wavelength in nanometres. That formula is quoted rather than derived, for the same reason the colour-matching functions are quoted: it is a fit to measurements of people, and this site quotes measurements and computes everything downstream of them.

Two dioptres is the difference between needing reading glasses and not. It is a larger error than most people’s uncorrected refractive error. And it is present in every eye, at all times, and cannot be corrected by spectacles, because it is not a fixed offset — it is a different offset at every wavelength simultaneously.

What that does to the image #

Defocus in dioptres is an abstraction. The quantity that matters is how large a blur it puts on the retina, and that depends on the pupil.

An eye with a pupil of diameter $p$ millimetres, defocused by $D$ dioptres, spreads a point source into a disc subtending approximately $p \times D \times 10^{-3}$ radians. Through a 3 mm pupil — an ordinary indoor pupil — the 0.83 dioptres of defocus at 450 nm becomes a blur circle of 8.5 minutes of arc.

Foveal acuity is about one minute of arc. So blue light, in an eye focused on yellow-green, arrives spread over a disc roughly eight times wider than the finest detail the retina can resolve. This is not a subtle degradation. It is most of the image gone.

Two everyday observations fall out of it immediately. Saturated blue text on a saturated red background, or the reverse, cannot both be sharp at once — the eye can accommodate on one or the other and the other shimmers. And deep blue objects at night, through a dilated pupil where the blur is doubled again, look soft in a way that has nothing to do with the light level.

The spacing of those three peaks is usually discussed as a question about discrimination: L and M overlap heavily, which is why red-green discrimination is fine-grained and why losing one of them is the commonest form of colour blindness. It is also, and less often, a question about optics — because a class whose sensitivity sits at the short end of the band is a class sitting where the dispersion curve is steepest.

The cone class that gets the worst of it #

The three cone classes sample different parts of the spectrum, so they receive different amounts of this blur. Weighting the defocus by each fundamental — so the number is the blur the class actually gets rather than the blur at its peak — gives the asymmetry.

L and M sit close together in the middle of the band, near where the eye is typically focused, and receive almost identical and almost negligible defocus. S sits far out in the short wavelengths where the dispersion curve is steepest, and receives 4.3 times as much.

That number is the case for something the retina actually does. S cones are sparse: they make up roughly seven per cent of the cone population, they are absent altogether from the very centre of the fovea, and they are arranged in a semi-regular lattice much coarser than the L and M mosaic. This is usually presented as a curiosity, or explained by reference to the developmental order in which the pigments appeared.

The optics offer a cleaner account. The image the S cones would be sampling has already been low-pass filtered by the eye’s own dispersion. There is very little high-spatial-frequency information left in the S channel for a denser mosaic to recover, so a dense S mosaic would be paying full metabolic and wiring cost for detail that is not there. A sparse one is close to free.

The argument is not proof — it is a consistency, and evolutionary just-so stories are cheap. What makes it worth stating is that the quantity it rests on is computable rather than asserted, and that the ratio came out at 4.3 without anybody choosing it.

The template matters to this argument because it separates two things that are easy to run together. Where a cone class is in the spectrum is set by its pigment, and the pigment’s peak is one number. How much defocus that class receives is set by the eye’s dispersion, which is a different curve entirely and knows nothing about pigments. The 4.3× figure above is what happens when those two independent facts are laid over each other, and neither was arranged with the other in mind.

Where the eye chooses to focus, priced #

Accommodating on 555 nanometres is one choice out of a continuum, and the alternatives can be costed rather than gestured at.

The eye focuses almost exactly where the detail is. Sweeping the accommodation wavelength and asking which value minimises the mean defocus received by L and M together, the answer is 565 nm, at 0.1809 dioptres. Focusing at 555 gives 0.1835 — one and a half per cent worse. So the eye sits within ten nanometres of the best available place for the two classes that carry the spatial image, on a band three hundred nanometres wide.

The compromise position exists, and it is a bad trade. The wavelength minimising the mean across all three classes is 535 nm. Moving there would take the S cones from 0.8066 dioptres down to 0.6912 — a saving of fourteen per cent — and would push L and M up from 0.1835 to 0.2206, which is twenty per cent worse. A fifth more blur where all the detail is, in exchange for a seventh less where there is almost none.

And the eye could make the S channel sharp if it wanted to. Accommodating at 450 nm leaves the S cones at 0.2070 dioptres, which is very nearly what L and M enjoy now. The optics do not prevent a sharp blue image. The eye declines to buy one, and the reason is arithmetic rather than anatomical.

What the sparsity would have to be #

The mosaic argument is a consistency, and a consistency can be given a number instead of a direction.

Through a three-millimetre pupil each channel’s mean defocus becomes a blur circle: 1.89 minutes of arc for L and M together, and 8.32 for S, a ratio of 4.40. Now suppose the retina samples each channel at the same multiple of what its own optics deliver — the same oversampling factor, whatever that factor turns out to be. Spacing must then scale with the blur, so the S lattice should be 4.40 times coarser than the L-and-M lattice, and geometry fixes its share of the mosaic at 4.9 per cent.

The measured share is about seven. In spacing terms the real mosaic gives a ratio of 3.64 where matched oversampling would ask for 4.40, so the S lattice is roughly a fifth finer than the optics require, which is the same statement as being about 1.4 times too dense.

That single number is both the argument’s strength and its limit. Two quantities computed from entirely unrelated things — a two-constant fit to the dispersion of ocular media, and a count of receptors in a retina — land within forty per cent of each other in density, across a thirteen-fold difference between the two cone populations. Forty per cent is not agreement, which is why this stays a consistency rather than becoming a derivation: an S share of five per cent would have fitted better and one of ten would not have refuted anything.

What this costs a colour measurement #

The aberration is a fact about images, and colorimetry does not measure images — it measures matches between two uniform fields. So the first question is whether any of this touches the numbers the rest of this site is built on, and the answer is that it touches one of them.

A colour match is made by adjusting a mixture until two adjacent fields become indistinguishable. If the two fields have different spectral compositions — which they do, because that is what a match is — then they are differently defocused, and the observer is comparing two patches that are not equally sharp. For a large uniform field this makes almost no difference, because a uniform field has no detail for blur to remove; the blur circle spreads each point into its neighbours, and its neighbours are the same colour.

That is why the effect is nearly absent from the tristimulus values and why the 1931 experiment survives it. It is also why the residual that does exist lives at the edge of the matching field, where the two halves meet, and why observers in those experiments were instructed to judge the boundary rather than the interior. The instruction is doing more work than it appears to.

The same reasoning explains why chromatic aberration never shows up in a metamer pair drawn as two swatches: two large flat patches are the one stimulus this defect cannot damage.

Where the model stops #

The chromatic eye is a reduced eye: one refracting surface, one index, one fitted curve. It does not describe an eye, it describes the wavelength dependence of an eye’s total power, and it does so as a two-parameter fit over a limited range.

Three things it does not carry. It has a pole at 214.1 nm and is meaningless below it, which the code refuses rather than extrapolating through. It says nothing about transverse chromatic aberration, which is a lateral displacement rather than a defocus, is small on-axis and grows into the periphery, and is a separate phenomenon with a separate literature. And it is a population fit, so it describes nobody: individual chromatic aberration varies, though much less than individual spherical refractive error does, which is itself an interesting fact about how tightly the dispersion of ocular media is constrained.

It also stops short of the thing a reader most wants to know, which is what any of this looks like. This site cannot show that. A figure demonstrating chromatic blur would have to control the reader’s accommodation, pupil size, viewing distance and display spectrum, and it can control none of them — the blur is a property of the reader’s own eye looking at the screen, not something that can be drawn onto the screen. The display is an unknown here in a more fundamental way than usual: the apparatus under discussion is downstream of the apparatus doing the discussing.

Accommodating in the green with a wide pupil is the low-light case, and a wide pupil is where defocus costs the most.

What was computed, and how #

Four assertions, each guarding a different way of getting this wrong.

The eye is in focus at exactly one wavelength, by construction, checked to 10⁻¹². This is the trivial one and it catches a sign error or a swapped argument, both of which produce a curve that looks entirely correct and is reflected about the axis or shifted along it.

Defocus falls monotonically with wavelength at every step of the grid — no turning point anywhere in the range. A fit producing one would be describing an anomalous dispersion nobody has measured.

The span across 400–700 nm is between 1.6 and 2.6 dioptres. Asserted as a band rather than a value, because the number depends on where the band is taken to end, and this site’s own 380–780 nm range is a choice made for unrelated reasons. What is not a choice is the order of magnitude.

The S cones receive more mean defocus than either other class, which is the claim the mosaic argument rests on and would be worth deleting the argument over.

The refusals: a wavelength below the fit’s pole, and — the one worth keeping — a demand that a twenty-nanometre band show two dioptres, which fails and must.

A small pupil is the bright-daylight case, where depth of focus is largest and the same dioptres cost least.

The observer arrives from an unexpected direction #

There is a connection here to the standard observer that is easy to miss and is not a coincidence.

The 2° colour-matching functions were measured on a small central field, which means through the fovea, which means through the macular pigment and with no rod contribution. The 10° functions were measured on a field that extends well outside the foveola. Those two fields differ in cone composition, in macular pigment density, and — relevant here — in what the optics deliver, because the blur that matters depends on where on the retina the image lands.

The largest disagreement between the two sets of functions is in the short wavelengths. That is where the macular pigment absorbs most, and it is also where the chromatic defocus is worst. The two effects are independent and they land in the same place, which is part of why the blue end is where every observer question on this site turns out to be sharpest.

Accommodating at the far red end is the extreme nobody chooses, and it makes the asymmetry of the curve unmistakable.

Why nobody notices #

Two dioptres of uncorrected aberration in an optical instrument would be immediately obvious. In the eye it is invisible to almost everyone, and the reasons are worth setting out because each of them is doing real work.

Accommodation lands on the middle. The eye focuses on where the luminance information is, which is where L and M are sensitive, which is the middle of the band. The two classes carrying nearly all the spatial detail receive almost no defocus, and the class receiving the most carries almost none of the detail.

Natural scenes have little saturated colour at high spatial frequency. Fine detail in the world is overwhelmingly achromatic — texture, edges, shadows — and broadband. A broadband edge has energy at every wavelength, so some of it is always in focus, and the visual system has the luminance channel it needs.

The pupil is usually small. Blur scales with pupil diameter, so bright conditions halve or quarter it relative to the dilated case. The effect is worst exactly when the light level is lowest, which is also when the rods are contributing and acuity is poor for unrelated reasons.

The visual system knows. Chromatic aberration is a stable, lifelong property of an individual eye, and the system is calibrated to it — which is why it can be used as an accommodation cue rather than merely suffered.

Where it does become obvious is precisely where those four break down: a saturated blue on a saturated red, at high spatial frequency, in dim light. Which is a description of a neon sign at night, and of a certain kind of badly-chosen slide.

Who found it, and when #

Newton knew the eye was chromatically uncorrected, or at least suspected it, and it mattered to him for a reason that turned out to be wrong: he concluded from the impossibility of correcting chromatic aberration in a single lens that refracting telescopes were fundamentally limited, and built a reflector instead. The achromatic doublet, which corrects it by combining crown and flint glass, was patented by John Dollond in 1758, having been invented rather earlier by Chester Moore Hall and kept quiet.

Measurements of the eye’s own chromatic aberration go back to Wollaston in the early nineteenth century and were put on a modern footing by Wald and Griffin in 1947 and by Bedford and Wyszecki in 1957. Thibos’s fit dates from 1992 and its virtue is that it is a model rather than a table: it has one functional form and two constants, so it can be asked what happens at a wavelength nobody measured, which is exactly what the figures here do.

The observation that S-cone sparsity might be optically rather than developmentally explained belongs to the 1990s work on the arrangement of the cone mosaic, and remains an argument from consistency rather than a demonstration.

There is one more piece of history worth keeping, because it inverts the usual order. Chromatic aberration was long treated purely as a defect — something the visual system must be overcoming, with the interesting question being how. The modern view is that it is at least partly a cue: the sign of the defocus tells an eye which way to accommodate, and an eye deprived of chromatic information accommodates more slowly and less accurately. A monochromatic world would be sharper and harder to focus in. That is a genuinely surprising trade, and it means the two dioptres are not simply a cost the eye pays.

Where this goes next #

The mosaic is the other half of this. The optics say the S channel carries little detail; the retina’s composition says the L:M ratio varies enormously between people — and the surprising fact is what that variation does and does not disturb.

Below this rung, the fourth receptor and the range where both systems run is where the eye stops being three sensitivities in a different direction, and the collapse from a spectrum to three numbers is the abstraction all of this is a correction to.