All Problems Dispersion and Absorption of Light
Problem 5.205
In some cases permittivity of substance turns out to be a complex or a negative quantity, and refractive index, respectively, a complex \(\left(n^{\prime}=n+i x\right)\) or an imaginary \(\left(n^{\prime}=i x\right)\) quantity. Write the equation of a plane wave for both of these cases and find out the physica meaning of such refractive indices.
5.205. Let us write the wave equation in the form \(A=A_{0} \mathrm{e}^{i(\omega t-k x)}\), where \(k=2 \pi / \lambda .\) If \(n^{\prime}=n+i x,\) then \(k=\left(2 \pi / \lambda_{0}\right) n^{\prime}\) and \[ A=A_{0} \mathrm{e}^{2 \pi x x / \lambda_{0} \mathrm{e}^{i\left(\omega t-2 \pi n x / \lambda_{0}\right)}} \] or in the real form \[ A=A_{0}{\mathbf{e}}^{\boldsymbol{x}^{\prime} x} \cos \left(\omega t-k^{\prime} x\right) \] i.e. the light propagates as a plane wave whose amplitude depends on \(x .\) When \(x<0,\) the amplitude diminishes (the attenuation of the wave due to absorption). When \(n^{\prime}=i x,\) then \[ A=A_{0} \mathrm{e}^{x^{\prime} x} \cos \omega t \] This is a standing wave whose amplitude decreases exponentially (if \(x<0)\). In this case the light experiences total internal reflection in the medium (without absorption).