Past exam of the medicine course of the University of Oxford / 1865 / First examination / Physics / 9

How do you account for the colours of a rainbow, of a soap-bubble, of mother-of-pearl ?

For a primary rainbow, sunlight is refracted into a water drop, internally reflected once, and refracted out. The refractive index varies with wavelength, producing optical dispersion. If $i$ is the incidence angle and $r$ the refracted angle, a spherical-drop ray has deviationA stationary deviation concentrates neighbouring rays into a bright angular region. For visible light in water, the primary bow lies around $42^{\circ }$ from the antisolar direction, with red outside violet. Each observer receives the suitable colour from drops on the corresponding cone; the bow is not a fixed coloured object attached to one particular group of drops. Further internal reflections produce secondary bows with different angles and colour order.

A soap film uses thin-film interference. Rays reflected from its two surfaces acquire an optical path difference $2nt\cos r$. With air on both sides of a nonabsorbing film, one of the surface reflections has a phase reversal. A constructive reflected condition is consequentlyDifferent thicknesses, angles and wavelengths therefore give different reflected colours. A sufficiently thin symmetric film becomes dark because its leading reflected contributions cancel. The colour is chiefly an interference effect, rather than a pigment carried by the soap.

Mother-of-pearl, or nacre, has a structured arrangement of mineral tablets and organic layers. Repeated refractive-index contrasts produce wavelength-selective reflection; surface grooves can also act as diffraction gratings. Thus both multilayer interference and surface diffraction can contribute. A representative repeating-layer condition is $2(n_1{d}_1\cos r_1+n_2{d}_2\cos r_2)=j\lambda$. Modern structural-colour measurements and modelling distinguish these two contributions instead of treating every nacre appearance as a single grating effect.^[ref] That microscopic account uses later structural evidence, not techniques available to an 1865 candidate.