Scattering

When elastically bound charged particles are exposed to electromagnetic waves, the particles are set into motion by the electric field. If the frequency of the wave equals the natural frequency of free vibrations of a particle, resonance occurs being accompanied by a considerable amount of absorption. Scattering, on the other hand, takes place at frequencies not corresponding to those natural frequencies of particles. The resulting oscillation is determined by forced vibration. In general, this vibration will have the same frequency and direction as that of the electric force in the incident wave. Its amplitude, however, is much smaller than in the case of resonance. Also, the phase of the forced vibration differs from the incident wave, causing photons to slow down when penetrating into a denser medium. Hence, scattering can be regarded as the basic origin of dispersion.

Elastic and inelastic scattering are distinguished, depending on whether part of the incident photon energy is converted during the process of scattering. In the following paragraphs, we will first consider elastic scattering, where incident and scattered photons have the same energy. A special kind of elastic scattering is Rayleigh scattering. Its only restriction is that the scattering particles be smaller than the wavelength of incident radiation. In particular, we will find a relationship between scattered intensity and index of refraction, and that scattering is inversely proportional to the fourth power of wavelength. The latter statement is also known as Rayleigh's law and shall be derived in the following paragraphs.

Fig. 2.6. Geometry of Rayleigh scattering

In Fig. 2.6, a simple geometry of Rayleigh scattering is shown. A plane electromagnetic wave is incident on a thin scattering medium with a total thickness L. At a particular time, the electric field of the incident wave can be expressed by

E(z) = E0 exp(ikz) , where E0 is the amplitude of the incident electric field, k is the amount of the propagation vector, and z denotes the optical axis. In a first approximation, we assume that the wave reaching some point P on the optical axis will essentially be the original wave, plus a small contribution due to scattering. The loss in intensity due to scattering is described by a similar relation as absorption, i.e.

where as is the scattering coefficient. Differentiation of (2.17) with respect to z leads to

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