Reflection and Transmission of Beams at a Dielectric Interface

Abstract

When complex values are assigned to the source coordinates in the expressions for the fields radiated by a line or point source in a homogeneous medium, the resulting fields have the form of a twoor three-dimensional Gaussian beam. This fact may be utilized to develop results for beam propagation and scattering in inhomogeneous regions from corresponding results for point or line source fields. After justifying the analytic continuation procedure to derive rigorous integral solutions for the (two-dimensional) beam fields reflected by and transmitted through a plane dielectric interface, asymptotic results are obtained by the saddle point method; it is found that the analytic continuation to complex source coordinates can be performed directly on the asymptotic expressions for reflected, refracted and lateral wave constituents of ordinary line source fields. Interpretation of the reflected fields reveals that they incorporate directly the lateral beam shift, a phenomenon usually regarded as a diffraction effect. When the beam is incident from the denser medium at an angle greater than the critical angle for total reflection, calculation of the evanescent transmitted fields requires special care since the corresponding results for an ordinary line source have not been well developed. A "local inhomogeneous plane wave" approximation is derived for the transmitted field in this case, together with physically meaningful restrictions on the range of validity of the approximation. These restrictions, and their interpretation, which have not previously been reported, are of importance in approximating the evanescent fields produced by total reflection in more general configurations not amenable to rigorous analysis.

Hexagonal dielectric resonators and microcrystal lasers

“…Figure 16 compares the semiclassical result in Eqs. (24)- (27) to the numerical data as function of n. While some resonances are very well described by the semiclassical approximation, some are only roughly described. The latter cases correspond to the strong fluctuations around the semiclassical hyperbola (24) already visible in Fig.…”

“…Ref. [27]), in that it happens above the critical angle of total internal reflection. Moreover, it has nothing to do with the angular Goos-Hänchen effect (see e.g.…”

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…38(b). For the case of reflection the beam shift is called angular Goos-Hänchen shift (Chan and Tamir, 1985;Ra et al, 1973) and for the case of transmission Fresnel filtering (FF) . There are subtle differences in the precise definition of FF and angular Goos-Hänchen shift which are of no relevance for the discussion here; see (Götte et al, 2013) for a comprehensive comparison.…”

Complex Analysis of the Induced Currents on a Perfectly Conducting Plane Under Complex Beam Incidence

“…This procedure has been successfully applied to a wide variety of electromagnetic problems, specially those dealing with Gaussian beams, [3,4]. (Let us state that there are other approachesto formulate Gaussian Beams [5]).…”

“…(Let us state that there are other approachesto formulate Gaussian Beams [5]). Examples [3] and [4] only use the analytical continuation as a mathematical tool, and do not include an in-depth study of the complex quantities which appear and the physical information underlying on them. Alternative ideas about using complex coordinates and spaces have been set forth in [6] and references therein.…”