Scattering
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A total monochromatic wave :math:`\vec \epsilon E_0 e^{i \vec k \cdot \vec x - i \omega t}` shined on an object is described using
.. math::
\vec E_{tot} = \vec E_{in} + \frac{E_0 e^{ikr-i\omega t}}{r} \vec f(\vec k).
Here the second term is the spherical scattered wave, while :math:`\vec f(\vec k)` is for shape of scattered wave.
The nature of this kind of scattered wave is that the incident wave induced the object to emit some radiation. We only consider the radiation part not the close field region.
The differential cross section is defined to be the probility of light being scattered per. For the case of scattering of electromagnetic wave is
.. math::
\frac{d\sigma}{d\Omega} = \frac{ r^2 \hat r \cdot \langle \vec S_{sc} \rangle }{\lvert \langle \vec S_{in} \rangle \rvert }.
For transverse wave,
.. math::
\hat r \cdot \langle \vec S_{sc} \rangle = \frac{c}{8\pi} \mathrm {Re} ( \vec E_{sc}^* \times \vec B_{sc} ),
and the final result of differential cross section becomes
.. math::
\frac{d\sigma}{d\Omega} = \lvert \vec f \rvert ^2 .