Total Internal Reflection in an Absorbing Medium

Abstract

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, \(\hat{\varepsilon }^{\text{Re}} + i\hat{\varepsilon }^{\text{Im}}\) 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 \(\frac{\pi }{2}\) with increasing \(\hat{\varepsilon }^{\text{Im}}\). 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 \(\hat{\varepsilon }^{\text{Im}}\). When \(\hat{\varepsilon }^{\text{Im}} = 0\), the phase of the reflection coefficient changes from 0° at the critical angle to −180° at φ = 90°. With increasing \(\hat{\varepsilon }^{\text{Im}}\), 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.