Suppose we write (for a given frequency)

(11.62) |

Then

(11.63) |

and the

Let's think a bit about : k = &omega#omega;v = &omega#omega;c n where: n = c/v = &mu#mu;&epsi#epsilon;&mu#mu;_0&epsi#epsilon;_0 In most ``transparent'' materials, and this simplifies to . Thus: k^2 = &omega#omega;^2c^2&epsi#epsilon;&epsi#epsilon;_0

Nowever, *now*
has real and imaginary parts, so
may
as well! In fact, using the expression for
in terms of
and
above, it is easy to see that:

Re Re | (11.64) |

and

Im Im | (11.65) |

As long as (again, true most of the time in trasparent materials) we can thus write:

(11.66) |

and &beta#beta;&ap#approx;(&omega#omega;/c) Re &epsi#epsilon;&epsi#epsilon;_0 This ratio can be interpreted as a quantity similar to , the fractional decrease in intensity per wavelength travelled through the medium (as opposed to the fractional decrease in intensity per period).

To find in some useful form, we have to examine the details of , which we will proceed to do next.

When
is in among the resonances, there is little we can do
besides work out the details of the behavior, since the properties of
the material can be dominated strongly by the *local* dynamics
associated with the nearest, strongest resonance. However, there are
two limits that are of particular interest to physicists where the
``resonant'' behavior can be either evaluated or washed away. They are
the *low frequency behavior* which determines the conduction
properties of a material far away from the electron resonances per se,
and the *high frequency behavior* which is ``universal''.