Anomalous dispersion and resonant absorption

When $\omega$ is approximately equal to $\omega_i$ the dispersion relation (4.18) reduces to

$$n^2 = n_i^{~2} + \frac{Ne^2 f_i/\epsilon_0 m}{\omega_i^{~2}-\omega^2 -{\rm i}\,g_i\,\omega\,\omega_i},$$ (642)

where $n_i$ is the average contribution in the vicinity of $\omega=\omega_i$ of all other resonances (also included in $n_i$ is the contribution 1 of the vacuum displacement current, which was previously written down separately). The refractive index is clearly complex. For a wave propagating in the $x$-direction

$${\bfm E} = {\bfm E}_0 \,\exp[{\rm i}\,(\omega/c)({\rm Re}(n)\,x-c \,t)] \exp[-(\omega/c)\,{\rm Im}(n)\,x].$$ (643)

Thus, the phase velocity of the wave is determined by the real part of the refractive index via

$$v = \frac{c}{{\rm Re}(n)}.$$ (644)

Note that a positive imaginary component of the refractive index leads to the attenuation of the wave as it propagates.

Let

$$a^2 = \frac{N e^2 f_i}{\epsilon_0 m\,\omega_i^{~2}},$$ (645)

$$x = \frac{\omega^2-\omega_i^{~2}}{\omega_i^{~2}},$$ (646)

$$y = \frac{{\rm Re}(n)^2 - {\rm Im}(n)^2}{a^2},$$ (647)

$$z = \frac{2\,{\rm Re}(n)\,{\rm Im}(n)}{a^2},$$ (648)

where $a, x, y, z$ are all dimensionless quantities. It follows from Eq. (4.20) that

$$y = \frac{n_i^{~2}}{a^2} - \frac{x}{x^2 + g_i^{~2}(1+x)},$$ (649)

$$z = \frac{g_i \sqrt{1+x}}{x^2 + g_i^{~2}(1+x) }.$$ (650)

Let us adopt the physical ordering $g_i\ll 1$. The extrema of the function $y$ occur at $x=\pm g_i$. It is easily demonstrated that

$$y_{\rm min} = y(x=g_i) = \frac{n_i^{~2}}{a^2} -\frac{1}{2g_i},$$ (651)

$$y_{\rm max} = y(x=-g_i) = \frac{n_i^{~2}}{a^2} + \frac{1}{2g_i}.$$ (652)

The maximum value of the function $z$ occurs at $x=0$. In fact,

$$z_{\rm max} = \frac{1}{g_i}.$$ (653)

Note that

$$z(x=\pm g_i) = \frac{1}{2g_i}.$$ (654)

Figure 5: Sketch of the variation of the functions $y$ and $z$ with $x$

Figure 5 shows a sketch of the variation of the functions $y$ and $z$ with $x$. These curves are also indicative of the variation of ${\rm Re}(n)$ and ${\rm Im}(n)$, respectively, with frequency $\omega$ in the vicinity of the resonant frequency $\omega_i$. Recall that normal dispersion is associated with an increase in ${\rm Re}(n)$ with increasing $\omega$. 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 $\omega_i$. 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 ${\rm Im}(n)$ is appreciable are called regions of resonant absorption. Anomalous dispersion and resonant absorption take place in the vicinity of the $i$th resonance when $\vert\omega-\omega_i\vert\stackrel {_{\normalsize <}}{_{\normalsize\sim}}O(g_i)$. Since the damping constants $g_i$ 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 $m/M$ (where $M$ 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.