Editing Cross section (physics) (section)

=== Dipole approximation for the scattering cross section ===

Let us assume that a particle supports only electric and magnetic dipole modes with polarizabilities <math display="inline">\mathbf{p} = \alpha^e \mathbf{E}</math> and <math display="inline">\mathbf{m} = (\mu \mu_0)^{-1}\alpha^m \mathbf{H}</math> (here we use the notation of magnetic polarizability in the manner of Bekshaev et al.<ref name="Bekshaev2013">{{cite journal | last=Bekshaev | first=A Ya | title=Subwavelength particles in an inhomogeneous light field: optical forces associated with the spin and orbital energy flows | journal=Journal of Optics | volume=15 | issue=4 | date=2013-04-01 | issn=2040-8978 | doi=10.1088/2040-8978/15/4/044004 | page=044004| arxiv=1210.5730 | bibcode=2013JOpt...15d4004B | s2cid=119234614 }}</ref><ref name="Bliokh2014">{{cite journal | last1=Bliokh | first1=Konstantin Y. | last2=Bekshaev | first2=Aleksandr Y. | last3=Nori | first3=Franco | title=Extraordinary momentum and spin in evanescent waves | journal=Nature Communications | publisher=Springer Science and Business Media LLC | volume=5 | issue=1 | date=2014-03-06 | issn=2041-1723 | doi=10.1038/ncomms4300 | page=3300| pmid=24598730 | arxiv=1308.0547 | bibcode=2014NatCo...5.3300B | s2cid=15832637 | doi-access=free }}</ref> rather than the notation of Nieto-Vesperinas et al.<ref name="Nieto-Vesperinas2010">{{cite journal | last1=Nieto-Vesperinas | first1=M. | last2=Sáenz | first2=J. J. | last3=Gómez-Medina | first3=R. | last4=Chantada | first4=L. | title=Optical forces on small magnetodielectric particle | journal=Optics Express | publisher=The Optical Society | volume=18 | issue=11 | date=2010-05-14 | pages=11428–11443 | issn=1094-4087 | doi=10.1364/oe.18.011428 | pmid=20589003 | bibcode=2010OExpr..1811428N | doi-access=free }}</ref>) expressed through the Mie coefficients as 
<math display="block">
\alpha^e = 4 \pi \varepsilon_0 \cdot i \frac{3 \varepsilon}{2 k^3} a_1, \qquad  \alpha^m = 4 \pi \mu_0 \cdot i \frac{3 \mu}{2 k^3} b_1.
</math> 
Then the cross sections are given by <math display="block">
\sigma_{\text{ext}} = \sigma_{\text{ext}}^{\text{(e)}} + \sigma_{\text{ext}}^{\text{(m)}} =
\frac{1}{4\pi \varepsilon \varepsilon_0} \cdot 4\pi k \Im(\alpha^e) + 
\frac{1}{4\pi \mu \mu_0} \cdot 4\pi k \Im(\alpha^m)
</math> <math display="block">
\sigma_{\text{sc}} = \sigma_{\text{sc}}^{\text{(e)}} + \sigma_{\text{sc}}^{\text{(m)}} = 
\frac{1}{(4\pi \varepsilon \varepsilon_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^e|^2 +
\frac{1}{(4\pi \mu \mu_0)^2} \cdot \frac{8\pi}{3} k^4 |\alpha^m|^2
</math> and, finally, the electric and magnetic absorption cross sections <math display="inline">\sigma_{\text{abs}} = \sigma_{\text{abs}}^{\text{(e)}} + \sigma_{\text{abs}}^{\text{(m)}}</math> are <math display="block">
\sigma_{\text{abs}}^{\text{(e)}} = \frac{1}{4 \pi \varepsilon \varepsilon_0} \cdot 4\pi k \left[ \Im(\alpha^e) - \frac{k^3}{6 \pi \varepsilon \varepsilon_0}  |\alpha^e|^2\right]
</math> and <math display="block">
\sigma_{\text{abs}}^{\text{(m)}} = \frac{1}{4 \pi \mu \mu_0} \cdot 4\pi k \left[ \Im(\alpha^m) - \frac{k^3}{6 \pi \mu \mu_0}  |\alpha^m|^2\right]
</math>

For the case of a no-inside-gain particle, i.e. no energy is emitted by the particle internally (<math display="inline">\sigma_{\text{abs}} > 0</math>), we have a particular case of the [[Optical theorem]] <math display="block">
\frac{1}{4\pi \varepsilon \varepsilon_0} \Im(\alpha^e) + \frac{1}{4\pi \mu \mu_0} \Im(\alpha^m) \geq \frac{2 k^3}{3} \left[ \frac{|\alpha^e|^2}{(4\pi \varepsilon \varepsilon_0)^2}  + \frac{|\alpha^m|^2}{(4\pi \mu \mu_0)^2} \right]
</math> Equality occurs for non-absorbing particles, i.e. for <math display="inline">\Im(\varepsilon) = \Im(\mu) = 0</math>.