Colour Science & Perception
what the eye settles, and what it cannot
Field

What light is

A spectrum is a function, not a colour. Illuminants, reflectance, blackbody radiation computed from Planck's law, and what has already been discarded before the eye is reached.

51 essays. Read in order: Light.

Four standard illuminants, and how little they have in common. Spectral power distributions for A, D50, D65, E, on one scale. Illuminant A rises steeply toward the red; the daylight illuminants carry the atmosphere's absorption structure; E is flat by definition. All four are ordinarily called white.

A spectrum is not a colour

part 1 Blackbody spectra from Planck's law, 2000 to 10000 K. Each curve is computed from Planck's law and normalised to its own peak. The peak moves toward shorter wavelengths as temperature rises. Only two of these radiators peak inside the visible band at all — a 2000 K source peaks at 1449 nm, far into the infrared, and merely rises toward the red across everything shown here.

Blackbody and the colour of temperature

part 2 One reflectance, two illuminants, two colours. A reflectance peaking near 610 nm, and the colours it produces under D65 and A. The object has not changed. The light has, and colour is a property of the pair.

The illuminant is half the answer

part 2 triphosphor fluorescent — three narrow phosphors plus the mercury lines, and the white it produces. The spectral power distribution of a triphosphor source, normalised to its own peak, and the colour a perfect white reflector takes under it: chromaticity (0.3379, 0.3389), correlated colour temperature 5258 K at Duv -0.0035. The white looks ordinary. The spectrum producing it does not.

A lamp is not a blackbody

part 3 A sample with two reflectance curves, and neither below one. The apparent reflectance of an optically brightened sample, measured under D65 and A. It exceeds 1 — the shaded band — which no reflector can do: more light leaves at these wavelengths than arrives at them, because the sample absorbs in the violet and re-emits in the blue. And the two curves differ, so the sample has no single reflectance to store. The effect drawn here is a floor: most of the excitation band lies below 380 nm, outside the range computed here.

Some paper is brighter than white

part 3 Luminous efficacy of a monochromatic source, under the 2° observer. Lumens per watt against wavelength. The curve is the luminous efficiency function scaled by 683, which is what luminous efficacy of radiation is — the 1979 redefinition of the candela fixed 555 nm at exactly 683 lm/W and everything else follows. Marked at 450 nm (26 lm/W), 500 nm (221 lm/W), 555 nm (683 lm/W), 600 nm (431 lm/W), 650 nm (73 lm/W). Under this observer the peak reaches 683 lm/W.

What a lamp cannot give back

part 4 Colour temperature, and the number nobody quotes beside it. The Planckian locus in the 1960 UCS diagram — the only diagram on which correlated colour temperature is well defined — with four sources and the perpendicular from each to its nearest point. The temperature is where the foot of the perpendicular lands; Duv is how long the perpendicular is. halophosphate sits 0.0246 off the locus at 4513 K, which is a visible green cast that its colour temperature does not mention.

The sun is not one illuminant

part 5 Mixtures of 2700 K and 6500 K, on the diagram colour temperature is defined on. Both sources are Planckian radiators, so both sit exactly on the locus. Every mixture of them lies on the straight line between them, because mixing is addition and chromaticity is a projection of it — and the locus is curved, so the line is a chord. The equal mixture sits -0.0064 off the locus at a correlated colour temperature of 3953 K: a light that is measurably pink, specified by a number that says nothing about it.

Two lamps do not average

part 4 D65, and a source built to have its chromaticity and nothing else. The smooth curve is the CIE's daylight reconstruction at 6504 K. The other is five Gaussian emission bands whose weights were solved so that the two agree in chromaticity to 8.5e-10 — closer than any instrument could tell them apart when looking at the lamps. Three numbers were matched and seventy-eight were not, and everything either source falls on will report the difference.

There is no D65 lamp

part 5 A 100 hertz drive, and whether anybody sees it. 3 cycles of a 100 hertz drive at 100 per cent modulation. The number beside each is how far above the threshold for seen flicker its loudest harmonic sits: above one and a stationary observer sees the flutter, below it and only something moving does. Fusion is at 60 Hz at 100 cd/m², and moves 12.5 Hz for every decade of light.

A lamp has a waveform

part 4 Colour temperature, and the number nobody quotes beside it. The Planckian locus in the 1960 UCS diagram — the only diagram on which correlated colour temperature is well defined — with four sources and the perpendicular from each to its nearest point. The temperature is where the foot of the perpendicular lands; Duv is how long the perpendicular is. halophosphate sits 0.0246 off the locus at 4513 K, which is a visible green cast that its colour temperature does not mention.

White is a region

part 3 Several sources on one scale. planck, led, narrowband, each normalised to its own peak and drawn on shared axes with the colour each produces beside it. Every one of these is sold as white light and every one is ordinarily described that way; what they have in common is a chromaticity, and very little else.

A screen is a poor lamp

part 4 The colour of a beam, against the angle it leaves at. A conformal converter — an even layer laid straight onto the die — makes light leaving at θ cross 1/cos θ times as much of it, so it is more completely converted and the beam is warmer at its edge. From the axis to 75° the correlated colour temperature falls by 556 K and the whole difference is ΔE00 16.4. The flat line is the same emitters under a dome, whose path length is the same in every direction by construction: no angular colour at all, exactly, which is what a remote converter is sold for.

A lamp has a direction

part 5 Dimmed to a fiftieth, three ways. Switching a lamp on and off faster than anyone can see scales the spectrum and changes nothing else, so its colour temperature is a horizontal line and its chromaticity moves by 7.3e-14 ΔE00 — exactly, to floating point. Reducing the drive current moves the die's peak and cools the junction, which moves the mixture: ΔE00 3.5 across the range. A filament has no spectrum of its own to move; it is a blackbody at whatever temperature the power leaves it at, and it falls 1672 K. Three methods, one instruction, three colours.

Three ways to dim a lamp

part 4 A lamp switched on, and what it is still doing minutes later. The junction warms from ambient to 80 °C with a time constant of 150 seconds, and three quoted slopes act as it does: the die's peak moves, the die loses efficiency, and the converter loses quantum yield. The output falls 25 per cent and the colour moves ΔE00 2.6. Nine tenths of the way takes 405 seconds. The vertical mark is the eye's own slow adaptation constant, 60 seconds, for scale.

A lamp switched on is not the lamp measured

part 5 A surface in a room, from the moment the light goes on. How far a blue surface is from the colour it will settle at, second by second, for an observer who walked in from the snow as the lamp was switched on. It starts ΔE00 27 away, is still 6.1 away after a minute — the point at which the eye is conventionally said to have adapted — and does not fall under a unit until 295 seconds.

The room settles after the eye does

part 6 A colour rendering index, computed for everybody instead of for one observer. Each bar spans what 120 eyes make of one lamp: the same reflectances, the same reference illuminant, the same arithmetic, different colour-matching functions. The mark is the standard observer's own answer, which is the number printed on the box. One lamp's spread is 3.8 points wide while the whole range the standard observer puts these lamps in is 1.8 — so a ranking quoted to a tenth of a point is a statement about one set of tables. Two of the marks fall outside the population's range entirely.

The index is one observer's opinion

part 6 The same census, sorted by where the change of light came from. Each row is a change of illumination. The pale bar is how far it moves an ordinary surface for an observer who does not adapt; the solid bar at its left end is what is left after the observer has applied the one gain adaptation gives them, which is the ratio of the two whites in the CAT16 basis and is not fitted to anything. Sorted by where the change came from. The two kinds of light that existed before electricity sit at the top and leave the smallest share of themselves behind; the discharge lamps are worse, and the worst of them is d65 to a triphosphor tube at 33 per cent.

Which lamp changes are free

part 7 Three ways to dim a lamp, and only one of them is free. What an adapted observer is left with, as the same lamp is taken down to one per cent by each of the three methods. Duty-cycle dimming lies exactly on zero at every depth: it scales the spectrum, a scaling is a gain in every basis, and adaptation removes all of it. Current dimming moves the pump and the phosphor apart and leaves 0.15 at a tenth. A filament follows the Planckian locus, which is the largest chromaticity change of the three and leaves 3.63 — the ordering by chromaticity and the ordering by what a person sees are not the same ordering.

Only one dimmer is invisible

part 7 Six functions of wavelength, and the six different places they stop. Every table this collection integrates against, drawn over the range the body that published it defined it on. The scale is logarithmic so that the ultraviolet and the near infrared both fit. The bottom row is the range used here before the infrared band was added, and it is the intersection of the two rows that matter for an eye looking at a reflector — which is the right answer only while everything in the integral is being multiplied together. The daylight basis runs 80 nanometres further down than that intersection, and it was published that way because the ultraviolet in daylight is what makes a brightened sheet of paper glow. The analytic row is drawn to the edge of the plot because it has no edge: Planck's law is a formula and is exact at every wavelength, which is why illuminant A needs no table at all.

The tables do not stop together

part 8 A fluorescent sample is a matrix, and a reflectance is only its diagonal. The Donaldson matrix of a brightened sheet: how much light leaves at each wavelength for light arriving at each wavelength. A reflecting surface has entries on the diagonal and nowhere else, which is exactly the statement that light leaving at 440 nanometres arrived at 440. The block off the diagonal is the fluorophore — it takes light between about 305 and 420 nanometres and returns it between 400 and 500, wherever in that band it was absorbed, which is why the block is a rectangle rather than a smear along the diagonal. A spectrophotometer that reports a reflectance is reporting the diagonal and folding the block into it at whatever weight its own lamp happened to give.

A reflectance is a diagonal

part 6 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

part 9 The colour is right long before the spectrum is. The colour error of the projection, against the number of readings. At twelve readings the fluorescent tube's colour is right to 0.48 ΔE00 while 66% of its spectrum is still unmeasured. That is the trap in one line: a reconstruction good enough to pass any colorimetric check will predict a match under a second illuminant that does not happen, because the part it got wrong is exactly the part a different lamp weights differently.

The colour is right first

part 9 Every change of light in the census, under three bases. Fourteen changes of illumination, each drawn three times: the residual left by a gain in the basis built from the dichromat confusion points, in CAT16, and in the basis that minimises the average. The ordering between the three is the same on nearly every row, and — the part an average hides — the worst row is the same row for every basis nobody fitted, which is two bounces off the same wall. The difficulty belongs to the change rather than to the choice of axes — except for the fitted winner, whose worst row is D65 to a triphosphor tube instead. What a fit buys is not a uniform improvement; it is the abandonment of the one change everybody else is beaten by.

Which changes of light pay for it

part 10 Every change of light in the census, under three bases. Fourteen changes of illumination, each drawn three times: the residual left by a gain in the basis built from the dichromat confusion points, in CAT16, and in the basis that minimises the average. The ordering between the three is the same on nearly every row, and — the part an average hides — the worst row is the same row for every basis nobody fitted, which is two bounces off the same wall. The difficulty belongs to the change rather than to the choice of axes — except for the fitted winner, whose worst row is D65 to a triphosphor tube instead. What a fit buys is not a uniform improvement; it is the abandonment of the one change everybody else is beaten by.

Everyone is beaten by the same wall

part 10 The family does have a worst case, at a band no pigment can cut. The worst change of light a painted wall can produce, at each band width, with the wall's centre wavelength, depth and base optimised at every point. The horizontal axis is logarithmic in the width. The curve rises as the band narrows, turns over at about 6.02 nanometres, and falls again — a band that narrow returns too little light to move the white much. The previous round's search reported no worst case because its box stopped at ten nanometres, marked, which is on the wrong side of the turn. The peak is 28.54 ΔE00 against 28.38 at that floor, which is 0.6 per cent higher: wrong in principle, right in practice to a fraction of a per cent.

A notch a pigment cannot cut

part 10 The winner survives the census's own construction; the middle of it does not. One row per perturbation of a constant the adaptation census is built from — the imaginary wall's centre wavelength, its width, its depth, its base, the macular filter's density and the two lens ages — each moved by an amount plausible for that quantity in its own units, up and down, and then all of them together. Each row shows where the five published transforms rank under it. Bradford holds the first column in all 14 rows. The second and third columns, which the table as built separates by six parts in a thousand, change places in 2 of them — so that ordering was never a fact about the transforms.

The census is a construction too

part 9 Which steps of the census ranking the test set actually resolves. A horizontal bar for each of the 13 adjacent pairs in the census's ranking, from the smallest mean residual to the largest. A bar's length is the gap between the two rows in ΔE₀₀; the whisker on its end is twice the standard error of that gap, computed as a paired difference because the same 125 surfaces score both rows. Where the whisker reaches back past zero the pair is not ordered by this test set, and 4 of the 13 are in that state — marked. The largest steps, at the two ends of the ranking, are twenty standard errors wide and are not in doubt at all. The smallest is four parts in ten thousand between two rows the table prints as different numbers.

Four steps the test set cannot order

part 11 Which fourth dimensions are expensive, and how fine is too fine to matter. Two curves on axes of how many half-cycles a cosine fourth basis function makes across the visible band, against what it costs the matrix theorem in ΔE₀₀, at a fixed ten per cent amplitude. Both curves touch zero at exactly one and two half-cycles: those are the family's own second and third basis functions, so a fourth coefficient along them adds no dimension and a change of light stays exactly a matrix. Between them the cost climbs, reaches a maximum, and — for the smooth source — falls away again, because structure finer than the scale on which three broad cone sensitivities differ integrates to nearly nothing. The two curves part company at the fine end. Under daylight-to-tungsten the cost has fallen by a factor of 2.2 from its peak; under daylight-to-a-triphosphor-tube it has barely fallen at all, because a source with three narrow emission lines has structure of its own at that scale for the surface's structure to beat against. The observer is identical in both curves.

A fourth dimension has a shape

part 11 The census as the surfaces stop being three-dimensional. A slope chart with three columns — a test set with no fourth reflectance dimension, one with a fourth dimension at ten per cent amplitude, and one at twenty — and a line per change of light. Almost every line rises: a surface with structure the observer's three channels cannot follow is a surface an adaptation gain handles worse. Two lines are drawn heavy. daylight to a three-primary display rises fastest, by 90 per cent, because a source made of three narrow lines is precisely the instrument that cannot see a fourth reflectance dimension. And daylight to a triphosphor tube falls — the only row that does — because a triphosphor tube already samples the spectrum at three places, so extra structure in the surface is partly averaged away rather than added. The order of the middle of the table is not the same at the two ends; the extremes do not move.

The row a fourth dimension improves

part 12 The adaptation census in six units, calibrated onto one scale. Each line is one of the fourteen changes of light in the adaptation census, drawn across the six units the results could have been published in. Every unit is multiplied by the single factor that best carries it onto ΔE2000 over a reference sample of surface pairs, so the vertical axis means the same thing in every column and a sloping line is a disagreement rather than a change of scale. The levels move by up to a factor of two. More to the point, the lines cross: ΔEok puts 10 of the 91 pairs of rows in the other order, and CAM16-UCS, the only appearance unit here, puts the fewest — 2.

The census in six units

part 12 A radiance factor, split into the part that was reflected and the part that was not. The two components of what leaves a coated press stock under M₁ — D50 including its ultraviolet. The lower band is the reflected component, which is what a reflectance curve means and is everything a reflectance-based model can hold. The band above it is light emitted at wavelengths it did not arrive at, and its total is decided by how much ultraviolet the source had rather than by anything about the sheet's colour. The line at one is the boundary a reflecting surface cannot cross; the sum reaches 1.15 at 430 nanometres.

The ultraviolet is half the product

part 12 Six functions of wavelength, and the six different places they stop. Every table this collection integrates against, drawn over the range the body that published it defined it on. The scale is logarithmic so that the ultraviolet and the near infrared both fit. The bottom row is the range used here before the infrared band was added, and it is the intersection of the two rows that matter for an eye looking at a reflector — which is the right answer only while everything in the integral is being multiplied together. The daylight basis runs 80 nanometres further down than that intersection, and it was published that way because the ultraviolet in daylight is what makes a brightened sheet of paper glow. The analytic row is drawn to the edge of the plot because it has no edge: Planck's law is a formula and is exact at every wavelength, which is why illuminant A needs no table at all.

The grid is a range, not an index

part 12 a fluorescent tube, mercury lines on a phosphor bed, with a 5-nanometre grid marked on it. The light drawn at a fifth of a nanometre, with the 5-nanometre tabulation points marked beneath. 38 of the 53 points carry more than a twentieth of the peak. What a summation over those points computes is not an approximation to the area under this curve; on a spectrum with features narrower than the spacing it is a different quantity, and the difference depends on where the points fall rather than on how many there are.

The index is a choice too

part 13 What an instrument's slit width does to a tabulated colour. The horizontal axis is the full width of a triangular slit, from zero — perfect point sampling — to twenty nanometres; the vertical is the distance from the true colour, logarithmic. For a smooth light the lines are flat: a slit narrower than any feature changes nothing. For a line spectrum they fall off a cliff at the left. Point-sampling a mercury line at five nanometres costs 1.26 ΔE₀₀ and integrating the same spectrum through a five-nanometre slit costs 0.014. A spectrometer does not sample a spectrum; it integrates one, and the blur everybody would remove if they could is what makes a five-nanometre table safe.

The slit is what makes it legal

part 14 What three fill-in rules cost when a five-nanometre table is read at one. A five-nanometre table read at one nanometre by three rules — hold the value, straight lines, a cubic through four points — each compared with the same tenth-nanometre reference. The dashed rule is the answer obtained by summing the table as it stands, at 0.00000 ΔE₀₀. Two of the three interpolations are worse than not interpolating. That is not a paradox: the summation's error is already small because the normaliser cancels most of it, and an interpolator introduces a shape the original curve did not have, which the cancellation cannot touch. A finer grid is not more resolution when the table is not finer.

A finer reading of a coarser table

part 14 How much the answer moves when the 5-nanometre grid is slid through one cell. Each bar is the spread of one light's colour across five grid origins, all at the same 5-nanometre step, in ΔE₀₀. A smooth light barely moves, and what movement it has is the end cells rather than the sampling. The fluorescent tube moves by 3.18 units and the laser projector by 35.0, because their emission lines are narrower than the step and whether a sample lands on one is a coincidence of arithmetic. This is the measurement that separates a quadrature error from an aliasing error, and no average over origins can substitute for it.

Where the grid starts

part 14 What each end of the 380–780 nanometre range costs, by light. Two bars per light, on a logarithmic axis: the upper is what extending the range down to 300 nanometres moves the answer, the lower what extending it up to 830 does. The asymmetry is the whole figure. A thermal source has about a fifth of its power outside this collection's range and almost all of it at the long end, where the observer is already zero; what costs money is the short end, where the observer is small but not zero and daylight is still strong. A light with no ultraviolet — an LED lamp, a laser — pays nothing at either end, which is the pairing again: a range only costs what the light puts in it.

Two ends and one is empty

part 14 The two tabulation choices over forty-two surfaces, under a 6500 K thermal radiator. Each column is one choice, measured over a family of forty-two analytic reflectances rather than on a single example: an absorption band of stated centre, width and depth. The four marks are the smallest, the median, the ninety-fifth percentile and the largest cost in ΔE₀₀, logarithmically. Under a smooth light the range is worth 9.1 times the step at the median, so a collection wanting one repair should widen its range rather than refine its step — and under a fluorescent tube the ranking reverses outright.

Which end to buy

part 15 The rectangle sum against the trapezoid sum, under a 6500 K thermal radiator. Colorimetry's summation is the rectangle rule at the tabulated points. The trapezoid rule differs from it by exactly one thing — half a cell at each end of the range — and the gap between these two lines is therefore that term and nothing else. At five nanometres it is a factor of 14.3, which means the number everybody calls a sampling error is mostly a truncation error wearing the step's clothes. On a light whose lines are narrower than the step the two rules agree to three decimal places, because there the error really is the sampling.

The endpoint term has a name

part 15 What a tabulation step costs, by light, on paper. The horizontal axis is the tabulation step in nanometres, from one to twenty; the vertical is how far the resulting colour is from the same integral taken at a tenth of a nanometre over the same range, in ΔE₀₀, on a logarithmic scale. Each line is one light. The three with no feature narrower than the step fall smoothly and stay below a tenth of a unit at five nanometres, which is the grid used throughout. The fluorescent tube and the laser projector do not fall at all: their lines are narrower than any step drawn here, so the answer depends on where the samples land rather than on how many there are. The sample is held at paper throughout.

A grid is not a resolution

part 15 The share of each light outside 380–780 nanometres, as power and as visual product. Two measurements of the same truncation. The upper bar is the fraction of the light's radiant power that lies outside the range; the lower is the fraction of the product of light, sample and observer — which is what a colour is made of. A thermal radiator puts a fifth of its power outside and about a thousandth of its colour, because the observer is zero where most of that power is. The gap between the two bars is the observer's own tails doing their job, and reading the upper number as though it were the lower is how a range gets argued about without being measured.

A lamp is two audits at once

part 15 A 2-nanometre notch, and what two five-nanometre grids record of it. The reflectance of a sample with a 2-nanometre notch at 552.3 nm, drawn finely from 535 to 570 nm. The dark ticks are the samples of a five-nanometre grid starting at 380 nm and the pale ticks those of the same grid started half a step later. The true minimum is 0.06; the first grid's deepest sample reads 0.70 and the second's 0.08. The sample has put a line into a calculation whose light has none.

The line can be in the sample

part 16 What a five-nanometre grid costs a steep-sided notch, against its width, under a 6500 K source. The cost of a five-nanometre grid starting at 380 nm, against a reference at two hundredths of a nanometre, for a sample with a flat-bottomed notch at 552.3 nm under a 6500 K thermal radiator, against the notch's width from 2 to 30 nm, for edges rising in 0.4, 2.2, 6.6 nm. With the steepest edges the cost is 0.02 at 10 nm, 1.99 at 12.5 and 0.03 at 20: it rises and falls with the step as its period and does not die away as the notch widens. With the softest edges it stays under 0.10 at every width.

The cost of a steep notch repeats every step

part 17 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

part 17 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

part 18 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

part 18 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

part 18 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

part 19 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

part 19 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

part 20

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