A colour that moves with the viewer

Every colour on this site so far has come from something absorbing. A pigment takes power out of the spectrum, a dye takes it out over a path length, a wall takes it out at every bounce. Ask what a surface is made of and the reflectance follows.

An oil film on wet tarmac has nothing in it that absorbs anything, and it is one of the most strongly coloured things anybody sees on an ordinary day.

The claim #

A thin film’s reflectance is a function of wavelength and viewing angle, because it is an interference condition on a path length. A pigment’s reflectance contains no path length and no angle, so it cannot vary with viewing direction. That is a clean, checkable separation between structural and pigmentary colour, and it is geometric rather than chemical.

Where the spectrum comes from #

Light meeting a thin film reflects twice: once at the top surface and once at the bottom. The second beam has travelled through the film and back, so it arrives with a phase delay

$$\delta = \frac{4\pi n d \cos\theta'}{\lambda}$$

with $n$ the film’s index, $d$ its thickness, and $\theta'$ the angle inside the film. The two beams then add, and whether they reinforce or cancel depends on $\delta$ — which depends on $\lambda$. So some wavelengths are returned strongly and others are cancelled, with no absorption anywhere. The film takes nothing out of the spectrum; it redirects it.

The angle is the part that matters here. As the film is tilted away from the viewer, $\theta'$ grows and $\cos\theta'$ shrinks, so every $\lambda$ satisfying a given interference condition gets smaller. Every maximum marches towards the blue.

Computed for a 340 nm film of index 1.45, the first-order maximum runs from 657 nm at normal incidence to 513 nm at 65° — 144 nm of movement, from a change in nothing but where the observer is standing. The chromaticity moves accordingly, and between 0° and 60° the difference is

$$\Delta E_{00} = 40.5$$

which is not a subtle shift. It is a different colour by any measure.

The check that was wrong, and why it matters #

The obvious way to verify “the peaks move towards the blue” is to find the reflectance maximum at each angle and confirm it decreases. That check was written, it failed, and the physics was right.

The reason is that a film shows several interference orders at once. For this film, maxima sit at $\lambda = 4nd\cos\theta'/(2m+1)$, which at normal incidence puts $m = 1$ at 657 nm and $m = 2$ at 394 nm — both inside the visible band. The global maximum was the 394 nm order at 0° and the 639 nm order at 20°, so a check written on the largest sampled value reported the peak moving 245 nm redward while every individual order was moving blueward exactly as predicted.

The repair is to track a named order rather than a summary statistic. That is the whole lesson and it generalises well past films: an argmax is not a tracker. Any time a claim is about how a feature moves and the check is written on the extremum of a sampled curve, the check is measuring the wrong thing the moment a second feature of comparable size exists.

It is worth noting what the wrong check would have cost. It refused a correct figure, which is the harmless direction — a false alarm rather than a silent pass. Had the film been thinner, with only one order in band, the check would have passed for years and then started failing when somebody changed a thickness parameter, with no indication that the check rather than the physics was at fault.

Why the two kinds of colour are cleanly separable #

The assertion in the generator makes the comparison directly: it computes the colour change for the film between 0° and 60°, and the colour change for a pigment over the same rotation. The pigment’s is exactly zero — not small, zero, because a reflectance with no angle in it cannot produce one.

That gives a test anybody can run without instruments. Tilt the sample. If the colour changes, the colour is structural; if it does not, it is pigmentary. The test is why the distinction survives in industries that care: goniochromatic or “colour-flop” finishes are specified and measured at multiple angles, and single-angle instruments simply cannot describe them.

There is a large practical consequence for measurement. An instrument reports a property of its own geometry, and for a pigment that geometry is a detail. For a film it is the answer: a 45°/0° spectrophotometer measuring an iridescent sample returns one point on a curve, and there is no way to tell from the number that a curve exists. Multi-angle instruments exist for exactly this, and they exist because the single-angle ones gave confident wrong answers.

Thickness and angle are the same parameter #

There is a symmetry in the interference condition that explains most of what a film looks like.

$\delta$ depends on the product $nd\cos\theta'$. So changing the thickness and changing the angle do the same thing, and a film of varying thickness viewed head-on shows the same sequence of colours as a film of uniform thickness tilted through a range of angles.

That is why an oil slick appears as bands. Its thickness varies smoothly across the puddle, so each band is a different point in the same sequence — and tilting the head moves the bands, because moving the angle is equivalent to moving along the thickness axis. The two effects are indistinguishable from one measurement, which is a genuine ambiguity rather than a limitation of the eye: a single reflectance spectrum cannot say whether a film is thick and steeply viewed or thin and viewed head-on.

The practical version: the sequence of colours a film goes through as it thickens from nothing is fixed and famous — grey, then white, then straw, then brown, blue, and on through progressively less saturated repeats. It is a one-dimensional path, and every point on it is reachable either by thickness or by angle. Which is why a soap bubble’s colours change as it thins and drains, and why the black spot that appears just before it bursts is the thickness at which the two reflections cancel across the entire visible band.

Where structural colour comes from in nature #

The mechanism is not exotic. It is how a great deal of biological colour works, and the reason is that it costs nothing to maintain.

A pigment is a molecule and molecules bleach. Structural colour is a geometry, so it persists as long as the structure does — which is why museum specimens of Morpho butterflies are as blue as living ones and why their pigmented neighbours have faded. The mechanism in Morpho is more elaborate than a single film, being a multilayer stack of chitin ridges, and the extension is straightforward: more layers make the interference sharper and the colour more saturated.

Peacock feathers, beetle elytra, the inside of an abalone shell, opal and the blue of some birds’ skin are all structural. The tell in every case is the angle dependence, and it is the reason iridescent animals are hard to photograph and harder to describe in a fan deck.

The engineered versions are the same physics deliberately. Anti-reflection coatings, dielectric mirrors, interference filters, security holograms and the pearlescent flake in automotive paint are all films or stacks chosen for their interference condition, and the ones designed to be invisible — anti-reflection coatings — are the same calculation with the thickness chosen to cancel rather than reinforce.

The security application is worth a note because it depends on precisely the property that makes structural colour hard to specify. A banknote’s colour-shifting ink cannot be photocopied or printed, because a copier records one angle and a printer lays down pigment — and pigment has no angle in it. The counterfeiting difficulty is the same fact as the measurement difficulty, which is a pleasant case of a limitation being sold as a feature.

What was computed, and how #

Two-beam interference, not the full Airy formula. The expression used sums two reflected beams with the correct Fresnel amplitudes at both interfaces and includes the multiple-reflection denominator, which is exact for a single film between two semi-infinite media. It is not a multilayer stack, and every caption says which film it drew.

The angle inside the film is computed, not the angle outside. Snell’s law is applied, and the function throws on total internal reflection rather than returning a meaningless value.

The colour is computed through a named observer, from the reflectance times an illuminant. These are not hues assigned by hand to interference orders; they are the same integral every other figure on this site does.

Two assertions, and the second is the interesting one. The film’s colour must change by more than 5 $\Delta E_{00}$ between the extreme angles, and a pigment’s must not change at all. Asserting only the first would demonstrate that films are angle-dependent without establishing what that is being contrasted against.

The argmax flipped because of the grid, not because of the physics #

A film shows several interference orders at once is the right diagnosis and it is not quite the mechanism. All four of the essay’s wavelengths reproduce exactly from the interference condition — 657.3 and 513.1 for the first order at 0° and 65°, 394.4 for the second at 0°, 638.8 for the first at 20° — and the reason the check reported a 244-nanometre redward jump is one step further in.

In a lossless film between identical media every interference maximum has the same reflectance. The condition is sin²(δ/2) = 1 at each of them, so the first order and the second are exactly as bright as each other, and nothing about their heights decides which one an argmax finds.

What decides it is the grid. On this collection’s five-nanometre sampling:

At 0° the second order lands nearer a sample point and at 20° the first one does, so the argmax follows the sampling rather than the light. The failing check was measuring which peak happened to fall closest to a grid point — a quantity with no physical content at all, changing between two angles for reasons that would move again if the grid were four nanometres or six.

That is a sharper version of the lesson and a more alarming one. An argmax is not a tracker is true of any curve with two comparable features; an argmax over equal features is not even a measurement, and its value is decided by the discretisation. A check written that way does not merely follow the wrong feature — it follows nothing.

The order leaves the band, and the check had to break by 25° #

There is a second reason the twenty-degree point is where the story changes, and it is physical rather than numerical.

The second order runs 394.4 at 0°, 391.6 at 10°, 383.3 at 20° and 377.3 at 25° — which is outside this collection’s own 380-nanometre floor. So between 20° and 25° the second order leaves the visible grid entirely, and after that there is only one order to find.

That means the wrong check was going to change behaviour at that angle whatever the sampling did, and it means the same check on a slightly different film would break somewhere else. A film thin enough to keep only one order in band would have passed it indefinitely; a film thick enough for three would have produced a sequence of jumps as each order crossed the floor in turn.

The check’s failure mode is a function of the film’s thickness and the grid’s lower bound, and neither of those is what the claim is about. That is the general shape of a check written on a summary: it depends on things the claim does not, so it fails and passes for reasons the claim cannot explain.

A thinner film of a lower index is the other end of the range a coating is made in, and the swing is worth measuring there rather than assuming it scales.

A higher index is less goniochromatic by one over n squared #

The three films drawn here differ in index as well as thickness, and the essay’s note that a higher index moves less can be given the scaling.

The whole angular effect is in cos θ′, and for small angles Snell’s law makes 1 − cos θ′ about sin²θ / 2n². So the movement goes as one over the square of the index, and a film at n = 1.6 travels 82 per cent as far as one at 1.45 for the same tilt in air.

At large angles the compression is stronger than the small-angle law suggests. At a 70-degree tilt the fractional shift is 29.2 per cent at n = 1.33, 23.8 at 1.45 and 19.1 at 1.60 — so a soap film (n = 1.33) is half again as goniochromatic as a high-index coating, at the same thickness and the same viewing.

That has a design consequence in both directions. A coating meant to shift colour wants the lowest index it can be made from, which is why the strongly flopping finishes use low-index layers and many of them; and a coating meant not to shift wants a high index, which is part of why an anti-reflection stack designed at normal incidence still works reasonably off-axis.

Where the model stops #

One film, no stack. The saturated colours of real iridescence come from multilayers, and a single film gives broad, pastel maxima. So these figures understate how saturated structural colour can be.

No dispersion in the film. The index is fixed. A real film’s $n$ varies with wavelength, which shifts the maxima slightly and does not change the angular behaviour.

Smooth, uniform, flat. A real oil slick varies in thickness across its area, which is why it appears as bands of colour rather than as one colour — the visual signature of a film is the thickness map, and a uniform patch cannot show it.

No polarisation. The two polarisations have different Fresnel amplitudes and therefore slightly different interference contrasts, and near Brewster’s angle the difference is large. The unpolarised average is used throughout.

No incoherent broadening. Very thick films wash out because the source’s coherence length is exceeded, and there is no coherence length in this model — so it would happily compute a strongly coloured 50 µm film, which does not exist.

No diffraction, so no gratings. A regularly spaced structure disperses light by a different mechanism, and an opal or a CD’s surface is a grating rather than a film. Both are structural colour and both are angle-dependent; only one is on this page.

And nothing underneath. The film here sits between air and air. A real oil slick sits on water and a real coating sits on a substrate, and the substrate’s index changes the sign of one reflection — which changes which wavelengths reinforce and shifts the whole colour sequence by half an order.

What the swatches are a summary of is a spectrum whose maxima slide, and three angles are enough to see both orders move together.

What the pictures cannot show #

The most striking thing about iridescence is that it changes as the reader moves, and a printed or rendered figure cannot move. A row of swatches at five angles is the honest static substitute, and it conveys the measurement while losing the phenomenon entirely.

Several of the swatches are outside the display’s gamut. Interference maxima are narrow, so the resulting colours are saturated, and where the display cannot reach them the figures hatch rather than clip — marking rather than lying, as everywhere on this site. Structural colour is exactly the case where that matters most, because it is among the most saturated colour there is.

The spectra are drawn as smooth curves on a 5 nm grid, and a sharp multilayer’s maxima can be narrower than that. A line narrower than the grid is an instrumental width rather than a real one, which is a limit this site’s whole wavelength range imposes.

The generalisation #

Two transferable points, and the first is about verification.

A summary statistic is not a tracker. Checking that a feature moves by watching an extremum works only while there is one feature. The correct check names the feature — by order, by index, by identity — and follows that. This is the same failure as measuring a distribution’s mode when it becomes bimodal, or following “the largest eigenvalue” through a crossing.

And a property can belong to the interaction rather than to either party. A film’s colour is not in the film and not in the light and not in the eye; it is in the relationship between a thickness, a wavelength and an angle. Asking “what colour is it” presupposes the answer belongs to the object, and for a whole class of materials it does not — which is the same shape as a room’s colour belonging jointly to its paint and its geometry, and as a translucent object’s belonging jointly to its dye and its thickness.

Who found it, and when #

Hooke described the colours of thin films in Micrographia in 1665 and got the mechanism substantially right — he attributed them to reflections from the two surfaces of the film. Newton took it further in Opticks, and “Newton’s rings” is his; he measured the thickness of films from their colours, which is a remarkable inversion to have made without a wave theory.

That absence is the interesting part. Newton’s corpuscular commitments left him without a mechanism for cancellation, and his explanation involved “fits of easy reflection and easy transmission” — a periodic disposition of the light, which is a wave in everything but name. Young supplied the interference account properly in 1801, and the thin-film colours were among his primary evidence.

The biological cases took much longer. That Morpho’s blue is structural rather than pigmentary was established in the early twentieth century — Michelson argued for it in 1911 — and the detailed multilayer structure had to wait for electron microscopy. The recurring difficulty was that a structural colour cannot be extracted: grind up the wing and the blue vanishes, because the colour was in the geometry, and a chemistry that works by dissolving things is poorly placed to study it.

Where the ladder goes next #

The rung above sets the ceiling on all of it: no surface can be more colourful than an optimal colour, a bound that follows from a reflectance being at most 1 and which structural colour approaches more closely than any pigment.

Two neighbours up there deal with the measurement rather than the material — why a scene has no white point, and why gloss changes what an instrument reports, which is the same geometry problem this page raises seen from the instrument’s end.