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The angle through which an optical element changes a light ray's propagation direction, with the chosen branch and convention specified. For the one-internal-reflection branch through a spherical drop, entry refraction, internal specular reflection, and exit refraction give $\delta =\pi +2\alpha -4\beta$, with $\sin \alpha =n\sin \beta$. The corresponding primary rainbow radius about the antisolar direction is $\pi -\delta$.

An atmospheric optical feature produced by interactions with ice crystals. A common $22^{\circ }$ halo comes from a $60^{\circ }$ prism path in hexagonal crystals; minimum deviation concentrates rays near its inner angular boundary. This differs from the liquid-water path of a rainbow.^[ref]

Wave dispersion means that different frequencies propagate with different phase velocities, so a wave packet generally changes shape. Equivalently the dispersion relation is not proportional to $k$ over the relevant band. Three natural examples are the colour separation in a rainbow, where the refractive index of water depends on wavelength; deep-water surface gravity waves, for which ${\omega }^2=gk$ and long waves have larger phase velocity; and capillary waves, whose short-wave limit is ${\omega }^2=(\gamma /\rho )k^3$ and whose shorter waves have larger phase velocity. Here $g$ is gravitational acceleration, $\gamma$ is surface tension, and $\rho$ is fluid mass density, with positive $k$ and negligible viscosity.

A distinct application is an optical prism spectrometer: the wavelength-dependent refractive index produces different refraction angles, separating spectral components for measurement. This exploits wave dispersion deliberately.

A common optical halo is produced by refraction through hexagonal atmospheric ice crystals. For a $60^{\circ }$ prism path and ice refractive index about $1.31$, the minimum deviation relation givesRandom suitable crystal orientations produce a ring around the Sun or Moon, with concentration near that minimum angle. Optical dispersion puts red near the inner edge. Other crystal paths or orientations produce other halos and arcs; the explanation should not treat every ring as the water-drop mechanism of a rainbow.^[ref]

A waterspout is a rotating atmospheric column over water. Converging air and an updraft can stretch and intensify existing rotation; cooling and condensation make part of the funnel visible, while strong surface winds raise spray. The visible funnel is not simply a solid column of sea water sucked intact to the cloud. Fair-weather waterspouts and thunderstorm-associated tornadic waterspouts have different developmental settings; a single thunderstorm-only explanation does not cover both.^[ref]

Hoar frost forms by deposition of atmospheric water vapour as ice on a sufficiently cold surface. The surface must be below freezing and below the relevant frost point. Radiative cooling on a clear quiet night can make a surface colder than air at ordinary thermometer height, so a positive reported air temperature does not rule out surface frost. Freezing previously formed liquid dew is frozen dew, a different formation route.^[ref]

Hail grows in a thunderstorm when ice embryos collect supercooled liquid droplets that freeze on them. A strong updraft increases residence time and can support growing stones; they fall out when their trajectories and weight take them out of adequate support. Clear and opaque ice layers reflect changing growth and freezing conditions. They are not proof that every layer required a complete repeated ascent and descent: horizontal movement and variations across an updraft matter too. Nor is the very cold cloud top necessarily the best growth region, because appreciable growth requires liquid water.^[ref]

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.

Regard the refraction angle as a function of the angle of incidence, $\beta =\arcsin (\sin \alpha /n)$. Implicit differentiation of Snell's law gives ${\beta }^{\prime }=\cos \alpha /(n\cos \beta )$. Differentiating the deviation angle gives ${\delta }^{\prime }=2-4{\beta }^{\prime }$. An interior stationary point therefore satisfiesThe stationary deviation of a primary rainbow ray is a minimum: differentiating $\cos \alpha =n\cos \beta {\beta }^{\prime }$ again and using Snell's law gives ${\beta }^{\prime \prime }=\tan \beta (({\beta }^{\prime }{)}^2-1)$. At the stationary point, ${\beta }^{\prime }=1/2$, and hence ${\delta }^{\prime \prime }=-4{\beta }^{\prime \prime }=3\tan \beta >0$ for a positive interior refraction angle. This minimum is responsible for the concentration of emerging light rays near the primary rainbow direction.

With the Sun behind the observer and raindrops ahead, the one-reflection branch sends many light rays toward directions close to the minimum deviation angle. Nearby values of the angle of incidence produce nearly the same outgoing direction there, forming a wave caustic. For water the resulting cone is approximately $42^{\circ }$ from the antisolar direction. The observer sees the intersecting drops as a circular rainbow, or the part of that circle visible above the ground.

Normal dispersion makes the refractive index larger for violet than for red light. At the minimum, increasing the refractive index increases the deviation angle, and therefore decreases the viewing angle $\pi -{\delta }_{\min }$. Red is on the outer edge of the primary rainbow, violet on the inner edge. The internal specular reflection is partial: with $n=4/3$, the relevant $\beta \simeq 40.2^{\circ }$ is below the critical angle $\arcsin (1/n)\simeq 48.6^{\circ }$. Diffraction and the finite angular size of the Sun smooth the geometrical concentration; the physical intensity does not actually become infinite.