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 (n′=n+ix)\left(n^{\prime}=n+i x\right) or an imaginary (n′=ix)\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.

Reveal Answer
5.205. Let us write the wave equation in the form A=A0ei(ωt−kx)A=A_{0} \mathrm{e}^{i(\omega t-k x)}, where k=2π/λ.k=2 \pi / \lambda . If n′=n+ix,n^{\prime}=n+i x, then k=(2π/λ0)n′k=\left(2 \pi / \lambda_{0}\right) n^{\prime} and A=A0e2πxx/λ0ei(ωt−2πnx/λ0) 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=A0ex′xcos⁡(ωt−k′x) 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.x . When x<0,x<0, the amplitude diminishes (the attenuation of the wave due to absorption). When n′=ix,n^{\prime}=i x, then A=A0ex′xcos⁡ωt A=A_{0} \mathrm{e}^{x^{\prime} x} \cos \omega t This is a standing wave whose amplitude decreases exponentially (if x<0)x<0). In this case the light experiences total internal reflection in the medium (without absorption).
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