A finer grid keeps the price of separating

A second slit buys a quarter priced the one arrangement of a spectrophotometer that keeps a measurement reusable. A sample’s reflectance measured through its own slit, and a lamp’s spectrum through another, can be multiplied together under any lamp at all, which is what a colour library is for. Measuring the reflected light through one slit instead — the product of lamp and sample, blurred once — is accurate, but it belongs to the lamp it was measured under. On the site’s five-nanometre grid, over sixty-eight notched samples near a fluorescent tube’s green mercury line, the best pair of separate slits beat one compromise slit by a quarter, and one slit on the product beat the pair by fifteen times.

That essay’s closing section asked whether the gap was a property of the grid. A finer grid is often cheaper than a different light path, and a one-nanometre grid needs almost no blur on a twelve-nanometre notch and little on a mercury line. Its prediction was sharp: at one nanometre the separable arrangement’s error would fall below the product’s at five, so a table measured finely through narrow slits would be both reusable and honest under a line lamp. Its alternative was that the covariance two separate slits throw away is large enough at any grid that no separable table survives a mercury line.

The census gives neither answer cleanly. It gives a third.

Near the old target, no nearer the new one #

On a one-nanometre grid under the tube, the best pair of separate slits — one nanometre on each — has a mean error of 0.019 over the sixty-eight notches, against 0.016 for one slit on the product at five nanometres. The prediction misses, by a fifth. But one slit on the product on the same one-nanometre grid has a mean error of 0.0007, twenty-nine times better than the pair. On every grid the separable arrangement costs a multiple of the product’s error — ×16 at five nanometres, ×15 at two, ×29 at one — and the multiple does not fall as the grid gets finer.

• The grid shrinks both arrangements. Five to one nanometre takes the tube’s best pair from 0.26 to 0.019, thirteen times better, and the product from 0.016 to 0.0007, twenty-three times.
• The tube’s separable table survives. At one nanometre its worst notch is 0.21 of a colour difference, inside any tolerance a colour library would write.
• The projector’s does not. At one nanometre its best pair’s mean is 0.14 and its worst notch 1.65, four times worse than the product at five and 391 times worse than the product on the same grid.
• The narrow notches decide it. On a one-nanometre grid the pair beats the five-nanometre product on notches twelve and twenty nanometres wide and loses on those five and eight wide.

The census on three grids #

The census is the one the two-slit essay ran, on three grids instead of one. Each notch is a smooth reflectance of 0.72 with a Gaussian notch of depth 0.66 cut into it, five, eight, twelve or twenty nanometres wide, centred every two nanometres across sixteen either side of the tube’s green mercury line at 546.1 nanometres. The same sixty-eight notches are placed across a three-laser projector’s green line at 532. For every grid, every pair of seven slit widths — one, two, three, five, eight, twelve and twenty nanometres — is tried with one width on the lamp and the other on the sample, and one slit on the product is tried at every width. Every error is a CIEDE2000 difference from the colour integrated on a tenth-of-a-nanometre grid with no slit at all.

All four lines fall as the grid gets finer, and they fall in parallel on a logarithmic scale. That is the whole shape of the answer. The tube’s best pair descends from 0.26 through 0.040 to 0.019; the tube’s product from 0.016 through 0.0026 to 0.0007. The pair at one nanometre arrives at the horizontal line where the product stood at five, and stops just above it. Meanwhile the product has gone on down, a decade and more, and the gap between the two solid-and-dashed pairs is as wide at the left of the figure as at the right.

The projector’s lines tell the same story more loudly. Its pair at five nanometres is off by 3.1 on average, because a five-nanometre grid steps straight over a 0.2-nanometre laser line and a separate slit on the sample has nothing to multiply it against. A finer grid rescues most of that — 0.17 at two nanometres, 0.14 at one — but the projector’s product falls faster still, to 0.0003.

What separating costs, measured as a multiple #

The ratio is the number that matters to someone choosing an instrument, because it says what reusability costs in accuracy at any given level of investment in grid. Under the tube it is ×16 at five nanometres, ×15 at two and ×29 at one. Under the projector it is ×99, ×81 and ×391.

The ratio does not fall, and at the finest grid it rises. The reason is in two slits are not one slit. A product measured through one slit keeps, in every window, the covariance between lamp and sample; two separate slits throw it away and multiply the averages. A finer grid makes every window narrower, so both the covariance and the other errors of a coarse table get smaller. But the other errors — the ones one slit on the product still makes, which are about how well a grid of windows integrates a smooth product — shrink with the square of the window, while the covariance lost across a narrow window at the edge of a line or a notch shrinks more slowly, because a line is still a line at one nanometre. So the product’s error falls faster than the pair’s, and the multiple grows.

The projector shows it most, because its lines are narrower than the finest grid tried. A one-nanometre window around a 0.2-nanometre laser line is one in which the lamp is almost all spike and the sample, if a notch edge crosses it, is sloping — the configuration that makes the largest covariance there is. Separating throws exactly that away.

Every pair, on the finest grid #

On a one-nanometre grid the best pair is the narrowest on both sides, and every step away from it along either axis costs. That is a change from the five-nanometre grid, where the best pair was five nanometres on the lamp and one on the sample: a line lamp wanted a slit at least as wide as the grid step so that the grid could see its lines, and a notch wanted one as narrow as possible. On a one-nanometre grid the two requirements coincide, because a one-nanometre slit on a one-nanometre grid is the widest the lamp needs and as narrow as the sample could want.

The census reports the same rule on every grid: the best pair has the grid’s own step on the lamp and one nanometre on the sample — five and one on a five-nanometre grid, two and one on a two-nanometre grid, one and one on a one-nanometre grid. The lamp’s slit is the grid step for the reason the slit is what makes it legal established: a slit narrower than the step leaves gaps between windows, and a line in a gap is lost. On the five-nanometre grid a one-nanometre slit on the lamp more than doubles the pair’s error, from 0.26 to 0.63. The sample’s slit can be narrower than the step without harm, because a notch is smooth enough that sampling it at points is nearly as good as averaging it.

Under the projector the same corner wins and the whole map is darker, meaning worse. The best pair is one nanometre on each side again, at 0.14, and nothing in the forty-nine does better. There is no arrangement of separate slits on a one-nanometre grid that brings a laser projector’s notched samples within a tenth of a colour difference on average.

Which notches still decide it #

The pair’s error falls with the notch’s width on either grid, and the product’s rises slightly. On a one-nanometre grid under the tube, the pair’s mean is 0.034 for five-nanometre notches, 0.022 for eight, 0.014 for twelve and 0.006 for twenty. The product at five nanometres runs the other way, from 0.010 for the narrowest notch to 0.024 for the widest, because a wider notch puts more of its slope inside each coarse window.

So the lines cross. For notches twelve nanometres wide and wider, a separable table on a one-nanometre grid beats one slit on the product at five; for eight and five, it loses. The prediction was right about wide notches and wrong about narrow ones, and the census’s mean — dominated by its narrowest cases — came down on the side of wrong by a fifth.

That matters for what a sample is. The line can be in the sample and a finer table is a worse table are reminders that the samples with narrow structure are interference filters and fluorescent dyes, not paints. A library of paints — whose reflectances rarely have features narrower than twenty nanometres — can be measured separably on a one-nanometre grid and used under a tube with less error than the product on a five-nanometre grid gave. A library of filters cannot.

The worst notch, which a tolerance has to cover #

A mean is a summary; a tolerance is a promise about the worst case. Under the tube on a one-nanometre grid, the pair’s worst notch is off by 0.21 of a colour difference. Under the projector it is 1.65 — a visible error, on a notch sitting across the green laser line. The product’s worst notch is under a tenth of a colour difference on every grid and both lamps; at one nanometre it is 0.001 under either.

That settles the question the two-slit essay asked in the terms it asked it. A separable table measured at one nanometre survives a mercury line: every notch within a quarter of a colour difference, a tolerance a colour library could state. It does not survive a laser: a reflectance measured for use under narrow-line illumination — a laser projector, and in practice a narrow-emitter LED panel — should be measured under that illumination, or stated with it, because no grid and no arrangement of separate slits recovers what separation loses there.

What reusability is worth, against what it costs #

The factor of fifteen has to be set against what separating buys, and what it buys is large. A library of a thousand paints measured separably, once, can be rendered under any lamp anyone later specifies — a new LED, a proposed tube, a museum’s retrofit — by multiplying tables. Measured as products, the same library has to be re-measured under every lamp, a thousand samples at a time, and a lamp that does not yet exist cannot be measured under at all. Three numbers cannot see a line is the reason a spectral library is kept rather than a table of colours in the first place: the colours change with the lamp, and only the spectra let the change be computed.

So the question an instrument maker faces is not whether the product arrangement is more accurate — it is, by fifteen times or more at every grid — but whether the separable arrangement is accurate enough, and for which lamps. The census answers that directly. Under a tube, on a one-nanometre grid, the separable table’s worst notch is a fifth of a colour difference, and a library that keeps its reusability pays that and nothing else. Under a laser projector the same table’s worst notch is a colour difference and two thirds, and the reusability is bought with an error a viewer can see.

A finer grid moves the boundary between those two cases, and it does not remove it. On a five-nanometre grid the separable table failed under the tube as well, with a worst notch of 1.75; at one nanometre the tube crossed into the safe side and the projector did not. A grid is not a resolution gave the rule for a single table — compare the lamp’s narrowest feature with the step — and the rule transfers to a pair: separating is safe where the lamp’s features are about as wide as the grid, and not where they are several times narrower.

What an instrument maker should take from it #

A finer grid is worth buying, and it does not buy reusability for free. Every grid improvement tried here improved the separable arrangement by a factor of ten or more, and it improved the product arrangement by more. Anyone choosing between the two should read the ratio, not the error: at every grid, separating costs fifteen times or more, and the fine grid makes it dearer.

The decision is about the lamps, not the grid. For line lamps whose features are about a nanometre wide — fluorescent tubes, most discharge lamps — a separable table at one nanometre is inside a working tolerance on these samples. For lamps whose features are narrower than any practical grid, it is not, and a narrow slit shrinks the error, not the bound found that the error bound a colour engine can compute will not warn it which case it is in.

And the grid does not change which slit goes where. The lamp’s slit should equal the grid step and the sample’s should be as narrow as possible, on every grid tried. One slit, two requirements found the two requirements at five nanometres; they are the same requirements at one, and they happen to meet there.

How the tables were made #

Each grid runs from 380 to 780 nanometres at its step. A table entry is the lamp, the sample or their product averaged through a triangular slit of the stated half-width centred on the grid point, integrated on a tenth-of-a-nanometre sub-grid; a slit of half-width equal to the step makes windows that tile. A colour is the sum over the grid of the lamp’s entry times the sample’s entry against the colour-matching functions, in CIELAB against the lamp table’s own white; for one slit on the product, the sample’s entry is the product’s entry over the lamp’s, which makes the sum the product’s own. The truth is the same sum on a tenth-of-a-nanometre grid from 380 to 780 nanometres with no slit. The observer is this collection’s analytic one, built from cone absorptances and evaluated at whatever wavelengths a grid asks for.

The best fixed pair on each grid is the one with the lowest mean error over the sixty-eight notches, and the multiple is its mean over the mean of one slit on the product at the grid’s own step.

What this leaves out #

The grid starts and ends where the site’s grid does. At 380 and 780 nanometres; the notches are all near 540 and the ends contribute nothing to their errors, but a sample with structure near the ends would bring two ends and one is empty into play as well.

The lamps are two, and constructed. A tube with 1.2-nanometre lines and a projector with 0.2-nanometre lines bracket most line lamps, but a narrow-emitter LED, whose features are ten to twenty nanometres wide, is closer to smooth, and a separable table would do better under it than under either lamp here.

And the slits are ideal triangles. A real spectrometer’s slit function has wings and is not quite the same shape across the spectrum. That changes the product arrangement’s error, which depends on how well the windows tile, more than the pair’s, which is dominated by the covariance separation loses.

Still open: whether a stated lamp class can stand in for the lamp #

The census says that a separable table is safe under a line lamp whose lines are about a nanometre wide and unsafe under one whose lines are much narrower. That suggests a middle course between measuring under every lamp and measuring under none: measure the product once under a representative of each class of lamp, store the ratio of the product’s colour to the separable colour as a correction attached to the reflectance, and apply the class’s correction when the sample is used under any lamp of that class.

The calculation is this census with the correction measured under one tube and applied under another — the same phosphors with lines at slightly different widths, or a different mix of the same mercury lines — and the question whether the corrected separable table beats the uncorrected one by most of the factor of fifteen. The prediction is that it does within a class and fails across one, because the covariance lost is a property of where the lamp’s lines sit against the sample’s structure, which a correction measured under one mercury lamp captures for every mercury lamp and captures for no laser.

A cheaper instrument moves both arms #

The habit is about comparing arrangements at equal cost, not an improved arrangement against an old benchmark.

The prediction set the fine-grid pair against the coarse-grid product, and on that comparison the pair very nearly won — a fifth short of the old target, ahead of it on wide notches. But the product was never going to stay on the coarse grid if the grid got finer; whatever makes the pair better makes the product better too, and here it made the product better by more. The honest comparison is at the same grid, and there the separable arrangement lost by fifteen times at every grid tried.

The failure mode is to improve one arm of a comparison and hold the other where it was. An upgrade that helps both arrangements should be judged by the ratio it leaves between them, and here the ratio went the wrong way.