Total Internal Reflection in an Absorbing Medium

We report the results of a rigorous calculation on the total internal reflection when the second medium is absorbing. A complex dielectric constant given by, (epsilon) over cap Re + i (epsilon) over cap (Im) is introduced to represent the effect of the absorption. Starting from Maxwell's equations, we show that the angle of refraction becomes progressively smaller than pi/2 with increasing (epsilon) over cap (Im). The reflection and transmission coefficients for both of s- and p-polarized incident beams are calculated on the same footing. The absolute value of the reflection coefficient becomes smaller than unity and begins to depend on the incident angle phi with increasing (epsilon) over cap (Im). When (epsilon) over cap (Im) = 0, the phase of the reflection coefficient changes from 0 degrees at the critical angle to -180 degrees at phi = 90 degrees. With increasing (epsilon) over cap (Im), the phase becomes less dependent on phi. From a calculation of the time average of the Poynting vector, we confirm that the law of conservation of energy holds at the boundary between the two media.