The eye weights where the light is not

A brightened sheet is described everywhere — in trade literature, in the word brightener itself — as being made brighter. The measurement says something narrower and more interesting.

Under a daylight source with its ultraviolet, a coated press stock returns 15 per cent more light than arrives at 430 nanometres. Its luminance, against the same sheet measured with the ultraviolet removed, is 0.31 per cent higher.

The same sheet under a lamp that supplies no excitation is the control for all of this, and it is a different object.

The claim #

A quantity of light means nothing until it is weighted, and the three weightings an eye applies are so unequal that where an effect sits decides what kind of effect it is. A brightener puts its light where the luminance weighting is a sixtieth of its peak and the short-wave weighting is at maximum, so it is a chromatic mechanism with a brightness name.

• The emission peaks at 435 nanometres. There ȳ is 0.0168 against 1.0 at 555 — a factor of 59 — while z̄ is 1.62, its own maximum region.
• So the emission lands almost entirely in Z. At 435 nanometres z̄ is 96 times ȳ, which is the largest ratio between two colour-matching functions anywhere in the visible band.
• The luminance gain is 0.31 per cent and the b* change is 7.4 units. One of those is invisible and the other is four times any tolerance a paper buyer would accept.
• The trick only works there. An emission band in the middle of the visible range would raise luminance and would also make the absorption band visible, so the sheet would gain brightness and lose colour.
• And “brighter than white” is therefore true of β_T and false of Y, which are two different questions the word brighter does not distinguish.

Three weightings, wildly unequal #

The collapse from a spectrum to three numbers is this collection’s founding fact, and it is normally discussed as a loss of information. It is also a reweighting, and the reweighting is severe.

The three colour-matching functions do not have the same shape, the same width or the same position. ȳ is by construction the photopic luminous efficiency function: a single hump peaking at 555 nanometres and falling to 0.0004 at 400. x̄ has two humps, a large one at 600 and a smaller one at 445. z̄ is a single narrow hump at 445 and is essentially zero above 560.

At the wavelength a brightener emits, those three take values 0.33, 0.017 and 1.62. A photon at 435 nanometres is worth ninety-six times as much to the Z channel as to the Y channel, and the imbalance is not an artefact of normalisation — the three functions are scaled so that equal-energy light gives equal tristimulus values, which is the condition that makes the comparison meaningful.

What the sheet actually gains #

The comparison that isolates the effect is the same sheet measured twice: once with the ultraviolet in the lamp and once with it removed. Everything else is held — the same substrate, the same geometry, the same observer — and the difference is the fluorescence and nothing else.

For a coated press stock the two measurements are

Y moves by 0.31 per cent and b* moves by 7.44 units. In CIELAB terms the lightness change is 0.11, which is a tenth of a just-noticeable difference and would be invisible in a side-by-side comparison; the blue-yellow change is more than seven, which is not far off the difference between a sheet of ordinary office paper and one that has been left in a window for a summer.

The share of the sheet’s luminance that was emitted rather than reflected is 0.36 per cent. On a heavily brightened sheet it reaches 0.44. So more than 99.5 per cent of what makes a brightened sheet bright is ordinary reflection, and the entire commercial value of the additive is in the last half of one per cent — spent where it buys colour rather than brightness.

The same arithmetic, at the other edge #

The mirror case is worth a sentence because it says the window is a window rather than a preference for the short wavelengths.

A material that absorbed in the near infrared and re-emitted in the deep red would be doing formally the same thing: taking light from a band the observer cannot see and returning it in one it can. Such materials exist — the up-converting phosphors used in security marking are the standing example, though they work by absorbing two photons rather than one — and nothing about them makes a sheet of paper read as whiter.

The reason is the same reason in reverse. At 680 nanometres ȳ is 0.017, almost exactly its value at 435, so the luminance gain would be as small. But the chromatic gain would be in the opposite direction: towards the long wavelengths, which is where paper’s own residual absorption already leaves it. A sheet emitting in the deep red would look yellower, which is the fault it is being sold as a cure for.

So the window is narrow at both ends and asymmetric in what it is worth. The eye’s blindness is symmetric — nothing is seen below 380 or above 780 — and the usefulness of borrowing from either side is not, because the direction the emission moves the chromaticity in is set by which side of the neutral point the substrate already sits on. Paper is yellow. That single fact decides which end of the spectrum a brightener has to work at, and the chemistry follows.

Bluer, and also slightly redder #

The table above holds one number that a reader expecting a pure blue shift would not predict: a* moves from 0.27 to 2.99. The sheet gets redder as well as bluer.

That is x̄’s second hump. The long-wave matching function is not a single peak; it has a smaller lobe at about 445 nanometres, where x̄ reaches 0.33 against its main peak of 1.06 at 600. So light at 435 raises X as well as Z, in the ratio 0.33 to 1.62 — about a fifth as much, and enough to move a* by nearly three units on a sample whose a* was otherwise a quarter of one.

The second lobe of x̄ is not the long-wavelength cone’s main sensitivity making a return. It is that cone’s own short-wave shoulder, plus the CIE’s requirement that all three functions come out non-negative — a constraint that has to put the shoulder somewhere rather than allow a negative excursion. So a brightened sheet reads slightly magenta partly because of a decision taken in 1931 about the sign of a table.

Which is why heavily brightened sheets are described as looking violet rather than blue. The chromaticity moves towards the short-wave end of the purple region, not straight down the blue-yellow axis, and the two coordinates move together in a fixed ratio decided entirely by where the emission band sits.

Why the whiteness formula looks the way it does #

The CIE whiteness formula is

W = Y + 800(xₙ − x) + 1700(yₙ − y)

and the coefficients look arbitrary until this essay’s arithmetic is in hand. Y arrives with a coefficient of one. The chromaticity departures arrive with coefficients of 800 and 1700.

That is not a preference for chroma over lightness; it is a unit conversion. Chromaticity coordinates run over a range of about 0.6 while Y runs over 100, so a coefficient of a thousand or so is what puts a visually comparable step in each term. What the formula is saying, once the units are normalised, is that a white sample’s perceived whiteness is roughly equally sensitive to its luminance factor and to how far it sits towards blue — which is a claim about human judgement, fitted to rankings, and it is why the formula rewards a brightener so heavily.

The two facts are the same fact from opposite ends. A brightener moves chromaticity and barely moves Y; the whiteness formula weights chromaticity heavily and Y lightly. Neither was designed against the other, and they meet at the point where a sheet gains forty points of whiteness for three tenths of a per cent of light.

Why the band has to be where it is #

The obvious improvement is to move the emission longer — to 500 or 550 nanometres, where ȳ is large — and gain real luminance. It fails for a reason that is exactly the Stokes shift.

The emission band is always to the long-wave side of the excitation band. Moving the emission to 550 means moving the excitation to something like 470, which is squarely in the visible: the sheet would then be absorbing blue-green light it can be seen absorbing, and would read as a strong yellow with a green glow on top. The absorption is a real loss at a wavelength the eye weights properly, and it would swamp the gain.

The constraint is therefore a window, and it is narrow. The excitation has to sit where the eye is blind, which after the lens’s own filter means below about 400 nanometres. The emission has to sit as far to the long-wave side of that as a Stokes shift can be pushed, which is 60 to 90 nanometres and lands at 430 to 460. That band is where every commercial brightener emits, and the reason is not chemistry preference — it is the only place the two constraints both hold.

The general lesson about weightings #

The three channels are so unequal that the same physical change produces qualitatively different reports depending on where in the spectrum it sits, and this is not a fact about brighteners.

A 1 per cent change in radiance at 555 nanometres is a 1 per cent change in luminance and almost no change in chromaticity, because the three functions are far apart in size there and Y dominates. The same 1 per cent at 435 is essentially a pure Z change: a chromatic shift with no luminance component at all. And the same 1 per cent at 380 or at 700 is nothing whatever — a change in a quantity the observer has no coefficient for.

So “how much light” is never the question. It is always “how much light, weighted how”, and the three answers available differ by two orders of magnitude across the band. Every argument on this site about what a metamer is, about which spectral structure survives, and about what a gain can remove is downstream of the same unevenness.

What was computed, and how #

The two measurements are a bispectral sample evaluated under the D50 source of a standard measurement condition, once with its ultraviolet and once with it cut at 400 nanometres. Both are integrated on the same grid, against the same 1931 functions, and normalised to their own lamp’s perfect diffuser — so the comparison holds everything but the excitation.

The luminescent share of luminance is computed by integrating the emitted radiance alone against ȳ and dividing by the total, which is the same decomposition the standards call β_L and β_S taken through the Y channel rather than left as spectra.

The values of the matching functions at 435 nanometres are read from the tabulated 1931 functions and not interpolated, because 435 is a grid point.

And the assertion that keeps the essay honest is a negative one. The radiance factor is required to exceed one over a band of at least twenty nanometres, and the luminance is required not to exceed the perfect diffuser’s. A version of the machinery that quietly produced a sheet brighter than white in the luminance sense would fail, because that is the claim this essay says is false and the figure would otherwise illustrate it.

Two more readings say how much of what the matrix carries survives being weighted, and what happens to it when the arithmetic stops too early.

Ninety-six is not the largest ratio, and the reason matters #

At 435 nanometres z̄ is 96 times ȳ, which is the largest ratio between two colour-matching functions anywhere in the visible band is checkable against the tabulated functions, and it is not the largest.

Reading the 1931 table on this collection’s own grid: at 445 the ratio is 60, at 435 it is 96, at 420 it is 161, and at 400 it is 171. The ratio rises monotonically towards the short-wave end, because ȳ collapses faster than z̄ does, and it goes on rising past every wavelength a brightener emits at. At the long-wave end it is unbounded in the other direction — z̄ is tabulated as zero above about 560, so x̄ over z̄ has no finite maximum at all.

So the emission is not sitting where the ratio is largest, and a design that chased the ratio would be wrong. At 400 nanometres z̄ itself is 0.0678; at 435 it is 1.623 and at 445 it is 1.783, which is its peak. The purity of the signal keeps improving towards the violet and the size of it collapses, by a factor of twenty-four between 445 and 400.

The right way to state what the brightener is doing is therefore the other way round. It emits where z̄ is large, which is 430 to 460, and the ninety-six-fold ratio to ȳ is a consequence of z̄’s peak happening to sit where ȳ has nearly given out. The two facts are not independent — z̄ is narrow and short-wave precisely because the short-wave cone is — but the objective is the numerator and the ratio is the report.

That also tightens the window argument. The Stokes shift puts the emission 60 to 90 nanometres above an excitation that must sit below 400, which lands it at 430 to 460; and z̄’s own peak is at 445, in the middle of that interval. The two constraints do not merely both hold in that band, they agree about where in it to sit, which is why every commercial brightener lands within fifteen nanometres of the same wavelength.

The two lamps do not separate all of the fluorescence #

The measurement is described as isolating the effect exactly — everything else is held, and the difference is the fluorescence and nothing else — and the essay’s own two numbers say it is 86 per cent of it.

The Y difference between the two conditions is 90.08 against 89.80, a rise of 0.311 per cent. The luminescent share of the sheet’s luminance, computed from the emitted radiance directly, is quoted at 0.36 per cent. Those are two measurements of the same thing and they differ by 0.05 percentage points, which is 14 per cent of the total fluorescence.

The gap is not an inconsistency; it is what a 400-nanometre cut leaves behind. The fluorophore’s excitation band has a tail reaching into the violet, so a lamp with its ultraviolet removed still excites some of it, and the ultraviolet-cut condition is a sheet with most of its fluorescence gone rather than all. The M1/M2 pair measures the ultraviolet-excited fluorescence, which is the quantity a measurement condition is defined to isolate and is not the same as the fluorescence.

That matters for one sentence of the argument and not for the rest. More than 99.5 per cent of what makes a brightened sheet bright is ordinary reflection uses the larger figure and is safe either way. The comparison table’s b* shift of 7.44 units is likewise a difference between two conditions rather than a total, so the sheet’s full chromatic debt to its brightener is a little larger than the table shows — by about a seventh, if the residual is chromatically like the rest of the emission.

What checks exactly #

Two of the arithmetic chains reproduce from the published numbers, which is worth recording beside the two that need restating.

The lightness pair follows from the luminance pair. Y of 90.08 and 89.80 give L* of 96.03 and 95.91 against the essay’s 96.03 and 95.92, so the normalisation really is to the lamp’s own perfect diffuser and the 0.11-unit lightness change is the cube-root compression of a 0.31 per cent luminance change, not an independent measurement.

And the mirror case at the far end holds to a per cent. ȳ at 680 nanometres is 0.0170 against 0.0168 at 435 — almost exactly its value is right to 1.2 per cent, which is closer than the sentence claims. A deep-red up-converter really would buy the same negligible luminance, and the asymmetry the essay draws from it is about direction alone.

Where the model stops #

Luminance is not brightness and this essay has been careful to say luminance. The Helmholtz–Kohlrausch effect means a chromatic stimulus looks brighter than an achromatic one of the same luminance, and a brightened sheet is more chromatic than an unbrightened one — so its apparent brightness gain is larger than 0.31 per cent, by an amount an appearance model rather than a colorimetric one has to compute.

The numbers are for one stated fluorophore. A brightener emitting at 460 rather than 435 would land where ȳ is 0.06 rather than 0.017, and the luminance gain would be about three times larger and still under one per cent.

And the 1931 functions are one observer. At 435 nanometres the disagreement between the 1931 and 1964 functions is proportionally larger than anywhere else in the band, so the exact factor of 96 between z̄ and ȳ is a property of which table is used.

Who found it, and when #

The luminous efficiency function was standardised in 1924, seven years before the colour-matching functions, and ȳ was made equal to it by construction — which is why luminance falls out of the Y channel and why this essay’s central number is a value in a ninety-year-old table.

That fluorescent whitening is a chromatic effect rather than a luminance one has been understood in the paper trade since the additives arrived, though it is usually stated the other way round: the trade knows that a brightener makes a sheet look cleaner rather than brighter, and that over-brightening produces a visible blue cast rather than glare. Those are the same observation without the arithmetic.

The arithmetic is not itself new; every whiteness formula encodes it. What is worth putting a number to is the ratio — 0.31 per cent of luminance against 7.4 units of b* — because it is the size of the mismatch between what the mechanism is called and what it does.

Where the ladder goes next #

If most of what a whiteness measurement reports is chromatic and the chromatic part is produced by the lamp’s ultraviolet, then most of the whiteness scale is the lamp rather than the sheet — which is a measurable statement and comes out at 82 per cent.

The other direction is the appearance question this essay deliberately did not answer. A sheet that gains no luminance and seven units of blue is being judged by an observer with a chromatic adaptation state, and what a brightened white looks like in a room that has adapted to it is a question colorimetry cannot reach.