(642) |

(643) |

(644) |

Let

(645) | |||

(646) | |||

(647) | |||

(648) |

where are all dimensionless quantities. It follows from Eq. (4.20) that

(649) | |||

(650) |

Let us adopt the physical ordering . The extrema of the function occur at . It is easily demonstrated that

(651) | |||

(652) |

The maximum value of the function occurs at . In fact,

(653) |

(654) |

Figure 5 shows a sketch of the variation of the functions and
with . These curves are also indicative of the variation of
and , respectively, with frequency
in the vicinity
of the resonant frequency . Recall that normal dispersion
is associated with an increase in with increasing .
The reverse situation is termed *anomalous dispersion*. It is
clear from the figure that normal dispersion occurs everywhere except in the
immediate neighbourhood of the resonant frequency . It is
also clear
that the imaginary part of the refractive index is only appreciable
in those regions of the electromagnetic spectrum where anomalous dispersion
takes place.
A positive imaginary component of the refractive index implies
that the wave is absorbed as it propagates through the medium,
so the regions
of the spectrum where is appreciable are called regions
of *resonant absorption*. Anomalous dispersion and resonant
absorption take place in the vicinity of the th resonance when
. Since the damping constants are,
in practice, very small compared to unity, the regions of the spectrum
in which resonant absorption takes place are strongly
localized in the vicinity
of the various resonant frequencies.

The dispersion relation (4.18) only takes electron resonances into account. Of course, there are also resonances associated with displacements of the ions (or atomic nuclei). The off-resonance contributions to the right-hand side of Eq. (4.18) from the ions are smaller than those from the electrons by a factor of order (where is a typical ion mass). Nevertheless, the ion contributions are important because they give rise to anomalous dispersion and resonant absorption close to the ion resonant frequencies. The ion resonances associated with the stretching and bending of molecular bonds typically lie in the infrared region of the electromagnetic spectrum. Those associated with molecular rotation (these resonances only affect the dispersion relation if the molecule is polar) occur in the microwave region of the spectrum. Thus, both air and water exhibit strong resonant absorption of electromagnetic waves in both the ultraviolet and infrared regions of the spectrum. In the first case this is due to electron resonances, and in the second to ion resonances. The visible region of the spectrum exists as a narrow window lying between these two regions in which there is comparatively little attenuation of electromagnetic waves.