The converter gives the red phosphor away

A camera turns raw readings into colour with matrices blended by the lamp’s colour temperature, and under a white LED the result is off by a colour difference or two. A small sensor beside the lens reads narrow channels as departures — how far each channel’s share of the light sits from what a smooth lamp of the camera’s measured colour would give — and the thread since one ratio reads a peak and a trough has asked how much of the camera’s error those departures carry.

LEDs built on the locus still fall off the line built thirty LEDs as a maker would: pumps at 440, 452 and 465 nm, red phosphor at four strengths, whites at 2,700, 4,000 and 6,500 K, each converter solved to put the white on the Planckian locus. No line in the departures read them all. A channel at each pump cannot see the recipe added channels at the pumps’ own wavelengths and found them buying little — and then found why. The camera’s error under these LEDs is simple in their recipe: within a pump, a line in red-phosphor strength, mireds and their product reads it within ±0.07, the 440 nm LEDs to ±0.043. The departures could read neither the red phosphor nor the temperature.

The temperature a sensor already has, from its colour channels; on the locus, an LED’s correlated colour temperature is exactly its colour. The red phosphor is the missing number, and its signature should be in the red: a garnet converter peaks between 530 and 600 nm and falls away through the red, while a red phosphor peaks near 630. The proposal was a ten-nanometre channel at 610 nm, where one is falling and the other rising, and the prediction was that it reads the red-phosphor strength within a tenth of its range at every pump and, with the temperature, the 440 nm LEDs’ error within ±0.08. The alternative was that the converter’s own tail mixes into every red channel and the recipe cannot be read from narrow departures at all.

No red channel reads it, and the converter does #

Within each pump, each LED left out, a line in the 610 nm departure and the mireds misses the red-phosphor strength by ±0.43 at 440 nm, ±0.36 at 452 and ±0.22 at 465, against a spread of ±0.36; no red departure from 600 to 680 nm does better than ±0.22. To hold the white on the locus, every added step of red phosphor moves the converter’s peak shorter, by 29 to 55 nm from none to the strongest that can be built. Within each pump the 600 nm departure and the mireds read that peak within ±5.2 nm of a ±21 nm spread. Over all thirty LEDs at once, the 600 nm departure, the mireds and their product read the red-phosphor strength to ±0.12; and the camera’s error, read in two stages — the red phosphor estimated that way, then the error from the estimated recipe within the LED’s pump — comes within ±0.078 at 440 nm, ±0.083 at 452 and ±0.058 at 465.

• The prediction fails within a pump. The 610 nm channel reads the red phosphor no better than its average at two pumps of three.
• The alternative is right about the mechanism and wrong about the conclusion. The converter does mix into every red channel — not as a tail, but as a compensation — and the recipe can still be read, through the converter.
• The error comes within the prediction’s ±0.08 at 440 nm by the two-stage route, though not by the one proposed.

The two LEDs are the same colour, by construction, and their spectra show what that costs. The red phosphor adds a hump near 630 nm; the converter, pushed from 579 to 529 nm, empties the band from 570 to 610 where the first LED put most of its light. The colour stays put because the eye’s three cones — and the camera’s colour channels — see the added red and the lost orange as nearly cancelling, and the converter’s move to the green keeps the balance. A channel at 610 nm sits exactly where the two changes meet: it gains from the red phosphor’s short flank and loses from the converter’s retreat.

Why the locus hides the red phosphor #

The locus solve has two numbers to set for each LED — how much of the pump the converter takes, and where the converter peaks — and two targets, the white’s position in the CIE 1960 plane. Adding red phosphor moves the white towards red, and the only way back, with the temperature held, is a converter that emits shorter. The converter’s peak falls by about 29 nm per unit of red phosphor at 2,700 K and about 50 at 4,000 K, and at 6,500 K the strong-red whites cannot be built at all because the converter would have to peak below 520 nm.

So among the LEDs a maker would build, red phosphor and a short converter come together, and every narrow channel in the red reads their difference. A departure is a channel’s share of the light against what a smooth lamp of the same colour gives; a smooth lamp of the same colour is the same for both LEDs above, and the two LEDs’ shares in a red channel differ by whatever the red phosphor adds there less whatever the converter’s retreat removes. At 610 nm those nearly balance, and no line in that departure can tell how much of either is present.

Within a pump, nothing reads it #

The census of red channels confirms it at every wavelength. Within a pump, with ten LEDs, a line in any one red departure and the mireds misses the red phosphor by between ±0.22 and ±0.43 against a spread of ±0.36 — never better than three fifths of the spread, and at 610 nm worse than the spread itself at two pumps. The best channel is the shortest, 600 nm, at about ±0.23 for every pump. It sits on the converter’s own long flank, where the converter’s position dominates and the red phosphor contributes least: it reads the converter, not the phosphor.

The converter is what the channel sees #

What the 600 nm departure does read is the converter’s peak, and well: within each pump, with the mireds, to about ±5 nm of a ±21 nm spread. At each colour temperature the thirty LEDs fall along one steep curve — a longer converter puts more of the light near 600 nm than a smooth lamp of the same colour, a shorter one less — with the three pumps barely separated on it.

And the converter’s peak, at a known temperature, is the red phosphor. The locus solve ties them: at each temperature, a given red phosphor gives one converter peak. A sensor that reads the converter and the temperature has read the red phosphor, through a relation the maker’s own design rule set. The within-pump lines missed it because ten LEDs are too few to learn a relation that bends with temperature; with the mireds’ product in the line and all thirty LEDs to learn from, it is there.

Read across the pumps #

The pumps do not change the relation between the converter and the red phosphor — the converter’s peak at a given recipe moves by two to five nanometres between pumps, against fifty for the red phosphor — so all thirty LEDs can teach the same line. Fitted over the other twenty-nine, the line estimates each LED’s red-phosphor strength to ±0.12, a third of its spread, half again better than any within-pump line. The estimates sort cleanly by strength: every LED with none estimated under 0.2, every LED with the full strength above 0.77, and the four strengths’ estimates do not overlap.

That is not a tenth of the range, which the prediction wanted, and it is not exact. But it is enough for what the red phosphor is needed for, which is the camera’s error.

Through the recipe to the error #

Reading the camera’s error in two stages — the red phosphor estimated from the 600 nm departure over all thirty, then the error from the estimated recipe within the LED’s own pump — brings the 440 nm LEDs to ±0.078, the 465 nm ones to ±0.058 and the 452 nm ones to ±0.083, pooled ±0.074. For the 440 and 465 nm LEDs that halves what any line in the departures managed. For the 452 nm LEDs it is worse than the departures’ own ±0.047, because the departures read those LEDs’ error directly, without the detour through a recipe estimated with error, and the detour adds its noise.

The prediction asked for ±0.08 at 440 nm and gets it, by a route it did not propose: not a channel that sees the red phosphor, but a channel that sees what the red phosphor did to the converter, read across enough LEDs to learn the maker’s rule. The error is still read with the pump known; two blue channels sort one pump of three, so the 440 and 452 nm LEDs would still be confused in practice.

A second stage again, and what it costs #

This is not the first time the thread has read a camera’s error in two stages. Two stages catch the tube and bill the lamps they misplace first placed a lamp in a family by its nearest neighbour and then read the error from that family’s line, and found the placement’s mistakes costing more than the family line bought; a classifier that abstains gives up the tube tried to make the first stage refuse when unsure. Those first stages chose among families, a discrete choice that is either right or badly wrong. This first stage estimates a number, the red-phosphor strength, and its mistakes are small and continuous: an LED estimated at 0.45 when it has 0.3 is read by the same line as its neighbours, a little off.

That difference is why the two-stage reading helps here where the family choice hurt there. A wrong family moved a lamp onto a line fitted for lamps unlike it; a wrong red-phosphor estimate moves an LED a short way along a line fitted for LEDs exactly like it. The cost is a blur rather than a jump, and at the 452 nm pump — where the departures already read the error directly — the blur is all there is: ±0.083 through the recipe against ±0.047 without it.

Why the red phosphor matters so much to the error is not computed here, and the likeliest reason is one the camera essays met early. A matrix is fitted under one light: a camera’s colour matrices are built under lamps with plenty of light in the long red, and a phosphor-converted LED without a red phosphor has little there, so the camera’s red channel is asked to do more of its work with less signal than its matrix expects. Adding red phosphor restores that light and the matrix’s assumption with it. The recipe line’s steep fall with red phosphor — up to a unit of error at fixed temperature — reads as that assumption being met, and if it is, that is why a sensor that knows the red phosphor knows most of what the camera will get wrong.

New LEDs miss the line their cousins drew found new LEDs that varied their phosphors missing the census’s line because the line had never seen a red phosphor vary; the two-stage reading here learns the red phosphor first, from the converter it displaced, and only then reads the error.

What a sensor designer should take from this #

On the locus, an LED’s recipe is constrained, and the constraint is information. A red phosphor and a short converter come together in every LED that sits on the locus, so a reading of either is a reading of both. The sensor cannot see the phosphor in the red, where the two cancel; it can see the converter on its own flank at 600 nm.

A good channel is where one cause dominates, not where two differ most. The 610 nm channel was proposed where the converter and the red phosphor differ most in shape, and that is exactly where their opposite changes cancel. The useful channel is where one of them — the converter — moves the light and the other barely reaches.

The camera’s error needs the recipe, and the recipe needs the maker’s rule. A sensor that reads LEDs off the locus, or a maker who balances the white another way, breaks the tie between converter and phosphor that the 600 nm channel relies on. LEDs varied in their phosphors fall off every line found the error a V about the locus for exactly such LEDs.

How the channels were read #

The thirty LEDs are the locus grid’s, their camera errors the census’s. The red departures are computed exactly as the census’s 450, 500 and 640 nm departures are, in ten-nanometre Gaussian channels at 600, 610, 620, 640, 660 and 680 nm; the 640 nm one is the census’s own. The converter’s peak is the locus solve’s second number. The within-pump lines are fitted on nine LEDs and read on the tenth; the pooled line on twenty-nine and read on the thirtieth. In the two-stage reading, the red phosphor is estimated for each held-out LED from the pooled line fitted without it, and the error from a line in the estimated red phosphor, the mireds and their product, fitted on the same pump’s other nine LEDs with their red phosphor estimated the same way. The temperature is the LED’s own; on the locus, a sensor’s colour channels report it.

What this does not reach #

A known pump. The error’s second stage uses each LED’s pump; the blue departures sort out the 465 nm LEDs and not the other two.

A maker’s rule. The converter solve here keeps every white exactly on the locus. Real LEDs sit within a tolerance of it, and a white a few thousandths off the locus loosens the tie between converter and phosphor.

One camera. The error is the census’s modelled camera’s; the recipe line is a fact about it.

Still open: whether a white off the locus breaks the tie #

The tie between the converter’s peak and the red phosphor holds because every white sits on the locus. Manufactured LEDs are binned within a few thousandths of it, by Duv, and a white 0.003 above the locus has a longer converter for its red phosphor than one on it.

The calculation is the grid rebuilt at Duv of −0.003 and +0.003 beside the locus — the converter solved for each — the red phosphor estimated from the 600 nm departure by the line learned on the locus, and the camera’s error read through it. The prediction is that the off-locus LEDs’ red phosphor is misestimated by about a fifth of its range, because a Duv of 0.003 moves the converter’s peak by roughly the amount a fifth of the red phosphor does, and that their error, read through the misestimate, falls back to ±0.12 — between the departures’ and the recipe’s. If the 600 nm line learned on the locus reads the off-locus LEDs as well, the departure carries the white’s Duv as well as its converter, and the tie is not what the reading relies on. Beside the locus the converter still gives the red away makes that calculation.

Reading a cause through what it displaces #

The habit here is about where to look for a quantity a design has hidden.

The red phosphor’s own light is in the red, and a channel was placed there to see it. The design rule that put every LED on the locus had already arranged for the red phosphor’s light to be hidden, by moving the converter to compensate. That move is the trace the phosphor leaves.

The failure mode is to look for a cause where its own signature is strongest, when a constraint has cancelled the signature there. Every constraint that holds an output fixed — a colour, a temperature, a balance — couples the inputs it holds it with, and a reading of one coupled input is a reading of the other. The test is to ask what else must change when the quantity changes, and to read that.