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Optical theorem

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In <a href="Physics" title="Physics">physics</a>, the optical theorem is a general law of <a href="Wave" title="Wave">wave</a> <a href="Scattering_theory" class="mw-redirect" title="Scattering theory">scattering theory</a>, which relates the zero-angle <a href="Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> to the total <a href="Cross_section_(physics)" title="Cross section (physics)">cross section</a> of the scatterer.<a href="#cite_note-1">[1]</a> It is usually written in the form

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<annotation encoding="application/x-tex">{\displaystyle \sigma ={\frac {4\pi }{k}}~\mathrm {Im} \,f(0),}</annotation>

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</math><img src="./cf1fceff36eded27ac5e1c26ac7d39ba7925c381.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.399ex; height:5.343ex;" alt="{\displaystyle \sigma ={\frac {4\pi }{k}}~\mathrm {Im} \,f(0),}" loading="lazy"></dd></dl>

where f(0) is the <a href="Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> with an angle of zero, that is the amplitude of the wave scattered to the center of a distant screen and k is the <a href="Wave_vector" title="Wave vector">wave vector</a> in the incident direction.

Because the optical theorem is derived using only <a href="Conservation_of_energy" title="Conservation of energy">conservation of energy</a>, or in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> from <a href="Conservation_of_probability" class="mw-redirect" title="Conservation of probability">conservation of probability</a>, the optical theorem is widely applicable and, in <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{\mathrm {tot} }}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{\mathrm {tot} }}</annotation>

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</math><img src="./24821a6b008fffc44e58a039cde0640663166a6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.661ex; height:2.009ex;" alt="{\displaystyle \sigma _{\mathrm {tot} }}" loading="lazy"> includes both <a href="Elasticity_(physics)" title="Elasticity (physics)">elastic</a> and inelastic scattering.

The generalized optical theorem, first derived by <a href="Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, follows from the unitary condition and is given by<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {n} ,\mathbf {n} ')-f^{*}(\mathbf {n} ',\mathbf {n} )={\frac {ik}{2\pi }}\int f(\mathbf {n} ,\mathbf {n} '')f^{*}(\mathbf {n} ',\mathbf {n} '')\,d\Omega ''}</annotation>

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</math><img src="./4cb84550818f1028e53dc1e2ce67c30212a76a34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:51.546ex; height:5.843ex;" alt="{\displaystyle f(\mathbf {n} ,\mathbf {n} ')-f^{*}(\mathbf {n} ',\mathbf {n} )={\frac {ik}{2\pi }}\int f(\mathbf {n} ,\mathbf {n} '')f^{*}(\mathbf {n} ',\mathbf {n} '')\,d\Omega ''}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {n} ,\mathbf {n} ')}">

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<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {n} ,\mathbf {n} ')}</annotation>

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</math><img src="./80a639157c8cec957535617448ef0cd239a95b08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.777ex; height:3.009ex;" alt="{\displaystyle f(\mathbf {n} ,\mathbf {n} ')}" loading="lazy"> is the scattering amplitude that depends on the direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation>

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</math><img src="./4a720c341f39f52fd96028dab83edd34d400be46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }" loading="lazy"> of the incident wave and the direction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} '}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} '}</annotation>

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</math><img src="./73db979d0673fbfd10891afdfdc8985e442ef9e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.509ex;" alt="{\displaystyle \mathbf {n} '}" loading="lazy">of scattering and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\Omega }">

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<annotation encoding="application/x-tex">{\displaystyle d\Omega }</annotation>

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</math><img src="./9a77777df83cbdbf96f0eaa6e74ba90eba9ce43b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.894ex; height:2.176ex;" alt="{\displaystyle d\Omega }" loading="lazy"> is the differential <a href="Solid_angle" title="Solid angle">solid angle</a>. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} =\mathbf {n} '}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} =\mathbf {n} '}</annotation>

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</math><img src="./f303554832f7d9374fb5ee1cd1384d19488cb44d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.754ex; height:2.509ex;" alt="{\displaystyle \mathbf {n} =\mathbf {n} '}" loading="lazy">, the above relation yields the optical theorem since the left-hand side is just twice the imaginary part of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {n} ,\mathbf {n} )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {n} ,\mathbf {n} )}</annotation>

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</math><img src="./490352b9f70c5531cc3490a8c451a4f744ccda46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.092ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {n} ,\mathbf {n} )}" loading="lazy"> and since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =\int |f(\mathbf {n} ,\mathbf {n} '')|^{2}\,d\Omega ''}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =\int |f(\mathbf {n} ,\mathbf {n} '')|^{2}\,d\Omega ''}</annotation>

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</math><img src="./7fc816cb9747b519eaf992ea26f44f284ac39fbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.005ex; height:5.676ex;" alt="{\displaystyle \sigma =\int |f(\mathbf {n} ,\mathbf {n} '')|^{2}\,d\Omega ''}" loading="lazy">. For scattering in a centrally symmetric field, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> depends only on the angle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

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</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation>

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</math><img src="./4a720c341f39f52fd96028dab83edd34d400be46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} '}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} '}</annotation>

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</math><img src="./73db979d0673fbfd10891afdfdc8985e442ef9e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.17ex; height:2.509ex;" alt="{\displaystyle \mathbf {n} '}" loading="lazy">, in which case, the above relation reduces to

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}</annotation>

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</math><img src="./61aaaabd97a106f3e88fce8b4df255c2161dc168.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.172ex; height:5.843ex;" alt="{\displaystyle \mathrm {Im} f(\theta )={\frac {k}{4\pi }}\int f(\gamma )f(\gamma ')\,d\Omega ''}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

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</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma '}">

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<annotation encoding="application/x-tex">{\displaystyle \gamma '}</annotation>

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</math><img src="./f0e2b9e9e12e56bd62af445be6803ec4843e919a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.965ex; height:3.009ex;" alt="{\displaystyle \gamma '}" loading="lazy"> are the angles between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {n} }</annotation>

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</math><img src="./4a720c341f39f52fd96028dab83edd34d400be46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {n} }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {n} '}">

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<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The optical theorem was originally developed independently by <a href="Wolfgang_Sellmeier" title="Wolfgang Sellmeier">Wolfgang Sellmeier</a><a href="#cite_note-3">[3]</a> and <a href="John_Strutt%2C_3rd_Baron_Rayleigh" title="John Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a> in 1871.<a href="#cite_note-4">[4]</a> Lord Rayleigh recognized the zero-angle <a href="Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> in terms of the <a href="Index_of_refraction" class="mw-redirect" title="Index of refraction">index of refraction</a> as

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<annotation encoding="application/x-tex">{\displaystyle n=1+2\pi {\frac {Nf(0)}{k^{2}}}}</annotation>

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</math><img src="./254e4928135032002d49874804f4e72a98ac997a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.141ex; height:6.009ex;" alt="{\displaystyle n=1+2\pi {\frac {Nf(0)}{k^{2}}}}" loading="lazy"></dd></dl>

(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 <a href="Hans_Bethe" title="Hans Bethe">Hans Bethe</a> and <a href="Frederic_de_Hoffmann" title="Frederic de Hoffmann">Frederic de Hoffmann</a>, after it had been known as a "well known theorem of optics" for some time.

<div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2></div>

The theorem can be derived rather directly from a treatment of a <a href="Scalar_(physics)" title="Scalar (physics)">scalar</a> <a href="Wave" title="Wave">wave</a>. If a <a href="Plane_wave" title="Plane wave">plane wave</a> is incident along positive z axis on an object, then the wave <a href="Scattering_amplitude" title="Scattering amplitude">scattering amplitude</a> a great distance away from the scatterer is approximately given by

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<annotation encoding="application/x-tex">{\displaystyle \psi (\mathbf {r} )\approx e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}.}</annotation>

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</math><img src="./2e0d3635ca3fc3d0e34f55b1d3bee4ead592654d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.015ex; height:5.676ex;" alt="{\displaystyle \psi (\mathbf {r} )\approx e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}.}" loading="lazy"></dd></dl>

All higher terms, when squared, vanish more quickly than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 1/r^{2}}</annotation>

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</math><img src="./bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> and for small angles, a <a href="Taylor_expansion" class="mw-redirect" title="Taylor expansion">Taylor expansion</a> gives us

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<annotation encoding="application/x-tex">{\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}\approx z+{\frac {x^{2}+y^{2}}{2z}}.}</annotation>

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</math><img src="./740a2366c383616dbba599f303218c5846ef0208.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.844ex; height:5.676ex;" alt="{\displaystyle r={\sqrt {x^{2}+y^{2}+z^{2}}}\approx z+{\frac {x^{2}+y^{2}}{2z}}.}" loading="lazy"></dd></dl>

We would now like to use the fact that the <a href="Intensity_(physics)" title="Intensity (physics)">intensity</a> is proportional to the square of the amplitude <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

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</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy">. Approximating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/r}">

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<annotation encoding="application/x-tex">{\displaystyle 1/r}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 1/z}</annotation>

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</math><img src="./eacfefd515fe6e44387f58670893730b20ed8604.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.413ex; height:2.843ex;" alt="{\displaystyle 1/z}" loading="lazy">, we have

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}|\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{aligned}}}</annotation>

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</math><img src="./6f02bd36ccee6388be9b1a2b4f41167d65c70361.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.868ex; margin-bottom: -0.303ex; width:60.077ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}|\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{aligned}}}" loading="lazy"></dd></dl>

If we drop the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/z^{2}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle 1/z^{2}}</annotation>

</semantics>

</math><img src="./693cade6f7cbec6fc0fbfe1998df3b97263ec492.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.469ex; height:3.176ex;" alt="{\displaystyle 1/z^{2}}" loading="lazy"> term and use the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c+c^{*}=2\operatorname {Re} {c}}">

<msup>

</mrow>

</msup>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c+c^{*}=2\operatorname {Re} {c}}</annotation>

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</math><img src="./a43f5bc516fe33795529ff0ba2a21142e8f2c99f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.693ex; height:2.509ex;" alt="{\displaystyle c+c^{*}=2\operatorname {Re} {c}}" loading="lazy">, we have

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</msup>

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<mrow>

<mfrac>

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</mfrac>

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<msup>

<msup>

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</msup>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle |\psi |^{2}\approx 1+2\operatorname {Re} {\left[{\frac {f(\theta )}{z}}e^{ik(x^{2}+y^{2})/2z}\right]}.}</annotation>

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</math><img src="./50b202d552e2ccc738d4f55b071285edf630aea7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.893ex; height:6.343ex;" alt="{\displaystyle |\psi |^{2}\approx 1+2\operatorname {Re} {\left[{\frac {f(\theta )}{z}}e^{ik(x^{2}+y^{2})/2z}\right]}.}" loading="lazy"></dd></dl>

Now suppose we <a href="Integral" title="Integral">integrate</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }">

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<annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation>

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</math><img src="./ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \infty }">

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<annotation encoding="application/x-tex">{\displaystyle \infty }</annotation>

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</math><img src="./c26c105004f30c27aa7c2a9c601550a4183b1f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.676ex;" alt="{\displaystyle \infty }" loading="lazy"> in x and y with negligible error. In <a href="Optics" title="Optics">optics</a>, this is equivalent to summing over many fringes of the <a href="Diffraction" title="Diffraction">diffraction</a> pattern. By the <a href="Stationary_phase_approximation" title="Stationary phase approximation">method of stationary phase</a>, we can approximate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\theta )=f(0)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(\theta )=f(0)}</annotation>

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</math><img src="./bfb14272482b7cf678f55ada4589e325be406e8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.527ex; height:2.843ex;" alt="{\displaystyle f(\theta )=f(0)}" loading="lazy"> in the below integral. We obtain

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<mfrac>

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<msup>

<msup>

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</msup>

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<annotation encoding="application/x-tex">{\displaystyle \int |\psi |^{2}\,dx\,dy\approx A+2\operatorname {Re} \left[{\frac {f(0)}{z}}\int _{-\infty }^{\infty }e^{ikx^{2}/2z}dx\int _{-\infty }^{\infty }e^{iky^{2}/2z}dy\right],}</annotation>

</semantics>

</math><img src="./c3737d6b6e79de73ecb14bc0535e9b2025dad0e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:61.384ex; height:6.343ex;" alt="{\displaystyle \int |\psi |^{2}\,dx\,dy\approx A+2\operatorname {Re} \left[{\frac {f(0)}{z}}\int _{-\infty }^{\infty }e^{ikx^{2}/2z}dx\int _{-\infty }^{\infty }e^{iky^{2}/2z}dy\right],}" loading="lazy"></dd></dl>

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 <a href="Gaussian_function" title="Gaussian function">Gaussians</a> and the definite integrals evaluated resulting in:

<mtr>

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</msup>

</mtd>

<mtd>

<mrow>

<mrow>

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<mfrac>

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</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

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</mtable>

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int |\psi |^{2}\,da&=A+2\operatorname {Re} \left[{\frac {f(0)}{z}}\,{\frac {2\pi iz}{k}}\right]\\&=A-{\frac {4\pi }{k}}\,\operatorname {Im} [f(0)].\end{aligned}}}</annotation>

</semantics>

</math><img src="./29d97adfaee1fe49b8806f77c1e9c87a35362f09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:35.15ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}\int |\psi |^{2}\,da&=A+2\operatorname {Re} \left[{\frac {f(0)}{z}}\,{\frac {2\pi iz}{k}}\right]\\&=A-{\frac {4\pi }{k}}\,\operatorname {Im} [f(0)].\end{aligned}}}" loading="lazy"></dd></dl>

This is the probability of reaching the screen if none were scattered, lessened by an amount <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (4\pi /k)\operatorname {Im} [f(0)]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (4\pi /k)\operatorname {Im} [f(0)]}</annotation>

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</math><img src="./b6de6cf8ce74db93e02830a33c28e5cef6dac7a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.384ex; height:2.843ex;" alt="{\displaystyle (4\pi /k)\operatorname {Im} [f(0)]}" loading="lazy">, which is therefore the effective scattering <a href="Cross_section_(physics)" title="Cross section (physics)">cross section</a> of the scatterer.

<ul><li><a href="S-matrix" title="S-matrix">S-matrix</a></li>

<li><a href="Kramers%E2%80%93Kronig_relations#Hadronic_scattering" title="Kramers–Kronig relations">Kramers–Kronig relations § Hadronic scattering</a></li>

<li><a href="Unitarity_(physics)#Scattering_amplitude_and_the_optical_theorem" title="Unitarity (physics)">Unitarity (physics) § Scattering amplitude and the optical theorem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=_GDBtD6qBeg">"Radar Cross Section, Optical Theorem, Physical Optics Approx, Radiation by Line Sources"</a> on <a href="YouTube_video_(identifier)" class="mw-redirect" title="YouTube video (identifier)">YouTube</a>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Landau, L. D., & Lifshitz, E. M. (2013). Quantum mechanics: non-relativistic theory (Vol. 3). Elsevier.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> 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 <a href="Franz_Ernst_Neumann" title="Franz Ernst Neumann">Franz Ernst Neumann</a>. Otherwise, little to nothing is known about Sellmeier.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> 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.

</li>

</ol></div></div>

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</style><cite id="CITEREFRoger_G._Newton1976" class="citation journal cs1"><a href="Roger_G._Newton" title="Roger G. Newton">Roger G. Newton</a> (1976). "Optical Theorem and Beyond". Am. J. Phys. 44 (7): 639–642. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976AmJPh..44..639N">1976AmJPh..44..639N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.10324">10.1119/1.10324</a>.</cite></li>

<li><cite id="CITEREFJohn_David_Jackson1999" class="citation book cs1"><a href="John_David_Jackson_(physicist)" title="John David Jackson (physicist)">John David Jackson</a> (1999). <a href="Classical_Electrodynamics_(book)" title="Classical Electrodynamics (book)">Classical Electrodynamics</a>. Hamilton Printing Company. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-30932-X</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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