The form of the dielectric constant

Let us investigate an electromagnetic wave propagating through a transparent, isotropic, non-conducting, medium. The electric displacement inside the medium is given by

$${\bfm D} = \epsilon_0 {\bfm E} + {\bfm P},$$ (631)

where ${\bfm P}$ is the electric polarization. Since electrons are much lighter than ions (or atomic nuclei), we would expect the former to displace further than the latter under the influence of an electric field. Thus, to a first approximation the polarization ${\bfm P}$ is determined by the electron response to the wave. Suppose that the electrons displace a distance ${\bfm s}$ from their rest positions in the presence of the wave. It follows that

$${\bfm P} = - Ne\,{\bfm s},$$ (632)

where $N$ is the number density of electrons.

Let us assume that the electrons are bound ``quasi-elastically'' to their rest positions, so that they seek to return to these positions when displaced from them by a field ${\bfm E}$. It follows that ${\bfm s}$ satisfies the differential equation of the form

$$m\,\ddot{\bfm s} + f\,{\bfm s} = - e {\bfm E},$$ (633)

where $m$ is the electron mass, $-f {\bfm s}$ is the restoring force, and $\dot{~}$ denotes a partial derivative with respect to time. The above equation can also be written

$$\ddot{\bfm s} + g\, \omega_0 \,\dot{\bfm s} + \omega_0^{~2} {\bfm s} =-\frac{e}{m} \,{\bfm E},$$ (634)

where

$$\omega_0^{~2} = \frac{f}{m}$$ (635)

is the characteristic oscillation frequency of the electrons. In almost all dielectric media this frequency lies in the far ultraviolet region of the electromagnetic spectrum. In Eq. (4.12) we have added a phenomenological damping term $g\,\omega_0\,\dot{\bfm s}$, in order to take into account the fact that an electron excited by an impulsive electric field does not oscillate for ever. In general, however, electrons in dielectric media can be regarded as high-Q oscillators, which is another way of saying that the dimensionless damping constant $g$ is typically much less than unity. Thus, an electron ``rings'' for a long time after being excited by an impulse.

Let us assume that the electrons oscillate in sympathy with the wave at the wave frequency $\omega$. It follows from Eq. (4.12) that

$${\bfm s} =- \frac{ (e/m)\,{\bfm E}}{\omega_0^{~2} -\omega^{2}- {\rm i}\,g\,\omega\,\omega_0}.$$ (636)

Note that we have neglected the response of the electrons to the magnetic component of the the wave. It is easily demonstrated that this is a good approximation provided that the electrons do not oscillate with relativistic velocities (i.e., provided that the amplitude of the wave is sufficiently small). Thus, Eq. (4.10) yields

$${\bfm P} = \frac{(Ne^2/m) \,{\bfm E}}{\omega_0^{~2} -\omega^{2}- {\rm i}\,g\,\omega\,\omega_0}.$$ (637)

Since, by definition,

$${\bfm D} =\epsilon_0\epsilon \,{\bfm E} = \epsilon_0 {\bfm E} + {\bfm P},$$ (638)

it follows that

$$\epsilon(\omega) \equiv n^2(\omega) = 1 + \frac{(N e^2/\epsi... ...m)} {\omega_0^{~2} -\omega^{2}- {\rm i}\,g\,\omega\,\omega_0}.$$ (639)

Thus, the index of refraction is frequency dependent. Since $\omega_0$ typically lies in the ultraviolet region of the spectrum (and since $g\ll 1$), it is clear that the denominator $\omega_0^{~2} -\omega^{2} - {\rm i}\,g\,\omega\,\omega_0\simeq\omega_0^{~2} -\omega^{2}$ is positive in the entire visible spectrum, and is larger at the red end than at the blue end. This implies that blue light is refracted more than red light. This is normal dispersion. Incidentally, an expression, like the above, which specifies the dispersion of waves propagating through some dielectric medium is usually called a dispersion relation.

Let us now suppose that there are $N$ molecules per unit volume with $Z$ electrons per molecule, and that instead of a single oscillation frequency for all electrons, there are $f_i$ electrons per molecule with oscillation frequency $\omega_i$ and damping constant $g_i$. It is easily demonstrated that

$$n^2(\omega) = 1 + \frac{N e^2}{\epsilon_0 m}\sum_i \frac{f_i}{ \omega_i^{~2} -\omega^2 -{\rm i}\,g_i\,\omega\,\omega_i},$$ (640)

where the oscillator strengths $f_i$ satisfy the sum rule,

$$\sum_i f_i = Z.$$ (641)

A more exact quantum mechanical treatment of the response of an atom, or molecule, to a low amplitude electromagnetic wave also leads to a dispersion relation of the form (4.18), except that the quantities $f_i$, $\omega_i$, and $g_i$ can, in principle, be calculated from first principles. In practice, this is too difficult except for the very simplest cases.

Since the damping constants $g_i$ are generally small compared to unity, it follows from Eq. (4.18) that $n(\omega )$ is a predominately real quantity at most wave frequencies. The factor $(\omega_i^{~2}-\omega^2)^{-1}$ is positive for $\omega <\omega_i$ and negative for $\omega >\omega_i$. Thus, at low frequencies, below the smallest $\omega_i$, all of the terms in the sum in (4.18) are positive, and $n(\omega )$ is consequently greater than unity. As $\omega$ is raised so that it passes successive $\omega_i$ values, more and more negative terms occur in the sum, until eventually the whole sum is negative and $n(\omega )$ is less than unity. Thus, at very high frequencies electromagnetic waves propagate through dielectric media with phase velocities which exceed the velocity of light in a vacuum. For $\omega\simeq \omega_i$, Eq. (4.18) predicts a rather violent variation of the refractive index with frequency. Let us examine this phenomenon more closely.