Abstract

The propagation constant $\beta =\omega \sqrt{{\mu }_0\epsilon }$ in the lossy or gain medium is a complex quantity, of which the imaginary part represents damping or growing amplitude. The propagation constant in the free space ${\beta }_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ is real, but the component of propagation constant parallel to the boundary surface is a complex quantity, because of the continuity of tangential components of the propagation vectors: β_1t= β_2t(generalized Snell’s law). Consequently, the normal component of propagation vector ${\beta }_{2n}={\left({\beta }_0^2-{\beta }_{2t}^2\right)}^{1/2}$ becomes complex. By using the wave impedances η_TM= β_n/ωϵand η_TE= ω_μ/β_nfor TM and TE waves, we obtain the complex reflection and transmission coefficients (generalized Fresnel formulas). In the case of total internal reflection at lossless media, β_2tis real and greater than β₀, and hence β_2nis imaginary. This is the well-known evanescent damping. In the case of lossy or gain medium, however, β_2nbecomes a complex quantity. The waves outside lossy or gain media are not perfectly evanescent but contain some radiating components. This phenomena which corresponds to the leaky surface waves will be compared with observation using an absorbing 90° prism.

© 1986 Optical Society of America