The glow's colour sets the blue ink and spares the orange

Office paper glows. An optical brightener in it absorbs the ultraviolet in daylight and emits the energy again as violet-blue light around 435 nm, which is why a brightened sheet can reflect more blue than falls on it and look whiter than white. Some paper is brighter than white found that glow; a proof cannot glow found what it does to a print. A tint printed as a pattern of ink dots lets the glow out through its open paper at full strength and through its ink only as far as the ink lets ultraviolet in, so a light tint of a strong ink does not have the hue of the strong ink: it turns, by up to nine degrees of Oklab hue for a red.

A light ink can cancel the glow at one strength found the fix. A diluted ink, printed at the larger coverage that matches the same lightness, changes the balance between open paper and ink film, and at one dilution the turn cancels. One light ink serves three papers within two degrees found that cancelling strength moving with the paper; a yellowing base leaves the ink where the glow put it found the paper’s base irrelevant to it. The glow alone sets the light ink found the premium stock, bleached until its glow equals the office sheet’s or the light sheet’s, needing those sheets’ inks within 0.0007, though base, whiteness and the ink’s shielding all differed: every ink’s strength is one curve in the glow’s share of the sheet’s luminance.

Every sheet in that census glows with one brightener, so its glow has one spectrum and differs from sheet to sheet only in amount. A different brightener, or one partly quenched by photo-yellowing products, emits a glow of a different colour, and a cancelling strength balances glow of one hue against ink of another. The prediction was that a brightener emitting fifteen nanometres longer, at 450 nm — nearer the blue and violet inks’ own hues — moves their strengths by more than a hundredth and the orange’s and red’s by under a fifth of that, so that the curve in glow holds for the warm inks and splits for the cool ones. If no ink moved by more than 0.003, the glow’s share would set the ink whatever its colour, and one measurement of a sheet’s fluorescence would suffice.

Same photons, three colours #

The test needs glows that differ in colour and nothing else. The premium stock was given its brightener with the emission peak moved to 420 and to 450 nm, the width, the excitation band and the quantum yield unchanged, at the two loadings that bleach the 435 nm brightener to the office and light sheets’ glow. Each moved sheet absorbs the same ultraviolet and emits the same number of photons as the sheet it is paired with.

At the office loading the three glows make up 0.27, 0.57 and 1.09 per cent of the sheet’s luminance; at the light loading 0.16, 0.33 and 0.64. The eye weighs the longer glows more, so the same photons nearly double their share of luminance from 435 nm to 450 and halve it from 435 to 420.

That pairing is the only one possible for the shorter glow. Matched by luminance instead, as the census sheets were, a 420 nm brightener cannot reach the office sheet’s share at all: at four times the premium stock’s own loading its glow is 0.55 per cent of the sheet’s luminance, under the office sheet’s 0.57. Its absorption saturates before the eye sees enough of what it emits.

The longer glow moves the cool inks #

With the emission at 450 nm the blue ink’s cancelling strength rises by 0.176 at the office loading and 0.122 at the light, to 0.984 and 1.000; the violet’s falls by 0.058 and 0.064. The orange’s moves by 0.0011 and 0.0007, and the red’s by 0.0014 and 0.0024. The prediction holds by a wide margin: the cool inks move by twenty-four to seventy times the warm inks’ largest move, where a factor of five was asked.

The cool inks move in opposite directions, and that is the first sign of what the glow’s colour is doing. A cancelling strength near one means the ink needs almost no diluting — the full-strength tint barely turns. The blue’s strength rising to one says a 450 nm glow leaves a blue tint’s hue nearly where the solid blue has it. The violet’s strength falling says a 450 nm glow turns a violet tint further than a 435 nm glow did, so it must be diluted more to cancel.

A glow at the blue’s own hue leaves the blue tint unturned #

A full-strength blue tint at ten-per-cent coverage turns by +1.22, +1.08 and +0.08 degrees of hue with the glow at 420, 435 and 450 nm; the violet by −1.02, −1.63 and −2.35; the orange by −5.64, −7.04 and −6.91. At the light loading the blue’s turn at 450 nm is under a hundredth of a degree.

So the blue’s cancelling strength goes to one for a reason that has nothing to do with the ink’s dilution: there is almost nothing to cancel. The turn a glow gives a tint is the hue difference between the light that comes out through the open paper — reflected light plus the full glow — and the light that comes out through the ink. When the glow’s own colour lies near the ink’s hue, adding glow to the open paper pushes the tint toward the ink’s hue rather than away from it, and the two contributions agree. A 450 nm glow has a chromaticity of 0.147, 0.042, bluer and greener than the 435 nm glow’s 0.155, 0.024, and a blue ink sits in that direction. The violet ink sits on the other side of the glow, so moving the glow toward the blue moves it away from the violet and the violet tint turns more.

The orange’s turn ignores a longer glow and shrinks with a shorter one #

The warm inks are where the prediction said nothing would happen, and at 450 nm nothing does. What is surprising is how little.

From 435 nm to 450 the glow’s share of luminance rises by a factor of 1.91 at both loadings, and the orange’s full-strength turn changes by 1.9 per cent at the office loading and 1.1 per cent at the light; the red’s by 1.4 and 2.0 per cent. From 435 nm to 420 the share falls to 0.48 of itself and the orange’s turn shrinks by 20 and 21 per cent, the red’s by 20 and 22.

If the warm turn were set by the glow’s luminance, doubling it should roughly double the turn, and halving it halve it. Neither happens: the turn barely notices a doubling and loses a fifth to a halving. The orange and red inks absorb strongly through the whole blue-violet region, so through their ink film almost no glow escapes at any of the three peaks, and the turn is set by the glow that escapes through the open paper — weighed, for its effect on hue, not by the eye’s luminance but by how far it pulls the tint’s chromaticity. A 450 nm glow is more luminous than a 435 nm one and nearly as far, in chromaticity, from an orange; a 420 nm glow is dimmer and deeper violet, and the turn loses a fifth. How those two changes divide that fifth between them is not separated here. Neither of those is the quantity the census’s curve was drawn in.

The curve in glow no longer reads the ink #

Two of the moved sheets have a glow share inside the premium curve’s range: the office loading at 420 nm, at 0.27 per cent, and the light loading at 450 nm, at 0.64. The first needs the orange at 0.596 where the curve reads 0.686, and the blue at 0.777 where it reads 0.897; the second the orange at 0.647 where the curve reads 0.543, and the blue at 1.000 where it reads 0.792. The misses run from 0.09 to 0.21, a hundred to three hundred times the agreement the bleached census sheets showed.

So the finding of the essay before was exact and narrow. The glow alone sets the light ink, and the glow’s share of luminance is the right single number for it, provided the glow has one spectrum; once the brightener changes, the share is no longer the variable, and a sheet’s ink cannot be read from a measurement of how much it fluoresces. It has to be read from what colour it fluoresces as well.

On the lighter loading the inks part the same way #

At the light sheet’s loading the blue needs 0.854, 0.877 and 1.000 with the glow at 420, 435 and 450 nm, the violet 0.776, 0.697 and 0.633, the orange 0.692, 0.646 and 0.647. Every pattern at the office loading repeats: the blue rising to one at the longer glow, the violet falling, the orange flat from 435 to 450 and higher at 420.

The repetition matters for a practical reason. A printer does not know a sheet’s loading; it knows, at best, a fluorescence measurement. If the glow’s colour moved the inks differently at different loadings, a correction for a brightener’s colour would need the loading too. It moves them the same way at both.

Why the ageing sheets never showed it #

The run of essays on this correction has aged its sheets in two ways, and neither could have found the glow’s colour. The brightener is being used up found daylight bleaching a sheet’s brightener month by month, and an ink for a fading sheet is set for its middle age held the base paper fixed while the brightener went, finding every bleached sheet’s cancelling strength a shade under the census line in glow. A bleached brightener is less of the same brightener: its glow keeps its spectrum and loses amount, so every bleached sheet lies on the curve in glow’s share and the census’s single variable stayed sufficient. The yellowed base that came next changed the paper and not the glow, and changed no ink.

What would move a glow’s colour in real paper is a different brightener or a quencher. Brighteners differ in where they emit, and the yellowing products of an aged sheet absorb short-wave light, which would take more from a glow’s violet side than from its blue. Neither appears in a census whose sheets differ only in loading. Adapting to a brightened sheet leaves the glow found that a reader adapting to the sheet’s white does not remove the glow’s effect on a tint, and the same holds here: every turn above is read by a reader adapted to its own sheet, and the blue ink’s turn vanishes at 450 nm for that reader, not for a fixed one.

So the cool inks’ sensitivity is a property the census could not have measured until a glow was allowed to change colour, and it is large. The 0.176 the blue’s strength moves between a 435 and a 450 nm glow is more than any paper-to-paper change in the census moved any ink.

What a print engineer should take from this #

Measure the glow’s colour, not only its amount. A fluorescence reading that gives the glow’s share of luminance sets a light ink only while every sheet uses the same brightener. A brightener emitting fifteen nanometres longer or shorter needs inks differing by up to 0.18 in strength from the inks a 435 nm brightener at the same loading needs.

The blue ink is the one to watch. A glow near the blue’s own hue makes its light ink unnecessary — a full-strength blue tint barely turns — and a shorter glow restores the need. The orange and red are nearly indifferent to a longer glow; under a shorter one the orange needs about 0.04 more strength and the red about 0.02.

A quenched brightener is not a weaker brightener. Photo-yellowing products that absorb the glow’s short wavelengths shift its effective colour toward the longer side as well as reducing it, and on this evidence the cool inks’ correction would move with the shift while the warm inks’ would follow only the amount.

How the sheets and inks were computed #

The premium stock is the census’s paper base with a brightener absorbing in a Gaussian band at 358 nm and emitting in a Gaussian band 25 nm wide, quantum yield 0.85, its emission peak set to 420, 435 or 450 nm, at the loading that bisection found to give the 435 nm brightener the office sheet’s glow share of 0.57 per cent and the light sheet’s of 0.33. Each sheet is illuminated by D65 extended to 300 nm and read through the 1931 observer, the glow’s share of luminance taken from its own tristimulus value. A tint at ten-per-cent coverage is a Yule–Nielsen halftone of the ink on the sheet, its open paper carrying the full glow and its ink film carrying the glow in proportion to the ink’s ultraviolet transmission; a reader adapted to the sheet by Bradford sees it in Oklab, and the turn is the tint’s hue less the solid ink’s. The cancelling strength is the dilution at which a matched-lightness light tint’s turn is nought, found by bisection.

What this does not reach #

One brightener shape moved, not real brighteners. Real stilbene brighteners differ in width and in the shape of their emission as well as in its peak; a pure shift isolates the colour from everything else, which is its purpose and its limit.

The warm inks’ mechanism is argued from the turns. That their turn follows the glow’s chromatic pull rather than its luminance is the reading of the numbers here; the turn is not decomposed into the open paper’s and the ink film’s parts.

Two loadings, eight pigments, one target. The ten-per-cent target and the census’s pigments are the ones the run of essays has used throughout.

Still open: whether one measured glow colour sets every ink #

The glow’s colour moves the cool inks strongly and the warm ones barely at the longer peak; at the shorter it moves all of them. A sheet’s glow can be described by two numbers — its share of luminance and its chromaticity — and the census holds sheets that vary each.

The calculation is a line in the glow’s share and its chromaticity coordinate y, fitted for each ink across the census sheets, the premium curve’s eleven loadings and the moved sheets at both loadings, read on each sheet left out. The prediction is that two numbers read every ink within 0.01 on every sheet, the blue’s strength following y and the orange’s following y only below 435 nm, because the glow’s chromaticity is what the turn depends on once its spectrum can move. If any ink misses by more than 0.02, its turn depends on the glow’s spectrum beyond its colour — on how much of the glow the ink film itself lets through at each wavelength — and a sheet’s ink needs its glow measured as a spectrum.

The glow’s pull on the white sets every ink made that calculation, on thirty sheets: the share and y miss by up to 0.07, and the glow’s pull on the adapted white reads every ink within 0.014.

A curve that had one variable fixed #

The habit here is about reading a finding’s scope from the variable its census held still.

The census found every ink set by the glow’s share of luminance to a thousandth. That was a true result about a census in which every glow had the same spectrum, and the essay that found it said so in its closing section. Moving the spectrum did what that section predicted for the cool inks and something it did not predict for the warm: it showed the share of luminance to be the wrong variable for every ink, once it is not the only one that varies.

The failure mode is to read a one-variable law found on a census that held its other variables fixed as a law in that variable. The test is to vary what the census held still, at a fixed value of the variable the law is written in, and see whether the law’s prediction survives.