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, ε^∧Re+iε^∧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 π/2 with increasing ε^∧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 φ with increasing ε^∧Im. When ε^∧Im=0, the phase of the reflection coefficient changes from 0° at the critical angle to −180° at φ = 90°. With increasing ε^∧Im, the phase becomes less dependent on φ. >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. (author)