References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Problem 9.23b.
In a dispersive medium, the permittivity depends on the frequency of electromagnetic radiation.
where there are electrons per atom with natural frequency and damping factor , and there are atoms per unit volume. Because is complex, the medium isn’t linear in the sense that the polarization is directly proportional to the applied field, but if we take both the polarization and field to be complex, then the medium is linear in the sense that
With this assumption, we can substitute the complex permittivity for the ordinary real permittivity in Maxwell’s equations and follow through the same steps to get the wave equation, which now becomes
Just as before, we can get plane wave solutions of the form
where is a complex wave vector
The actual real and imaginary parts of are complicated expressions since is a sum of complex numbers, but we can use the shortcut notation
giving (assuming is polarized in the direction):
The intensity of the radiation is proportional to so the intensity falls off according to as we penetrate the medium. The absorption coefficient is defined as
and gives a measure of the reciprocal of the distance at which the intensity is attenuated.
We can write the complex permittivity in 1 as
If we stay away from the resonant frequencies, where , the sum terms are quite small so we can approximate them in 5 by the first order term in a Taylor expansion. If we also take as is true of most materials, and use , we get
Using for small , we get
From 7 the speed of the wave is
so the index of refraction is
and the absorption coefficient is
If we stay away from resonances, the damping term becomes insignificant so the index of refraction is approximately
If the frequency of the wave is significantly less than all the resonant frequencies we can further approximate this using for small :
In a vacuum, so
Eqn 22 is known as the Cauchy formula, although Cauchy had many equations named after him (particularly in the area of complex variable theory), so the name is easily confused with other formulas. The parameter is the coefficient of refraction and is the coefficient of dispersion. The more usual form of Cauchy’s equation seems to be (I couldn’t find any sources that gave the formula in the form used by Griffiths).
Example Applying this model to hydrogen at 0 C and atmospheric pressure (that is, standard temperature and pressure, or STP), the number of electrons per molecule of is . In the previous post, we found that the resonant frequency is . At STP, an ideal gas occupies , so the number density is
The parameters are
The experimental values quoted by Griffiths are
so the values from the model are at least around the right order of magnitude.