A lopsided band does not break the trade

A fourth emitter spends the gap it fills put two wants of a lamp designer on one axis. A three-emitter LED lets a pigment that cannot change its reflectance faster than over forty nanometres reach most of its ideal colour solid, because the lamp is nearly dark where a slow reflectance does its changing. Adding a fourth emitter to render colours better takes some of that darkness back. The essay swept the fourth emitter’s position and found the two wants peaking in different places: at 580 nanometres the lamp rendered at a fidelity index of 88 and the pigment lost more than a per cent of its reach in 109 of the solid’s 307 directions; at 540 it rendered at 81 and cost 65.

That essay ended on a suspicion about itself. Every fourth emitter it tried was a Gaussian — symmetric, twenty-four nanometres wide, one power. A gap is spent when something is put in the middle of it, and rendering is bought by having light where there was none, but not necessarily in the middle. A band with a sharp edge on one side and a long skirt on the other could fill part of an amber gap and leave the rest dark. The proposal named its prediction and its alternative: “the frontier moves and does not disappear” — or, if some asymmetric band renders like the 580 emitter and costs like the 540 one, “the whole conflict was an artefact of insisting the emitter be symmetric, which is a thing about the model and not about lamps.”

The prediction holds. The details around it do not.

A few directions, and no escape #

Freed to be lopsided, a fourth emitter moves the frontier between rendering and a slow pigment’s reach by between one and seven of the solid’s three hundred directions. To render at a fidelity index of 88 costs 103 directions with the best symmetric band and 97 with the best skewed one, against the 540-nanometre emitter’s 65. No band of either family renders like the 580 emitter and costs like the 540.

• Freeing width and power moved the frontier as far as freeing the shape. The earlier sweep’s one width and one power needed 109 directions to render at 88; a symmetric band of any width and power needs 103; a lopsided one 97.
• The asymmetry that helps leans the other way from the proposal. Every skewed band on the frontier rendering at 85 or better but one has its long half on the red side, filling the space between the amber and the red emitter and leaving the green-to-amber gap dark.
• What the fourth band renders is the reds, not the white. Rebalancing the three emitters alone moves the lamp’s white from under 1,800 to over 12,800 kelvin and never renders above 80.6. The fourth band’s work shows up in the colour shifts of the orange and red test samples.

Two families, searched alike #

The fair test of the proposal is not a lopsided band against the earlier essay’s one symmetric band. It is a lopsided family against a symmetric family given every other freedom the lopsided one has. So both families are swept over the same grid: centres every five nanometres from 510 to 610, powers of 0.15, 0.3 and 0.5 against the other emitters’ 0.55 to 0.70, and widths of 6, 12, 20, 32 and 50 nanometres. A symmetric band uses one width on both sides. A skewed band — a split Gaussian, the simplest shape with an offset and an asymmetry — uses one width on its blue side and another on its red. That gives 315 symmetric lamps and 1,260 skewed ones.

Each lamp gets two scores. Its fidelity index, the collection’s CAM16-UCS rendering measure against a reference illuminant at the lamp’s own colour temperature, over twelve test samples; and its cost to a slow pigment, the number of directions of the object-colour solid in which a reflectance that cannot change faster than over forty nanometres falls more than a per cent short of the ideal reach, the census three lines spare a slow pigment introduced.

The two step lines run together from end to end. The skewed family’s is above the symmetric one’s nearly everywhere, which it has to be — a split Gaussian with equal halves is a Gaussian, so the skewed family contains everything the symmetric one could do but for the grid’s exclusion of the equal pairs — and the gap between them is never more than a point and a half of fidelity or seven directions of cost. Both lines climb steeply from the three-emitter lamp’s corner: the first few points of fidelity are nearly free, and the last few, from 86 to 90, cost thirty or forty directions whichever shape is allowed.

That climb is the trade the earlier essay found, and it is still there. A lamp designer asking for more rendering pays in a slow pigment’s reach at the same rate whether the fourth emitter is symmetric or not.

What each freedom saves #

Read along the horizontal axis, the figure is a price list. The earlier sweep, confined to a 24-nanometre band at power 0.3, pays 87 directions to render at 84 and 109 to render at 88, and cannot render at 89 at all. A symmetric band of free width and power pays 82 and 103, and reaches 90 for 106. A skewed band pays 77 and 97, and reaches 90 for 101.

So the first freedom, width and power, bought as much as the second, shape. At 88 it is six directions each. The proposal’s diagnosis — that a symmetric emitter was the constraint — was half the story: the earlier sweep was confined in more ways than one, and loosening any of them moves the frontier a little.

And no amount of either reaches the 540 emitter’s cost at a high rendering. The dashed line is 65 directions, what the 540-nanometre emitter cost while rendering at 81. At 88 and above the cheapest skewed band costs thirty directions or more beyond it. The proposal’s escape — a band that renders like the amber emitter and costs like the green one — would sit at the right of this figure on the dashed line, and the nearest lamp to that corner is thirty-two directions above it.

Which way the useful bands lean #

The skewed bands that render well lean red. Of the twelve on the frontier rendering at 85 or better, eleven have a longer half on the red side than the blue, by factors of 1.6 to 8. They sit between 575 and 590 nanometres, with a sharp blue edge and a skirt running towards the red emitter at 625. The bands that lean blue are, with one exception at 590 nanometres, cheap ones between 545 and 570, rendering in the low eighties.

The proposal pictured the opposite: a band centred near 560, “whose skirt fills the near side of the amber span and leaves the far side dark.” Its geometry was reasonable — the amber span runs from the green emitter at 528 to the red at 625, and a skirt reaching back into it from a band near 560 would fill its near side. The census does not want that span filled at all.

The skewed lamp keeps the gap between 540 and 575 exactly as dark as the three emitters left it. Its fourth band rises over twenty nanometres to its peak at 585 and runs its long red half across the dip between 590 and 620, where the symmetric band of the same centre falls back almost to nothing. That is where it saves its six directions. The gap has to be dark, not the line narrow found that what a slow pigment needs from a line lamp is darkness where its reflectance changes, and under the three emitters the directions of the solid they spare put their edges unevenly: more than half of them between 480 and 515 nanometres, about one in five between 540 and 575, and one in seven between 590 and 620. A lopsided band spends light where fewer of the spared edges are, and that is the whole of its advantage — a small one, because the dip it fills is not empty of edges, only emptier.

What the fourth band renders #

The proposal carried a second hypothesis inside its prediction: that the rendering the far amber positions buy “is partly a shift of the white point, and a shift of the white point cannot be had without power where the white needs it.” A lamp’s white does move when a fourth emitter is added. The three emitters as built sit at 7,500 kelvin and well above the Planckian locus, a Duv of +0.033, which is a visibly greenish white. The cheapest symmetric lamp rendering at 88 sits at 5,060 kelvin and a Duv of +0.016; the cheapest skewed one at 3,910 kelvin and +0.001, nearly on the locus.

Moving the white with the three emitters already there does not render. Over 150 rebalancings of their powers, the white ranges from 1,800 to 12,800 kelvin and from 0.053 below the locus to 0.093 above it — far further than any fourth emitter moves it — and the fidelity index never exceeds 80.6. The best rebalancings are not those nearest the locus; they are greenish whites 0.05 to 0.08 above it. And the four-emitter lamps that render at 89 or better all have whites between the locus and 0.024 above it, but so do many that render in the seventies. Nearness to the locus is common to good and poor lamps alike.

That is not a surprise once the index is read: it compares each lamp with a reference at the lamp’s own colour temperature, so a white that has moved is compared with a reference that has moved with it. What the index punishes is a lamp that renders samples differently from its own reference, and a lamp with nothing in the amber or red renders the orange and red samples differently from anything.

The fourth band’s work is at the red end of the samples. Under the three emitters the test samples whose reflectance peaks at 600, 625 and 650 nanometres shift by 4.8, 6.2 and 5.1 from their reference colours; under the cheapest symmetric lamp rendering at 88, by 2.7, 2.8 and 1.0. The samples at the blue end barely move. Three emitters leave a reflectance that peaks in the orange with a single narrow red line to reflect, and the fourth band gives it something to reflect in the amber — which is exactly why a fourth band has to sit where the solid’s slow edges are.

The skewed lamp renders those reds slightly worse, 3.6 and 3.1 against 2.7 and 2.8, and makes it up on other samples. That is its trade: a band whose sharp side is at 575 lights less of the orange than a symmetric band at the same centre, and costs a slow pigment less for the same reason.

What a lamp designer should take from it #

The conflict is real. A fourth emitter spends the gap it fills found it with one shape of band, and it survives every width, power and asymmetry tried. Rendering the orange and red samples needs light between the green and red emitters, and a slow pigment needs that stretch dark. No arrangement of a fourth band has both.

The shape buys a few per cent, and it is the red-leaning shape. A designer who wants a four-emitter lamp to be kind to slow pigments should put its fourth band’s sharp edge on its blue side and let its skirt run into the red emitter. That saves five or six directions of the solid at a high rendering — two per cent of it — or, held at the same cost, renders up to a point and a half better.

And the white is a separate choice. A designer can put a four-emitter lamp’s white nearly anywhere by balancing its powers, as the rebalanced three-emitter lamps show; what the fidelity index rewards is the fourth band’s presence in the amber, and what a slow pigment charges for is the same presence. There is no D65 lamp and a lamp is not a blackbody are where the white of an emitter lamp was first treated as a design variable; this census says it is not the variable the trade turns on.

How the lamps were scored #

The three emitters are this collection’s narrow-band LED: Gaussians at 455, 528 and 625 nanometres, 20, 33 and 18 nanometres wide, at powers 0.55, 0.62 and 0.70. A fourth band is added on the collection’s five-nanometre table. A split Gaussian of centre c, blue width b and red width r is the Gaussian of full width b below c and of full width r above it, meeting at its peak.

The fidelity index is the collection’s own: the mean CAM16-UCS shift of twelve test samples, each a smooth raised-cosine bump in reflectance 220 nanometres across, centred every 25 nanometres from 400 to 675, between the lamp and a reference at the lamp’s correlated colour temperature — Planckian below 5,000 kelvin, daylight above — scaled so that a shift of one unit costs 5.5 points. The slow-pigment cost is the census of the limits assume a pigment that switches instantly under an arbitrary spectrum: over every sixth direction of a Fibonacci sweep of the sphere, the best reflectance whose slope is limited to a full swing in forty nanometres is found by dynamic programming, and a direction counts as lost when it reaches less than 99 per cent of the ideal support. The frontier at each cost is the best fidelity index any lamp of the family reaches at or below it.

What this leaves out #

The family is still a family. A split Gaussian has two widths; a real phosphor band has a shape set by its chemistry, often with a long red tail and a steep blue side — which is, as it happens, the red-leaning shape the census preferred. A phosphor-converted fourth band would be worth scoring with its measured shape rather than a split Gaussian.

The three emitters are held. A designer free to move them too could shift the whole trade; the question here was only whether the fourth emitter’s shape was the constraint.

And the pigment is a slope limit. A sharp edge is bought with depth found that a band-shape limit on the pigment, three absorption bands none narrower than forty nanometres, reaches further than a slope limit in most directions. A slow pigment of that kind would charge the fourth band differently, and the frontier here is the slope limit’s.

Still open: whether a band-shape pigment moves the frontier more than a band-shape lamp #

This census changed the lamp’s shape and held the pigment’s model. The other half is the reverse. A sharp edge is bought with depth found that a pigment made of absorption bands draws sharper edges than its bands’ widths, when it is loaded heavily enough, and so reaches further than a slope limit of the same width. Its extra reach is in the directions whose edges it can sharpen, and those are the directions a fourth emitter costs.

The calculation is this frontier with the pigment modelled by absorption bands rather than by a slope limit, at the forty-nanometre band width and the loading a sharp edge is bought with depth found typical, for the same symmetric and skewed fourth emitters. The prediction is that the band-shape pigment moves the frontier further than the lopsided lamp did — by twenty or thirty directions at a high rendering rather than six — because the conflict is between the lamp’s light in the amber and the pigment’s ability to switch there, and a pigment that switches faster gives up less. If so, the trade is decided more by what the pigment is than by what the lamp is, and a lamp designer’s best move is to know the pigments it will light.

A suspicion about the search is worth running #

The habit is about testing whether a finding belongs to the world or to the search that produced it.

The earlier essay’s trade could have been an artefact: one width, one power, one symmetric shape, swept over position. The right response was to widen the search in every direction it had been confined, and to give the alternative family the same freedoms as the original, so that whatever moved could be credited to the right freedom. Done that way, the trade survived, two freedoms each bought a little, and the one expected to matter most leaned the opposite way from the guess.

The failure mode is to test the suspected constraint alone. A lopsided band tried against one fixed symmetric band would have looked like a larger improvement than it is, because it would have been crediting the shape with what width and power were doing.