Some paper is brighter than white

Every object colour on this site is a reflectance multiplied by an illuminant, wavelength by wavelength, and then integrated. That model has an assumption buried so deep in it that it is almost never stated: light leaving the surface at 440 nm arrived at the surface at 440 nm.

For most materials that is exactly true. For the paper this sentence would be printed on, it is not.

Which two lamps are chosen decides how far apart the two curves come out, and never whether there are two of them.

A reference that nobody has ever built is the clean case, because whatever the sheet does under it belongs to the sheet rather than to somebody’s lamp.

Two more pairs make the same point with a reference that is not a lamp at all, which is where the arithmetic is cleanest.

What fluorescence does to the model #

A fluorescent material absorbs at one wavelength and re-emits at a longer one, which the three-number collapse has no way to record. The energy arrives in the ultraviolet or the violet and leaves in the blue, so the light departing at 440 nm was never at 440 nm on the way in, and how much of it there is depends on how much ultraviolet the illuminant had — a quantity the reflectance curve does not mention and cannot.

The consequence is that a fluorescent sample has no reflectance. Reflectance is a fraction: the ratio of what leaves to what arrives, at each wavelength independently. A fluorescer routes energy between wavelengths, so the ratio at 440 nm depends on what arrived at 380, and a function of one wavelength cannot express that.

What such a sample has instead is a matrix: how much leaves at each wavelength for light arriving at each wavelength. Donaldson measured such matrices in 1954, and a bispectral measurement — an instrument that illuminates at one wavelength and measures at another, across the whole grid — is what a fluorescent sample actually requires. It is slow, the instruments are expensive, and almost nobody does it.

Reflectance above one #

The most direct symptom is a quantity that cannot exist. Measure a brightened sample against a perfect diffuser under daylight, and the ratio comes out above 1: more light leaves at those wavelengths than arrives at them.

This is not a violation of energy conservation, and the accounting is worth doing. Energy arrives across the whole spectrum including the ultraviolet, and leaves in the visible. Integrated over everything the sample receives and emits, less leaves than arrives — some is lost as heat, and the emitted photons are longer-wavelength and therefore individually less energetic than the ones absorbed. What is violated is the per-wavelength accounting, which is the only accounting the reflectance model does.

The quantity that is being reported when a number above 1 appears is properly called a radiance factor, and the distinction between it and reflectance is precisely the one this essay is about. Software that stores it in a field called “reflectance” and clamps it to 1 — which a great deal of software does — has thrown away the effect and kept a plausible number.

This is deliberate, and it is everywhere #

Optical brightening agents are added to paper, textiles, and detergent on purpose. Roughly all white office paper and most white fabric is treated, which means the display’s white and the page’s white are unlike each other in one more way.

The reason is that the alternative is worse. Paper pulp is yellowish; bleaching it further is chemically aggressive and weakens the fibre. Adding a fluorescer that absorbs ultraviolet and emits blue puts back exactly the short-wavelength light the yellowing removed, so the sheet reads as neutral — and slightly more than neutral, because it returns more blue than it received.

“Whiter than white” is a laundry advertisement and also a literal description of the mechanism. A fluorescer makes a fabric reflect more visible light than falls on it, which no amount of bleaching can do.

The commercial consequence is that paper white is a standing argument in printing. A proof made on brightened stock and viewed under a lamp with ultraviolet looks different from the same proof under a lamp without, and neither is wrong. Standard viewing booths specify their ultraviolet content for exactly this reason, and specify it separately from everything else about the illuminant.

The failure in the form it actually takes #

Nobody looks at a sheet of paper and concludes its reflectance exceeds unity. What happens is quieter and considerably more expensive.

A sample is measured on an instrument under one light. The measurement is stored as a reflectance curve. Somebody later uses that curve to predict the colour under a different light — which is exactly what a reflectance curve is for, and exactly what this one cannot do.

Over five ΔE is many times any tolerance a supplier would be held to. And nothing about the stored curve says it cannot be carried between illuminants — it is a column of numbers in a file, indistinguishable in form from a curve for an ordinary pigment.

Why the effect here is understated #

The numbers on this page are a floor rather than an estimate, and the reason is a limit of the site rather than of the physics.

A commercial optical brightener absorbs between roughly 340 and 380 nm. Every spectrum on this site runs from 380 to 780, so only the tail of that excitation band is visible to the calculation and the rest is simply missing. What survives is daylight lifting the sample’s peak apparent reflectance about 15% above where tungsten leaves it — and rather more than that in the part of it that is actually fluorescence, which the next section separates out. With the ultraviolet included it would be substantially more again.

The truncation is kept rather than fixed, and declared rather than quietly worked around. Widening the range would make every spectrum on the site a different length, and the fix for one essay would touch the observer functions, the illuminants, the metamer construction and every figure downstream. Stating the limit costs a paragraph; changing the range costs the site’s consistency.

The lift is larger than the peak makes it look #

That comparison is between two peaks, and it flatters the tungsten lamp for the reason a ratio of two varying quantities usually does.

Under D65 the apparent reflectance peaks at 1.4515; under illuminant A, at 1.2650. That is the fifteen per cent. But 0.82 of each is the sheet’s own base reflectance, which has nothing to do with fluorescence, and the part that does is what is left: 0.632 against 0.445, which is 42 per cent more. Measured against the impossible part alone — how far each curve rises above one — it is 0.452 against 0.265, 70 per cent more. Three correct numbers answering three questions, of which the middle one is the question a printer is asking.

Two more say why even 42 per cent understates the physics. The band from 380 to 420 nanometres carries 9.49 per cent of D65’s power across this site’s range and 1.41 per cent of illuminant A’s, so daylight delivers five times the excitation in absolute terms. The lift differs by only 42 per cent because a radiance factor divides by the lamp, and tungsten’s own output at the emission peak is about 28 per cent of daylight’s. A lamp that is weak in the blue makes a small fluorescent addition look large, which is the same denominator that made the peak comparison look small. Nothing cancels here; one quantity is being reported three ways.

The energy accounting, done #

The paragraph above appealed to the ultraviolet outside this site’s range to keep the books balanced. It does not need to.

Integrated over 380 to 780 nanometres alone, the sample returns 0.9007 of the energy falling on it under D65, 0.8686 under D50 and 0.8320 under illuminant A. So the apparent reflectance peaks at 1.45 and the sample is still a net absorber by a tenth, on the site’s own grid, with no appeal to a band the grid cannot see. Per-wavelength accounting and total accounting are different books, and only the first is broken.

The Stokes shift is in the model’s numbers too, and it is worth checking, because a caricature that got it wrong would draw an equally plausible curve. The energy-weighted mean of the excitation band under daylight is 402.9 nanometres and the emission peaks at 438, so a photon leaves carrying 0.9198 of the energy it arrived with. The model’s stated efficiency of 0.85 is an energy efficiency, which at that shift implies a quantum yield of 0.924 — roughly one photon out for every 1.08 absorbed. Commercial brighteners are quoted near 0.9. So the caricature is not only two bands in the right places; the number joining them is in the right place as well.

And the prediction error has a shape as clean as its size. Of the 5.38 units between what the sample does under illuminant A and what a curve measured under D65 predicts, 5.33 are in b*, 0.76 in a* and four hundredths in lightness. Ninety-nine per cent of the failure lies on the blue-yellow axis, which is the axis the brightener was added to move — so the error is not a general drift. It is the brightener’s own contribution, delivered under a lamp that never produced it.

Two things this makes visible about the ordinary case #

The interesting thing about a broken model is what it reveals about when the model works.

Reflectance is an approximation with a domain. It is exact for materials whose interaction with light is absorption and scattering at each wavelength independently. That covers most pigments, most dyes and most natural surfaces, which is why the approximation is good enough to be treated as a definition. It is not a definition.

The illuminant was always half the answer, and here it is more than half. For an ordinary surface, the illuminant and the reflectance multiply and the roles are symmetrical: change either and the colour changes. For a fluorescer the roles are not symmetrical, because the illuminant determines how much fluorescence there is as well as what it multiplies. The sample’s behaviour is a function of the light in a way that a reflectance cannot capture even in principle.

Paper white, and the argument it causes #

The commercial consequence has a name in printing, and it is one of the longest-running disagreements in the industry.

A proof and a press sheet have to match. Both are printed on paper, both papers are brightened, and the two are rarely brightened to the same degree — proofing stock and production stock are different products from different mills. So the substrate contributes a difference that is not in either the ink or the press, and the difference depends on how much ultraviolet the viewing booth emits.

Standard viewing conditions therefore specify ultraviolet content separately from everything else about the illuminant, which is unusual: nothing else in colour specification carries a separate term for one part of the spectrum. It is there because that part of the spectrum is not being reflected, it is being converted, and the conversion is not in any reflectance curve.

The move to LED lighting has made this sharply worse. A phosphor-converted white LED has essentially no ultraviolet and very little violet, so brightened paper under one fluoresces far less than under daylight — the same sheet, visibly different, for a reason that is not in its reflectance curve and not in the lamp’s colour temperature. Print shops that changed their lighting found their proofs stopped matching, and the cause was in neither the proof nor the press.

What a colour management system does about it #

Nothing, mostly, and understanding why is worth a paragraph.

An ICC profile characterises a device by measuring patches and storing the relationship between values sent and colours produced. If the substrate fluoresces, the measurement captures the fluorescence as it was under the measuring instrument’s lamp — and the profile then carries that lamp’s ultraviolet content as an invisible assumption baked into every entry.

The profile is not wrong. It is correct for the illuminant it was measured under, and it says nothing about which one that was. A profile measured on an instrument with a UV-included lamp and one measured with a UV-cut filter, on the same brightened stock, are different profiles for the same paper — and instrument manufacturers ship both, precisely because the choice matters and neither is right.

That is the same structural problem as the reflectance that cannot be stored, arriving one layer up. A colour management pipeline assumes device behaviour is a fixed relationship, and for a fluorescent substrate it is a relationship that depends on the light.

What was computed here #

The fluorescer is a two-band caricature of a Donaldson matrix: absorb across a stated excitation band, re-emit into a stated emission band at a stated quantum efficiency. It is not a measured material and is stated not to be.

Three assertions run in the gate.

The apparent reflectance must exceed 1. Written as the violation rather than as a tolerance, because the violation is the finding. If it did not exceed 1, nothing would be being re-emitted and the essay would have no subject.

It must depend on the illuminant. The same sample under daylight and under tungsten must differ by a stated margin — because the whole claim is that a fluorescer has no illuminant-independent reflectance, and an implementation that had accidentally made the emission a fixed addition rather than a function of what was absorbed would pass the first check and fail this one.

And the prediction must fail. Measure under D65, store the apparent curve, predict under illuminant A, compare with a direct calculation. The gap must exceed a supply tolerance. This is the assertion that matches what actually goes wrong in practice, and it is the one worth having: the first two establish that something odd is happening, and only the third establishes that it costs anything.

Where the model stops #

The caricature has two bands. A real Donaldson matrix is a full grid, with excitation and emission overlapping in complicated ways and multiple fluorophores in a real sample. Nothing here would predict a specific brightened paper’s behaviour.

There is also no re-absorption. Light emitted by a fluorescer inside a scattering medium can be absorbed again before it leaves, which real paper does and this model does not.

And phosphorescence — the same mechanism with a long delay — is absent entirely. For steady illumination the distinction does not matter; for anything strobed or pulsed, including a great many LED sources, it does.

What the pictures cannot show #

The hero figure plots apparent reflectance and marks the impossible region above 1. It cannot show the reader a sample that is brighter than white, and the reason is the one this site keeps arriving at.

A swatch on this page is produced by the reader’s display, and the display’s maximum is white. A brightened sheet under daylight returns more blue than a perfect diffuser would, and reproducing that on a screen would require a pixel brighter than the screen’s white — which is the definition of what the screen cannot do.

So the effect is legible on this page only as a curve crossing a line. The demonstration requires a sheet of office paper and a source with some ultraviolet in it, and it is worth doing: the paper looks obviously, startlingly blue-white beside an unbrightened sheet, and the difference is much larger than the plotted curves suggest.

Why it is not simply banned #

Given the measurement trouble, a reasonable question is why brighteners are not just left out of anything that has to be colour-managed.

Some stocks do leave them out, and they are sold at a premium as such. The reason they are the exception is that the alternative is a yellower sheet, and a yellower sheet reduces the gamut of everything printed on it. Every ink is a subtractive filter over the substrate, so the substrate’s own colour is the ceiling: a print on a slightly yellow sheet cannot reach the blues a print on a brightened one can, and cannot reach the same white either.

That is the gamut argument in its most physical form. The brightener genuinely expands what the medium can show, at the cost of making the medium’s behaviour depend on the light it is viewed under, and both halves of that trade are real.

The equivalent trade appears in textiles, where a brightened white fabric is whiter than any unbrightened one can be, and in detergent, where the brightener is deposited on the fabric by the wash and accumulates. In every case the effect is genuinely desirable and genuinely breaks the measurement model, and the industries have settled on managing the consequence rather than giving up the effect.

Who found it, and when #

Stokes described fluorescence in 1852 and gave the rule that emission is at longer wavelengths than absorption, which is why the shift carries his name. The mechanism — absorption to an excited state, non-radiative loss, emission from a lower state — is why the rule holds and why the effect cannot run backwards without a second photon.

Optical brightening agents were introduced commercially in the 1930s and became near-universal in paper and detergent within two decades. The colour-measurement problem arrived immediately afterwards and has never been fully solved in routine practice.

Donaldson published the bispectral measurement method in 1954, and the matrix is named after him. The instruments to do it properly exist, are expensive, and are found in standards laboratories rather than in the places where colour disputes actually happen — which is why a reflectance curve for a brightened sample is still the normal thing to find in a file, with nothing in the file to say it should not be trusted.

Where this goes next #

The lamps whose ultraviolet content decides how much of this happens are a lamp is not a blackbody. The instruments that have to measure it, and what they report instead, are what the instrument reports. And the model this breaks — colour as reflectance times illuminant — is the illuminant is half the answer.