Atmospheric refraction is a consequence of molecular scattering, which is rarely stated given the historical accident that before light and matter were well understood refraction and scattering were locked in separate compartments and subsequently have been sequestered more rigidly than monks and nuns in neighboring cloisters.
Consider a beam of light propagating in an optically homogeneous medium. Light is scattered laterally to the beam, weakly but observably, and more strongly in the same direction as the beam (i.e., the forward direction). The observed beam is a coherent superposition of incident light and forward-scattered light excited by the incident light. Although real refractive indices are often defined by ratios of phase velocities (see Sec. 3.5.1), we may also look upon the real refractive index as a parameter that specifies the phase shift between an incident beam and forward-scattered light. The connection between incoherent scattering and refraction, coherent scattering, can be divined from the expression for the refractive index n of a gas and that for the scattering cross section as of a gas molecule:
where N is the number (not mass) density of gas molecules, k is the wavenumber of the incident light, and a is the polarizability of the molecule (i.e., induced dipole moment per unit incident inducing electric field). The appearance of the polarizability in Eq. (8.37) but its square in Eq. (8.38) is the clue that refraction is associated with electric fields whereas scattering is associated with electric fields squared. Scattering without qualification often means incoherent scattering in all directions. Refraction, in a nutshell, is coherent scattering in a particular direction.
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