A reflectance is a diagonal

The reflectance model is so nearly universal that its assumption is almost never written down. A surface is described by one number per wavelength — the fraction of what arrives at λ that leaves at λ — and every calculation of an object colour on this site multiplies that curve by an illuminant and integrates.

The assumption is that the two λs are the same λ. It is a strong statement about a physical process and it is true of almost every material anyone paints, prints or weaves. It is false of the one that is in nearly every sheet of white paper sold.

The claim #

The complete description of what a surface does to light is a square matrix, not a curve. A reflectance is that matrix’s diagonal, and it is a complete description only for materials whose off-diagonal entries are zero.

• The general object has two indices, one for the wavelength going in and one for the wavelength coming out. Donaldson measured such matrices in 1954 and the name stuck.
• A fluorophore’s block is rank one, not a smear along the diagonal, because everything it absorbs comes back in the same emission shape wherever in the excitation band it was absorbed. That is Kasha’s rule, and it is what makes the arithmetic tractable.
• The two components have separate names in the standards. β_S is what was reflected and β_L is what was emitted; their sum β_T is what an eye receives, and only the first is a property of the sample alone.
• The emission costs energy even at perfect efficiency. The emitted photon is longer in wavelength than the absorbed one, so a fluorophore with a quantum yield of exactly one returns 15.7 per cent less power than it took. Fluorescence is not free light.
• And an instrument reporting a reflectance is reporting the diagonal with the block folded into it, at a weight decided by its own lamp — which is the reason the number it prints is not a property of the sample.

Two indices instead of one #

Write the surface as an operator. Light arrives with a spectral distribution E(λ) and leaves with a distribution L(μ), and the most general linear relation between them is

L(μ) = ∫ D(μ, λ) E(λ) dλ

with D a function of two wavelengths. An ordinary reflector has D(μ, λ) = ρ(λ) δ(μ − λ): everything on the diagonal, nothing anywhere else, and the operator collapses to a multiplication by a single curve. That collapse is what makes a reflectance a reflectance, and it is why the entire apparatus of object colorimetry works with one array per surface.

The collapse is a physical fact about how light interacts with most pigments: a photon is absorbed and its energy becomes heat, or it is scattered and keeps its energy. Neither route changes its wavelength. What a fluorophore does instead is absorb the photon into an excited electronic state, let some of the energy leak away as vibration, and then emit a new photon from the bottom of that state — at a longer wavelength, because energy was lost on the way down.

The emission does not remember where it came from. All the vibrational levels of the excited state relax to its lowest one faster than the emission happens, so a photon absorbed at 320 nanometres and one absorbed at 400 produce identical emission spectra. That is Kasha’s rule, and its consequence for the matrix is exact: the off-diagonal block is an outer product of one excitation profile and one emission profile — a rank-one term, and one column of it is every column of it up to a scale.

What the standards call the two halves #

The nomenclature is worth learning because it is the only place the distinction is normally made explicit.

β_S, the reflected radiance factor, is what would leave if the fluorophore were absent — the diagonal’s contribution, and the thing a reflectance curve means. β_L, the luminescent radiance factor, is the block’s contribution expressed the same way: emitted radiance divided by the radiance a perfect diffuser would return at that wavelength. β_T = β_S + β_L is what actually leaves and what an eye or a camera receives.

Two properties of that decomposition decide everything downstream.

β_S is a property of the sample; β_L is not. Doubling the ultraviolet in the lamp leaves β_S untouched and roughly doubles β_L, so the sum changes with the lamp. A number that changes with the lamp is not a description of a surface, and calling it a “reflectance” is a category error that gets made every day.

β_L is not bounded by one. A reflectance cannot exceed unity because a surface cannot return more light at a wavelength than arrived at that wavelength. A radiance factor can, because the light did not arrive at that wavelength. For a coated press stock under D65 with its ultraviolet the sum reaches 1.24, and under the warmer D50 a measurement standard specifies, 1.15 — and the fact that this is impossible for a reflector is the whole content of some paper being brighter than white.

Two more readings of the same object say that the diagonal and the block off it are separable rather than two names for one thing.

Two sheets that share a diagonal and differ in everything else is the sharpest form of the same point, and it needs no arithmetic to read.

The Stokes shift, and the energy it costs #

The emitted photon is longer in wavelength than the absorbed one — Stokes observed this in 1852, in a solution of quinine under sunlight through a prism, and named the whole phenomenon fluorescence after fluorspar.

The consequence for the arithmetic is that quantum yield and energy yield are different numbers, and only the first is what a chemist quotes. A quantum yield of 0.85 means eighty-five photons out per hundred in. Each of those photons carries hc/λ, and λ has grown from 358 nanometres to 435, so each carries 82 per cent of what the absorbed one did.

Computing the whole thing properly means doing the absorption and emission in photon flux and converting back to power at both ends, which is what this collection’s fluorophore does. Run with a quantum yield of exactly one — a perfect fluorophore, emitting a photon for every photon it takes — the power returned is 84.3 per cent of the power absorbed. The missing 15.7 per cent went into the vibrations of the molecule, which is to say into heat.

So fluorescence is not free light and the sheet is not violating anything. The energy books balance at every wavelength; what fails is only the wavelength-by-wavelength bookkeeping a reflectance assumes. A brightened sheet under daylight is dimmer overall than the same sheet with a perfectly white non-fluorescent coating would be — a sheet that would in any case be outside what a real pigment can reach, and looks whiter, because the light it does return is in the band where blue is decided and the band it took it from is one nobody can see.

The energy loss is a property of the lamp as well as the fluorophore #

The two figures given for the Stokes cost do not agree with each other, and the gap between them is the interesting part.

Taken at the band centres, an absorbed photon at 358 nanometres returning at 435 carries 82.3 per cent of the energy it had. Correcting for the two bands’ widths — the photon-weighted mean of 1/λ over a Gaussian, which is what the energy ratio actually needs — moves that by four parts in ten thousand, to 82.26 per cent. It is not a width effect.

The computed figure is 84.3 per cent, two points higher. Solving backwards, a return of 0.843 against an emission at 435 nanometres implies that the absorbed photons averaged 366.7 nanometres rather than 358 — 8.7 nanometres to the red of the excitation band’s own centre.

That displacement has one available cause, and it is the illuminant. Daylight’s power climbs steeply across the excitation band; the fluorophore absorbs the product of its own profile and whatever arrives, so the photons it actually takes are drawn disproportionately from the long-wave side of its band. Nine nanometres of shift on a band with a standard deviation of 22 is about four tenths of a standard deviation, which is the right order for a source rising as fast as D65 does below 400 nanometres.

So the 15.7 per cent loss is not a constant of the fluorophore. Under a lamp with more short-wave ultraviolet the absorbed photons would average shorter, each would carry more energy, and the fractional loss would be larger; under a lamp whose ultraviolet is confined to the long edge of the band the loss would be smaller. It is one more quantity in this essay that reads as a property of the sample and is a property of the sample and the light together — which is the essay’s own thesis arriving in the one place it was not being applied.

What the two β_T figures conceal #

The same dilution runs through the headline pair. A coated stock reaches β_T = 1.24 under D65 and 1.15 under D50, which is a difference of 7.8 per cent and reads as a modest sensitivity to the lamp.

It is not, because both numbers contain the same β_S. Only the luminescent half moves, so the ratio between the two lamps’ luminescent terms is (1.24 − β_S)/(1.15 − β_S), and that is a strongly increasing function of the base:

Across any plausible base the luminescent term moves by a quarter to a half between the two lamps, against the 7.8 per cent the reported figures show. The reflected half is acting as ballast: it is the larger of the two terms, it does not move, and it drags the visible sensitivity down by a factor of three to six.

That is worth stating wherever β_T is quoted, because it inverts how the numbers read. A pair of figures eight per cent apart invites the conclusion that the lamp is a second-order concern. The quantity the lamp acts on moved by a third, and it only looks small because it is being reported inside a sum with something that did not move.

And a matrix that stops at 380 is the wrong matrix #

One arithmetic point about the two-monochromator cost, since the essay gives it as eighty-one times eighty-one readings instead of eighty-one.

Eighty-one is the visible band on a five-nanometre grid, 380 to 780. The excitation profile this essay models is centred at 358 and runs well below that, which is exactly the truncation the neighbouring figure is about. A matrix confined to the visible band would miss the brightener’s excitation entirely and would report a rank-one block excited by almost nothing.

The measurement that captures it needs excitation columns from 300 nanometres — ninety-seven of them — against eighty-one emission rows: 7,857 readings, ninety-seven times a single scan rather than eighty-one. A small correction to a cost estimate, and it is the same boundary the rest of the essay turns on: the reason the operator is expensive is not that it has two indices, it is that one of them runs outside the band everything else on this site is defined over.

An unbrightened sheet under the ultraviolet-excluded condition is the case where the emitted component should be smallest, and it is the control the split needs.

Why the instrument reports the diagonal anyway #

A spectrophotometer illuminates the sample and reads the returned radiance band by band. What it prints is β_T(λ) under its own lamp, and it prints it in the column headed reflectance — which is one more thing the instrument reports that is a property of the instrument.

That is not a defect of any particular instrument; it is what a single-monochromator instrument can do. Measuring the matrix properly means controlling the incident wavelength and the detected wavelength independently — two monochromators, one before the sample and one after — and stepping both. On a five-nanometre grid across the visible band that is eighty-one times eighty-one readings instead of eighty-one, and the machine that does it is a laboratory instrument rather than a production one.

There is a cheaper route that is used, and it is worth stating because it is where the two-monochromator problem becomes a one-monochromator one: illuminate with a broadband source and filter it. Two readings — one with the full source and one with the ultraviolet cut — separate β_S from β_T, because the second reading has no excitation in it. That is exactly what the M₂ measurement condition is for, and it recovers the two components without ever forming the matrix.

What it cannot do is predict the sheet under a third lamp, because β_L scales with how much excitation that lamp supplies and the two readings do not say what the excitation profile was. Predicting under an arbitrary source needs the columns, and the columns need the second monochromator.

What was computed, and how #

The fluorophore here is two Gaussian bands and a quantum yield. Its absorption profile is centred at 358 nanometres with a standard deviation of 22, its emission at 435 with 25, and the amount it absorbs saturates in the loading through a Beer–Lambert exponential rather than rising linearly — which is why doubling the brightener in a sheet buys much less than twice the glow.

The operator is never formed as a ninety-seven by ninety-seven array, because it does not need to be. Rank one means the whole block is emission(μ) × absorbed, with absorbed a single scalar: the total photon flux the fluorophore took from the incident light. Forming the matrix would be a demonstration rather than a computation, which is what the figure at the top of this essay is.

The substrate is a level with an exponential rise in absorption towards the short wavelengths, which is cellulose and its residual lignin. Six stocks are built from it with loadings from zero to 2.8, spanning an unbrightened sheet to a laundered shirt.

The three assertions that keep it honest. The emission band must lie to the long-wave side of the excitation band, or the model would be inventing energy. A perfect fluorophore must return less power than it took, and the loss is required to fall between ten and fifty per cent — a bound wide enough that passing it means the photon arithmetic is right rather than that the tolerance was fitted. And the substrate must be a genuine reflectance everywhere, refusing any level above one, because a base that reflects more than it receives would put the impossibility in the wrong term.

Under an incandescent lamp the ultraviolet is unspecified rather than excluded, which is the condition most measurements outside a laboratory are actually made in.

Where the model stops #

Kasha’s rule is a rule and not a theorem. It holds well for the stilbene brighteners this is a model of, and there are fluorophores where it fails — where emission depends on excitation wavelength, and the block is genuinely two-dimensional rather than rank one. For those the arithmetic here understates the difficulty rather than overstating it.

The excitation profile is stated, not measured. A real Donaldson matrix is measured on a bispectrophotometer and no two commercial brighteners have the same one. Everything in this essay that is a shape survives that; the exact peak of 1.24 does not, and belongs to a stated band centred at 358 nanometres.

Reabsorption is not modelled. In a thick or heavily loaded layer some of the emitted light is absorbed by another brightener molecule before it escapes, which distorts the emission’s short-wave edge and is the reason very heavy loadings go green. The model here saturates the absorption and not the emission, so it predicts diminishing returns without predicting the colour shift that accompanies them.

And the operator is assumed linear. At the intensities anybody measures paper at, it is. Under a focused laser it is not, and the whole framework changes.

Who found it, and when #

Stokes described the wavelength shift in 1852 and gave the phenomenon its name. Kasha’s rule dates from 1950. Donaldson published the two-monochromator measurement and the matrix that carries his name in 1954, working at the National Physical Laboratory on exactly the problem that had made whiteness measurement unrepeatable.

The commercial pressure came first and the metrology followed. Optical brightening agents went into detergents and paper through the 1940s and 1950s, and by the time the CIE was writing whiteness formulae the samples being ranked were fluorescent, the instruments were not, and two laboratories with different lamps got different answers about the same sheet.

The vocabulary of β_S, β_L and β_T is the CIE’s, and its existence is the admission: a discipline that had one symbol for what a surface does needed three the moment the surfaces stopped being reflectors.

The generalisation #

The pattern is one that shows up wherever a model is a special case that stopped announcing itself.

A reflectance is a diagonal operator. That is a restriction, and while every sample obeys it the restriction is invisible — there is no experiment that distinguishes “the operator is diagonal” from “the operator is a curve”, because nobody has an operator to compare against. The vocabulary collapses to match: the word reflectance comes to mean both the physical quantity and the data structure, and the assumption is carried in the shape of the array rather than in any sentence.

Then a sample arrives that breaks it, and the first symptom is not an error. It is a number that will not repeat between instruments — which reads as a calibration problem, gets chased as one, and is a category error being reported by the only channel available.

The diagnostic worth carrying: when two correctly working instruments disagree about a sample and neither is drifting, the quantity being measured may have more indices than the report has columns.

Where the ladder goes next #

If the reported number depends on the lamp, then the lamp is part of the specification, and what an instrument brings with it is the next question — four standard answers, of which a printing specification has to name one.

The other direction is what the operator makes possible that a reflectance forbids. Two sheets can have identical diagonals and different blocks, which means they match under any light with no ultraviolet in it and separate under any light that has some — a stronger statement than two spectra making one colour. That pair is not ordinary metamerism, and no change of light puts it back together.