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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.^{[1]} It is usually written in the form

where f(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 k 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^{[2]}

where is the scattering amplitude that depends on the direction of the incident wave and the direction of scattering and is the differential solid angle. When , the above relation yields the optical theorem since the left-hand side is just twice the imaginary part of and since . For scattering in a centrally symmetric field, depends only on the angle between and , in which case, the above relation reduces to

where and are the angles between and and some direction .

The optical theorem was originally developed independently by Wolfgang Sellmeier^{[3]} and Lord Rayleigh in 1871.^{[4]} Lord Rayleigh recognized the zero-angle scattering amplitude in terms of the index of refraction as

(where N 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.

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

All higher terms, when squared, vanish more quickly than , and so are negligible a great distance away. For large values of and for small angles, a Taylor expansion gives us

We would now like to use the fact that the intensity is proportional to the square of the amplitude . Approximating as , we have

If we drop the term and use the fact that , we have

Now suppose we integrate over a screen far away in the *xy* plane, which is small enough for the small-angle approximations to be appropriate, but large enough that we can integrate the intensity over to 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, we can approximate 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:

This is the probability of reaching the screen if none were scattered, lessened by an amount , which is therefore the effective scattering cross section of the scatterer.

**^**"Radar Cross Section, Optical Theorem, Physical Optics Approx, Radiation by Line Sources" on YouTube**^**Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.**^**The original publication omits his first name, which however can be inferred from a few more publications contributed by him to the same journal. One web source says he was a former student of Franz Ernst Neumann. Otherwise, little to nothing is known about Sellmeier.**^**Strutt, J. W. (1871). XV. On the light from the sky, its polarization and colour. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 41(271), 107-120.

- Roger G. Newton (1976). "Optical Theorem and Beyond".
*Am. J. Phys*.**44**(7): 639–642. Bibcode:1976AmJPh..44..639N. doi:10.1119/1.10324. - John David Jackson (1999).
*Classical Electrodynamics*. Hamilton Printing Company. ISBN 0-471-30932-X.