Usually, when we consider optical scattering, we imagine that we have a **monochromatic plane wave** incident upon a **polarizable medium** embedded in
(for the sake of argument) free space. The **target** we imagine is a
``particle'' of some shape and hence is mathematically a (simply) connected
domain with compact support. The picture we must describe is thus

The incident wave (in the absence of the target) is thus a pure plane wave:

(14.1) | |||

(14.2) |

The incident wave

(14.3) | |||

(14.4) |

In these expressions, and , while are the polarization of the incident and scattered waves, respectively.

We are interested in the **relative power distribution** in the
scattered field (which should be proportional to the incident field in a
way that can be made independent of its magnitude in a linear
response/susceptibility approximation). The power radiated in direction
with polarization
is needed per unit intensity in the
incident wave with
. This quantity is expressed as

(14.5) |

[One gets this by considering the power distribution:

(14.6) |

as usual, where the latter relation steps hold for transverse EM fields 7.1 and 7.2 only and where we've projected out a single polarization from the incident and scattered waves so we can discuss polarization later.]

This quantity has the units of area () and is called the **differential cross-section**:

(14.7) |

In quantum theory a scattering cross-section one would substitute ``intensity'' (number of particles/second) for ``power'' in this definition but it still holds. Since the units of angles, solid or not, are dimensionless, a cross-section always has the units of area. If one integrates the cross-section around the solid angle, the resulting area is the ``effective'' cross-sectional area of the scatterer, that is, the integrated are of its effective ``shadow''. This is the basis of the optical theorem, which I will mention but we will not study (derive) for lack of time.

The point in defining it is that it is generally a property of the scattering
target that linearly determines the scattered power:

(14.8) |

We need to use the apparatus of chapter 7 to handle the vector polarization
correctly. That is, technically we need to use the Stokes parameters or
something similar to help us project out of **E** a particular polarization
component. Then (as can easily be shown by meditating on:

(14.9) |

(14.10) |

(14.11) |

(14.12) |

From this we immediately see one important result:

(14.13) |

To go further in our understanding, and to gain some useful practice against the day you have to use this theory or teach it to someone who might use it, we must consider some specific cases.