A finer table is a worse table

The cost of a steep notch repeats every step found that a flat-bottomed notch with steep sides costs a coarse grid far more than a Gaussian one, and ended by asking what a real filter’s own table would do: a published transmittance at a nanometre, resampled onto five by the interpolation a colour engine actually uses, would say where on the wave real filters fall.

Every notch considered here so far has been a Gaussian — the line can be in the sample introduced it and the cost of a steep notch repeats every step gave it edges. A real one is not either.

The grid steps over it, and the rule does not matter #

An interference filter’s features are a nanometre or two wide and a five-nanometre grid steps over them. What that costs is several colour differences, the three fill-in rules give the same answer to three decimal places, and a measurement through a slit at the same step is two orders of magnitude better.

• A notch filter of finesse 30 has stop bands 1.5 nanometres wide spaced 22.9 apart, and resampling it from a one-nanometre table onto five costs 4.93 colour differences.
• The three rules agree to 0.003 of that — point sampling, straight lines and a cubic are the same answer, because none of them is ever handed a sample inside a feature.
• A band-pass filter costs 15.06, and there a cubic does overshoot: 0.073 past a transmittance of one, which is a table that cannot be a filter.
• The same filter measured through a five-nanometre slit costs 0.027 — 180 times better on the same grid.
• The worst case is not the sharpest filter. The cost rises to 5.28 at features 4.6 nanometres wide and falls again, because a feature far narrower than the step is one the grid misses entirely.

What a real coating looks like #

A thin-film interference filter is a stack of quarter-wave layers, and its transmittance is the Airy function of a Fabry–Pérot etalon to a good approximation: T = 1 / (1 + F sin²(δ/2)), with the round-trip phase δ = 4π n d / λ set by the cavity’s optical thickness and the wavelength. Two numbers describe it. The order — how many half-waves fit in the cavity — sets how far apart the pass bands are: a free spectral range of the centre wavelength divided by the order, which at order 24 and 550 nanometres is 22.9. The finesse sets how narrow each band is: the free spectral range divided by the finesse, which at 30 is 1.53.

So a real filter is periodic and its features are narrower than the grid’s step, and both of those are different from a Gaussian. A Gaussian notch has one feature and no sidebands; its width is a free parameter swept here from 48 nanometres down to one; and at every width it is a single smooth thing that a grid either resolves or does not.

An etalon has features the grid cannot resolve at any realistic finesse, and it has them repeatedly across the visible. That is why it is worth measuring separately rather than inferred from the Gaussian sweep.

The rule is not the question #

The obvious variable, once a table has to be read between its points, is which rule reads it. It turns out to be the least important thing in this essay.

Point sampling gives 4.930, straight lines 4.929 and a cubic 4.932. They are the same answer, and the reason is that they never disagree: a rule only matters where the values it is interpolating between differ, and the grid’s points all land in the filter’s flat regions because the features are a third the size of the step. Every rule is interpolating between two nearly equal numbers, and every rule gets the same nearly equal number.

That is a negative result of the kind a finer reading of a coarser table is about, and it is useful. A colour engine’s choice of interpolation is not the question here, and a laboratory changing its engine’s setting from linear to cubic to improve a filter’s computed colour is adjusting something that cannot move.

What does move the answer is whether the table came through a slit: 0.027 against 4.93, on the same grid, with the same rule.

Except when a cubic leaves the range #

There is one thing a rule can do that the others cannot, and it shows up on the filter whose shape gives the grid something to interpolate across.

A band-pass filter is the same etalon read the other way round: mostly opaque, with narrow pass bands. Its costs are much larger — 15.06 colour differences at a finesse of 20 — because the grid is now missing almost all of the light the filter passes rather than missing a little of what it blocks.

And on its pass bands’ shoulders the grid does occasionally land where the values differ, which is where a cubic behaves differently from a straight line. A cubic through four points overshoots 0.073 past a transmittance of one. A straight line between two points cannot leave the range its endpoints are in, and does not — exactly zero.

That is worth having because it is the one defect here a check could catch without a reference. A filled table whose values leave the physical range has been filled by the wrong rule, whatever its colour difference says, and a colour engine could refuse it. Nothing else in this essay is detectable from the data alone.

The worst case is not the sharpest filter #

A reader expecting the cost to rise with the filter’s sharpness would be wrong, and the shape of the curve says why.

The cost rises from 3.82 at features 9.2 nanometres wide to 5.28 at 4.6 and falls to 4.41 at 0.9. The maximum is where the feature is about the size of the grid’s step.

The mechanism is the one where the grid starts measured for a line lamp, arriving in the sample. A feature much narrower than the step is one the grid either lands on or misses, and mostly misses — so a very sharp filter is tabulated as though its stop band were not there at all, which is wrong by a fixed amount that does not grow as the band narrows. A feature about the size of the step is one the grid half-catches, differently depending on where it lands, and half-catching is the worst kind of catching.

So sharpening a filter past a certain point does not make its computed colour worse, which is a small and genuinely surprising consolation, and it comes with the obvious corollary: a filter’s colour computed on a coarse grid is wrong by an amount that says almost nothing about how sharp the filter is.

What the slit is actually doing #

The 180-fold difference between a sampled table and a measured one deserves its mechanism spelled out, because “the slit low-passes before sampling” is a sentence that can be said about almost anything.

A five-nanometre grid can represent features down to about ten nanometres and no finer; that is what a sampling interval means. The filter has features of 1.5. Something has to happen to those 1.5-nanometre features before they meet the grid, and there are exactly two possibilities: they are removed, or they are aliased.

Point sampling aliases them. Each grid point returns whatever the filter happens to be doing at that exact wavelength, which for a periodic filter is a nearly arbitrary point in its cycle, so the tabulated filter is a filter with a different — and arbitrary — shape. That is why the answer depends on where the grid’s origin falls, and why the three fill-in rules make no difference: they are all interpolating a signal that is already wrong.

A slit removes them, and removes them correctly: it replaces the filter’s value at each grid point with its average over a window, so a stop band narrower than the window contributes its whole depth weighted by how much of the window it occupies. The tabulated filter is then a genuinely blurred filter rather than an arbitrary one, and a blurred filter’s colour is close to the real filter’s, because a colour is an integral and the blur has preserved the integral.

That is the whole of it, and it is the reason the advice inverts. A fine table is a sampled signal whose features will be aliased at the next step; a coarse measurement is an integrated signal whose features have already been folded in.

The two curves together say the peak is a property of the grid rather than of the filter. A designer choosing a filter cannot avoid it by choosing a sharper coating and cannot avoid it by choosing a gentler one; what decides where a particular filter falls on the curve is the working grid’s step, which the designer does not choose at all. The one thing a designer could do about it is publish the filter’s shape rather than its samples — the cavity’s order and finesse are two numbers, they determine the transmittance exactly, and a colour engine given them could integrate rather than interpolate.

The two rules for what makes a filter hard are therefore independent and both are about the grid. A feature far below the step is missed; a feature near the step is half-caught. Neither is about the coating’s quality, and a catalogue that sorted filters by steepness would sort them by neither.

A finer table is more dangerous than a coarser measurement #

The result that matters for practice inverts an assumption nobody states.

A filter maker publishes a transmittance table, and a good maker publishes it at a fine step — a nanometre or better — because a fine table carries more of the filter. A colour engine given that table and a five-nanometre working grid resamples it, and the resampling is where the information is lost. The fine table’s extra content does not survive the resampling; it is discarded, and discarded badly.

A spectrophotometer measuring the same filter at five nanometres does something different. Its slit integrates the filter over several nanometres before any tabulation happens, so the features’ energy is in the tabulated numbers rather than between them. Its table is coarser and carries more of what the colour needs.

0.027 against 4.93 is the size of that difference — the same reversal the slit is what makes it legal found for a lamp, arriving in the sample — and it is the whole of the practical advice: for a filter whose features are narrower than the working grid, a measurement at the working grid beats a published table at a finer one. That is the opposite of what a careful laboratory would assume, and the reason it is true is that a slit is a low-pass filter applied before sampling and a fine table is a sampled signal that still has to be re-sampled.

The one arrangement better than either is the one already recommended twice here: keep the fine table and blur it once, against the lamp it will be used with, which reproduces the one-slit answer and is what two slits are not one slit established. A fine table is not useless; it is useless if it is resampled, and it is exactly what is needed if it is integrated.

How the filter and the resampling were computed #

The filter is a Fabry–Pérot etalon of stated order and finesse, evaluated in closed form. A notch is 0.88 × (1 − 0.95 A) with A the Airy function, and a band-pass is 0.02 + 0.9 A; the two are the same cavity read with its layers arranged for the opposite purpose. The default is order 24 and finesse 30, which gives a free spectral range of 22.9 nanometres and features 1.53 wide.

The table is the filter evaluated at the five-nanometre grid’s own points — which is what resampling a one-nanometre table onto five amounts to, since a one-nanometre table’s value at a grid point is the filter’s own value there. It is read back at one nanometre by each of the three rules and the colour summed there, against the filter evaluated in closed form and summed at a tenth of a nanometre.

The slit comparison replaces the table with the filter integrated through a triangular slit of five nanometres at each grid point, and changes nothing else. The overshoot is the largest amount by which a filled table’s values leave the interval a transmittance occupies.

What this leaves out #

An etalon is an idealisation of a coating. A real filter is a stack with a designed passband shape, ripple in its pass region and a blocked range outside it, and its sidebands are suppressed rather than periodic. What survives is the scale of its features relative to a five-nanometre grid, which is the whole of the argument; the exact numbers are this cavity’s.

The lamp is smooth throughout. Under a line lamp the filter’s features and the lamp’s would also have to be paired, which is the covariance problem two slits are not one slit measured, and it would be added to everything here rather than replacing it.

And the working grid is five nanometres. A one-nanometre working grid would carry the filter’s features and would cost almost nothing — the problem is entirely the convention, and the convention exists because five nanometres is a choice that was made for smooth samples under smooth lamps.

Still open: what a filter maker’s published table actually contains #

Everything here treats a published table as the filter’s own values at the table’s own points, which is the best case. What a maker actually publishes is a measurement, made on a spectrophotometer with its own slit, and that changes the question in a direction worth knowing.

If the published table came through a slit narrower than its step, it is a sampled signal and everything here applies. If it came through a slit as wide as its step, it is already integrated — and resampling it onto a five-nanometre grid is then a second, coarser sampling of an already-blurred signal, which is a different and less bad arrangement.

The measurement that would settle it is a reading of the maker’s own specification rather than a computation: the instrument, the slit width and the interval, for a dozen catalogue filters. The prediction is that most will state the interval and not the slit, which is the omission what the instrument reports is about, and that where both are stated they will be equal — because that is the ordinary instrument design, and it would mean published filter tables are safer than this essay’s worst case and still not safe on a grid four times coarser than they were measured at.

A finer sample is not more information if it is going to be resampled #

The habit is about where in a chain the sampling rate matters.

More samples are more information, and the instinct to prefer a finer table is right. What the instinct misses is that information survives only as far as the next sampling step: a signal sampled finely and then sampled coarsely retains what the coarse sampling retains, and nothing about having been fine first helps. The fine step bought nothing because it was spent before the bottleneck.

The move is to find the coarsest sampling in the chain and ask what happens before it. A low-pass applied before that step preserves energy; one applied after it preserves nothing that was already lost; and a fine table with no low-pass anywhere is a signal that will be aliased at whatever the coarsest step turns out to be.

The failure mode is to judge data by its own resolution. A one-nanometre table is better data than a five-nanometre one and produces a worse answer, because the pipeline it enters has a five-nanometre step in it and the five-nanometre table came through a slit.