A lamp has a waveform

Every lamp on this site is a spectrum: eighty-one numbers, one per five-nanometre band, and nothing else. That description has been complete for daylight and for a hot filament since the site’s foundation, because both are steady on any timescale a person has.

It is not complete for a light-emitting diode, whose spectrum this site constructs rather than tabulates. A diode has no thermal mass worth the name, so it does whatever its driver does — and the commonest driver switches it fully on and fully off, at somewhere between a hundred and a few thousand times a second, and dims it by shortening the on-time.

The spectrum is unaffected. The time average is what a spectroradiometer reports, and every colorimetric claim on this site stands. What is added is a second axis, and three thresholds on it that lie three orders of magnitude apart.

The claim #

A lamp’s modulation is invisible, visible, or visible only on something moving — and which of the three depends on what the observer is doing rather than on the lamp.

Three thresholds, and the third is derived rather than measured:

• Flicker, seen directly in a stationary field, stops at fusion — sixty hertz at an office light level, and twelve and a half hertz higher for every decade of light.
• The stroboscopic effect, seen on something moving under the lamp, survives far past it. A hand waved under a kilohertz lamp shows a comb of separate images.
• The phantom array, seen when the eye moves, reaches into the kilohertz. A point source glanced past becomes a dotted line.

The third is where the derivation is. A modulation at f hertz, on an image moving at v degrees a second across the retina, lays down a pattern of f/v cycles per degree. Whether that pattern is visible is a question the spatial contrast sensitivity function answers — a function measured on stationary gratings, by people who were not thinking about lamps, and imported here without a constant fitted.

A kilohertz drive during a saccade at 350 degrees a second lands at 2.86 cycles per degree, which is within a factor of two of the frequency the eye is most sensitive to. The effect nobody can believe — a lamp flickering a thousand times a second, visibly — is the peak of the spatial contrast sensitivity function, seen from an unfamiliar direction.

The three thresholds, in one table #

The same waveform, a full-depth square drive, judged three ways:

The first column collapses over a factor of two in frequency. The other two do not collapse at all until the retinal pattern passes the acuity limit, which for a saccade is at 17.5 kilohertz and for a waved hand at 10.

That is the finding, and it is not a small correction to the flicker literature: it says the frequency above which a lamp is safe from all three effects is not sixty hertz or a hundred and twenty, but the acuity limit times the fastest motion anybody makes — three orders of magnitude higher.

The worst frequency is the speed times four #

The table has a shape in it that is easy to read past. Down the last two columns the ratio rises from fifty hertz to a thousand and then falls again by ten thousand — 118, 121, 133, 312, 168 for a saccade. A stroboscopic effect that gets worse as the lamp gets faster is the opposite of every rule of thumb in the trade, and it is not an anomaly. It is the ratio $f/v$ walking up the spatial sensitivity curve, over its peak, and down the far side.

Which means the curve’s landmarks can be read backwards. The spatial function this argument imports has two of them — a peak at 4 cycles per degree and an acuity limit at 50 — and dividing by nothing and multiplying by a speed turns each into a frequency:

The right-hand column reproduces the two ceilings quoted earlier, which is the check that the arithmetic is the same arithmetic. The column worth reading is the third.

Every frequency a driver is actually built at is in it. Mains-derived lamps modulate at 100 and 120 hertz; switching drivers sit between a few hundred hertz and a few kilohertz. The first of those is on top of the worst frequency for smooth pursuit and the second is on top of the worst frequency for a saccade — so the entire industrial range lands where the eye’s spatial machinery is most sensitive, for exactly the two motions a person makes without deciding to.

And the standard remedy makes it worse. The advice given when a lamp is reported as flickering is to raise the drive frequency, on the reasoning that flicker fusion is at sixty hertz and everything above it is safe. It is safe, in the first column: at a thousand hertz the stationary ratio has fallen below a ten-thousandth. In the third column the same change takes the saccade ratio from 121 to 312, because it has moved the pattern from 0.35 cycles per degree towards the peak at four. A lamp moved from a hundred hertz to a kilohertz is two and a half times more visible to a moving eye and ten thousand times less visible to a still one, and only one of those is measured by anything on the datasheet.

The honest version of the rule of thumb is therefore two-sided, and the table says where each side lies. Below about twenty-five hertz nothing is safe from anything. Between there and a few kilohertz, raising the frequency trades stationary visibility for moving visibility at a rate that depends on which motion is in question. And the frequency above which a lamp is genuinely done with all three effects is the bottom of the right-hand column for the motions that occur in the room — 10 kilohertz where hands move, 17.5 where eyes do.

That last figure is the one this model states and the practical literature does not reach, and the limits below say why it should be read as an upper bound rather than a specification.

Why this is a derivation and not an analogy #

The step worth being careful about is what a moving image does to the photons.

During a saccade, a lamp’s image sweeps across the retina. Each retinal position receives light only while the image is passing it, so the quantity absorbed at position x is proportional to the lamp’s output at the moment the image was there — which is t = x/v. The temporal waveform is written onto the retina as a spatial pattern, at 1/v degrees per second of it, and the writing happens before any neural filtering.

The temporal filter therefore does not remove the pattern. It acts on each position’s brief transient, and what is left in the photon count is a faithful spatial copy of the drive. So the question of whether the copy is visible is a spatial question, and the spatial machinery answers it.

The one place the argument needs a hand is at the top. The spatial sensitivity function used here is a smooth exponential roll-off that never quite reaches zero, so at a hundred per cent modulation it would keep reporting a visible pattern hundreds of cycles per degree past anything the optics or the cone mosaic can carry. A hard ceiling at the acuity limit is imposed for that reason, it is the only such intervention in the file, and the reason is written where it is used.

The two numbers a lamp is sold by #

A lamp’s temporal behaviour reaches a specification as percent flicker — the Michelson contrast of its waveform — and sometimes as a flicker index, the share of a cycle’s area above the cycle’s own mean.

Neither contains a frequency, and the frequency decides everything in the table above.

Percent flicker cannot see a duty cycle. A driver switching a diode fully on and fully off reads 99.8 per cent at every dimming level, because the maximum and minimum never move. The index reads 0.50 at half brightness and 0.89 at a tenth. A lamp reported as unchanged by dimming is therefore routinely much worse dimmed, and the number on the datasheet is the one that cannot tell — a figure of merit that has stopped discriminating, which is the failure the rendering index has in a different direction.

And the frequency doubling nobody expects. A lamp run from a rectified supply with no smoothing modulates at twice the supply frequency, because both halves of the mains cycle produce light. A fifty-hertz country’s lamps flicker at a hundred and a sixty-hertz country’s at a hundred and twenty, which is why footage shot in one and lit by the other bands — the two rates beat, in the same arithmetic that makes two halftone screens beat.

What was computed, and how #

Five landmarks, and the transfer functions are solved from them — the peak, the two fusion frequencies, the depth of the low-frequency dip and the threshold at the peak, each with the range the literature reports. The solve reproduces every constraint to 10⁻⁹.

A waveform is decomposed before it is judged. A square wave carries a third of its fundamental at three times the rate and a fifth at five times, so a drive can be over threshold at a frequency it contains no energy at the name of. Every harmonic goes through the sensitivity function and the largest response decides.

The stroboscopic half imports the spatial file and adds nothing. strobe computes f/v, hands it to the spatial visibility function, and returns the ratio. assertTheCeilingScalesWithSpeed requires the ceiling to be exactly proportional to the velocity, which it is to nine decimal places — because the ratio is f/v and nothing else.

And the velocities are quoted. Fixational drift at half a degree a second, smooth pursuit at twenty, a waved hand at two hundred, a saccade at three hundred and fifty, the tip of a fan blade at nine hundred. These are the only quoted numbers in the stroboscopic half; everything else comes out of a function measured for another purpose, on the site’s standing rule that measurements of people are quoted and everything downstream is computed.

Where the model stops #

The predicted ceiling is an order of magnitude above what is reported. This model says a saccade carries the effect to 17.5 kilohertz at full modulation, and the practical literature worries about frequencies up to about three. The gap is real and has two named causes: no driver produces a hundred per cent modulation at twenty kilohertz, and the acuity limit at full contrast is optimistic. The model’s number is an upper bound on a stimulus nobody builds.

The worst-frequency table is two landmarks and a multiplication. The peak of the spatial function is quoted at 4 cycles per degree with a reported range of 2 to 6, so every entry in the third column carries that range multiplied through it — the saccade’s worst frequency is 1.4 kHz on the quoted value and anywhere between 0.7 and 2.1 on the range. The ordering of the rows and the fact that the industrial range sits inside them survive the whole of that spread; the individual numbers do not.

A saccade’s velocity is not constant. It accelerates and decelerates over a few tens of milliseconds, so the retinal pattern is a chirp rather than a grating, with its frequency lowest at the ends.

There is no rod contribution. The stroboscopic effect is strongest in dim light, where fusion is lowest and the rods are working, and this model has neither term.

And there is no position in the field. Flicker is more visible in the periphery — the reason a tube flickers most when it is not being looked at — and the model has no eccentricity in it.

The generalisation #

The sentence worth carrying: a lamp’s flicker frequency is not a property with a threshold; it is a ratio, and the denominator is a speed.

Every rule of thumb in lighting practice fixes the denominator by accident. Above a hundred and twenty hertz is fine assumes the observer and everything they are looking at are stationary, which is true of a wall and false of a running machine, a bicycle wheel, a falling stream of water, a hand, and the observer’s own eyes several times a second.

There is a corollary that is sharper than the rule and easier to act on. Because the denominator is a speed and the sensitivity function has a peak rather than a cliff, every lamp has a worst frequency for every observer, and it is that observer’s speed times four cycles per degree. A specification cannot avoid all of them at once, since the speeds in a room span three orders of magnitude. What it can do is say which motion it is written for — a machine shop cares about the tool, an office about the eye — and then choose a frequency on the far side of that motion’s peak rather than merely above sixty hertz.

The surprising connection is with halftone screens. Both are cases where a pattern’s visibility is decided by where its energy sits relative to a sensitivity function that peaks in the middle rather than at the bottom — and in both the practical answer is to push the energy up rather than down. A printer moves a screen’s fundamental past the eye’s limit by making the ruling finer; a lighting engineer does the same thing by raising the drive frequency. The difference is that a printer’s observer is not moving, and a lighting engineer’s is.

Who found it, and when #

The stroboscopic effect predates electric light: Faraday and Plateau were making wheels appear to stand still with slotted discs in the 1830s, and the effect became an industrial problem the moment gas discharge lighting reached machine shops in the twentieth century, where a lathe chuck could appear stationary while turning.

The phantom array is more recent as a named phenomenon — described in the 1990s and studied seriously since light-emitting diodes made high-frequency modulation ordinary — and it is the one that made high-frequency drives worth measuring at all.

The two figures of merit are older than the lamps they are now applied to. Percent flicker and flicker index were defined for gas-discharge and fluorescent lighting, where the waveform is roughly sinusoidal and both numbers behave; applied to a square drive with a variable duty cycle, one of the two stops discriminating entirely.

What has not changed is practice. A lamp reaches a specification as a percent flicker and a frequency, with no light level, no motion, and no observer in the statement.

What the pictures cannot show #

A page cannot flicker, and it especially cannot flicker at a kilohertz. Every figure here is a drawing of a waveform and of what a model says about it; the reader’s own experience of a lamp is the evidence, and the honest form of every claim is conditional on a light level and a speed.

And the one effect a reader can check needs no apparatus at all. Glancing quickly from one side of a room to the other, past any small bright light-emitting-diode source — a standby indicator, a bicycle light, a car tail light — produces a dotted trail if the source is modulated and a smooth one if it is not. That is the phantom array, it is a measurement anybody can make in a second, and this page cannot make it.

Where the ladder goes next #

The nearest unfinished piece is the chromatic half of the stroboscopic argument. A colour-sequential source writes a coloured pattern onto the retina during a saccade — the coloured fringes anybody who has looked around a room lit by such a projector has seen — and the chromatic spatial sensitivity functions this site already has would say at what speed and frequency the fringes stop being resolvable. That is a computation with no new machinery in it.

The second is the join to the light level. Fusion moves with adaptation and so does acuity, in opposite directions, so the safe frequency for a stationary observer rises with the light while the safe frequency for a moving one falls. Neither this essay nor any specification has both.