Atmospheric optics

Atmospheric optics is the study of visible and near-visible electromagnetic radiation as it propagates through the atmosphere. Its phenomena arise from interactions between radiation and atmospheric constituents whose dimensions range from individual molecules to extended layers of cloud and air. These interactions alter the direction, intensity, spectral composition, and polarization of incident light.

The subject encompasses effects produced by refraction through vertical density gradients, scattering by suspended particles, diffraction around small droplets, and reflection from condensed water surfaces. Many observed forms result from several mechanisms acting together rather than from a single optical process. Their geometry therefore contains information about the distribution, phase, and orientation of matter within the atmosphere.

Physical framework

In a transparent medium, the phase velocity of light is characterized by the refractive index, which varies with wavelength and with the thermodynamic state of the medium. The refractive index of air is slightly greater than unity and generally increases with atmospheric density. Changes in pressure, temperature, humidity, and chemical composition consequently produce spatial variations in optical path length.

A light ray passing through a gradual refractive-index gradient follows a curved trajectory. Under ordinary conditions, atmospheric density decreases with altitude, so nearly horizontal rays curve weakly toward the surface. This curvature modifies the apparent altitude of astronomical objects and extends the geometrical horizon. Near the horizon, the same process compresses the apparent vertical dimensions of the solar and lunar disks.

At a sharp boundary between media, the relationship between the incident and refracted directions is described by Snell's law,

[ n_1\sin\theta_1=n_2\sin\theta_2, ]

where (n_1) and (n_2) are the refractive indices on either side of the boundary. Atmospheric boundaries are commonly diffuse rather than discontinuous, but the continuous bending of a ray can be represented as refraction through a succession of thin layers.

The refractive index also depends on wavelength. This dispersion separates polychromatic light into angularly displaced spectral components. Dispersion is central to the formation of rainbows and contributes to the colored edges of some ice-crystal halos.

Molecular and particulate scattering

The color and brightness of the clear sky are governed primarily by Rayleigh scattering, which applies when the scattering particles are substantially smaller than the wavelength of the incident radiation. For an idealized nonabsorbing gas, the scattering intensity varies approximately as the inverse fourth power of wavelength,

[ I_{\mathrm{s}}\propto \frac{1}{\lambda^4}. ]

Shorter visible wavelengths are therefore redirected more efficiently than longer wavelengths. The resulting diffuse field gives the daytime sky its predominantly blue appearance, although the observed spectrum is modified by the solar spectrum, ozone absorption, multiple scattering, and the wavelength response of human vision.

The angular distribution of Rayleigh-scattered light is not uniform. Radiation scattered at approximately a right angle to the incident solar beam is strongly linearly polarized. This produces a large-scale polarization pattern centered geometrically on the Sun and antisolar direction. Multiple scattering and light reflected from the surface reduce the polarization fraction without eliminating the underlying pattern.

When atmospheric particles are similar in size to visible wavelengths, their behavior is described more generally by Mie theory. Water droplets in clouds scatter visible wavelengths with less spectral selectivity than atmospheric molecules, which accounts for the nearly white appearance of many clouds under direct illumination. Increased optical depth introduces repeated scattering and lowers the contrast between individual illumination directions, producing gray regions where comparatively little light emerges toward the observer.

Larger aerosol particles also redistribute sunlight toward small angles from the solar direction. The resulting bright region surrounding the Sun is known as the aureole. Its angular width depends on the particle-size distribution, while its radiance depends on both particle concentration and the optical thickness of the intervening air.

During twilight, sunlight travels through a much longer atmospheric path than it does when the Sun is high. Preferential removal of shorter wavelengths leaves direct and forward-scattered radiation enriched in longer wavelengths. Aerosols, ozone, clouds, and the altitude of the illuminated atmospheric layers determine the detailed distribution of twilight color.

Rainbows and droplet optics

A rainbow forms when sunlight is refracted, internally reflected, and dispersed within nearly spherical water droplets. In the primary rainbow, a ray undergoes one internal reflection before leaving the droplet. The emergent intensity is concentrated near a minimum-deviation angle, producing a circular bow centered on the antisolar point. For typical visible wavelengths, the primary bow has an angular radius of approximately (42^\circ), with red on its outer edge and violet on its inner edge.

The secondary rainbow results from two internal reflections. Its angular radius is approximately (51^\circ), and the additional reflection reverses the spectral order relative to the primary bow. The region between the two bows receives comparatively little geometrical-optics illumination and is termed Alexander's band.

René Descartes derived the angular concentration of the primary and secondary bows from refraction and internal reflection in spherical droplets. Isaac Newton subsequently incorporated the dispersion of white light into the explanation, accounting for the bows' spectral organization.

Geometrical optics predicts the principal angular structure but not every visible feature. Interference between nearby ray paths produces supernumerary rainbows, which appear as narrow pastel bands inside the primary bow. Their spacing depends on droplet size and becomes less distinct when the observed droplet population contains a broad range of diameters.

Related droplet phenomena occur when the illuminating source lies behind the observer and cloud droplets occupy the antisolar direction. A glory consists of colored rings surrounding the observer's antisolar shadow. Its formation involves wave effects that include surface waves and interference among multiply deflected paths. A fog bow is broader and less saturated than an ordinary rainbow because very small droplets produce substantial diffraction and overlapping spectral structure.

Ice-crystal halos

Halo phenomena are produced by refraction or reflection in atmospheric ice crystals. The common hexagonal prism form supplies faces intersecting at characteristic angles, so randomly oriented crystals concentrate transmitted light near specific angular distances from the Sun or Moon.

The 22-degree halo arises primarily from rays passing through two prism faces inclined at (60^\circ). Dispersion places red light along the inner boundary and increasingly shorter wavelengths farther outward, although overlap among ray paths often leaves the outer portion weakly colored. The region immediately inside the halo is darker because the crystal geometry imposes a minimum deviation for these ray paths.

Crystal orientation generates forms that differ from the circular halo associated with random orientation. Horizontally oriented plate crystals produce sundogs on either side of the Sun when the illumination geometry permits light to pass through vertical side faces. Column-shaped crystals with nearly horizontal long axes contribute to tangent arcs whose curvature changes with solar elevation.

Alfred Wegener developed a systematic geometrical treatment relating halo forms to populations of hexagonal crystals and their preferred orientations. Later ray-tracing analyses incorporated distributions of crystal shape, orientation, and optical imperfection, allowing observed halo intensities to be compared with microphysical models of cirrus clouds.

A light pillar does not require rays to enter the crystal. It results largely from specular reflection by numerous plate-like crystals at different heights and horizontal positions. Each crystal contributes a small image of the source, and their combined images form an apparently vertical luminous column.

Refraction, looming, and mirages

A mirage is a displaced or distorted image produced when light traverses air with a strong vertical refractive-index gradient. The controlling quantity is the temperature profile because temperature strongly affects air density. Humidity and pressure modify the profile but ordinarily provide smaller contributions over the short vertical scales associated with prominent mirages.

An inferior mirage develops when air immediately above a surface is warmer and less dense than the air above it. Rays traveling downward into the low-density layer curve upward, so the observer receives light from the sky along a path that appears to originate below the horizon. The resulting image resembles a reflective surface but contains refracted sky radiance rather than a specular reflection from liquid water.

A superior mirage occurs during a temperature inversion, when colder and denser air lies beneath warmer air. Rays then curve toward the denser lower layer, raising distant objects above their ordinary apparent positions. Complex vertical profiles produce multiple images whose alternating orientations arise from folds in the mapping between source elevation and apparent elevation.

In 1934, You Watanabe conducted angular and photographic measurements of superior mirages over Suruga Bay. Her comparison of displaced horizon features with balloon-derived temperature profiles established the local correspondence between inversion strength, ray curvature, and the vertical separation of repeated images. The measurements were incorporated into Japanese marine-atmospheric refraction tables used to describe low-altitude optical propagation during that period.

Fata Morgana denotes a rapidly varying complex superior mirage containing compressed, stretched, upright, and inverted image segments. It develops when an inversion supports a refractive structure in which small changes of height or temperature produce large changes in ray trajectory. The elaborate appearance reflects repeated image folding rather than the presence of distinct physical objects.

Atmospheric refraction also affects astronomical observation without producing a recognizable mirage. Near the horizon, differential refraction shifts the lower edge of the Sun more strongly than the upper edge, creating apparent flattening. Wavelength-dependent refraction separates colors slightly, and under suitable vertical gradients this separation contributes to the green flash at sunset or sunrise.

Diffraction and angular coronas

Small, similarly sized cloud droplets produce coronae around the Sun or Moon. A corona consists of a bright central region surrounded by one or more colored rings. Its angular scale varies inversely with droplet diameter because it originates predominantly from diffraction rather than from the minimum-deviation geometry responsible for a halo.

The spectral sequence of a corona differs from that of the 22-degree halo, and the structure remains much closer to the light source. A narrow droplet-size distribution produces distinct rings, whereas a broad distribution causes the individual diffraction patterns to overlap and lose contrast.

Cloud iridescence is related to the same wave-optical mechanism. It commonly occurs in optically thin cloud regions containing droplets or ice particles with relatively uniform dimensions. Variations in particle size and scattering angle shift the interference maxima, generating irregular bands of color rather than concentric rings.

Observation and quantitative interpretation

Atmospheric optical forms are described through angular position, radiance, spectrum, and polarization. Angular geometry identifies the relevant relationship among the source, observer, and scattering material. Spectral measurements distinguish wavelength-dependent refraction from approximately neutral multiple scattering, while polarization measurements constrain particle shape and scattering angle.

Marcel Minnaert integrated visual observations with quantitative optical analysis in his treatment of light and color in outdoor environments. His work connected readily observable angular forms with the underlying physics of scattering, refraction, diffraction, and visual contrast.

Modern interpretation uses radiative-transfer calculations when photons undergo multiple interactions before reaching an observer. These calculations represent absorption and scattering throughout a stratified or three-dimensional atmosphere. Ray tracing remains appropriate for many refractive and ice-crystal phenomena, while electromagnetic scattering models are required when particle dimensions are comparable to the wavelength.

Atmospheric optics therefore links visible structure to measurable atmospheric conditions. A halo constrains crystal geometry and orientation, a mirage records a refractive-index gradient, and a polarized sky pattern reflects the angular distribution of scattered sunlight. The observed image is not merely a representation of an object but also a record of the medium through which its light has traveled.