Scattering of Light

the change in the characteristics of a flux of optical radiation (light) on its interaction with matter. These characteristics include the spatial distribution of intensity, the frequency spectrum, and the polarization of the light. Frequently only the change in the direction of the light propagation caused by the spatial inhomogeneity of the medium and perceived as a nonintrinsic glow in the medium is called the scattering of light.

A consistent description of the scattering of light is possible within the framework of the quantum theory of the interaction of radiation with matter, based on quantum electrodynamics and quantum concepts of the structure of matter. In this theory, a single scattering event is considered as the absorption by a particle of matter of an incident photon with an energy ħω, a momentum ħ k, and a polarization μ followed by the emission of a photon with an energy ħω’, a momentum ħ k’, and a polarization μ’. Here ħ is Planck’s constant, ω and ω’ are the photon frequencies, and each of the quantities k and k’ is a wave vector. If the energy of the emitted photon is equal to the energy of the absorbed photon (ω = ω’), the scattering is called Rayleigh, or elastic, scattering. When ω ≠ ω’, the scattering is accompanied by a redistribution of energy between radiation and matter and is called inelastic scattering.

In many cases, a description of the scattering of light within the framework of the wave theory of radiation proves adequate (seeRADIATION, ELECTROMAGNETIC and OPTICS). From the standpoint of this theory, called the classical theory, an incident light wave excites forced oscillations of the electric charges in the particles of the medium, and these oscillations become sources of secondary light waves. Here, the interference of light between the incident and secondary waves plays a decisive role.

The differential scattering cross section dσ is the quantitative characteristic of scattering in both the classical and quantum descriptions. It is defined as the ratio of the radiation flux dl scattered into a small element of solid angle dΩ, to the value of the incident flux I₀ (dσ = dI/I₀). The total scattering cross section σ is the sum of dσ over all dΩ; the cross section is usually measured in square centimeters. In elastic scattering, it may be considered that σ measures an area that does not transmit light in the direction of the light’s original propagation. The classical description of scattering often uses a scattering matrix, which relates the amplitudes of the incident wave and the waves scattered in all possible directions and makes it possible to account for a change in the state of polarization of the scattered light. The scattering indicatrix—a curve that shows the difference in the intensities of light scattered in different directions—is an incomplete but graphic representation of the scattering.

Because of the abundance and diversity of the factors that determine the scattering of light, it is extremely difficult to develop a simultaneously unified and detailed method of describing it for different cases. Idealized situations with different degrees of adequacy for the phenomenon itself are therefore considered.

The scattering of light by an individual electron is to a high degree of accuracy an elastic process. The cross section is independent of frequency (Thomson scattering) and is equal to σ , where r₀ = e²/mc² is the classical radius of the electron (which is much shorter than the wavelength of light), e and m are the electron charge and mass, and c is the speed of light in a vacuum. The scattering indicatrix of nonpolarized light in this case is such that twice as much light is scattered forward or backward, that is, at angles of 0° and 180°, than is scattered at an angle of 90°. The scattering of light by individual electrons is a process common in astrophysical plasmas; in particular, it is responsible for many phenomena in the solar corona and the coronas of other stars.

The main feature of the scattering of light by an individual atom is the strong frequency dependence of the scattering cross section. If the frequency ω of the incident light is small compared to the frequency ω₀ of the natural oscillations of the atomic electrons (atomic absorption line), then σ ~ ω⁴, or λ^-4, where λ is the wavelength of the light. This relation, found on the. basis of the concept of the atom as an electric dipole oscillating in the field of the light wave, is called Rayleigh’s law. The cross sections increase sharply near atomic absorption lines (ω (ω≈ ω₀), reaching very large values σ ≈ λ² ~ 10¹⁰ cm² at resonance (ω = ω₀). Because of a number of peculiarities in the resonant light scattering, the phenomenon is called resonance fluorescence. The indicatrix of the scattering of unpolarized light by atoms is analogous to the indicatrix described for free electrons. The scattering of light by individual atoms is observed in rarefied gases.

In the scattering of light by molecules, lines of inelastic scattering (frequency-shifted lines) appear in the scattered spectrum in addition to Rayleigh (unshifted) lines, in contrast to the case of atomic scattering. The relative shifts are ǀω -ω’ǀ/ω ~ 10^-3–10^-5, and the intensity of the shifted lines is only 10^–3–10^–6 times the intensity of the Rayleigh lines. (For inelastic scattering by molecules, see.)

The scattering of light by small particles accounts for a broad class of phenomena that may be described on the basis of the theory of the diffraction of light by dielectric particles. Many characteristic peculiarities of scattering by particles can be traced within the framework of the rigorous theory worked out for spherical particles by the British scientist A. Love (1889) and the German scientist G. Mie (1908, Mie’s theory). When the radius of the sphere r is much shorter than the wavelength of the light λ_n in the matter composing the sphere, scattering by the sphere is analogous to the nonresonant scattering of light by an atom. The scattering cross section and intensity in this case are strongly dependent on r and on the difference between the dielectric constants ∈ and ∈₀ of the materials of which the sphere and the surrounding medium are composed: σ ~ λ_n^-4r⁶(∈ – ∈₀)²(Lord Rayleigh, 1871). As r increases to r ~λ_n or more (with the condition that ∈ > 1), sharp maxima and minima appear in the scattering indicatrix. The cross sections increase rapidly near the Mie resonances (2r= mλ_n, m = 1, 2, 3, …) and become equal to 6π r², forward scattering is intensified and back scattering is diminished, and the dependence of the polarization of the light on the scattering angle becomes much more complicated.

The scattering of light by large particles (r≫ λ_n) is considered on the basis of the laws of geometrical optics and taking into account the interference of rays reflected and refracted at the surfaces of the particles. An important feature of this case is the angular periodicity of the scattering indicatrix and the periodic dependence of the cross section on the parameter r/λ_n. The scattering of light by large particles accounts for aureoles, rainbows, halos, and other effects that occur in aerosols, mists, and elsewhere.

The scattering of light by media consisting of a large number of particles differs significantly from scattering by individual particles. This is connected, first, with the interference between waves scattered by the individual particles and with the interference between such waves and the incident wave. Second, in many cases effects of multiple scattering, in which the light scattered by one particle is scattered again by others, are important. Third, the interaction of the particles with one another precludes the assumption that their motions are independent.

In 1907, L. I. Mandel’shtam showed that for the scattering of light in a continuous medium, the disruption of the medium’s optical homogeneity, in which the refractive index of the medium is not constant but varies from point to point, is a basic necessity. In an unbounded and completely homogeneous medium, waves elastically scattered by individual particles in all directions not coincident with the direction of the primary wave are mutually “damped” as a result of interference. In addition to the boundaries of the medium, optical inhomogeneities include inclusions of foreign particles or, in their absence, fluctuations of density, anisotropy, and concentration that arise as a result of the statistical nature of the thermal motion of the particles.

If the phase of a scattered wave is determined unambiguously by the phase of the incident wave, the scattering is said to be coherent; otherwise, it is incoherent. By historical tradition, scattering by an individual molecule or atom is often said to be coherent if it is Rayleigh scattering and incoherent if it is inelastic scattering. This division is merely conventional; Rayleigh scattering, like the Raman scattering, may be an incoherent process. A rigorous solution to the problem of coherence in scattering is closely connected with the concept of quantum coherence and radiation statistics. In incoherent light scattering, the phases of the secondary waves are random with respect to one another because of the irregular, random distribution of inhomogeneities in the medium; during interference, total mutual damping of waves propagating in an arbitrary direction therefore does not occur. This causes the sharp difference in the spatial distribution of coherently and incoherently scattered light.

The scattering of light by thermal fluctuations—called molecular scattering—was first pointed out by M. Smoluchowski in 1908. He developed the theory of molecular scattering by rarefied gases in which the position of each particle may be considered, with good accuracy, to be independent of the positions of the other particles. This is the cause of the randomness of the phases of waves scattered by each particle. The interaction of the particles with one another may be ignored in some cases. This makes it possible to assume that the intensity of light incoherently scattered by an ensemble of particles is the simple sum of the intensities of the light scattered by the individual particles. The total intensity is proportional to the density of the gas.

In optically thin media (seeOPTICAL THICKNESS), scattering retains many features inherent in scattering by individual molecules or atoms. In optically dense media, multiple scattering becomes extremely significant. Thus, in the earth’s atmosphere the scattering cross section for sunlight by density fluctuations is characterized by the same relation σ ~ λ^–4 as nonresonant light scattering by individual particles. This explains the blue color of the sky; the atmosphere scatters the high-frequency (blue) component of the spectrum of the sun’s rays much more strongly than the low-frequency (red) component. The process of scattering in resonance fluorescence, when a large number of particles are located in the volume λ³, is extremely complex. Under these conditions, collective effects become decisive; light scattering can occur in a manner unusual for a gas, for example, by acquiring the character of metallic reflection from the surface of the gas. A complete theory of resonance fluorescence has not been worked out.

Molecular scattering by pure solid and liquid media containing no impurities differs from the nonresonance scattering of light by gases because of the collective nature of the fluctuations in the refractive index. The fluctuations, in turn, result from fluctuations in the density and temperature of the medium when there is sufficiently strong interaction of the particles with one another. In 1910, A. Einstein developed a theory of the elastic scattering of light by fluids, using Smoluchowski’s ideas as a starting point. This theory was based on the assumption that the dimensions of the region’s optical inhomogeneities in the medium are small in comparison with the wavelength of light. Near the critical points of phase transitions (seeCRITICAL STATE), the intensity of the fluctuations increases greatly and the dimensions of the regions of the inhomogeneities become comparable to the wavelength of light. This results in a sharp intensification of scattering by the medium—critical opalescence, which is complicated by the phenomenon of multiple scattering.

Additional causes of the scattering of light are fluctuations of concentration in solutions and surface fluctuations at the interface of two immiscible fluids (L. I. Mandel’shtam, 1913). Phenomena similar to critical opalescence occur near the critical points (the precipitation point in the case of solutions, the separation point in the case of immiscible fluids).

The motion of regions of inhomogeneities of the medium leads to the appearance of frequency-shifted lines in scattered light spectra. A typical example is scattering by elastic density waves—hypersound. (SeeBRILLOUIN SCATTERING for a detailed description.)

The above discussions deal with the scattering of light of comparatively low intensity. In the 1960’s and 1970’s, the study of the scattering of extremely strong light fluxes, in which characteristic differences proved to be inherent, became possible after the development of very powerful sources of optical radiation with a narrow spectral range (lasers). Thus, for example, in resonance scattering of intense monochromatic light by an individual atom, a spectral doublet appears instead of the Ray-leigh line. In this case, the light is scattered by an atom whose state has already been changed by the action of the strong electromagnetic field of the light wave. Another peculiarity of the scattering of intense light consists in the intense nature of so-called induced processes in matter, which sharply change the characteristics of the scattering. (SeeINDUCED LIGHT SCATTERING and NONLINEAR OPTICS for detailed descriptions.)

The phenomenon of the scattering of light is used extensively in the most varied types of research in physics, chemistry, and different fields of technology. Scattered light spectra make it possible to determine molecular and atomic characteristics of substances and their elastic and relaxation constants. In many cases, these spectra are the sole source of information on forbidden transitions in molecules (seeFORBIDDEN LINES). Many methods of determining the dimensions, and sometimes the shape, of small particles are based on light scattering. This is especially important, for example, in the measurement of atmospheric visibility and the investigation of polymer solutions. The processes of induced light scattering lie at the basis of active spectroscopy and are used extensively in lasers with tunable frequency.