The complex index of refraction
We want to look now at the consequences of our result, Eq. (32.33). First, we notice that $\alpha$ is complex, so the index $n$ is going to be a complex number. What does that mean? Let’s say that we write $n$ as the sum of a real and an imaginary part: \begin{equation} \label{Eq:II:32:35} n=n_R-in_I, \end{equation}  where $n_R$ and $n_I$ are real functions of $\omega$. We write $in_I$ with a minus sign, so that $n_I$ will be a positive quantity in all ordinary optical materials. (In ordinary inactive materials—that are not, like lasers, light sources themselves—$\gamma$ is a positive number, and that makes the imaginary part of $n$ negative.) Our plane wave of Eq. (32.21) is written in terms of $n$ as \begin{equation*} E_x=E_0e^{i\omega(t-nz/c)}. \end{equation*}  Writing $n$ as in Eq. (32.35), we would have \begin{equation} \label{Eq:II:32:36} E_x=E_0e^{-\omega n_Iz/c}e^{i\omega(t-n_Rz/c)}. \end{equation}  The term $e^{i\omega(t-n_Rz/c)}$ represents a wave travelling with the speed $c/n_R$, so $n_R$ represents what we normally think of as the index of refraction. But the amplitude of this wave is \begin{equation*} E_0e^{-\omega n_Iz/c}, \end{equation*}  which decreases exponentially with $z$. A graph of the strength of the electric field at some instant as a function of $z$ is shown in Fig. 32–1, for $n_I\approx n_R/2\pi$. The imaginary part of the index represents the attenuation of the wave due to the energy losses in the atomic oscillators. The intensity of the wave is proportional to the square of the amplitude, so \begin{equation*} \text{Intensity}\propto e^{-2\omega n_Iz/c}. \end{equation*}  This is often written as \begin{equation*} \text{Intensity}\propto e^{-\beta z}. \end{equation*}  where $\beta=2\omega n_I/c$ is called the absorption coefficient. Thus we have in Eq. (32.33) not only the theory of the index of refraction of materials, but the theory of their absorption of light as well. In what we usually consider to be transparent material, the quantity $c/\omega n_I$—which has the dimensions of a length—is quite large in comparison with the thickness of the material.