In physics, the optical theorem is a general law of wave scattering theory, which relates the zero-angle scattering amplitude to the total cross section of the scatterer. It is usually written in the form

where (0) is the scattering amplitude with an angle of zero, that is the amplitude of the wave scattered to the center of a distant screen and is the wave vector in the incident direction.

Because the optical theorem is derived using only conservation of energy, or in quantum mechanics from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics includes both elastic and inelastic scattering.

The generalized optical theorem, first derived by Werner Heisenberg, follows from the unitary condition and is given by

is the differential solid angle, in which case, the above relation reduces to

.

History

The optical theorem was originally developed independently by Wolfgang Sellmeier and Lord Rayleigh in 1871. Lord Rayleigh recognized the zero-angle scattering amplitude in terms of the index of refraction as

(where is the number density of scatterers), which he used in a study of the color and polarization of the sky.

The equation was later extended to quantum scattering theory by several individuals, and came to be known as the Bohr–Peierls–Placzek relation after a 1939 paper. It was first referred to as the "optical theorem" in print in 1955 by Hans Bethe and Frederic de Hoffmann, after it had been known as a "well known theorem of optics" for some time.

Derivation

The theorem can be derived rather directly from a treatment of a scalar wave. If a plane wave is incident along positive z axis on an object, then the wave scattering amplitude a great distance away from the scatterer is approximately given by

and for small angles, a Taylor expansion gives us

We would now like to use the fact that the intensity, we have

, we have

Now suppose we integrate over a screen far away in the xy in x and y with negligible error. In optics, this is equivalent to summing over many fringes of the diffraction pattern. By the method of stationary phase in the below integral. We obtain

where A is the area of the surface integrated over. Although these are improper integrals, by suitable substitutions the exponentials can be transformed into complex Gaussians and the definite integrals evaluated resulting in:

, which is therefore the effective scattering cross section of the scatterer.

See also

• S-matrix

References