What a lamp cannot give back

A watt is a watt. A lumen is a watt that happened to arrive at a wavelength the eye is good at.

That is the whole of photometry, and the exchange rate between the two units is a curve. Everything a lighting engineer argues about — efficacy, efficiency, lumens per watt, the number on the box — is that curve integrated against a spectrum, and the curve is one this site already has, because it is ȳ, one of the three colour-matching functions.

The ceiling is a property of an observer rather than of physics, and the standard supplies two observers.

How steeply the ceiling falls away from its peak is the other thing worth reading off the same curve, and it is steeper than the shape suggests.

A white source has to cover a span rather than a point, so the number that matters to it is an average over a stretch of that curve.

Two more readings say that the ceiling is a property of an observer and that the number everybody quotes is one point on it.

The constant is a definition, and this matters #

683 lm/W is not the result of an experiment. In 1979 the candela was redefined as the luminous intensity of a source emitting monochromatic radiation of frequency 540 THz with a radiant intensity of 1/683 watt per steradian, and that act made photometry a defined scaling of radiometry rather than a parallel discipline with its own standards.

The consequence is worth stating plainly, because it is the reason this essay can exist on a colour site at all. Every lumen anybody has ever quoted is a radiometric quantity multiplied by a curve and a constant. There is no separate photometric apparatus, no second class of instrument, no independent chain of traceability. A lumen is a fact about watts and about ȳ — which means it inherits every choice buried in the three numbers a spectrum collapses to, including the one about whose eyes.

540 THz is 555.17 nm in vacuum and about 555 nm in air. The difference matters to metrologists and to nobody here, but pretending it away would be the kind of small dishonesty this site tries not to accumulate.

Where the ceiling comes from #

If the exchange rate is 683 lm/W at 555 nm and lower everywhere else, then no spectrum can beat 683, and the only spectrum that attains it is a monochromatic source sitting exactly on the peak.

That bound is trivial to state and surprisingly hard to feel. It says that the most efficient possible light source — the theoretical best, with no losses of any kind, converting every electrical watt into radiation with perfect fidelity — would be a green laser. It would render every object in the room as a shade of green, and it would be, by the only metric photometry offers, unimprovable.

Real lamps are compared against this number constantly and the comparison is nearly meaningless, because the interesting constraint is not the ceiling. It is the much lower ceiling that applies once the light also has to be white.

Equal-energy white comes out at 180 lm/W. D65 reaches 204 and D50 206, both slightly better than flat because daylight carries somewhat less power in the deep red and deep violet than a flat spectrum does. Illuminant A — a tungsten filament at 2856 K, computed here from Planck’s law rather than tabulated — manages 154.

The three spectra differ in the only way that matters to the arithmetic: how much of their power sits under the peak of the efficiency curve rather than out at the ends. The tube and the LED put theirs in a few narrow places chosen for exactly that reason. The radiator cannot choose; its shape is fixed by its temperature.

Every one of those numbers is under a third of the ceiling, and the reason is structural rather than technological. A spectrum the eye will accept as white has to put substantial power into the short and long ends of the band, where ȳ is small. Those watts are very nearly free of lumens. They are not waste in any engineering sense — they are what makes the light white, and white is a property of the whole spectrum rather than of any part of it — but the photometric accounting cannot see the difference.

What the accounting cannot see #

The ceiling is a fact about the efficiency curve. The reason nobody builds the green laser is a fact about surfaces, and it is invisible to every quantity so far.

A surface reflects the light that falls on it. If no power falls in the band a surface reflects, that surface returns nothing, and it does not matter how efficacious the source was. The lumens went somewhere else.

The sweep is the argument. As the band narrows from 200 nm to 30 nm the efficacy rises by better than a factor of two, and the fraction of the source’s own power that a broad red reflectance can send back falls by better than a factor of five. These are not two effects that happen to correlate. Power moved towards the peak of ȳ is, necessarily and by the same arithmetic, power moved out of the bands that every surface which is not green reflects.

The red-return figure here is deliberately crude — a broad reflectance, a ratio of integrals, no observer model on the far side. A proper fidelity index is a different computation, is already on this site, and involves rendering a set of test samples under the source and under a reference and measuring the colour differences. What the crude version buys is that the mechanism is visible in one line of arithmetic instead of being buried in a standard.

Three marks are enough to carry the shape of the ceiling if they are put where a lamp designer would actually put a primary.

The number that survives the trade #

There is a temptation, having established the opposition, to conclude that efficacy and rendering are simply exchangeable and that a lamp designer picks a point on a curve. That is nearly right and the “nearly” is where the whole industry lives.

The opposition is real at fixed spectral shape. It is not a strict trade-off across all shapes, because the efficiency function is not the only structure in the problem — the test samples’ own reflectances have structure too, and a source can put its power in the bands those samples reflect rather than in the bands between them. That is exactly what a triphosphor lamp does and why it beats a halophosphate one at both quantities at once.

The expansion phase of this site tried to assert that ranking and had to withdraw it. The claim failed under every retuning of the two spectra that kept both faithful to their known structure, and the lesson was recorded rather than tuned away: a ranking between two constructed spectra is a fact about the constructions. What survives is the structural claim — narrow the band at fixed shape and the trade bites — and the measured statement that how saturated the test samples are decides the verdict.

What was computed, and how #

Every number above comes from two objects the site already had.

The first is ȳ, taken from the 1931 colour-matching functions and normalised to its own peak, which is what V(λ) is. There is no separate luminous efficiency table here; asking for one would be asking for a second copy of a curve that is already load-bearing in half the figures on this site.

The second is the integral

$$K(S) = 683 \cdot \frac{\int S(\lambda) V(\lambda)\,\mathrm{d}\lambda}{\int S(\lambda)\,\mathrm{d}\lambda}$$

evaluated as a rectangle sum on the site’s 5 nm grid from 380 to 780 nm. The grid step cancels between numerator and denominator, and it is written out rather than cancelled, because the cancellation only holds while both integrals run over the same band — a future caller integrating the numerator over the visible and the denominator over everything would silently be computing a different quantity.

Three assertions guard the result. A monochromatic source at 555 nm must come out at exactly 683, which is what the candela was defined to make true and which fails immediately if the grid, the normalisation or the integral is wrong. No spectrum swept across every wavelength on the grid, nor any of the standard illuminants, may exceed it. And equal-energy white must land under a third of it, which is the structural claim rather than a stored number.

Where the model stops #

The efficacies above are efficacies of the light, not of the lamp, and the gap is enormous for exactly the source most people picture when they think of a light bulb.

This site’s spectra run from 380 to 780 nm. A tungsten filament at 2856 K radiates overwhelmingly outside that band, and every integral here is blind to it.

This figure was drawn twice. The first version asserted that the visible share rises with temperature, which is what everybody expects, is true from 2000 K to about 6000 K, and is false across the rest of the range a real star or studio lamp occupies. Nothing but the assertion objected — the bar chart underneath it looked entirely reasonable, and would have shipped.

What replaced it is the stronger statement, and the one that actually explains incandescence’s fate: the visible share has an interior maximum at about 6950 K, it reaches 49.6% there, and no temperature whatever gets a thermal radiator past half. A filament cannot be run at 6950 K; tungsten melts at 3695 K. So the practical ceiling for incandescence is the 10.7% at 2856 K, and it is a ceiling imposed by materials science on top of one imposed by Planck’s law. Every efficient lamp ever built is non-thermal, and this is why.

The two ceilings compose badly. A tungsten lamp radiating 10.7% of its power in the visible, at 154 lm/W of that visible power, delivers about 16 lm/W overall — which is roughly what a tungsten lamp delivers.

That figure has not moved much in a century, and it cannot, because both of the constraints behind it are physics rather than engineering. Filament lamps were improved for a hundred years by better vacuums, better fills, coiled and coiled-coil geometries, halogen regeneration cycles that permit higher temperatures — and the whole accumulated effort bought a factor of about two. Non-thermal sources arrived and bought a factor of ten in a decade, because they are not standing under either ceiling.

Six marks across the whole band, under the ten-degree observer, say how much of the answer is the observer and how much is the shape of the curve.

The observer is a choice here too #

V(λ) is ȳ from the 2° observer. A photometric quantity computed against the 10° functions is a different quantity, and the site’s standing point about the observer arrives here in the one place it is nearly always assumed not to apply.

The difference is small for a broadband source and large for a narrow one, for a reason that is visible in the curves: the two efficiency functions disagree most on the steep flanks, and a broadband source averages the disagreement away. Measured on this site’s own machinery, D65’s efficacy moves by 0.1% between the two observers. A monochromatic source at 470 nm moves by 104% — the 10° functions make it more than twice as luminous, because the 10° observer is substantially more sensitive in the blue.

That is not a rounding difference and it is not academic. Narrowband blue emitters are the pump in essentially every white LED manufactured, which means the photometric rating of the most common light source on earth depends materially on which of two standard observers is used to compute it. The specification says which. Almost nothing else does. There is a longer argument about that choice waiting at the top of this ladder, and this is the cheapest available illustration of why it is not a technicality.

Who found it, and when #

The efficiency function came first and the constant came last, which is the reverse of how the story is usually told.

Measurements of the eye’s relative sensitivity across the spectrum were being made through the nineteenth century, by flicker photometry and by step-by-step brightness matching, and the CIE adopted a standard photopic curve in 1924 — seven years before the colour-matching functions that ȳ would turn out to be part of. The 1931 system was then constructed so that ȳ equalled the 1924 curve, deliberately, which is why luminance falls out of XYZ as a single coordinate and is the single most useful structural property the system has.

The candela’s definition moved several times after that: from a platinum-point blackbody, to the 1979 frequency-and-watt definition that fixed 683, to the 2019 SI revision that recast it as a fixed numerical value of the luminous efficacy $K_{cd}$ of 683 lm/W exactly. The number has not changed since 1979. What changed is that it stopped being a consequence of an artefact and became a defined constant, which is why the assertion above can demand exactness rather than a tolerance.

The 1924 curve is also known to be wrong in the blue — substantially, by up to a factor of ten below 460 nm — and the Judd–Vos corrections of 1951 and 1978 exist to fix it. The CIE has never replaced V(λ) in the definition of the lumen, because doing so would change the numerical value of every photometric measurement ever made. That is a defensible decision and it means the exchange rate between watts and lumens is known to be slightly wrong and is kept anyway.

Three ceilings, and the largest is not physics #

The tungsten figure is assembled above from two constraints described as both physics rather than engineering. Multiplied out, the gap between a filament lamp and the defined ceiling factorises into three terms, not two, and the largest of the three is the one the sentence before it correctly calls materials science.

The whole shortfall is 683 lm/W against about 16.5, a factor of 41.4. It splits cleanly:

Their product is 41.4 to the last digit, which is the check that the three are independent rather than three readings of one thing.

The materials term is the biggest, and it is the only one of the three that a different universe of chemistry could relax. Planck’s ceiling is a property of thermal radiation and the whiteness ceiling is a property of the eye; the melting point of tungsten is a fact about a particular metal, which happens to have the highest melting point of any metal and still falls three thousand kelvin short of where a thermal radiator would want to run. Nothing known gets close — the best refractory carbides sublime around four thousand kelvin — so the term is unrelaxable in practice while being contingent in principle, which is a different kind of ceiling from the other two and worth distinguishing from them.

It also settles a question the section leaves open. A reader might reasonably wonder whether incandescence was defeated by the infrared or by the requirement to be white. It was defeated by the infrared, by about a factor of nine against the whiteness term’s four and a half — the two radiative terms together are ×9.4, and they are what a non-thermal source escapes. An LED still pays the ×4.44 for being white; it pays neither of the others, which is the whole of its advantage.

The trade is not one lumen for one unit of rendering #

The four-band sweep is presented as an opposition, and it is worth reading as a rate rather than a direction, because the rate is steep.

From a 200-nanometre band to a 30-nanometre one, efficacy rises by a factor of 2.10 and the share a red surface returns falls by a factor of 5.68. Taking the ratio of the logarithms, each doubling of efficacy costs a factor of 2.34 in the exponent — so the second half of the journey to the ceiling costs about five times as much rendering as it buys in lumens.

That asymmetry is the reason the trade is not a matter of taste. If the exchange were one for one, a designer could pick any point and defend it. At an elasticity above two the flat end of the curve is where every sensible design sits, and the narrow end is only reachable by a source nobody wants.

The 30-nanometre band makes the point from the other side: it is already at 96.6 per cent of the 683 ceiling. The remaining 3.4 per cent of the theoretically available lumens is what a designer would be buying by going monochromatic, and it costs the last of the red return. There is almost nothing left to win at the narrow end and almost everything left to lose, which is a stronger statement than the trade-off framing makes and is visible in the same four numbers.

Two smaller things the numbers say #

Daylight’s advantage over a flat spectrum is not slight. D65 at 204 and D50 at 206 against equal-energy’s 180 are 13.3 and 14.4 per cent better, and illuminant A at 154 is 14.4 per cent worse. Those are the same size as the entire gap between two of this collection’s published illuminants, and slightly better than flat undersells a spectrum shaped by an atmosphere into carrying a seventh more lumens per watt than a spectrum shaped by nothing.

And the observer’s reach spans three orders of magnitude. D65’s efficacy moves by 0.1 per cent between the two standard observers and a 470-nanometre line moves by 104 per cent — a ratio of about a thousand between the two cases, decided entirely by how wide the emitter is. The phrase the standard observer is doing no work at one end of that range and almost all of it at the other, and the pump wavelength of every white LED made is at the wrong end.

Where this goes next #

The trade measured here is between efficacy and one crude proxy for rendering. The proper fidelity computation is one rung below on this ladder and is what a lighting specification actually cites; the structure of the illuminant is what decides which surfaces a source flatters and which it flattens.

The blue-sensitivity difference between the two observers is the thread worth pulling. It shows up as a 104% swing in the efficacy of a 470 nm line, which is the pump wavelength of a white LED, and the same disagreement is what makes narrowband displays a metamerism problem rather than a colorimetry one. Both are the same fact about two curves, arriving in two industries that do not talk to each other.