Energy Flux and Density of Nonuniform Electromagnetic Waves with Total Reflection

Abstract

Analytic expressions are obtained for the energy fl ux and density of refracted nonuniform waves produced during total refl ection at the boundary between two isotropic media for the general case of elliptically polarized incident light. The average values are determined as functions of the parameters of the adjoining media and the angle of incidence. The cases of linearly and circularly polarized incident waves are examined in detail. An explicit general expression relating the energy fl ux and density of these waves for arbitrarily polarized incident light is obtained. Introduction. Nonuniform waves are damped (or amplifi ed in the case of inversion media) plane electromagnetic waves for which the planes of equal phase and equal amplitude are not parallel to one another (1). These waves arise in transparent media during total internal refl ection of light and in absorbing (amplifying) media for obliquely incident light. The properties of nonuniform waves are important in a number of cases of practical importance, such as in studies of the propagation of electromagnetic waves in light guides (waveguides), whose operation is based on the phenomenon of total internal refl ection. Nonuniform electromagnetic waves are characterized by the fact that their polarization curves for the vectors E and H can be different, i.e., E and H have different polarizations (1). It is also important to note that the phase velocity of these waves in a medium depends on the angle of incidence. Studies of nonuniform waves at the interfaces of dielectric media have recently attracted special interest in connection with their possible use in high-effi ciency thin-fi lm polarization beam splitters, as well as in optical fi lters and antirefl ection coatings with a large numerical aperture and spectral width (1-4). Another important area in research on nonuniform waves is related to the development of so-called all-angle refl ectors (5). Extending the capabilities and improving the characteristics of devices of this kind will require studies of the features of nonuniform waves under general conditions of elliptical (including the special case of circular polarization), and not just linear, polarization. Although the general theory of nonuniform electromagnetic waves was developed back in the 1960's, research on the behavior of these waves at the interfaces of different media has not lost its importance. Thus, knowledge of the properties and behavior of nonuniform electromagnetic waves at the interfaces of media, in particular their energy characteristics, is of practical, as well as theoretical, interest. This paper is a study of the energy fl ux and density of nonuniform refracted waves which arise under the conditions of total internal refl ection at the interface of two isotropic media as functions of the parameters of the adjoining media and the angle of incidence for the general case of polarized incident radiation. In particular, explicit analytic expressions for their average values in the case of a linearly polarized wave with an arbitrary oscillation azimuth, as well as in the case of circularly polarized incident radiation. To establish the dependence of the energy fl ux and density of a nonuniform wave on the parameters of the adjoining media and the angle of incidence, we begin with the solutions of the corresponding boundary problem for Maxwell's equations. We write the electric fi eld vectors of incident (E1), refl ected ( c 1 E ), and transmitted (E2) plane waves with a harmonic time dependence (~e -i�& t ) in the form