Either disc can be the wide one

The instruction is in every standard that deals with translucent material, and it is always given as a rule of thumb — light more of the sample than is being measured. The arithmetic behind it says something the rule does not, which is that the reverse instruction works exactly as well.

The claim #

The aperture error is a product of two apertures, and widening either of them to infinity makes it exactly zero.

• A measurement has two discs: the one that is lit, and the one that is looked at. The usual account treats only the first.
• With both at two millimetres, a translucent slab reads 51.75 per cent of its own reflectance.
• With the lit disc widened to four hundred, it reads 100.00 per cent.
• With the measured disc widened instead, and the lit one left at two, it also reads 100.00 per cent.
• And the symmetry is not a coincidence. It is the same reciprocity that makes an integrating sphere exact, in a different variable.

Where the symmetry comes from #

The kernel says how much of the light entering at one point comes back out at each distance. What an instrument collects is that kernel, weighted by how much of the lit region lies at each distance from each point of the measured region, and divided by what a perfect diffuser would give in the same arrangement.

Written out, the weight at separation r is the area the two discs share when their centres are r apart. That expression is symmetric in the two radii — the intersection of a disc of radius a with one of radius b does not care which is which — and the reference is the area they share at zero separation, which is the smaller of the two.

So make either radius large. If the lit disc is enormous, every point of the measured disc sees the whole plane’s worth of returning light, and the reading is the kernel’s total. If the measured disc is enormous, every photon that comes back anywhere is collected, and the reading is again the kernel’s total. The two limits are the same limit approached from opposite sides.

Numerically, on a slab with a four-millimetre diffusion length: both discs at two millimetres gives 51.75 per cent; a four-hundred-millimetre lit disc with a two-millimetre measured one gives 100.00; a two-millimetre lit disc with a four-hundred-millimetre measured one gives 100.00.

Why the usual explanation is not wrong #

The standard account — that light leaks out of the illuminated spot and is lost — describes the first case correctly. Light does leak, and widening the illuminated area does mean that light leaking out of the middle is replaced by light leaking in from the edges.

What it does not describe is the second case, where nothing about the leaking has changed and the reading is still exact. There the fix is that the detector is collecting the leaked light rather than the sample being given more of it.

This matters practically because the two arrangements are not equally available. An instrument’s illuminated area is set by its optics and is usually the harder of the two to change; its measured area is set by a port and can often be opened. An operator who knows only the first version of the rule will conclude that a sample cannot be measured properly with the instrument to hand, when in fact opening the detector’s field of view would do it.

It also matters because it identifies which of the two factors is being killed. Either factor being zero makes a departure zero, and the aperture’s factor here is the shortfall of the weight against one. Both instructions set that shortfall to zero; they simply do it with different hardware.

The two discs an instrument actually has #

It is worth being concrete about the hardware, because the symmetry is easy to state and easy to mistake for a formality.

A hand-held spectrophotometer has a lamp — often a ring of them, or a sphere — that illuminates a patch of the sample, and an optical path that collects light from a patch defined by a port and a lens. In the common designs the two patches are nominally the same size, which is exactly the arrangement the arithmetic says is worst: with both discs at the same radius the shared-area weight falls fastest with separation, so the shortfall is largest for a given amount of glass.

Instruments intended for translucent work already break that symmetry, and they break it in the direction the standards recommend: the illuminated patch is a few millimetres larger than the measured one. The arithmetic here says the other asymmetry would serve as well, and that is not idle — a bench instrument with an integrating sphere illuminates the sample through the sphere, which is to say over an area much larger than any port, and therefore satisfies the condition automatically by a route nobody describes this way.

That gives a small and testable prediction. A sphere instrument and a 45°/0° instrument measuring the same translucent sample should disagree by more than their geometries alone account for, because the sphere is also satisfying the aperture condition and the directional instrument is not. The two geometries already disagree by eight ΔE₀₀ on a dark gloss sample; on a translucent one the disagreement should carry a second term with a completely different cause.

What the standards actually say #

ASTM E2214 and the CIE’s publications on translucent materials both give the instruction, and both give it as guidance rather than as a quantity: use the largest aperture the specimen permits, illuminate an area larger than the measured one, and state the backing.

That is three instructions and they correspond exactly to three separate departures. The first addresses the size of the measured disc. The second addresses the size of the lit disc. The third addresses the finite thickness of the sample, which is a fourth term this collection does not model.

None of the three is given a number, and there is a good reason: the number depends on the sample. A rule of thumb that would be correct for coated paper — where a four-millimetre radius already recovers 98.1 per cent — is badly wrong for a pigmented plastic at 91.6 per cent and useless for marble at 63.2, and the loss is not even flat across the band. A rule with no sample in it cannot have a number in it, which is why the standards are written as advice, and why the honest replacement is a measurement rather than a better rule.

What was computed, and how #

The two-aperture reading is one radial integral: the kernel against the shared area of two discs, over separations from a fraction of a micron to the sum of the two radii, on a logarithmic grid.

Three properties of it are asserted rather than assumed. The reading is below the kernel’s total for every finite pair, because the weight is at most one everywhere and the kernel is positive. It rises monotonically as either radius grows. And the two wide limits agree with the total to within two per cent at four hundred millimetres, which is where the assertion is set — the residual there is the kernel’s own tail rather than an error in the arrangement.

The reference deserves its own sentence. A reflectance factor is a ratio to what a perfect diffuser would return under the same illumination, and a perfect diffuser’s kernel is concentrated at the point of entry. Under this arrangement its reading is the area the two discs share at zero separation, which is the smaller of them — so a measured disc larger than the lit one does not dilute the reading with unlit surroundings, because the reference is diluted identically. Getting that reference wrong is the easiest way to produce a plausible and meaningless number here, and it is checked by requiring an opaque sample to read exactly one at every aperture pair.

How wide is wide enough #

Widening a disc to infinity is exact and unbuildable, so the practical question is how far is far enough.

Bisecting for the radius at which a measurement recovers 99 per cent of a sample’s own reflectance:

A range of forty-four to one, and only the first is inside what an instrument is actually built with. A four-millimetre aperture is adequate for coated paper and a factor of eighty short for wax.

Those six numbers have one number behind them. Divided by each material’s own edge width — the distance over which its reading climbs from a tenth to nine tenths across a lit boundary — they come out at 35.7, 33.1, 32.5, 32.6, 32.3 and 32.9.

An aperture has to be about thirty-three edge widths across, to within five per cent, over materials whose edge widths span a factor of forty-eight and whose required apertures span forty-four.

That is the useful form of the rule, because an edge width can be measured from a photograph and an aperture requirement cannot. Photograph the sample across a shadow boundary, measure the distance over which the reading settles, multiply by thirty-three, and the result is the aperture that would read the material to within one per cent. For coated paper that is eight millimetres and every instrument is comfortable. For anything light passes through it is a number no instrument has.

Where the model stops #

The symmetry is exact for the model and rests on two of its properties.

The kernel is radially symmetric. For a fibrous or brushed material it is not, and the shared-area weight would then depend on the orientation of the two apertures as well as their sizes. The symmetry between lighting and looking would survive — it comes from the convolution rather than from the symmetry of the kernel — but the numbers would not.

The kernel is also assumed not to reach the edge of the specimen, which is the same assumption as the semi-infinite slab wearing different clothes, and it is why a translucent object beside a shadow has no reflectance at a point.

And the illumination is uniform across the lit disc. Real instruments have a beam profile that falls off at the edges, which is a soft aperture rather than a hard one, and a soft aperture recovers slightly less at a given nominal size. That is a correction to the numbers rather than to the structure.

The third limit is the one that would actually bite in a laboratory: the sample has to be larger than the lit disc. Widening the illumination on a specimen the size of a coin runs out of specimen, and then the arrangement is no longer the one modelled here — it is the edge problem, with the sample’s own boundary playing the part of the shadow.

The generalisation #

There is a transferable rule here about instructions that are stated asymmetrically.

An instruction of the form make A large is often a disguised statement about a product A × B, with B fixed by the apparatus and therefore invisible. Whenever that is so, the instruction has a hidden twin — make B large — which is equally correct and usually unstated, because whoever wrote the instruction could not vary B.

Colour measurement is full of instructions of that shape. Warm the lamp up is about a product of the lamp’s drift and the measurement’s duration, and the lamp’s half of it has been measured here. Use a large field of view is about a product of the observer’s field size and the sample’s angular extent, and the observer’s half is two standard functions. In both cases one factor belongs to the apparatus and one to the world, and the instruction names whichever the writer could change.

Finding the twin is worth the effort for two reasons. It is sometimes the cheaper of the two, as it is here. And it identifies the actual quantity, which lets somebody say how large is large enough: not as large as possible, but large enough that the shared-area weight is within a stated tolerance of one, which is a number a sample and a tolerance decide between them.

This collection has met the same shape in the fluorescent departure, where the instruction specify the lamp’s ultraviolet has the twin use a sample with no brightener in it — and the twin is useless there, because the sample is the thing being measured rather than a choice. The twin being useless is itself information: it says the factor belongs to the world rather than to the apparatus.

What the aperture pair is choosing between is visible as a colour and as a limit, and both are worth having beside the arithmetic.

What a reader can check #

The symmetry is visible with a phone torch and a piece of white plastic — a chopping board, a lampshade, a bar of soap.

Press the torch flat against the plastic and look at the lit spot from the other side: the bright patch is much larger than the torch’s opening, and its edges are the kernel. Now cover most of the torch with a card so that only a small hole is lit, and the patch is smaller and dimmer but still much wider than the hole.

The first arrangement is a wide lit disc and a narrow looked-at one; the second is the reverse if the eye is fixed on a small area. In both, what reaches the eye is the same convolution, and in both the total light is conserved. Nothing about the material has changed between them, and the only thing that changed is which end of the pair was widened.

The demonstration also shows why the effect is invisible on ordinary opaque things. A sheet of coated paper under the same torch has a bright spot with a sharp edge, because its kernel is a fifth of a millimetre wide and the eye cannot resolve the difference between that and nothing.

Who found it, and when #

The practical rule is old and comes from industry rather than from theory. Paper, plastics and textiles all arrived at it independently in the middle of the twentieth century, in each case after inter-laboratory comparisons that disagreed by more than anybody could explain.

The theory it corresponds to is older still: the shared-area weight is a two-dimensional autocorrelation, and the observation that the measured value is a convolution of the kernel with the instrument’s aperture is the same statement optics makes about a point-spread function and a detector. What is unusual in this case is that the sample supplies the point-spread function and the instrument supplies the detector, which is the reverse of the arrangement in imaging — and that reversal is why a camera has the opposite version of this problem.

What it would take to state the rule properly #

A rule with a number in it is available and would say something like: the illuminated and measured radii shall be such that the shared-area weight exceeds 0.99 over the diffusion lengths of the specimen.

That is unusable as written, because the diffusion length is what nobody knows. But it can be turned round. Measure the specimen at two apertures; if the two readings agree to within the tolerance the specification is written in, the wider one is adequate and the rule is satisfied. If they do not, the specimen is translucent by the standard’s own criterion and needs a wider port or a different instrument.

That is a rule whose test is the same operation as its remedy, which is the property worth having. It also produces a number nobody currently records: the difference between the two readings is a measurement of the sample’s translucency, in the same unit as its colour, and it is exactly the quantity a specification would need to say whether two laboratories can be expected to agree. Today that number is discarded, because instruments report one reading per sample and there is no field for a second.

Where the ladder goes next #

The obvious use of the symmetry is not to satisfy it but to break it deliberately. Two readings at two different aperture pairs give two numbers per band, and the kernel has two parameters per band, so a laboratory with one instrument and two ports has enough information in principle to recover the sample’s transport as well as its colour.

Whether that recovery is stable is an entirely separate question, and this collection’s own history says to be careful: a matched count of measurements and unknowns is not the same as a well-conditioned inversion, and the difference is usually several orders of magnitude in the noise.