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Attenuation by a complex $\epsilon $

Suppose we write (for a given frequency)

\begin{displaymath}
k = \beta + i \frac{\alpha}{2}.
\end{displaymath} (9.116)

Then
\begin{displaymath}
\mbox{\boldmath$E$}_\omega(\mbox{\boldmath$x$}) = e^{i k x} = e^{i \beta x} e^{-
\frac{\alpha}{2} x}
\end{displaymath} (9.117)

and the intensity of the (plane) wave falls off like $e^{- \alpha
x}$. $\alpha$ measures the damping of the plane wave in the medium.

Let's think a bit about $k$:

\begin{displaymath}
k = \frac{\omega}{v} = \frac{\omega}{c} n
\end{displaymath} (9.118)

where:
\begin{displaymath}
n = c/v = \frac{\sqrt{\mu\epsilon}}{\sqrt{\mu_0\epsilon_0}}
\end{displaymath} (9.119)

In most ``transparent'' materials, $\mu \approx \mu_0$ and this simplifies to $n = \sqrt{\epsilon/\epsilon_0}$. Thus:
\begin{displaymath}
k^2 = \frac{\omega^2}{c^2}\frac{\epsilon}{\epsilon_0}
\end{displaymath} (9.120)

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

\begin{displaymath}
\mbox{Re~} k^2 = \beta^2 - \frac{\alpha^2}{4} = \frac{\omega^2}{c^2}
\mbox{\rm Re~} \frac{\epsilon}{\epsilon_0}
\end{displaymath} (9.121)

and
\begin{displaymath}
\mbox{Im~} k^2 = \beta \alpha = \frac{\omega^2}{c^2} \mbox{\rm Im~}
\frac{\epsilon}{\epsilon_0} .
\end{displaymath} (9.122)

As long as $\beta^2 >> \alpha^2$ (again, true most of the time in trasparent materials) we can thus write:

\begin{displaymath}
\frac{\alpha}{\beta} \approx \frac{\mbox{\rm Im~}
\epsilon(\omega)}{\mbox{\rm Re~}
\epsilon(\omega)}
\end{displaymath} (9.123)

and
\begin{displaymath}
\beta \approx (\omega/c) \sqrt{\mbox{\rm Re~} \frac{\epsilon}{\epsilon_0}}
\end{displaymath} (9.124)

This ratio can be interpreted as a quantity similar to $Q$, the fractional decrease in intensity per wavelength travelled through the medium (as opposed to the fractional decrease in intensity per period).

To find $\alpha$ in some useful form, we have to examine the details of $\epsilon(\omega)$, which we will proceed to do next.

When $\omega$ 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''.


next up previous contents
Next: Low Frequency Behavior Up: Dispersion Previous: Anomalous Dispersion, and Resonant   Contents
Robert G. Brown 2014-08-19