Concept

Bandpass — where it appears

The width of the slice of spectrum an instrument reports at each sample point, as distinct from the interval between points. A slit wider than the structure being measured smooths a real spectrum, and a lamp with narrow lines is where the two disagree most.

Named by 20 essays across 2 fields — each of them below, with the objects they name alongside it.

A 20 nm bandpass, and what it does to the sample. The true reflectance, the same reflectance as a 20 nm instrument reporting every 20 nm returns it, and the slit function responsible, drawn at its own scale around 550 nm. The reconstruction is what every calculation downstream will use, and it differs from the truth by ΔE 2.97 under D65. Nothing in the file the instrument writes marks which values were measured and which were interpolated.

What the instrument reports

A spectrophotometer sees a sample through a slit of finite width, at a finite number of wavelengths. What it writes down is the truth convolved with its own bandpass, and nothing in the file says which values were measured and which were invented.

matching · Gamut

What a coarse wavelength grid costs, by source. Colour error against grid size, for four sources through one reflectance. Daylight survives every grid tested: 0.67 ΔE00 even at 40 nm. A source with lines in it does not — the narrowband source reaches 16.2. The grid is not a property of the arithmetic; it is a claim about what the light has in it.

Five nanometres is a choice

Every integral here is taken in five-nanometre steps, and the interval has never had to be defended. Coarsening it to twenty costs daylight two hundredths of a colour difference and a fluorescent tube six and a half — and which way of coarsening is used decides a further factor of six.

matching · Gamut

An instrument, as the only thing it really is. The 3 filters a bank of that size puts across the visible range, each drawn against wavelength. Everything the instrument can report about a spectrum is 3 numbers — the integral of the light against each of these — so the set of spectra it cannot tell apart is everything orthogonal to all 3 of them, which is 78 dimensions of the 81 this site works in. Three of these is a colorimeter in spirit; the eye is three of them too.

Three numbers cannot see a line

An instrument that returns three filtered readings of a spectrum determines a three-dimensional projection of it and is exactly blind to the other seventy-eight. On daylight that costs almost nothing; on a fluorescent tube, three quarters of the lamp lies in the part no reading reaches, and adding filters recovers it slowly.

A 12-nanometre notch at 546.1 nm under a fluorescent tube, tabulated five ways. The cost against a tenth-nanometre reference, on a five-nanometre grid, of a notched sample under a fluorescent tube, mercury lines on a phosphor bed, when the two factors of the colour are tabulated as points, when the lamp alone is measured through a five-nanometre slit, when the sample alone is, when each is measured through its own slit, and when the light the sample reflects is measured through one slit. The costs are 2.535 for both sampled at points, 0.379 for the lamp through a slit, 2.724 for the sample through a slit, 0.727 for both through their own slits, 0.019 for the product through one slit. The tick on the two-slit bar is the same two tables summed at a tenth of a nanometre, 0.749: what the separate slits leave is not the grid's.

Two slits are not one slit

A spectrometer's slit is what makes a coarse table honest, for a lamp and for a notched sample alike. But a colour is a sum over the product of the two, and a notch measured through one slit and a lamp measured through another are not the product measured through a slit. Under a smooth light the difference is nothing. Under a fluorescent tube a notch on the mercury line comes out 0.73 colour differences off from two slits — worse than no slit at all for some notches — and 0.02 off from one slit on the reflected light.

Six ways of tabulating one notch on one mercury line. A 12-nanometre notch centred on a fluorescent tube's 546.1 nm line, its colour computed on a five-nanometre grid six ways, on a logarithmic scale. Point sampling costs 2.54 colour differences and two separate slits 0.727. Sharpening both blurred tables with the published three-term correction takes it to 0.188 — a real improvement, four times better — and one slit on the light the sample actually reflects gives 0.019. The correction recovers the part of the damage that is a blur, and the part that is left is not a blur.

A linear repair for a bilinear loss

The Stearns correction sharpens a table blurred by a triangular slit, and the obvious question was whether applying it to a lamp's table and a sample's restores the product the two of them are wrong about. It restores four fifths of the damage and cannot touch the rest: a three-term filter is linear, the covariance two separately blurred tables discard is bilinear in the two factors, and no linear operator applied to each factor separately produces a bilinear term. What is left is ten times the one-slit answer, at every position of the notch.

One instrument, one slit, two requirements. The slit's width swept, with three measurements on a logarithmic scale: a smooth sample under the line lamp, where only the lamp's structure is at stake; the notched sample under a smooth lamp, where only the notch is; and the real case, both at once. The lamp wants a slit of 5 nanometres and the sample wants 1, and each wants what it wants for the same reason: a slit should spread a feature the grid cannot resolve and leave one it can. The real case is best at 3 nanometres — which is neither requirement's answer — and costs 0.247 there, an order of magnitude more than either requirement alone.

One slit, two requirements

A line lamp wants a wide slit, because a wide slit spreads a line where a coarse grid can see it. A notched sample wants a narrow one, because a wide slit fills the notch the grid could have resolved. An instrument has one slit. Measured on the two requirements separately the best widths are five nanometres and one; measured on the two together the best is three, which is neither — and it costs twelve times what the lamp alone would cost and thirty-four times what the sample alone would.

An interference notch filter, and where a five-nanometre grid lands on it. The transmittance of a Fabry-Pérot etalon of order 24 and finesse 20, drawn at a fifth of a nanometre, with the standard grid's points marked. Its features are 2.29 nanometres wide and spaced 22.9 apart, so the grid steps over them: between two adjacent grid points the transmittance rises and falls completely, and neither point records it. That is what a real coating looks like, and a Gaussian notch — which is what this collection's earlier work used — is a much gentler object.

A finer table is a worse table

A real interference filter is not a Gaussian notch. It is an etalon, with pass bands a nanometre or two wide spaced twenty-three apart, and a five-nanometre grid steps over them. Resampled from the maker's one-nanometre table its colour is out by five colour differences under every fill-in rule — the three rules agree to three decimals, because none of them is ever handed a sample inside a feature. The same filter measured through a five-nanometre slit is out by 0.03.

Every pair of slits, over 68 notches. The colour error, in ΔE₀₀ from the truth, for every pair of slit widths — the lamp's table blurred through the width down the side, the sample's through the width across — on 68 notches, the mean over all of them under a fluorescent tube. Circle area follows the error. The best pair is 5 nm on the lamp and 1 nm on the sample, at 0.26; the best single slit, on the diagonal, is 5 nm at 0.36. One slit on the reflected light, at 5 nm, averages 0.016.

A second slit buys a quarter

A line lamp wants a five-nanometre slit and a notched sample a one-nanometre slit, so an instrument with a slit for each should do much better than one with a single compromise. Over sixty-eight notches under a fluorescent tube, it does better by 28 per cent. One slit on the reflected light does fifteen times better than the best pair, and an oracle choosing the best pair for every notch is still six times worse. The error was never that the factors were flattened; it was that they were flattened separately.

Three bounds against the error they bound, over 68 notches under a fluorescent tube. Each notch placed across by its actual colour error from blurring the lamp and the sample separately, and up by a bound on that error, both on logarithmic scales; the dashed diagonal is where a bound equals the error, and a valid bound sits above it. Cauchy–Schwarz with the true window variances is above the diagonal on every notch, a median 14.7 times the error. Estimated from the blurred tables it falls below on 6 of 68, as low as 0.45 of the error. The Bhatia–Davis bound from the tables and declared ranges is above on every notch and a median 196 times the error.

The tables cannot bound what they discarded

A colour computed from a lamp's blurred table and a sample's blurred table is wrong by the covariance the two blurs threw away, and Cauchy–Schwarz bounds a covariance by two variances. With the true variances the bound always holds and sits fifteen times above the error. With variances read from the tables it fails on six of sixty-eight notches under a fluorescent tube and twenty-two under a laser projector — on the line, where the error is largest. A blurred table does not carry the width of a line, and the covariance depends on it.

What declaring a narrowest feature buys, and where it stops being true. The median looseness of a Cauchy–Schwarz bound whose lamp variance is bounded by a declared narrowest feature, against the width declared, for a fluorescent tube and a three-laser projector. Each lamp's own Bhatia–Davis bound — the peak declared and nothing else — is the upper dashed line, and the bound with the true variances is the lower one. The marks are the width each lamp's lines actually have. Declaring it truly takes the tube from ×196 to ×86 and the projector from ×30 to ×14. The open circles are declarations the lamp does not meet, where the bound falls below the error.

A declared width buys a factor of two

A colour engine given two separately blurred spectral tables cannot bound its own error from them, and the bound that always holds — the peak declared and nothing else — sits a median 196 times above the error under a fluorescent tube. Adding one number, the width of the lamp's narrowest feature, brings that to 86. It never fails on any declaration the lamp truly meets, it fails on 47 of 68 notches on one it does not, and its rank correlation with the error it bounds is 0.27.

Narrowing the slit makes the declared-width bound looser, not tighter. The median ratio of the declared-width bound to the error it bounds, over sixty-eight notches under each lamp, with the slit and the grid narrowed together from five nanometres to one and each lamp declaring its true line width; the dotted lines are the bound with both variances known. Under the tube the declared bound goes from ×86 at five nanometres to ×253 at one, while the true-variance bound stays between ×12 and ×15. Under the projector the declared bound lies on its true-variance bound at every slit.

A narrow slit shrinks the error, not the bound

A colour engine given two separately blurred spectral tables can bound its own error if the lamp declares its narrowest feature, and under a fluorescent tube that bound sits 86 times above the error at a five-nanometre slit. Narrowing the slit was supposed to close the gap, because a comb of narrow features stops fitting in a narrow window. It does the opposite. At one nanometre the error falls 27-fold and the bound only sevenfold, so the bound is 253 times loose and certifies no notch within a colour difference. The looseness was never the comb. It is the lamp's one declared peak, charged to every window.

The ceiling a table reads for each window, against the lamp it came from. The fluorescent tube between 525 and 590 nanometres, with the green and yellow mercury lines, and for each window of a one- and a five-nanometre slit the highest the lamp could be there according to the table's own entries. The dashed line is the tube's peak, which the global declaration charges to every window. At one nanometre the ceiling follows the lamp, a median 2.05 times each window's true peak; at five it sits on the global peak for twenty nanometres either side of each line.

A narrow table can vouch for its own peaks

The bound a colour engine can put on its bandpass error is loose because a lamp declares one peak for its whole spectrum and the bound charges it to every window. A table measured through a slit about as narrow as the lamp's lines can do better: no feature of the declared width can be blurred below a computable fraction of its height, so the table's own entries cap how tall the lamp can be in each window. At one nanometre that ceiling is never below the truth and sits twice above it, where the global peak sat twenty times above. The bound halves, and for the first time it certifies notches within a colour difference. At five nanometres it buys nothing at all.

A finer grid shrinks both arrangements, and the gap between them stays. The mean error over sixty-eight notches of the best fixed pair of separate slits and of one slit on the reflected light at the grid's own step, on grids of one, two and five nanometres, under the tube and the projector. The tube's best pair falls from 0.256 to 0.019 and reaches the product's five-nanometre error of 0.016 without passing it; the product on the same fine grid is 0.0007. Under the projector the pair at one nanometre is 0.135.

A finer grid keeps the price of separating

A sample's reflectance and a lamp's spectrum measured through separate slits can be reused under any other lamp; measured together through one slit on the reflected light, they are fifteen times more accurate and belong to that one lamp. A finer grid was expected to close the gap, because a one-nanometre table barely blurs anything. It closes most of the distance to the old answer and none of the distance to the new one. On a one-nanometre grid the best separate pair under a fluorescent tube is as good as one slit on the product used to be at five, and one slit on the product on the same fine grid is twenty-nine times better again.

A correction measured under one tube, carried to seven other lamps. For each lamp, the mean error over sixty-eight notch filters of the separable table (upper bar), of the separable table corrected by the ratio of product to separable colour measured under the census tube (middle), and of the product measured under the lamp itself (lower). Under the census tube the correction is the product exactly; under tubes with other line widths or a moved phosphor bed it removes nine tenths of the error; under a tube with its lines re-mixed, 63 per cent; under the laser and the LEDs it makes things worse.

A correction travels with its lines

Keeping a sample's reflectance and a lamp's spectrum as separate tables costs fifteen times the error of measuring their product under a fluorescent tube. The middle course proposed was to measure the product once under a representative tube, store the ratio of product to separable colour with the sample, and apply it under any tube. Under tubes whose lines are narrower or wider, or whose phosphors sit elsewhere, it removes nine tenths of the error. Under a tube with the same lines in a different mix it removes under two thirds, and less the further the mix moves. Under an LED or a laser it makes things worse — a white LED five times worse than no correction. A class of lamps, for this purpose, is a mix of lines.

How well the table's entries around the 546 nm line fit each width of line. For the tube's green mercury line, 1.2 nm wide, the seven table entries around it fitted by a line of each width from 0.2 to 12 nm through the same slit on a sloping background, centre, height and background free: the best fit's miss. Through a one-nanometre slit the miss has a sharp minimum at the true width and only widths near it pass half a per cent; through two, a broader dip; through three and five the curve is flat from the narrowest width tried up to about two, and any of them passes.

A narrow table declares its own lines

A bound on a colour table's bandpass error needs one number from the lamp's maker: its narrowest feature, 1.2 nm for a fluorescent tube. Through a slit about as narrow as that feature, the table's entries around each line carry its width. Measured to half a per cent, a one-nanometre table's entries admit only line widths within four per cent of the truth, and a bound built on the narrowest of them is within two per cent of the bound on the maker's number. Through two nanometres the table still declares a usable width. Through three it cannot rule out a line five times narrower, and the bound it supports is the global one. The failure is as sharp as predicted, and never unsafe.

A pair is refused once it is split by about a grid step. Two 1.2 nm lines centred on 578 nm and split by 0.3 to 4 nm, tabulated through slits of one, two and three nanometres on grids of the same step, and the least miss of any single line fitted to the entries round the peak. Each slit's curve crosses half a per cent between three quarters of its own step and one full step: a one-nanometre table refuses a pair split by 0.9 nm, a two-nanometre table one split by 1.8, a three-nanometre table one split by 3.

A table can refuse a pair it cannot resolve

A narrow spectral table reads each of its peaks as one line and declares the line's width from the entries round it, and the assumption under that is that every peak is a line. Split the mercury tube's yellow into the real pair at 577 and 579 nm and the table notices: through one and two nanometre slits no single line fits the entries within half a per cent, so the table refuses to read a width there. Swept apart, a pair is refused once it is split by about the grid step and admitted when it is closer — always as one wider line, up to 3.3 times too wide. Through three nanometres the pair reads as a 3.2 nm line, and a 3 nm europium band beside it reads the same.

Wings raise the width a table declares, and it follows the line's half maximum. The width the tube's table declares for itself at half a per cent — the narrowest width any of its peaks admits — as every mercury line is given Lorentzian wings from nothing to half its core's width, through one and two nanometre slits, beside the Voigt line's own full width at half maximum. With no wings both slits declare under 1.2 nm; with wings a quarter of the core's, 1.36 through 1 nm and 1.41 through 2 nm, against a half-maximum width of 1.37. Through one nanometre the table declares nothing at the widest wings: every line is refused.

A line's wings widen the width a table declares

A narrow table declares its lamp's narrowest line width by fitting each peak with a Gaussian, and the worry was a line whose profile is not Gaussian: a mercury line broadened by collisions has Lorentzian wings on its core, every line shares them, and a Gaussian fit can only take the wings in by widening. It does. With wings a quarter of the core's width, the table declares 1.36 nm for a 1.2 nm core — close to the winged line's own half-maximum width, 1.37. The prediction called that the unsafe direction. It is not: the ceiling and the variance the earlier bound builds on the declared width never fall below the winged lamp's own, because through a slit a winged line behaves like the wider line the table says it is.

Where the ceiling on a self-absorbed line holds, and where it does not. In the windows within four nanometres of the 546 nm line, the smallest ratio of the ceiling the tables put on a window's peak to the absorbed line's true peak, built on the narrowest width the table admits for the line and on the emitted 1.2 nm, through each slit. On the admitted width the ceiling holds to a depth of three and falls below the peak at 3.5 (×0.981) and 4 (×0.885) through two nanometres. On 1.2 nm it holds through two nanometres at every depth.

A reversed line is read as one line

A strong mercury line in a dense discharge absorbs its own centre on the way out: at an optical depth of one its top is flat, and beyond that it dips, until at four the line is two humps with a trough eighty per cent deep between them. The prediction was that a one-nanometre table refuses such a line from a depth of about one and a two-nanometre table reads it as a Gaussian about the width of its flat top, safely. Both hold, and the second only up to a point. A two-nanometre table admits the line at every depth to four, and from three and a half the width it admits is too wide for the humps: the ceiling built on it falls eleven per cent below the line's true peak. The only line that could make a declared width unsafe was one narrower at its peak than its entries suggest — and a reversed line is exactly that.

Where a two-nanometre table's ceiling on a notched line fails. In the windows within four nanometres of the 546 nm line, the ceiling built on the narrowest width a two-nanometre table admits for the line, as a multiple of the absorbed line's true peak, behind columns a whole, a half and a third as wide as the line. The uniform column's ceiling crosses below one at a depth of 3.42 and ends where the table refuses the line; the half and third columns' cross at 2.34 and 2.35 and are never refused, falling to ×0.73 and ×0.80 at a depth of five.

A notch in a line is never refused

A dense discharge's green mercury line loses its centre to the atoms between the glow and the tube wall, and those atoms are cooler than the glow, so their absorption is narrower than the line it cuts. With the absorbing layer half or a third as wide as the line, a one-nanometre table still refuses the line from an optical depth of one, exactly where it refused the broad dip. A two-nanometre table never refuses it at all: to a depth of five, with ninety-eight per cent of the centre gone, one Gaussian fits its entries within a seventh of a per cent. And the ceiling built on the width it reads fails from a depth of about two and a third — a whole unit sooner than behind a column as wide as the line — and never recovers, falling to three quarters of the true peak.

Where a reversed tube's table declares a width its ceiling cannot stand on. With every mercury line behind its own column, in proportion to its strength, the ceiling built on the width the whole table declares — the narrowest any of its peaks admits — as a multiple of the reversed lamp's true peak in the worst window, through one and two nanometres behind columns as wide as the lines and half as wide. The one-nanometre lines stop where the table refuses every peak and declares nothing. Through two nanometres behind the cooler column the table never stops declaring, and its declaration is unsafe from a depth of two and a half, falling to ×0.69.

A reversed tube declares from its least reversed line

A spectral table declares its lamp's line width as the narrowest width any of its peaks admits, and that declaration survived every reversed line so far because the tube's other lines were left unreversed to set it. A real discharge reverses its strong lines together. With every mercury line behind its own column, in proportion to its strength, the prediction was that a one-nanometre table would refuse every strong line and declare from the weak violet line, and a two-nanometre table would declare an unsafe width from a green-line depth of two and a half. The violet line is never read as a peak at all. The yellow line, the least reversed of the three that are, sets the declaration at every depth and widens it as it reverses. Through one nanometre the table refuses the blue and green lines and goes on declaring from the yellow — unsafely, in the violet's window, the one line the table never checked. Through two nanometres behind a cooler layer it refuses nothing, and from two and a half its declaration fails in the blue line's window, down to seven tenths of the true peak.

Named alongside it

The objects these essays reach for when they reach for this one.

Spectrophotometry Spectral structure Wavelength grid Instrument Worst case Fluorescent Declared input Bound Integration Measurement uncertainty Sampling Measurement condition