Absorption
Perhaps you have noticed something a little strange about the last form (Eq. 31.20) we obtained for our dispersion equation. Because of the term $i\gamma$ we put in to take account of damping, the index of refraction is now a complex number! What does that mean?  By working out what the real and imaginary parts of $n$ are we could write \begin{equation} \label{Eq:I:31:21} n=n'-in'', \end{equation} where $n'$ and $n''$ are real numbers. (We use the minus sign in front of the $in''$ because then $n''$ will turn out to be a positive number, as you can show for yourself.) We can see what such a complex index means when there is only one resonant frequency by going back to Eq. (31.6), which is the equation of the wave after it goes through a plate of material with an index $n$. If we put our complex $n$ into this equation, and do some rearranging, we get \begin{equation} \label{Eq:I:31:22} E_{\text{after plate}}= \underbrace{\vphantom{E_0}e^{-\omega n''\,\Delta z/c}}_{\text{A}} \underbrace{e^{-i\omega(n'-1)\,\Delta z/c} E_0e^{i\omega(t-z/c)}}_{\text{B}}. \end{equation} The last factors, marked B in Eq. (31.22), are just the form we had before, and again describe a wave whose phase has been delayed by the angle $\omega(n' - 1)\,\Delta z/c$ in traversing the material. The first term (A) is new and is an exponential factor with a real exponent, because there were two $i$’s that cancelled. Also, the exponent is negative, so the factor is a real number less than one. It describes a decrease in the magnitude of the field and, as we should expect, by an amount which is more the larger $\Delta z$ is. As the wave goes through the material, it is weakened. The material is “absorbing” part of the wave. The wave comes out the other side with less energy. We should not be surprised at this, because the damping we put in for the oscillators is indeed a friction force and must be expected to cause a loss of energy. We see that the imaginary part $n''$ of a complex index of refraction represents an absorption (or “attenuation”) of the wave. In fact, $n''$ is sometimes referred to as the “absorption index.”  We may also point out that an imaginary part to the index $n$ corresponds to bending the arrow $E_a$ in Fig. 31–3 toward the origin. It is clear why the transmitted field is then decreased. Normally, for instance as in glass, the absorption of light is very small. This is to be expected from our Eq. (31.20), because the imaginary part of the denominator, $i\gamma_k\omega$, is much smaller than the term $(\omega_k^2 - \omega^2)$. But if the light frequency $\omega$ is very close to $\omega_k$ then the resonance term $(\omega_k^2 - \omega^2)$ can become small compared with $i\gamma_k\omega$ and the index becomes almost completely imaginary, as shown in Fig. 31–5(b). The absorption of the light becomes the dominant effect. It is just this effect that gives the dark lines in the spectrum of light which we receive from the sun. The light from the solar surface has passed through the sun’s atmosphere (as well as the earth’s), and the light has been strongly absorbed at the resonant frequencies of the atoms in the solar atmosphere. The observation of such spectral lines in the sunlight allows us to tell the resonant frequencies of the atoms and hence the chemical composition of the sun’s atmosphere. The same kind of observations tell us about the materials in the stars. From such measurements we know that the chemical elements in the sun and in the stars are the same as those we find on the earth.