A second slit buys a quarter

One slit, two requirements found that a spectrometer’s slit is asked to do two opposite things. A fluorescent tube’s mercury lines want a slit of five nanometres, wide enough to spread each line where a five-nanometre grid can see it. A sample with a twelve-nanometre notch wants a slit of one, narrow enough not to fill the notch in. One instrument has one slit, and on one notch the best compromise was three nanometres at a cost of 0.248 colour differences — twelve times what the lamp alone needed and thirty-four times what the sample alone needed.

It ended on the obvious repair. Instruments with selectable slits exist, so measure the lamp through one width and the sample through another. The prediction was that the best pair would sit near each requirement’s own best, five and one, and deliver much better colour than any single width. The alternative it named was that the pair would barely help, which would mean the error was never about how wide the slits were but about the two factors being blurred apart.

The alternative is what happens.

The right pair, and a small gain #

Over sixty-eight notched samples under a fluorescent tube, the best fixed pair is five nanometres on the lamp and one on the sample, exactly as predicted. It averages 0.256 colour differences, against 0.358 for the best single slit — 28 per cent better — and its worst notch is 1.75 against 2.60. One five-nanometre slit on the reflected light averages 0.016, with a worst of 0.029.

• A pair is a fraction better than one slit, not a factor: between 1.1 and 1.7 times, depending on how wide the notch is.
• One slit on the product is fifteen times better than the best pair at the mean and sixty times at the worst notch.
• An oracle choosing the best of forty-nine pairs separately for every notch averages 0.106, which no instrument could do and which is still six times the product’s error.
• The single notch the earlier essay measured picks a different pair, three on the lamp and five on the sample, by a cancellation that makes it the best pair for only six of the sixty-eight.
• Under a three-laser projector the ordering is the same and the gap wider: 4.27 for one slit, 3.11 for a pair, 0.032 for one slit on the product.

Sixty-eight notches instead of one #

The earlier measurement used one notch, twelve nanometres wide and centred exactly on the 546.1-nanometre mercury line. That is the worst position for a notch and a natural one to choose, and it is also a single case, where errors from different sources can cancel by accident. A slit width chosen on it is chosen for that notch.

So the census here has four notch widths — 5, 8, 12 and 20 nanometres — each centred at every second nanometre from sixteen below the line to sixteen above it: seventeen positions each, sixty-eight notches in all. For each notch the lamp’s table is blurred through one of seven triangular slits and the sample’s through one of the same seven, tabulated on the five-nanometre grid and compared with the truth computed at a tenth of a nanometre. That is forty-nine arrangements per notch. Beside them is the arrangement two slits are not one slit already found to work: a single slit on the reflected light, which blurs the product of lamp and sample rather than each factor.

The figure at the top of the page is the census’s mean for every pair, lamp’s slit down the side and sample’s across. Its smallest circle sits where the prediction put it, at five and one. The single slits run down the diagonal, and the best of them is five nanometres at 0.358.

The pair the prediction named #

The predicted pair is right, and it is worth seeing why before seeing how little it buys.

A mercury line is 1.2 nanometres wide, and a grid is not a resolution is why that ratio to the step decides what a slit must do. The five-nanometre grid needs it spread across at least one step, or the sum depends on where the line falls between grid points; where the grid starts measured that dependence at 3.18 colour differences for this tube. The earlier measurement found five nanometres best for the lamp alone, and the census agrees: for every sample slit from one to eight nanometres, the lamp’s best slit is five.

A notch wider than a step or two is already resolvable on a five-nanometre grid, and any slit fills it in. Averaged over the census, the sample’s best slit is the narrowest on offer, one nanometre, with the lamp’s at five.

Choosing each factor’s slit for its own requirement is therefore the natural design, and the census confirms it as the best fixed choice. It wins on its own for 27 of the 68 notches, and no other pair wins more than six.

The other forty-one notches are spread over seventeen pairs. The best pair for a particular notch depends on where the notch sits relative to the line, and a slit width is fixed before anyone knows that.

One notch picks a cancellation #

The earlier measurement’s single notch is one of the forty-one.

On that notch the best pair is three nanometres on the lamp and five on the sample, at 0.183. The census’s pair, five and one, costs 0.393 there, worse than the single three-nanometre slit’s 0.248. The table is not smooth. Moving the sample’s slit from three to five nanometres improves the error from 0.247 to 0.183, and moving it on to eight makes it 0.637.

That pattern is a cancellation. A three-nanometre slit on the lamp leaves the line partly aliased and a five-nanometre slit on the sample partly fills the notch, and on this notch the two errors happen to offset. Walk the notch along and the offset goes: across the census, three and five is the best pair for only six notches. The winner on one case is a coincidence of two errors, and a slit chosen on it would be chosen for a coincidence.

That is the argument for a census before a recommendation, and a mean is not a worst case is the argument for reporting both statistics from it. The census’s pair is the best of the forty-nine on the mean and on the worst notch. The single case’s pair is best on six notches and costs 0.43 at the mean.

A notch walked across the line #

The three arrangements that matter can be followed along one notch width as the notch moves.

The pair is below the single slit at 15 of 17 positions, which is what makes it the better fixed choice. Both curves swing by more than a factor of ten as the notch moves, and both are at their worst with the notch on the line or beside it, where one slit reaches 0.73 and the pair 0.41. The swing is the pairing error: the notch and the line overlap inside a slit window, and how much depends on where the notch is.

One slit on the product stays between 0.016 and 0.022 at every position. It does not swing, because it never separates the two factors. The light reaching the slit has already been reflected, notch and line together, so whatever they do to each other inside the window is in what the slit integrates.

Why a pair cannot close the gap #

A linear repair for a bilinear loss gave the mechanism for sharpening, and it applies unchanged to widths.

A colour is a sum over wavelength of the lamp times the sample. Blurring a product through a slit gives the product of the two blurs plus the covariance of the two factors inside the slit window. Blurring each factor separately and then multiplying keeps the first term and discards the second. The covariance is large exactly where a narrow feature in one factor meets a narrow feature in the other — a line inside a notch — and it is zero where either factor is flat across the window.

A pair of slits changes how much each factor is blurred. It changes the first term, and a good choice makes that term as accurate as each factor allows. It cannot put back the second term, because no choice of widths for two separate blurs produces a quantity that depends on both factors together. That is the same algebra that stopped a three-term sharpening filter, applied to the width instead of the kernel.

So the best a pair can do is bounded below by the covariance each notch carries, and the census measures the bound directly.

The best single slit averages 0.358; the best fixed pair 0.256; the best single slit chosen separately for every notch 0.180; the best pair chosen separately for every notch 0.106; one slit on the product 0.016. The oracle rows are not instruments. They are what a spectrometer would deliver if it knew each notch’s position in advance and switched slits to suit, and they are included to measure the floor rather than to propose it.

The floor is the point. Even the oracle pair is six times worse than the product, because every pair of separate blurs discards the covariance and the product keeps it. What a selectable slit buys is the step from 0.358 to 0.256. What a different arrangement of the light path buys is the step from 0.256 to 0.016, which is almost all of it.

How the gain depends on the notch #

The census splits by notch width, and the split shows which case a pair helps most.

At a five-nanometre notch a pair barely helps: 0.564 against 0.502. A notch as narrow as the grid’s step is poorly served by any slit on the sample, and the lamp’s slit is five in both arrangements, so the difference between them is small. At eight and twelve nanometres the pair’s advantage grows to 1.6 and 1.7 times, and at twenty it is 1.7 again, with every error smaller because a wide notch has little covariance with a line.

One slit on the product runs the other way, from 0.010 at five nanometres to 0.024 at twenty. It is small at every width, and why it grows with the notch is not separated here. Even at its worst width it is three times better than the pair, and at five nanometres it is fifty times better. The narrow notches are where separate blurring is most wrong, and they are where the product slit is best.

Under a laser projector #

The same census under a three-laser projector tests the argument on lines narrower still — 0.2 nanometres — with a notch walked across the 532-nanometre green line.

One slit averages 4.27 colour differences, with a worst notch of 25; the best pair, again five on the lamp and one on the sample, 3.11 with a worst of 24.5; the oracle pair 1.63; one slit on the product 0.032, with a worst of 0.080. The ordering is identical and the gaps are larger. A laser line has almost no width, so the covariance between it and a notch in the same window is nearly the whole of the product. Separating the factors throws away most of what there is to measure, and at the worst notch a pair of slits recovers almost nothing over one.

A pair saves 27 per cent at the mean here, about what it saved under the tube. The product slit saves a hundredfold.

How the census was computed #

The fluorescent tube is the collection’s standard model: broad phosphor bands with mercury lines of 1.2 nanometres at 404.7, 435.8, 546.1 and 578.0. The laser projector is three gaussian lines of 0.2 nanometres at 465, 532 and 638. A notched sample is the notch used throughout these essays on the slit: a gaussian dip of a stated full width at half maximum, 0.66 deep, in a flat reflectance of 0.72.

A slit is triangular with a stated full width at half maximum, integrated at a tenth of a nanometre. For a pair, the lamp’s function is blurred through one slit and the sample’s through the other, each is tabulated on the five-nanometre grid from 380 to 780, and the colour is computed from those two tables with the white taken from the blurred lamp. For the product arrangement, lamp times sample is blurred through one slit and divided by the lamp blurred through the same slit, which is what an instrument measuring the reflected light and a reference white through one slit reports; where a laser’s blurred output is zero the quotient is set to zero, since no light is reflected there. The truth is the same colour computed on a tenth-of-a-nanometre grid with no slit. Errors are CIEDE2000.

What this leaves out #

A real instrument’s slit is not a triangle. A spectrometer’s instrument function comes from the entrance and exit slits together with the grating’s dispersion and the optics’ aberrations, and it has tails a triangle does not. Tails spread a line further than the stated width suggests, which would move the lamp’s best slit but not the algebra: any pair of separate blurs discards the covariance, whatever their shapes.

The product arrangement has costs this model does not charge. Measuring the reflected light directly means measuring the sample under the lamp it will be seen under, rather than measuring the sample once and computing its colour under any lamp. That flexibility is why what the instrument reports is usually a reflectance rather than a radiance, and a notch in a sample is exactly the case where that flexibility costs colour.

The samples are notches. A sample with a narrow peak, a fluorescent sample, or one with an interference filter’s periodic structure — a finer table is a worse table — would give different numbers and the same ordering, since the ordering follows from the algebra rather than the shape.

And the grid is five nanometres. On a one-nanometre grid every error here shrinks and the gap between separate and joint blurring shrinks with it, because a finer grid needs less blur.

Still open: whether a narrower grid makes a pair enough #

The arrangement that works needs the lamp at measurement time. The arrangement that keeps a sample’s reflectance reusable, a pair of separate slits, loses about fifteen times more colour on a five-nanometre grid. The question worth measuring is whether a finer grid brings the separable arrangement within a tolerance, since a finer grid is often cheaper than a different light path.

The calculation is this census with the grid step as a third variable: one, two and five nanometres, each with its own best pair and its own product slit. The prediction is that at one nanometre the separable arrangement’s error falls below the product’s at five, because a one-nanometre grid needs almost no blur on a twelve-nanometre notch and less on a line. If it does, a table measured at one nanometre with narrow slits is a reusable reflectance that is also honest under a line lamp. If it does not, the covariance is large enough even at one nanometre that no separable table survives a mercury line, and a reflectance measured for use under fluorescent light should be stated with the light.

A second setting helps only as much as its error allows #

The habit is about the repair a trade-off seems to invite.

When one setting serves two requirements badly, the obvious move is to give each requirement its own setting. It feels like removing the trade, and sometimes it does. It removes only the part of the error that came from sharing the setting. If some of the error came from separating the two things in the first place, giving each its own setting keeps that part intact and makes it look like the irreducible floor of the problem.

The move is to price the arrangement that does not separate them before pricing the settings. Here that was one slit on the reflected light, and it put the floor fifteen times below where the best pair of settings could reach, so the choice between one slit and two turned out to be a small part of the error.

The failure mode is to optimise the settings on one case. The single notch the earlier measurement used had a best pair that was a cancellation, better than the census’s pair on that notch and worse on sixty-seven others. A setting chosen from one case only fits that case.