Optical theorem

 ( Scattering Theory ) 

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

$\backslash sigma=\backslash frac\{4\backslash pi\}\{k\}~\backslash mathrm\{Im\}\backslash ,f(0),$

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, $\backslash sigma\_\backslash mathrm\{tot\}$ includes both elastic and inelastic scattering.

The generalized optical theorem, first derived by Werner Heisenberg, follows from the unitary condition and is given byLandau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.

$f(\backslash mathbf\{n\},\backslash mathbf\{n\}\text{'})\; -f^*(\backslash mathbf\{n\}\text{'},\backslash mathbf\{n\})=\; \backslash frac\{ik\}\{2\backslash pi\}\; \backslash int\; f(\backslash mathbf\{n\},\backslash mathbf\{n\}$)f^*(\mathbf{n}',\mathbf{n})\,d\Omega''

where $f(\backslash mathbf\{n\},\backslash mathbf\{n\}\text{'})$ is the scattering amplitude that depends on the direction $\backslash mathbf\{n\}$ of the incident wave and the direction $\backslash mathbf\{n\}\text{'}$of scattering and $d\backslash Omega$ is the differential solid angle. When $\backslash mathbf\{n\}=\backslash mathbf\{n\}\text{'}$, the above relation yields the optical theorem since the left-hand side is just twice the imaginary part of $f(\backslash mathbf\{n\},\backslash mathbf\{n\})$ and since $\backslash sigma=\backslash int|f(\backslash mathbf\{n\},\backslash mathbf\{n\}$)|^2\,d\Omega. For scattering in a centrally symmetric field, $f$ depends only on the angle $\backslash theta$ between $\backslash mathbf\{n\}$ and $\backslash mathbf\{n\}\text{'}$, in which case, the above relation reduces to

$\backslash mathrm\{Im\}\; f(\backslash theta)=\backslash frac\{k\}\{4\backslash pi\}\backslash int\; f(\backslash gamma)f(\backslash gamma\text{'})\backslash ,d\backslash Omega\text{'}\text{'}$

where $\backslash gamma$ and $\backslash gamma\text{'}$ are the angles between $\backslash mathbf\{n\}$ and $\backslash mathbf\{n\}\text{'}$ and some direction $\backslash mathbf\{n\}\text{'}\text{'}$.

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

$n\; =\; 1\; +\; 2\backslash pi\; \backslash frac\{Nf(0)\}\{k^2\}$

(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.

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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

$\backslash psi(\backslash mathbf\{r\})\; \backslash approx\; e^\{ikz\}+f(\backslash theta)\backslash frac\{e^\{ikr\}\}\{r\}.$

All higher terms, when squared, vanish more quickly than $1/r^2$, and so are negligible a great distance away. For large values of $z$ and for small angles, a Taylor expansion gives us

$r=\backslash sqrt\{x^2+y^2+z^2\}\backslash approx\; z+\backslash frac\{x^2+y^2\}\{2z\}.$

We would now like to use the fact that the intensity is proportional to the square of the amplitude $\backslash psi$. Approximating $1/r$ as $1/z$, we have

$\backslash begin\{align\}$

   |\psi|^2 &\approx \left|e^{ikz}+\frac{f(\theta)}{z}e^{ikz}e^{ik(x^2+y^2)/2z}\right|^2 \\
            &= 1+\frac{f(\theta)}{z}e^{ik(x^2+y^2)/2z}+\frac{f^*(\theta)}{z}e^{-ik(x^2+y^2)/2z}+\frac{|f(\theta)|^2}{z^2}.
   \end{align}
     

If we drop the $1/z^2$ term and use the fact that $c+c^*=2\backslash operatorname\{Re\}\{c\}$, we have

$|\backslash psi|^2\; \backslash approx\; 1+2\backslash operatorname\{Re\}\{\backslash left\backslash frac\{f(\backslash theta)\}\{z\}e^\{ik(x^2+y^2)/2z\}\backslash right\}.$

Now suppose we Integral 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 $-\backslash infty$ to $\backslash infty$ 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 $f(\backslash theta)=f(0)$ in the below integral. We obtain

$\backslash int\; |\backslash psi|^2\backslash ,dx\backslash ,dy\; \backslash approx\; A\; +2\backslash operatorname\{Re\}\backslash left\backslash frac\{f(0)\}\{z\}\backslash int\_\{-\backslash infty\}^\{\backslash infty\},$

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:

$\backslash begin\{align\}$

   \int |\psi|^2\,da &= A + 2\operatorname{Re}\left[\frac{f(0)}{z}\,\frac{2\pi i z}{ k}\right] \\
                     &= A - \frac{4\pi}{k}\,\operatorname{Im}[f(0)].\end{align}
     

This is the probability of reaching the screen if none were scattered, lessened by an amount $(4\backslash pi/k)\backslash operatorname\{Im\}f(0)$, which is therefore the effective scattering cross section of the scatterer.

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See also

• S-matrix

• John David Jackson (1999). 047130932X, Hamilton Printing Company. 047130932X

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