Anomalous Dispersion, and Resonant Absorption

Figure 9.1: Typical curves indicating the real and imaginary parts of $\epsilon /\epsilon _0$ for an atom with three visible resonances. Note the regions of anomalous (descending) real dispersion in the immediate vicinity of the resonances, separated by large regions of normal (ascending) dispersion.

The $\gamma_i$ are typically small compared to the oscillator frequencies $\omega_i$. (Just to give you an idea, $\gamma_i \sim 10^{9}$ sec$^{-1}$ to $\omega_i \sim 10^{15}$ sec$^{-1}$ for optical transitions in atoms, with similar proportionalities for the other relevant transitions.) That means that at most frequencies, $\epsilon(\omega)$ is nearly real

Suppose we only have a few frequencies. Below the smallest $\omega_i$, all the (real) terms in the sum are positive and Re $\epsilon(\omega) > 1$. As we increase $\omega$, one by one the terms in the sum become negative (in their real part) until beyond the highest frequency the entire sum and hence Re $\epsilon(\omega) < 1$.

As we sweep past each ``pole'' (where the real part in the denominator of a single term is zero) that term increases rapidly in the real part, then dives through zero to become large and negative, then increases monotonically to zero. Meanwhile, its (usually small) imaginary part grows, reaching a peak just where the real part is zero (when $\epsilon(\omega)$ is pure imaginary). In the vicinity of the pole, the contribution of this term can dominate the rest of the sum. We define:

Normal dispersion

as strictly increasing Re $\epsilon(\omega)$ with increasing $\omega$. This is the normal situation everywhere but near a pole.

Anomalous dispersion

as decreasing Re $\epsilon(\omega)$ with increasing $\omega$. This is true only near a sufficiently strong pole (one that dominates the sum). At that point, the imaginary part of the index of refraction becomes (relatively) appreciable.

Resonant Absorption

occurs in the regions where Im $\epsilon$ is large. We will parametrically describe this next.