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| In [[physics]], the '''optical theorem''' is a general law of [[wave]] [[scattering theory]], which relates the forward [[scattering amplitude]] to the total [[cross section (physics)|cross section]] of the scatterer. It is usually written in the form
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| :<math>\sigma_\mathrm{tot}=\frac{4\pi}{k}~\mathrm{Im}\,f(0),</math>
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| 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]], <math>\sigma_\mathrm{tot}</math> includes both [[Elasticity (physics)|elastic]] and inelastic scattering. Note that the above form is for an incident [[plane wave]]; a more general form discovered by [[Werner Heisenberg]] can be written
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| :<math>\mathrm{Im}~f(\bold{\hat{k}}', \bold{\hat{k}})=\frac{k}{4\pi}\int f(\bold{\hat{k}}',\bold{\hat{k}}'')f(\bold{\hat{k}}'',\bold{\hat{k}})~d\bold{\hat{k}}''.</math>
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| Notice that as a natural consequence of the optical theorem, an object that scatters any light at all ought to have a nonzero forward scattering amplitude. However, the physically observed field in the forward direction is a sum of the scattered and incident fields, which may add to zero.
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| == History ==
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| The optical theorem was originally discovered independently by Wolfgang von Sellmeier and [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] in 1871. Lord Rayleigh recognized the forward [[scattering amplitude]] in terms of the [[index of refraction]] as
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| :<math> n = 1+2\pi \frac{Nf(0)}{k^2}, </math>
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| (where N is the number density of scatterers)
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| 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 publication. 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 ==
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| The theorem can be derived rather directly from a treatment of a [[scalar (physics)|scalar]] [[wave]]. If a [[plane wave]] is incident on an object, then the wave amplitude a great distance away from the scatterer is approximately given by
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| :<math>\psi(\bold{r}) \approx e^{ikz}+f(\theta)\frac{e^{ikr}}{r}.</math> | |
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| All higher terms, when squared, vanish more quickly than <math>1/r^2</math>, and so are negligible a great distance away. Notice that for large values of z and small angles the [[binomial theorem]] gives us
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| :<math> r=\sqrt{x^2+y^2+z^2}\approx z+\frac{x^2+y^2}{2z}. </math> | |
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| We would now like to use the fact that the [[intensity (physics)|intensity]] is proportional to the square of the amplitude <math>\psi</math>. Approximating the r in the [[denominator]] as z, we have
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| :<math>|\psi|^2=|e^{ikz}+\frac{f(\theta)}{z}e^{ikz}e^{ik(x^2+y^2)/2z}|^2</math>
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| :<math>=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}.</math>
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| If we drop the <math>1/z^2</math> term and use the fact that <math>A+A^*=2~\mathrm{Re}~A</math> we have
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| :<math>|\psi|^2\approx 1+2~\mathrm{Re}\left(\frac{f(\theta)}{z}e^{ik(x^2+y^2)/2z}\right).</math>
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| Now suppose we [[Integral|integrate]] over a screen in the ''x''-''y'' plane, at a distance which is small enough for the small angle approximations to be appropriate, but large enough that we can integrate the intensity from <math>-\infty</math> to <math>\infty</math> with negligible error. In [[optics]], this is equivalent to including many fringes of the [[diffraction]] pattern. To further simplify matters, let's approximate <math>f(\theta)=f(0)</math>. We quickly obtain
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| :<math>\int |\psi|^2~da \approx A +2~\mathrm{Re}\left(\frac{f(0)}{z}\int_{-\infty}^{\infty} e^{ikx^2/2z}dx\int_{-\infty}^{\infty} e^{iky^2/2z}dy\right)</math>
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| where ''A'' is the area of the surface integrated over. The exponentials can be treated as [[Gaussian function|Gaussians]] and so
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| :<math>\int |\psi|^2~da=A+2~\mathrm{Re}\left(\frac{f(0)}{z}\frac{i2z\pi}{k}\right)</math>
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| :<math>=A-\frac{4\pi}{k}~\mathrm{Im}~f(0),</math>
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| which is just the probability of reaching the screen if none were scattered, lessened by an amount <math>(4\pi/k)~\mathrm{Im}~f(0)</math>, which is therefore the effective scattering [[cross section (physics)|cross section]] of the scatterer.
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| == References ==
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| * {{cite journal | author=R. G. Newton| title=Optical Theorem and Beyond | journal=Am. J. Phys | year=1976 | volume=44 | pages=639–642| doi=10.1119/1.10324|bibcode = 1976AmJPh..44..639N | issue=7 }}
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| * {{cite book | author=John David Jackson | title=Classical Electrodynamics | publisher=Hamilton Printing Company | year=1999 | isbn=0-471-30932-X}}
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| [[Category:Scattering theory]]
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| [[Category:Scattering, absorption and radiative transfer (optics)]]
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| [[Category:Quantum field theory]]
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| [[Category:Physics theorems]]
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Myrtle Benny is how I'm called and I feel comfy when people use the full title. For a whilst I've been in South Dakota and my parents live nearby. Doing ceramics is what adore performing. Hiring is his occupation.
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