The line can be in the sample

Which end to buy reduced three tabulation decisions — the step, the range and the rule for the values between — to one comparison: the width of the light’s narrowest feature against the step. A smooth light is safe at five nanometres, a light with features a few times the step is safe if its table is not resampled, and a light with lines narrower than the step is not safe on any grid. That essay said where its own rule would fail. A sample with structure narrower than the step would put a calculation in the unsafe regime with the light having nothing to do with it, and every sample it tested was smooth.

The governing feature is the product’s #

A sample with a narrow notch puts a calculation in the aliased regime under any light, and under a line lamp even a notch much wider than the step is fragile where its edge meets a line.

• A two-nanometre notch under a 6500 K thermal source costs 2.14 colour differences at five nanometres and spreads 2.02 across five grid origins.
• A five-nanometre notch costs 0.27, and from eight nanometres up the cost is two hundredths or less.
• Under a fluorescent tube the cost does not vanish as the notch widens: 1.20 at twelve nanometres and 1.38 at twenty, with origin spreads of 3.62 and 5.16.
• A twelve-nanometre notch walked across the tube’s 546.1 nm line spreads 8.13 when it sits on the line, and 0.17 at its quietest.

A notch a pigment cannot cut #

Pigments and dyes absorb over bands a hundred nanometres wide or more, because their absorption comes from electronic transitions broadened by their molecular environment. A notch a pigment cannot cut found exactly that limit in the widths of surfaces that can occur in a painted room. Every surface in the tabulation measurements was smooth for that reason, and the rule built on them read the light’s features and not the sample’s.

Some reflectances and transmittances are not made of pigment. An interference filter’s notch is decided by layer thicknesses and can be two nanometres wide; a laser-line filter is made to be that narrow; a structural colour’s reflection band can be ten. They are ordinary objects in instruments, in safety eyewear, in coatings and in nature, and a colour is computed from them on the same five-nanometre grid as a painted wall.

What the grid records #

A sum on a five-nanometre grid evaluates the sample at eighty-one wavelengths. A notch two nanometres wide falls between two of them, or on one, depending on where the grid starts.

The notch at 552.3 nm has a true reflectance of 0.06 at its centre and 0.72 away from it. A grid starting at 380 nm samples 550 and 555, both outside the notch, and records a deepest value of 0.70 — it has missed the notch almost entirely. A grid starting half a step later samples 552.5, inside it, and records 0.08. The same sample, the same step, the same light, and one grid sees a notch while the other sees almost a flat reflectance.

At five nanometres wide the notch is as wide as the step. The first grid’s deepest sample reads 0.35 and the second’s 0.06: still different, but both see a notch. By eight nanometres every grid origin places at least one sample well inside the notch, and the sum becomes a fair estimate of the integral.

Width against cost, under five lights #

The cost of the grid on a notched sample can be measured against a tenth-nanometre reference over the same range, as the smooth samples were, and repeated across notch widths under each light.

Under a tungsten lamp, a 6500 K source and a phosphor white LED the curves are nearly the same: about 1.2 colour differences at a one-nanometre notch, 2.1 to 2.4 at two, 1.7 to 1.9 at three, about 0.3 at five, and two hundredths or less from eight up. Under a three-emitter LED the costs are about half of those. Under the fluorescent tube the curve falls to 0.20 at five nanometres and then rises again — 0.46 at eight, 1.20 at twelve, 1.38 at twenty, 1.58 at forty-eight.

The cost at one origin is partly luck — a one-nanometre notch costs less than a two-nanometre one at the 380 nm origin because that grid misses the narrower notch more completely. The robust quantity is how far the answer moves when the origin does.

Under the three smoothest lights the spread is 2.7 to 3.0 at one nanometre, 2.0 to 2.2 at two, 1.6 to 1.8 at three, 0.3 at five and two hundredths or less from eight up. That is the aliasing band of the tabulation rule exactly, occupied by a sample under a light with no features. Under the fluorescent tube the spread is 2.5 at one nanometre, 0.67 at five — and then 0.86 at eight, 3.62 at twelve and 5.16 at twenty.

The sum does not know which factor had the feature #

The three smooth lights give nearly the same curve, and the reason is the restated rule in miniature.

A colour is a sum over the product of the light and the sample, and the sum is taken after the product is formed. Nothing in it records whether a narrow feature belonged to the light or to the sample. A two-nanometre notch in a sample under a smooth lamp and a two-nanometre dip in a lamp’s spectrum falling on a smooth sample are the same kind of product — a narrow feature multiplying a broad one — and a five-nanometre grid mis-samples both by the same arithmetic.

Across the few nanometres the notch occupies, the tungsten lamp, the 6500 K source and the phosphor LED are each close to flat, so each acts there as little more than a scale factor, and the notch’s cost is set by the notch. The fluorescent tube is the one light that is not flat there. Its mercury line sits 6.2 nm from the notch’s centre, which is inside the reach of a notch twelve nanometres wide and well outside the reach of one two nanometres wide. That is why the tube’s curve keeps company with the others for narrow notches and parts from them as the notch widens into the line.

Where a notch meets a line #

The fluorescent tube’s curve rises with notch width for a reason that shows what the product rule means.

The tube has a mercury line 1.2 nm wide at 546.1 nm, which the grid already cannot hold on its own: a smooth red pigment under that tube already moves by 3.18 colour differences as the grid slides, and the notched sample, with its notch well away from the line, by one to one and a half. A notch changes how much of the line’s power the sample returns, and a notch whose edge sits on the line changes it steeply — so the product of the sample and the light has a line whose height depends on exactly where the notch’s edge falls, and the grid’s sampling of that line depends on the origin.

Walked across the line, a twelve-nanometre notch’s spread rises from about 1.4 at 530 nm to 8.13 at 546 nm, falls to 0.17 at 557 nm and returns to about 1.6. A notch more than twice as wide as the step is the most fragile sample in the set when it sits on a line, and the rule read on the light alone — a 1.2 nm line, so unsafe — or on the sample alone — a twelve-nanometre notch, so safe — would get neither the size nor the position right.

The rule, restated #

The decision procedure’s comparison was stated as a property of the light because every sample it met was smooth. Stated for any sample, it is the width of the narrowest feature in the product of the light and the sample, against the step.

Under a smooth light that is the sample’s narrowest feature; under a line lamp with a smooth sample it is the line; with both, it is whichever is narrower, and where a sample’s edge crosses a line the product can have a sharp feature even when neither factor is sharp at that place. The three-band classification survives with that substitution: above about four the sum is safe, between one and four it is safe if not resampled, below one it is not safe at all.

Narrow structure is ordinary where interference makes the colour #

A pigment cannot cut a two-nanometre notch; a stack of thin layers can, and so can a single film once it is thick enough. The colour is in the thickness for a soap film, and the same interference gives a film’s reflectance a series of fringes whose spacing narrows as the film thickens: for a film of index 1.33 the fringes sit nearly sixty nanometres apart at two micrometres thick and under six apart at twenty. A thick coating read through a five-nanometre table is a notched sample many times over.

The same is true of the filters instruments and displays are built from — dichroic mirrors, anti-reflection coatings, the infrared-cut filter in a camera — and of the notch filters spectroscopists use to block a laser line. Each has a transmittance its maker usually tabulates finely, and each is summed at points whenever its colour is computed. A table’s step records where values were written down, not how finely the structure in them was resolved — a grid is not a resolution — and resampling a fine table to five nanometres before summing it is how a correctly measured notch becomes an aliased one. The maker’s fine table is the honest object; the resampled one is a convenience that has quietly changed the sample.

What an instrument does about it #

A spectrometer does not sample a spectrum at points. It integrates through a slit, and the slit is what makes it legal: a five-nanometre slit blurs a line enough that a five-nanometre table holds it honestly. The same argument applies to a notched sample measured by an instrument — the instrument’s own bandpass blurs the notch before any table is written.

The trouble is a sample whose reflectance is known as a formula or a fine table and is summed at points, which is what a calculation does and an instrument does not. A computed colour of an interference filter, a structural colour or a laser-safety coating on a five-nanometre point grid is in the aliased regime whatever light it is computed under, and the repair is the same as for a line lamp: integrate the product through a bandpass at least as wide as the step, or compute on a grid fine enough to hold the notch.

Three numbers keep no record of it #

Once the sum is taken the notch has left no trace in the answer. A computed colour is three tristimulus values, and nothing in three numbers says that one came from a sum that caught a notch and another from a sum that missed it. Three numbers cannot see a line, and they cannot see whether a notch was sampled either.

That decides what a calculation should report. A colour computed at one grid origin carries no evidence of its own error, and the cost against a fine reference is available only to somebody who has computed the fine reference. The spread across origins needs no reference: slide the grid a nanometre at a time, recompute, and see whether the answer moves. For a smooth sample under a smooth light it barely moves. For a two-nanometre notch it moves by two colour differences, and for a twelve-nanometre notch on a mercury line by eight. The test costs five sums and it is the one check a calculation can run on itself. It does not say what the right answer is, only whether the grid is deciding it, which is the question the tabulation rule exists to answer.

The null the whole measurement rests on #

One sample costs nothing on any grid under any light, exactly, and it is the notch with no depth.

A flat reflectance computes to the same colour on every grid, because the white it is judged against is summed on the same grid and the two sums differ only by a constant factor, which divides out. A notch of zero depth is a flat reflectance, and its computed cost is under 10⁻¹³ under all five lights at all five origins — the arithmetic’s own floor. A neutral has no grid established that identity for neutral samples; here it is what makes every non-zero cost above attributable to the notch rather than to the calculation itself.

What was computed, and how #

The sample is a constant reflectance of 0.72 with a Gaussian notch of depth 0.66 and a stated full width at half maximum, centred at 552.3 nm unless walked. Each light is a closed-form spectrum: two thermal radiators, a phosphor LED, a three-emitter LED, and a fluorescent tube modelled as a phosphor bed with four mercury lines 1.2 nm wide. The observer is given in closed form, so it can be evaluated at any wavelength.

The reference colour is the sum at a tenth of a nanometre from 380 to 780 nm, with the white summed on the same fine grid. The cost is ΔE₀₀ between the reference and the colour summed on a five-nanometre grid over the same range with its own white, at the 380 nm origin; the spread is the range of that cost across origins offset by one to four nanometres.

Where the measurement stops #

The notch is Gaussian. A real interference filter’s notch has a flat bottom and steep sides, which makes the sampling worse at a given width, and ripple outside the notch, which adds features of its own.

The reference is a tenth-nanometre sum, twelve times finer than the tube’s lines and twenty times finer than the narrowest notch; it is converged for these shapes, and would need to be finer for a laser line.

And only point sampling is measured. An instrument with a slit, or a calculation that integrates through one, changes every number here, and the size of that change on notched samples is not computed.

The habit #

The habit is about a rule derived on a class of inputs that happened to share a property.

A rule stated in terms of one factor — the light — because every case examined had the other factor — the sample — well-behaved is a rule with a hidden condition. It is right on everything it was built on and wrong the first time the other factor is not well-behaved, and the failure has the same shape as the cases the rule already knew about.

The move is to restate the rule in terms of what the mathematics actually involves. A sum of a product cares about the product.

The failure mode is to check a new case against the rule’s stated factor and pass it. A smooth light does not make a calculation safe when the sample has the line.

Who noticed it first #

That sampled integrals of narrow spectral features alias is the Nyquist condition, and that interference filters and structural colours have narrow features is standard. That instruments integrate through a bandpass for exactly this reason is part of every standard on spectral measurement.

That the decision procedure for tabulation defects must read the product rather than the light, and the specific interaction where a notch’s edge crosses a mercury line, are computed here.

Still open: a notch through a slit #

The repair for a line lamp was a slit, and it is the obvious repair here. How wide a slit a notched sample needs, whether a notch’s steep sides need a wider one than a line’s, and what the residual is at the position where a notch meets a line, is the same bandpass calculation run on the product — and it would say whether computed colours of interference coatings can be trusted on the standard grid at all.