The ultraviolet is half the product

Three of the four departures in this round are pairings whose value depends on how their two factors are aligned. One is not, and the exception has a cause worth knowing.

The claim #

The fluorescent departure is exactly proportional to the light in the fluorophore’s excitation band, and to nothing else about the lamp.

• Scaling the lamp below 400 nanometres by a factor scales the emitted light by the same factor, to twelve figures, over scales from zero to twice the real lamp.
• The relation is affine rather than proportional, and the intercept is 17.7 per cent — what the dye still takes from the visible part of its own excitation band when everything below 400 is cut.
• The reason is Kasha’s rule. A molecule relaxes to its lowest excited state before it emits, so the emission spectrum does not depend on where in the band the photon was absorbed.
• That makes the off-diagonal part of the response rank one: one spectrum belonging to the sample, times one number belonging to the light.
• And a rank one pairing has no alignment to vary, which is why this is the only departure whose size really is a product of two magnitudes.

What a fluorescent sample does #

A reflecting surface returns light at the wavelength it arrived at. A fluorescent one absorbs a photon at one wavelength and emits at a longer one, so its full description is a matrix rather than a curve: how much comes out at each wavelength, for light going in at each wavelength.

The matrix Donaldson measured in 1954 has a diagonal, which is the ordinary reflectance, and a lower triangle, which is everything the reflectance model cannot hold. For an optical brightener in paper the triangle is concentrated: absorption in a band around 358 nanometres, emission in a band around 435, and nothing anywhere else.

That concentration is the whole reason the departure is tractable. A general matrix has eighty-one times eighty-one entries and would need a bispectral instrument to fill. This one has a shape.

Kasha’s rule, and what it does to the algebra #

The shape is not an approximation for convenience. It is a consequence of how excited molecules behave, and it has a name.

A molecule that absorbs a photon is promoted to some vibrational level of an excited electronic state. Before it can emit, it loses the vibrational energy to its surroundings — a process some thousands of times faster than emission — and arrives at the lowest vibrational level of the lowest excited state. It emits from there. So the emission spectrum is the same whatever the absorbed photon’s energy was, which is Kasha’s rule, stated in 1950.

Written as a matrix, that says the off-diagonal part factorises: an excitation spectrum a(λᵢ) describing what the molecule takes, times an emission spectrum e(λₒ) describing what it gives back, with a quantum yield in between. The matrix is rank one.

The consequence for this round’s arithmetic is direct. Every other departure is an inner product whose value depends on how two functions line up, and the alignment is what refused the round’s prediction. Here the sample’s factor has only one direction, so there is nothing for the light to be aligned with or against: the pairing collapses to a single number, ⟨a, S⟩, multiplied by a fixed spectral shape.

The measurement that shows it #

Scaling only the part of a D50 lamp below 400 nanometres, and holding everything above it, gives the emitted photon count as a function of the scale factor. At scale 0, 0.25, 0.5, 1 and 2 the counts fall exactly on a straight line — the largest departure from it is under a part in a million million, which is the arithmetic’s own floor rather than a measurement.

That exactness is worth pausing on. It is not that the relation is approximately linear over a small range; it is linear over a range that includes zero and twice the real lamp, because emission is an integral of the lamp against a fixed spectrum and an integral is linear in its integrand. No other departure in the round has that property, and the reason is that no other departure’s sample factor is one-dimensional.

The intercept is the interesting number. At scale zero — the whole lamp below 400 nanometres removed — the sample is still emitting 17.7 per cent of what the full lamp excites, because the excitation band’s long tail reaches into the visible. A cut filter at 400 nanometres does not turn the fluorescence off; it turns about five sixths of it off.

Why the intercept makes M2 a condition rather than a control #

The graphic arts standards name four measurement conditions. M0 is a tungsten lamp with unspecified ultraviolet; M1 is D50 including its ultraviolet; M2 has a cut filter below 400 nanometres; M3 adds cross-polarisation.

The obvious reading of M2 is that it measures the sample not fluorescing, so that M1 minus M2 is the fluorescence and M2 is the sample’s true reflectance. The intercept says otherwise. Under M2 the dye is still excited, at about a sixth of its M1 rate, so M2 is a measurement of a partially-fluorescing sample rather than of a non-fluorescing one.

That is why the four conditions are four different quantities rather than four degrees of care. Each names a lamp; each therefore fixes the light-side factor at a different value; and none of them makes the factor zero. A sample with no dye in it makes the sample-side factor exactly zero, and that is the only condition in this round’s table that is an identity rather than a limit.

What the sample’s factor is, and why it saturates #

The light’s half of the product is the lamp’s power in the excitation band. The sample’s half is what the dye takes of it, and that half does not behave linearly at all.

Doubling the brightener loading does not double the emission, because the first molecules have already absorbed the photons the second ones would have taken. Beer’s law puts that in as an exponential — the absorbed fraction is one minus exp(−cd), which is nearly linear when the loading is small and flattens once the band is optically thick.

The consequence is visible across the stocks in this collection’s table. A lightly brightened sheet at a loading of 0.35 shows 3.95 ΔE₀₀ between the conditions with and without ultraviolet; at 0.8 it is 6.07; at 1.2 it is 6.98; at 2.0 it is 7.66; at 2.8 it is 7.60 — which is lower than the sheet with less dye in it. Adding brightener past a point buys nothing and eventually buys a cast, because the emission band is not white and piling more of it on moves the sheet off the neutral axis.

So the product form holds in the light and not in the sample. That is not a failure of the pairing — the pairing is about the departure at a given sample, and the sample’s factor is whatever it is — but it is a warning about reading the two halves symmetrically. One of them is a property of an instrument and scales; the other is a property of a material and saturates.

What it means for the whiteness a mill sells #

The commercial fact underneath all of this is that a paper mill sells whiteness, and whiteness measured under a lamp with ultraviolet is a different quantity from whiteness measured without.

The consequence, computed in this collection, is that most of a sheet’s whiteness is the lamp rather than the sheet. This essay adds the reason it is exactly proportional: the emission is linear in the lamp’s ultraviolet, so a sheet’s advantage over its competitor scales with whatever ultraviolet the buyer’s shop happens to have.

That makes brightener an unusual product. Its benefit is not a property of the thing sold; it is a property of the thing sold multiplied by a property of the buyer’s premises, and neither party normally measures the second. A shop lit by an LED with no ultraviolet at all receives none of what was paid for.

What was computed, and how #

The fluorophore is a two-band model: a Gaussian absorption centred at 358 nanometres with a width of 22, a Gaussian emission at 435 with a width of 25, a quantum yield of 0.85, and a loading that saturates — doubling the dye does not double the emission, because the first molecules have already taken the photons.

The emitted photon count is the lamp, times the substrate’s reflectance, times the fraction the dye absorbs, integrated in photon units across the whole wide grid from 300 nanometres. Photon units rather than power units, because a fluorophore counts photons: one absorbed photon gives at most one emitted photon, whatever its energy, and the energy difference is lost to heat.

The linearity test scales the lamp below 400 nanometres and refits the line through the endpoints, then measures how far the intermediate points sit from it. Nothing about the arrangement is tuned to produce linearity; it comes out because the integral is linear and the model’s saturation is in the loading, which is held fixed.

The measurement condition arithmetic uses this collection’s existing implementation, which computes each condition’s lamp — including the sharp-cut filter’s finite edge — rather than switching between tabulated spectra.

Two thirds of the intercept is the filter #

The intercept is attributed to one cause — the excitation band’s long tail reaches into the visible — and the stated model says the tail supplies about a third of it.

Taking the model’s own absorption band, a Gaussian at 358 nanometres of width 22, weighting it by D50 in photon units across the wide grid, and cutting perfectly sharply at 400: what survives is 5.8 per cent of the full excitation, not 17.7.

The rest is the filter’s own edge, which the implementation is explicit about including. Working backwards, reproducing 17.7 per cent needs a roll-off with a ten-to-ninety transition of about seventy nanometres — which is very soft for something called a sharp-cut filter, and softer than any real one.

Either way the conclusion about M2 gets stronger rather than weaker. If most of the intercept is the filter, then M2 is not merely a partially-fluorescing measurement — it is a measurement whose fluorescent content depends on which filter the instrument has. Two instruments both claiming M2, one with a 20-nanometre edge and one with 70, would leave the dye excited at 12 and 18 per cent of its M1 rate. That is a disagreement of the same kind as the M0 lamp’s unspecified ultraviolet, inside a condition defined to remove it.

And it makes the intercept the least portable number here. It is the figure the M2 argument leans on hardest, and it is the one that depends most on a component nobody has specified — a filter edge, in a model whose other parameters are all properties of the dye.

The saturation is exactly Beer’s law #

The five loadings are offered as evidence that the sample’s factor saturates, and they fit the exponential exactly.

ΔE = 7.75 × (1 − exp(−1.97 c)) reproduces the first four to a root-mean-square of 0.07 — 3.86 against 3.95, 6.15 against 6.07, 7.02 against 6.98, 7.60 against 7.66 — with two parameters fitted to four points.

Two numbers fall out that the table does not carry.

The ceiling is 7.75 ΔE₀₀. That is what the departure would be at infinite brightener loading, and it is the number a mill’s chemistry is asymptotically buying.

Half of it is reached at a loading of 0.35, which is the lightest stock in the collection’s own table. So the table’s first row is the half-saturation point and its last four are all in the flat part: 79, 91, 98 and 100 per cent of the ceiling.

The commercial reading is sharp. A coated printing paper at a loading of 1.2 is already at 91 per cent of what any amount of brightener could do, and doubling the dye to 2.4 buys eight points. The marginal return per unit loading is 15.3 ΔE₀₀ at zero and 1.4 at 1.2 — a factor of eleven — so the last increment of brightener in an ordinary sheet is worth a tenth of the first.

The cast is 0.12, and it is the only thing off the curve #

The fifth point is the one the essay flags: 7.60 at a loading of 2.8, lower than the 7.66 at 2.0.

Against the fitted curve the fall is small. Beer’s law predicts 7.72 at that loading, and the measurement is 7.60 — 0.12 below, which is one and a half per cent of the ceiling.

So adding brightener past a point buys nothing and eventually buys a cast is right about the direction and the cast is a tenth of a ΔE₀₀ rather than a visible penalty. What the top of the table actually shows is saturation with a small correction, not a turning point: the curve has flattened, the last two loadings differ by 0.12 in the fitted model and by 0.06 in the measurement, and both are inside what the rest of the table’s rounding permits.

That is worth saying because the turning point is the more memorable claim and the flattening is the larger effect. Between a loading of 1.2 and 2.8 the departure moves by 0.6 ΔE₀₀ in total, most of it up and a tenth of it back down, on a sheet carrying more than twice the dye. The reason to stop adding brightener is that it stops working, not that it starts hurting.

Where the model stops #

Three limits, and the first is the one that would matter most in a laboratory.

The dye is one band. Real brighteners are mixtures, often deliberately, and a second dye with a different excitation band would make the sample factor two-dimensional. The matrix would then be rank two, and the departure would acquire an alignment term like the other three.

Saturation is in the loading and not in the light. At high enough intensity a fluorophore’s excited state population saturates and the emission stops being linear in the lamp. That threshold is far above anything a spectrophotometer produces, and it is a real limit on the claim rather than a hypothetical one — it is why fluorescence measurements with laser sources need a different model.

And Kasha’s rule has exceptions. Azulene is the standard one, emitting from a higher excited state, and a few other molecules do the same. Nothing in a paper mill does, but the rule is a strong generalisation rather than a law.

The generalisation #

The transferable observation is about when a correction can be given as a single number.

A correction that is rank one can be published as one coefficient, because its shape is fixed and only its magnitude varies. A correction of higher rank cannot: it has a direction as well as a size, and reporting the size alone loses the part that decides the sign.

Colour measurement has both kinds and treats them the same way, which is where the trouble comes from. Fluorescence is rank one and is properly handled by a single specified condition. The room’s directional structure is not rank one, and specifying a single number about it — a gloss reading, say — cannot fix the departure, because two rooms with the same number can move a sample in opposite directions.

The test is available before any measurement is made. Ask whether the mechanism has one channel or many. One absorbing band and one emitting band is one channel. A hemisphere of directions is many. A kernel over a plane is many. And the number of channels decides whether a specification can be written as a number or has to be written as a procedure.

A building supplies the other half of the same factor, and unlike the sky it is a choice somebody made.

The last filter in the chain is the observer, whose optics close the band from the other side and whose table stops for that reason rather than for a tabulator’s.

Who found it, and when #

Donaldson’s matrix is from 1954 and was measured at the British Paper and Board Industry Research Association, which is exactly where the problem was: brighteners had arrived, and a paper’s whiteness now depended on the lamp it was measured under.

Kasha’s rule is from 1950, from photochemistry rather than from colour, and it is one of those results whose consequences in a neighbouring field are larger than anybody intended. Without it a fluorescent sample would need a full bispectral measurement to be characterised at all; with it, an excitation spectrum and an emission spectrum suffice, and those are what a fluorimeter reports.

The M-conditions are from ISO 13655, revised in 2009, and their arrival is a good illustration of how long a departure can be known before it is specified: fifty-five years after Donaldson, and about fifteen after brightened stock became universal in printing.

The one place the rank matters for a figure #

A rank one departure can be drawn as two curves; a higher-rank one cannot be drawn at all.

That is not a presentational nicety. The excitation spectrum and the emission spectrum, side by side, are a complete description of what this dye does — anybody with those two curves can compute the sample under any lamp. The equivalent for the directional index would be a function of four variables, and there is no arrangement of ink that shows one.

This collection’s figure rules have a version of that constraint in them: a figure has to say what it cannot show. For the fluorescent departure the answer is nothing — the two curves are the whole object. For the room the answer is three of the four dimensions, and every polar plot there is a slice with a caption saying so.

Rank is therefore the property that decides whether a departure can be published as a picture, as a number, or only as a procedure. It is worth checking early, because it determines what the eventual result can look like.

Where the ladder goes next #

The obvious extension is the second dye. A two-band brightener would make the sample factor rank two and would give this departure the alignment term the other three have — and it would then be possible to construct two lamps with the same total ultraviolet that excite the same sheet differently, which is a claim a laboratory could test in an afternoon.

The other open item is the one this collection has been unable to reach for seven rounds: nothing here is measured. The brightener is a two-band caricature with a stated shape, and while its structure follows from Kasha’s rule, its widths and its peak do not follow from anything. A real excitation spectrum would change the intercept, and the intercept is the number this essay leans on hardest.