One light ink serves three papers within two degrees

Paper sold as bright white usually contains an optical brightener, a dye that absorbs ultraviolet and re-emits it as blue, and some paper is brighter than white found that such a sheet can return more visible light than falls on it. A reader adapts to the sheet, and adapting to a brightened sheet leaves the glow found that adaptation removes the sheet’s colour but not all of what its glow does to a pale tint printed on it: ink’s vehicle absorbs the ultraviolet, so the inked part of a tint does not glow while the bare paper beside it does, and the tint’s hue turns by several degrees that no adaptation can discount.

A run of essays then tried to shrink that residue by changing how the ink is laid down. The rim was never most of the glow found that even the coarsest screen leaves nine tenths of it, because the residue belongs to the bare paper’s share and not to the dots’ edges. A light ink can cancel the glow at one strength changed the ink instead. A light ink is the same pigment at a fraction of its strength, printed over more of the paper to give the same colour, so that more of the sheet sits under vehicle and less of it glows. On brightened office paper every one of eight pigments had one strength at which the residue passed through nought, from 0.42 for the red to 0.81 for the cyan, and an ink set at that strength held the pale ramp within a degree and a quarter.

A printer does not print on one paper. An office stock, a lightly brightened sheet and a premium sheet heavy with brightener may all pass through the same press with the same inks, and a brightener fades with light and time, so even one stock is several papers over its life. The prediction was that the cancelling strength falls as the brightener loading rises, because a sheet that glows more has more bare paper to cover, and that one ink set for the office sheet leaves the premium sheet’s pale tints within two degrees and the light sheet’s within one — far better than the full-strength ink on any of them. The alternative named was a cancelling strength that moves by more than a fifth between sheets, which would make a light ink set for one paper a full-strength mistake on another.

One ink, both sides of nought #

On the census’s three brightened sheets — light, office and premium, with brightener loadings of 0.35, 0.8 and 2 and glow making 0.33, 0.57 and 0.84 per cent of their luminance — every pigment’s cancelling strength at a ten-per-cent target falls at each step of brightener: the orange’s from 0.646 to 0.558 to 0.509, the yellow’s from 0.790 to 0.641, the green’s by under one per cent. Printed at the office sheet’s cancelling strength on the other two, the light inks overshoot on the light sheet, turning seven of eight pigments’ five-per-cent residues past nought the other way, and fall short on the premium sheet for all eight. From five to twenty per cent, every residue on either sheet is under 1.8 degrees, where the full-strength ink leaves up to 5.8 on the light sheet and 12.6 on the premium.

• The prediction’s direction holds. A brighter sheet wants a weaker ink, for every pigment.
• Its tolerance holds for the premium sheet and not for the light one. Both sit within two degrees; the light sheet does not sit within one.
• The alternative holds for two pigments and does no harm. The orange’s and the yellow’s strengths move by about a fifth between the light and premium sheets, and the office ink still removes four fifths of the residue on both.

Drawn pigment by pigment, the office ink’s residues on the three sheets line up in the same order every time. On the office sheet each sits near nought; on the light sheet it sits on the side the full-strength ink’s residue does not, and on the premium sheet on the side it does. The orange’s five-per-cent residue is +1.17 degrees on the light sheet, −0.30 on the office and −1.80 on the premium; its full-strength residues on the same sheets are −4.4, −7.2 and −10.0. The ink set for the middle sheet is too weak for the lightly brightened one and too strong for the heavily brightened one, and errs by about the same amount each way.

That symmetry is the useful result. A printer who must choose one ink for several stocks can choose it for the middle of the range and expect the error on either side to be similar in size and opposite in sign, and small — under two degrees on the palest tints, where the full-strength ink’s error is four to twelve.

The residue a brighter sheet leaves #

Before asking what the ink must do, it is worth seeing how much it must undo. Every pigment’s full-strength residue grows at each step of brightener: the red’s from −5.8 degrees on the light sheet to −9.3 on the office and −12.6 on the premium, the orange’s from −4.4 to −10.0, the cyan’s from +1.4 to +3.5. On every pigment the premium sheet’s residue is between two and two and three quarters times the light sheet’s, around the ratio of their glows, 0.84 per cent of luminance against 0.33.

The residue grows with the glow because it is made of the glow. The tint’s bare paper glows and its inked paper does not; the reader’s white includes the sheet’s full glow, so a tint that glows less than its colour implies is seen turned. A sheet with more brightener widens the gap between the glowing and the non-glowing parts of every tint, and the pale tints, which are mostly bare paper, show it most. The light ink must therefore cover more of a brighter sheet, and covering more of the sheet at the same colour means a weaker ink.

How much weaker each pigment’s ink must be #

The fall is not the same for every pigment. The orange’s cancelling strength falls from 0.646 on the light sheet to 0.509 on the premium, by 21 per cent; the yellow’s by 19, the violet’s and the blue’s by about 14, the cyan’s by 11, the red’s by 11, the magenta’s by 6, and the green’s — 0.437 to 0.433 — by under one. The prediction expected a fall; the alternative’s threshold of a fifth is crossed by the orange and nearly by the yellow, and by no other pigment.

The size of the fall does not follow the size of the strength. The blue and the cyan cancel at nearly the same strength on the light sheet, 0.88 and 0.87, and fall by 14 and 11 per cent; the orange cancels at 0.65 and falls by 21, the green at 0.44 and falls by under one. What the fall does follow is how much more paper each pigment’s ink must cover on the brighter sheet, which the next figure draws: the orange’s coverage grows most and its strength falls most, the green’s coverage hardly grows and its strength hardly moves. Why the green’s coverage is so nearly fixed — its cancelling ink covers 1.97 to 1.99 times the target’s paper on all three sheets — the census reports rather than derives.

More paper under the vehicle #

The mechanism is in the coverage. At its cancelling strength on each sheet, the orange’s light ink covers 1.39 times the target’s paper on the light sheet, 1.57 on the office and 1.69 on the premium; the yellow’s 1.22, 1.35 and 1.47; the blue’s 1.08, 1.14 and 1.20. A weaker ink is printed over more of the paper to give the same colour, and the extra paper it covers is paper that no longer glows.

The office sheet’s own ink, carried to the other two sheets, keeps the office sheet’s coverage on both — 1.57 for the orange on all three — because a tint’s coverage is matched on a sheet without fluorescence, where the three papers differ only slightly in colour. So on the light sheet the office ink covers more paper than that sheet’s own cancelling ink would, and puts out more glow than the light sheet has to put out: the overshoot. On the premium sheet it covers less than the premium ink would, and leaves some glow standing: the shortfall.

The vehicle’s own ultraviolet absorbance works slightly against this. On the premium sheet it passes 16.5 per cent of the light that would have excited the brightener, against 12.2 per cent on the light sheet, so an inked patch on the premium sheet still glows a little more. The coverage effect is the larger, and every pigment’s strength falls regardless.

Along the pale ramp #

The cancelling strength is set at one target, ten per cent, and a printer prints a ramp. On the office sheet the orange’s residue runs from −0.30 degrees at five per cent through nought at ten to +0.70 at twenty and +2.6 at forty; the light sheet’s runs about a degree and a half above it at every target, the premium sheet’s about a degree and a half below. The red behaves the same way and further: on the premium sheet −1.39 at five per cent and +5.6 at forty.

The three sheets’ ramps are nearly parallel, offset rather than diverging, which is why a single ink keeps all three within two degrees from five to twenty per cent. At forty per cent every sheet’s residue has drifted upward together, because a light ink set to cancel at ten per cent over-covers a darker tint on any sheet; a light ink can cancel the glow at one strength found the same drift on the office sheet alone, and a forty-per-cent tint is far enough from the bare paper that its residue was small to begin with.

The glow, not the loading #

Plotted against the share of each sheet’s luminance that is glow, the cancelling strengths fall along nearly straight lines — the orange’s from 0.646 at 0.33 per cent glow to 0.509 at 0.84, the yellow’s along a parallel, the green’s flat. The brightener loadings, 0.35, 0.8 and 2, are not evenly spaced in glow: the premium sheet carries two and a half times the office sheet’s brightener and only half again its glow. The ink’s strength follows the glow, not the loading.

That matters for a fading sheet. The brightener is being used up found a brightened sheet’s glow falling with exposure as the brightener bleaches. On these lines a premium sheet faded to the office sheet’s glow wants the office sheet’s ink, and the office ink on a faded premium sheet is not a compromise at all. A printer setting light inks for a stock that will be read for years should set them for the glow the sheet will have, not the glow it leaves the mill with.

Why the screen could not do this #

The earlier essays reached for the screen first, and the reason it failed is the reason the light ink works. A thin dot edge lets the glow back in found that a dot’s thin rim of vehicle lets part of the glow through; stochastic dots hold the glow at every tint and squares of four print a pale tint like a 150-line screen moved the dots’ arrangement around and found the residue barely moving. Every screen of a given tint leaves the same share of paper bare, because the tint’s coverage is fixed by its colour, and the bare share is what glows.

A light ink is the one change that moves the bare share at a fixed colour. Its weaker pigment needs more coverage to reach the same tint, so less paper is bare, and that is a lever on exactly the quantity the residue is made of. The finding here is that the lever’s setting depends on how much the bare paper glows — and that, because the residue changes sign smoothly through its zero, an ink set for one paper is still a good setting for its neighbours.

What a printer should take from this #

One light ink per pigment can serve a range of brightened stocks. Set for the middle sheet, it overshoots on a lightly brightened one and falls short on a heavily brightened one by under two degrees on the palest tints, where the full-strength ink errs by four to twelve.

Set it by glow, not by brightener content. The cancelling strength follows the share of the sheet’s light that is glow, which saturates with loading and falls as the brightener fades.

Orange and yellow are the pigments to watch. Their strengths move by a fifth across the census’s sheets; the green’s does not move at all. A printer who can afford two light inks for the warm pigments and one for the rest covers the range more closely than one ink for all.

How the tints were computed #

Each sheet is the census’s brightened stock under D65 extended to 300 nm: its reflected light, its glow and the share of the brightener’s excitation the ink’s vehicle passes, all from the fluorophore model of the earlier essays, reduced to the visible. A tint of coverage a is the Yule–Nielsen mixture with n = 3 of the ink on the sheet’s reflected light and the bare sheet, with the sheet’s glow added in full from bare paper and through the vehicle’s share from inked paper. A light ink of strength f is the pigment’s absorbance scaled by f, its reflectance on the 0.9 white that white times the pigment’s ratio to it raised to f, and its coverage the one that matches the full-strength target’s Oklab lightness on the same sheet without fluorescence. The reader is fully adapted to the sheet in hand by Bradford, and the residue is the tint’s hue turn against the full-strength solid on the brightened sheet, less the turn the target makes on the non-fluorescent one. The cancelling strength is found by bisection between full strength and the weakest strength that reaches the target.

What this does not reach #

Three sheets. The census holds three brightened stocks. Real stocks vary in their fluorophore’s spectrum as well as its loading, and a sheet with a different brightener could fall off the lines in the last figure.

A uniform film. Each tint is a mixture of bare and fully inked paper, with no rim and no screen geometry; the earlier essays found the screen changing the residue little, but the light ink’s extra coverage puts its dots closer together, and a screen at that coverage was not computed.

A reader fully adapted. Every residue is for a reader adapted to the sheet in hand. A reader moving between two stocks under one lamp is adapted to neither, and sees the sheets’ own colours as well.

Still open: whether one ink serves the sheet’s whole life #

A sheet’s glow falls as its brightener bleaches, and the lines above say the cancelling strength follows the glow. A printed piece read on the day it is printed and a year later is, for these purposes, two papers.

The calculation is the premium sheet with its brightener bleached by a quarter, a half and three quarters of its loading, each pigment’s cancelling strength read on each, and the residue the fresh sheet’s ink leaves at each stage. The prediction is that the strength rises back along the line in the last figure as the glow falls — reaching the office sheet’s strength when the glow reaches the office sheet’s — and that the fresh premium ink stays within two degrees until half the brightener is gone. If the strength does not track the glow on a bleached sheet, the brightener’s loading and spectrum matter separately, and an ink set for a sheet’s glow is set for one day of its life. An ink for a fading sheet is set for its middle age makes that calculation.

A middle that errs both ways #

The habit here is about choosing a compromise.

A light ink set for one paper and used on others will be wrong on them, and the useful question is not whether it is wrong but how: by how much, and in which direction. Here the direction was opposite on either side of the paper it was set for, and the size about equal, so that the choice of the middle paper bounded the error on all three.

The failure mode is to read a quantity’s movement across conditions as the size of the error a fixed setting would make. The cancelling strength moved by a fifth, which sounds like a fifth of an ink’s worth of error. The residue it left moved by under two degrees, because the residue is flat near its zero and the setting sat in the middle. The test is to compute the residue a fixed setting leaves in each condition, not the setting each condition would choose.