One slit, two requirements

The slit is what makes it legal established that a spectrum integrated through a slit before it is tabulated can be summed honestly on a coarse grid, because the blur puts a narrow feature’s power where the grid can see it. The line can be in the sample found the same problem in the other factor and proposed the same repair: a slit for the sample too.

Both are right, and the two requirements do not want the same slit.

Opposite requirements, and a compromise that is neither #

A wide slit spreads a lamp’s lines where a coarse grid can see them and fills a sample’s notch that the grid could have resolved. One instrument has one slit, so it is making a trade rather than a setting — and the width that serves the trade is not the width either requirement would have named.

• The lamp’s lines are best served by a slit of 5 nanometres, where a smooth sample under the tube costs 0.020 colour differences, against 1.498 at a slit of 1.
• The sample’s notch is best served by a slit of 1 nanometre, where a notched sample under a smooth lamp costs 0.007, against 0.263 at a slit of 20.
• The two together are best served by 3, which is neither, and cost 0.248 there.
• That is twelve times the lamp’s own best and thirty-four times the sample’s, so the trade’s price is not a rounding.
• Giving either requirement its own way is worse than the compromise: 0.727 at the lamp’s 5 nanometres and 0.795 at the sample’s 1.

Why the two want opposite things #

A slit’s width is the one setting that decides both, and it decides them in opposite directions for the same reason.

A lamp’s mercury line is 1.2 nanometres wide and the grid’s step is five — a ratio a grid is not a resolution makes the deciding one. Point-sampled, the line falls between two grid points or on one, and the sum either misses its power entirely or counts all of it at one wavelength — which is the aliasing the slit exists to prevent, and which where the grid starts showed swinging by eight colour differences as the grid’s origin slides. A slit several nanometres wide spreads the line across several grid cells, and the sum sees its whole power in roughly the right place. Wider is better, up to the point where the slit starts smearing the lamp’s own broad structure.

A sample’s notch is twelve nanometres wide and its edges are steep. Point-sampled on a five-nanometre grid it is nearly right, because a twelve-nanometre feature is resolvable at five. A slit several nanometres wide fills the notch in — takes light from either side of it and puts it into the bottom — so the tabulated sample is shallower than the sample. Narrower is better, all the way down.

So one setting is being asked to be large for one factor and small for the other, and neither requirement has a reason to yield: both are about avoiding a specific, computable error.

Each requirement is cheap when it is alone. 0.020 for the lamp at 5 nanometres, 0.007 for the sample at 1. The real case at its own best width costs 0.248, which is more than either by an order of magnitude, and the width that achieves it is 3 — a width neither requirement would have chosen.

The optimum moves with the sample #

A setting that has to be chosen once would at least be tolerable if the right choice were the same for everything. It is not.

At a twelve-nanometre notch the best single slit is 3 nanometres; at five it is 2; at twenty it is 5. And the sample’s own requirement moves with it in a way that is worth reading: for a wide notch the sample wants the narrowest slit available, and for a notch narrower than the grid’s own step the sample wants a wide one — because a two-nanometre notch is already below what the grid can resolve, and a slit that spreads it is helping rather than hurting.

So the sample’s requirement is not simply narrow. It is narrow enough to resolve the feature and wide enough to spread what cannot be resolved, which is the same rule the lamp’s requirement follows, applied to a different feature. The two are the same principle pointing opposite ways because their features are on opposite sides of the grid’s step.

That makes the setting worse than a trade. A trade has one answer; this has one answer per sample, and an instrument measuring a shade library measures every one of them through the same slit. The best a laboratory can do with a fixed instrument is choose the width that suits the narrowest feature it expects to meet, which is the shade it will meet least often.

The compromise is not a compromise between the two answers #

That last point is the one worth dwelling on, because it is not what a trade usually looks like.

A trade between two requirements normally lands between their two answers, and 3 is indeed between 1 and 5. But it is not there because it splits the difference. Giving the lamp its own way costs 0.727 on the real case and giving the sample its own way costs 0.795 — both worse than the 0.248 at three, and both worse than each other’s best by amounts that have nothing to do with how far each is from three.

The reason is that on the real case the two errors do not add. A slit of five nanometres makes the lamp legal and fills the notch, and the filled notch then sits under a properly spread line — so the error is a pairing error of the kind two slits are not one slit measured, and its size depends on how much structure each factor has left, which is a product rather than a sum. At three nanometres both factors keep some structure and the pairing error is smaller than it is when either factor has been flattened.

So the best single width is the one that leaves the least product of residual structures, and that is a different optimisation from either requirement’s own. It also explains why the curve is not monotone on either side: the real case is 0.795 at one nanometre, 0.791 at two, 0.248 at three, 0.727 at five, and rises steadily after.

The three notch widths together give the rule. The best single slit is the widest one that still resolves the narrowest feature either factor has, and it is the lamp’s answer when the sample has no narrow feature and the sample’s when the lamp has none. The conflict is only a conflict when both have one.

What a narrow notch does to the whole argument #

A notch narrower than the grid’s own step is the case the trade behaves most strangely in, and it is worth a figure of its own because the strangeness is instructive.

Below about five nanometres the notch is no longer a feature the grid could have resolved, so the sample’s requirement flips: a slit that spreads the notch is doing for the sample exactly what a slit does for the lamp’s line. At a two-nanometre notch both factors want a wide slit, and the compromise is 2 nanometres costing 2.08 colour differences — the largest price in this essay, from the case where the two requirements agree.

That is the clearest evidence that the trade is not really between the lamp and the sample. It is between each feature and the grid, and the lamp and the sample only disagree when their features happen to fall on opposite sides of the grid’s step. Two factors with features on the same side agree about the slit and are both badly served, because a grid that cannot resolve either is a grid that will pair them wrongly whatever the slit does.

What the grid would buy #

The conflict is between a slit and a fixed grid, and the grid is the part that is a convention.

Halving the grid’s step from five nanometres to two changes the arithmetic throughout: the lamp’s line no longer needs as much spreading, the sample’s notch is resolvable with much less margin, and the two requirements move closer together. On a two-nanometre grid the twelve-nanometre notch’s best single slit is 2 nanometres and it costs 0.126 — half what the five-nanometre grid’s best could reach.

Half, and not nothing. The trade survives refinement because the pairing error survives refinement: two slits are not one slit measured that the separately blurred tables’ error is not the grid’s and stays where it is however finely the sum is taken. What the finer grid buys is a narrower slit, and a narrower slit leaves each factor more of its own structure, and more structure left is a smaller product of residuals — which is the mechanism this essay is about, improved rather than removed.

So the practical order of repairs is the one the numbers give. Measuring the product through one slit removes the problem, and is available whenever the instrument can see lamp and sample together. Refining the grid halves it, and is available whenever the instrument’s own slit is narrower than its reporting interval. Choosing the slit for the sample in hand is the last thing left when neither is available, and it is what this essay measures.

What it costs to state, and what it costs not to #

The practical reading is about what a measurement report contains.

An instrument’s slit width is a property of the instrument and is usually in its specification. A five-nanometre slit at a five-nanometre interval is the ordinary design, which is a design chosen for the lamp’s requirement and not for the sample’s — reasonably, since a spectrophotometer measures samples under its own lamp and its own lamp is smooth — an instrument brings its own light is the essay about what else that decides.

The arrangement this is about is one where the two tables come from different instruments, which is the ordinary arrangement for a colour calculation: a sample measured on a spectrophotometer and a lamp measured on a spectroradiometer, combined by a colour engine that knows neither instrument’s slit. In that arrangement there is no single slit at all — there are two, and they are whatever the two instruments happened to be — and the computation above is the best case rather than the typical one.

And the honest report is a number rather than a setting. An instrument that recorded its slit width and its sampling interval alongside every table would let a colour engine compute the bound this essay computes, for the actual pair in hand, and refuse rather than deliver a colour it cannot support. Nothing in any table format carries the slit width today, and what the instrument reports is the standing essay about what a measurement omits.

How the two requirements were separated #

The lamp’s requirement is measured by putting a smooth sample under the structured lamp, so that only the lamp’s lines are at stake. The sample used is a broad Gaussian absorption rather than a flat reflectance, and that is not cosmetic: a flat sample’s colour is the lamp’s own white divided by itself, the white normalisation cancels the lamp entirely, and every slit width scores exactly zero — which reports that the lamp’s lines cost nothing rather than that the test cannot see them. That is the same degeneracy a neutral has no grid names, arriving as a bug in a measurement rather than as a result.

The sample’s requirement is measured by putting the notched sample under a smooth thermal source, so that only the notch is at stake. The real case is both at once.

In every case both factors are integrated through a triangular slit of the stated full width at half maximum before being tabulated on the five-nanometre grid, and the colour is compared in ΔE₀₀ against the unblurred product summed at a tenth of a nanometre. The slit widths swept are 1, 2, 3, 5, 8, 12 and 20 nanometres, which brackets the instruments in use.

What this leaves out #

The slit is triangular in every case. A real instrument’s slit function is a convolution of its entrance and exit slits with its own aberrations, and it is closer to a triangle than to a rectangle or a Gaussian; a different shape moves the optimum and does not remove the conflict, because the conflict is between spreading and filling and every slit does both.

The interval is held at five nanometres throughout. An instrument that could narrow its slit and its interval together would be making a different trade, and a much better one — the conflict here is between a slit and a fixed grid, and the fixed grid is a convention rather than a limit of anything.

And the optimum is over seven widths rather than continuous. The three-nanometre minimum is a minimum among the widths tested, and the true optimum is somewhere near it; nothing in the argument depends on its exact position, and the cost at it is a lower bound on what a single slit can achieve rather than an exact one.

Still open: whether a two-slit instrument would pay for itself #

The conflict is between one setting and two requirements, and the obvious repair is two settings. A spectrophotometer with a selectable slit is not exotic — research instruments have had them for decades — and the question is whether the colour it buys is worth the mechanism.

The measurement that would answer it is a small extension of this one. Sweep the two slits independently rather than together, so that the lamp is measured through one width and the sample through another, and find the best pair. The prediction is that the best pair is close to each requirement’s own best — five nanometres for the lamp and one for the sample — and that the colour it delivers is much better than the 0.248 a single slit reaches, because the pairing error that dominates the compromise comes from both factors having been flattened.

The interesting possible outcome is the other one. If the best pair is not much better than the best single width, then the pairing error is not about the two factors being flattened but about their being flattened separately, and no arrangement of slits helps — which would point back at the one repair that is already known to work, and would say that a selectable slit is a convenience rather than a solution.

A single setting serving two requirements is a trade, and it has a price #

The habit is about a parameter that appears in two places.

A setting chosen for one purpose usually has another purpose somewhere else in the same calculation, and the second purpose is easy to miss because the setting has a name that belongs to the first. A slit width is described as a resolution, which is what it is to the lamp; to the sample it is a blur, and blur and resolution want opposite things.

The move is to measure each requirement alone before measuring them together. The three curves are the whole method: two of them say what each requirement would choose, the third says what is actually achievable, and the gap between the third’s minimum and the other two’s is the price of the trade — a number nobody has unless the three are measured separately.

The failure mode is to optimise the setting against the whole calculation and report the answer. Three nanometres is the right answer and it is uninformative on its own: it does not say that there is a conflict, it does not say what the conflict costs, and it looks like a tuned parameter rather than like the evidence that one instrument is doing two jobs.