An instrument brings its own light

A spectrophotometer has a lamp in it. That is so obvious as to be beneath mention, and for a reflecting sample it is: the instrument divides by its own lamp and reports a ratio, and any lamp with power everywhere gives the same answer.

For a fluorescent sample it does not divide out, and the standard that says so names four different lamps rather than one.

The claim #

The four measurement conditions of ISO 13655 are not four levels of rigour. They are four questions, and a specification that does not name one of them has not finished being written.

• M₀ is an incandescent lamp with unspecified ultraviolet, so two M₀ instruments need not agree with each other. It is the legacy condition and a great deal of installed equipment is still it.
• M₁ is D50 including its ultraviolet content, specified. It is the only one of the four that measures a sample the way a person under daylight would see it.
• M₂ excludes the ultraviolet with a cut filter below 400 nanometres, so the sample is measured not fluorescing. That is a deliberate choice and it is the right one for some purposes.
• M₃ is M₂ with cross-polarisation, which removes the interface reflection — a change of geometry rather than of spectrum, and the only one of the four differences that shows up on an unbrightened sheet.
• On a coated press stock M₁ and M₂ are ΔE00 6.98 apart, and on a heavily brightened one 7.66. On the unbrightened control, 0.01.

What the four actually specify #

Two independent things vary across the four and it is worth separating them before reading any number.

The spectrum of the source. M₀ is “incandescent”, which in practice means a tungsten lamp of whatever colour temperature the instrument’s designer chose, with whatever ultraviolet it happens to have — the standard does not pin it. M₁ pins the source to D50 including the ultraviolet part of the D50 definition, which is the whole point of the condition. M₂ takes D50 and cuts everything below 400 nanometres.

The polarisation. M₀, M₁ and M₂ say nothing about it. M₃ adds crossed polarisers before and after the sample, which extinguishes the specular interface reflection — the four per cent of light that bounces off the surface without ever meeting a pigment.

So the four are two axes with three points on one and two on the other, and only four of the six combinations are named. That asymmetry is the standard admitting what it is for: M₃ exists because the printing industry wanted a way to read wet ink without the gloss, and the ultraviolet question and the gloss question were solved in the same document because they arrived in the same decade.

The control row, and what it establishes #

Every number in this essay is read against the top row of the census, which is a sheet of the same base with no brightener in it.

For that sheet the four conditions give ΔE00 0.27 between M₀ and M₁, 0.26 between M₀ and M₂, and 0.009 between M₁ and M₂. The last of those is the important one: with no fluorophore present, adding or removing the ultraviolet changes nothing, because the sample has no way to use it. Nine thousandths of a unit is the numerical noise of a colour-difference formula.

M₃ is the exception and its difference is a real one. Against M₂ it is 1.13 apart, and the mechanism is entirely in the luminance: Y falls from 85.65 to 81.41, a drop of 4.95 per cent, which is the interface reflection being extinguished. That is the pedestal the measuring geometry found, taken out by an optical trick instead of by an angle, and it is present on every sample whether or not anything fluoresces.

So the census separates cleanly. Differences involving M₃ are about the surface. Differences between M₀, M₁ and M₂ are about the ultraviolet, and they are zero unless the sample is fluorescent.

The pedestal is a subtraction, not a percentage #

The 4.95 per cent is the accidental form of that number. Measured across all six stocks, the drop in Y from M₂ to M₃ is 4.2346 units on every one of them, identical to five figures — on the unbrightened control, on the coated press stock, and on the shirt whose base is the darkest of the six. The percentage runs from 4.71 to 5.18 only because the denominators do.

That is what an interface reflection is. The four per cent that bounces off the surface never meets a pigment, so it carries no information about the sample and is scaled by nothing the sample does; it is a constant added to every reading a non-polarised condition takes. Removing it is a subtraction, and a subtraction reported as a percentage of a varying quantity looks like a variable when it is not.

The same measurement settles the asymmetry the condition is built on. Cross-polarisation is supposed to remove the reflected component and leave the emitted one alone, because a fluorescent molecule has forgotten which direction its photon arrived from. The luminescent contribution to Y under M₂ and under M₃ agrees to five decimal places on all six stocks — 0.05064 both times on the coated press stock, 0.08882 both times on the shirt. So the rise in the luminescent share that M₃ produces is entirely its denominator shrinking. The numerator is untouched, which is the assumption checked rather than assumed.

What the disagreement costs #

On the brightened rows the M₁-against-M₂ column runs 3.95, 6.07, 6.98, 7.66 and 7.60, rising with brightener loading and then falling slightly on the most heavily loaded sample because its base is darker.

Those are large numbers. A print tolerance on a critical colour is around 2; a paper specification’s tolerance on the sheet’s own colour is usually stated in whiteness points rather than ΔE, but the equivalents are in the same region. A single sheet measured by two instruments that are both working correctly and both traceable can differ by three to four times the tolerance the specification is written to.

The failure mode this produces in practice is not a disagreement — it is a silent agreement between people using the same instrument, and a disagreement that appears at the moment somebody upgrades. A press that has always used M₀ equipment and a paper supplier whose data sheets are M₁ have been quoting different numbers for the same stock for years, and both sets are right.

The ranking, which is what the number is for #

Nobody reads a whiteness or a paper colour on its own. It is read against another sheet’s, to decide which is whiter, and that makes the sharper question available: can the measurement condition change the order?

It can. Ranking the six stocks by CIE whiteness gives one order under M₁ and a different one under M₂: office paper and a laundered white shirt change places, and they change places between M₀ and M₂ as well.

The mechanism is simple once stated. The shirt has a heavier brightener loading and a darker base; the office paper has a lighter loading and a brighter base. Under a lamp with ultraviolet the loading dominates and the shirt wins; with the ultraviolet removed the loading contributes nothing and the base decides, so the paper wins.

A buyer choosing the whiter of two samples gets a different answer depending on which instrument the supplier used, with both instruments in calibration and both data sheets correct. That is the argument for naming the condition, and it is stronger than any argument from the size of the difference — a difference can be budgeted, and a reversal cannot.

How little of the sheet is doing this #

The disagreement is large and the thing producing it is very small, and the two are worth putting beside each other.

On the coated press stock the fluorescent emission is 0.362 per cent of the sheet’s luminance under M₁. That is the whole of what M₂ removes, and it is worth 6.98 ΔE00. Across the five brightened stocks the share runs 0.169, 0.293, 0.362, 0.439 and 0.471 per cent — under half a per cent everywhere, against differences of four to eight units.

The leverage can be stated exactly. Dimming the same sheet until it is 6.98 ΔE00 away from itself, with no change of hue, takes 27.0 per cent of its luminance. The brightener buys the same difference with 0.36 per cent, a factor of seventy-four, and the reason is that the emission lands at the blue end of the band, where the luminance response is nearly nothing and the chromatic response is not.

The decomposition says the same thing in three numbers. Between M₁ and M₂ the coated stock moves by 0.11 in lightness, 2.72 in a* and −7.45 in b*. Under two per cent of the difference is lightness; the rest is a shift along the blue-yellow axis, which is why the effect is invisible to a photometer and unmissable on a sheet of paper.

It is also why the disagreement cannot be calibrated away with a gain. A gain acts on luminance, and there is almost no luminance in the disagreement to act on.

Whether any of this is an artefact of the two-degree observer is a fair question, and the whole census can simply be recomputed against the ten-degree one.

What the ultraviolet does to the whole scale #

The reversal is the sharpest failure and it sits inside a larger one the ranking figure does not show: the ultraviolet is most of the spread.

Across the six stocks the CIE whiteness runs from 81.80 to 128.97 under M₁, a range of 47.2 points. Under M₂ the same six run from 81.83 to 93.61, a range of 11.8. Three-quarters of the scale that separates these sheets from one another disappears with the lamp’s short-wave power, and what is left has to distinguish six stocks in a quarter of the room.

That is the mechanism behind the reversal rather than a fact beside it. Compressing a scale by four moves every pair closer to a tie, and the pairs ranked by a small margin under one condition are the ones that can cross under another. The shirt and the office paper are 7.75 points apart under M₁ and 2.32 the other way under M₂. Under M₀ they are 0.07 points apart, which is not a ranking at all: a legacy instrument does not put those two sheets in an order so much as decline to.

A second pair is close behind. The shirt and the lightly brightened sheet are 20.8 points apart under M₁ and 0.54 apart under M₂, which is inside the repeatability of a real instrument — so on that pair the condition does not decide the order, it decides whether there is one.

Why M₂ is not simply the wrong answer #

The temptation is to conclude that M₁ is correct and M₂ is a legacy compromise, and that is not right either.

M₂ measures a repeatable quantity, in the sense an instrument’s own report can be. The reflected component is a property of the sample and nothing else; two M₂ instruments agree to their own precision, and an M₂ number today and the same number in five years differ only by whatever the sample has actually done. An M₁ number carries the instrument’s ultraviolet calibration, which is the hardest part of the instrument to keep stable and the part that drifts as the lamp ages.

And M₂ is the right question for some work. A textile dyer matching two dyed fabrics wants to know about the dyes, which sit on a substrate that is the white point everything is divided by; the brightener in it is a common factor that adds noise to the comparison and no information. A conservator assessing a print behind a filter is looking at a sample that is not fluorescing, so M₂ is not an approximation for them — it is the condition.

The honest statement is that the two conditions answer different questions and both questions are asked. What is wrong is not choosing M₂; it is quoting a number without saying which question it answers, and then comparing it to a number that answers the other.

The ranking survives the same substitution, which matters more than the census does because a ranking is what a buyer actually acts on.

What a workflow does with four answers #

The practical question is which condition to use, and the standards answer it in a way worth reading carefully because it is a division of labour rather than a recommendation.

M₁ is specified for exchange. A profile, a characterisation data set and an aim value are all things two parties have to agree on, and M₁ is the condition under which two instruments in different buildings agree with each other about a fluorescent substrate. That is what a standard is for.

M₂ survives inside a process. A press checking its own consistency hour by hour cares about repeatability rather than about agreeing with anybody, and the ultraviolet calibration is the part of an instrument that drifts. A shop that measures M₂ internally and quotes M₁ externally is not being inconsistent; it is using each condition for what it is good at.

M₃ is a press-side condition and rarely leaves. Its whole purpose is reading ink while it is still wet, which is a measurement nobody outside the pressroom needs.

And M₀ is a legacy that has to be named to be retired. Its numbers are not wrong, they are unspecified — two M₀ instruments can differ from one another as much as an M₀ and an M₁ do, and no amount of care on either side closes that.

The failure the four conditions were introduced to prevent is therefore not a measurement error at all. It is a silent disagreement: two parties both correct, both traceable, quoting numbers that were never comparable, in a column with the same heading.

What was computed, and how #

The four conditions are built as a source and a filter. M₀ is illuminant A on the wide grid, which is Planck’s law at 2856 K and therefore has a computed rather than assumed ultraviolet content. M₁ is D50 reconstructed from the CIE daylight basis, including the sixteen bands below 380 nanometres the basis is published for. M₂ is that source through a logistic cut edge at 400 nanometres with a four-nanometre width, which is a real sharp-cut filter rather than a step.

M₃ applies the same filter and additionally removes the Fresnel interface reflection from the reflected component only. That asymmetry is deliberate and physical: fluorescent emission is unpolarised by the time it leaves, because the molecule that emits it has forgotten which direction the photon arrived from. Cross-polarisation therefore changes the ratio between the two components rather than scaling the answer, and on a brightened sheet it raises the luminescent share.

Each sample is normalised to its own condition’s perfect diffuser, so what is compared is the sample rather than the exposure.

The assertion is a contrast rather than a level. The conditions must agree on the unbrightened sheet and disagree by at least five times as much on the worst brightened one. A version of the machinery in which all four conditions differed on everything would pass a threshold on the second half and fail the first, which is what makes the pair a test of the mechanism.

Where the model stops #

M₀ is modelled as illuminant A and the standard does not say that. It says incandescent. A real M₀ instrument might be at 2856 K or at 3200, with a glass envelope of unspecified transmittance; the point of the condition is that it is not specified, and this collection has had to pick something to compute with. The M₀ numbers here are therefore one instance of a family, and the family’s spread is the argument against the condition.

The polariser is ideal. Real crossed polarisers pass a little of the specular component and attenuate the rest of the light by more than half, which costs signal and therefore precision — a trade the standard mentions and this model does not.

And the substrate’s fluorescence is the only fluorescence. Real printing inks are not fluorescent but some pigments are, and a few spot colours are sold precisely because they are. Nothing here models an ink that glows.

Who found it, and when #

ISO 13655 in its 2009 revision introduced M₀, M₁ and M₂; M₃ came with it. The reason the revision was needed is documented in the standard’s own introduction and is exactly the story above: brightened substrates had become universal, instruments differed in their ultraviolet content, and press-side and supplier-side measurements of the same sheet did not agree — with the proof caught in the middle of the argument.

The underlying metrology is older. The CIE’s recommendations on measuring fluorescent specimens go back to the 1970s, and the two-monochromator method that settles the question outright is Donaldson’s from 1954. What the 2009 revision did was not solve the problem; it made the problem nameable, so that a data sheet could say which of four things it meant.

The choice of D50 rather than D65 for M₁ is a printing-industry decision and it has the side effect of understating the phenomenon relative to daylight: D50 carries about half the short-wave power D65 does, so a sheet’s M₁ whiteness is well below its outdoor whiteness. That is deliberate — the condition is meant to represent a viewing booth, not a window.

The generalisation #

The pattern is a measurement whose apparatus was a free variable for as long as the answer did not depend on it.

Every reflectance instrument has a lamp and nobody wrote down which, because for reflectances the lamp divides out. The moment a class of samples arrived for which it does not, the free variable became a specification variable — and the transition is invisible, because the instruments did not change and the numbers they print are in the same column with the same heading.

The diagnostic is to ask what the apparatus contributes that the sample cannot use. For a reflector, a lamp’s ultraviolet contributes nothing, so it is free. For a fluorophore it contributes directly, so it is a parameter. The rule generalises to any measurement with an excitation in it, and the failure it names is a parameter that was allowed to stay implicit for so long that its name had to be invented after the fact.

Where the ladder goes next #

If the condition can change the ranking, the obvious repair is to correct one condition’s measurements to the other’s — and the best possible 3×3 cannot do it, because the correction needed is proportional to how much paper is showing rather than to what the patch reflects.

The other direction is the number these measurements are made to produce. If the four conditions disagree about whiteness by tens of points, then most of the whiteness scale is the lamp rather than the sheet, which comes out at 82 per cent and makes a data sheet’s headline figure a joint property of two things.