A butterfly's electric blue holds no blue pigment. Stack a few dozen transparent layers, alternating high and low index, and the structure alone reflects a whole band of colour — and tilts it bluer as you turn it. This is structural colour: colour from geometry, not dye. The band you see is a photonic band gap, and this bench proves it two independent ways.
The stack, in cross-section. Alternating high-index (bright) and low-index (dark) layers, each a quarter-wave thick, drawn to scale. Incoming white light enters at your viewing angle; the stack throws back its band colour. Each interface reflects a sliver; when they all add in phase, the band is born.
The proof. The green curve is the reflectance R(λ) computed by the transfer-matrix method over the whole finite stack. The shaded band is the photonic band gap predicted independently by Bloch band theory — the wavelengths where |½·tr M_cell| > 1 (the dashed white curve, clamped to the strip). They coincide: the bright band is the gap.
A bench, not a feat — the nine Feats of Light are complete without it.
Most colour is pigment: a molecule absorbs some wavelengths and you see the rest. Structural colour is different — nothing is absorbed. Stack many thin, transparent layers alternating between a high refractive index n_H and a low one n_L, each a quarter-wave thick (n·d = λ₀/4), and every interface reflects a faint echo. At the design wavelength those echoes all arrive in phase and add up; the stack becomes a near-perfect mirror for a whole band of colours — and passes everything else. That band is what you see. It is the same physics behind a Morpho butterfly, a peacock feather, opal, a beetle's shell, and the sheen on a soap bubble's big cousin.
This is a one-dimensional photonic crystal. Treat the endless periodic stack with Bloch's theorem, exactly as a solid-state physicist treats electrons in a crystal: one unit cell (an H layer plus an L layer) has a 2×2 transfer matrix M_cell, and a wave can only travel through when cos(K·Λ) = ½·tr(M_cell) has a real solution. Wherever |½·tr M_cell| > 1 there is no real Bloch wavenumber K — light simply cannot propagate — and that forbidden range is the photonic band gap, the stop band, the reflected colour. Its centre is the Bragg wavelength λ₀ = 4·n_H·d_H and its width is exactly Δλ/λ₀ = (4/π)·asin((n_H−n_L)/(n_H+n_L)) — wider for a bigger index contrast.
The claim, kept honest. The green reflectance curve comes from multiplying the transfer matrix of the whole finite stack, interface by interface — it knows nothing about bands. The shaded gap comes from the trace of a single unit cell — it knows nothing about the finite stack. A built-in self-test proves the bright reflectance band sits exactly where the band theory forbids propagation, that the gap centre lands on the Bragg wavelength and its width on the closed form (to machine precision, in the frequency domain where the gap is truly symmetric), and that more periods drive the mirror toward perfect. Two independent routes, one answer.
The signature of structure. Tilt the stack and the band slides bluer — the optical path through each layer shortens with angle, so the Bragg condition shifts to shorter wavelengths. Pigment can never do that; its colour is fixed by chemistry. That angle-dependent shimmer — the way a Morpho's blue pales toward violet as it banks — is the unmistakable fingerprint of colour made from geometry. Turn the angle slider and watch the swatch shift.