Suppose we write (for a given frequency)
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:
As long as (again, true most of the time in trasparent materials) we can thus write:
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''.