Nonspecular reflection from a lossy dielectric

International audienceNonspecular effects for reflection from a lossy dielectric are studied under the paraxial approximation. It is shown that lateral shifts larger than 100 wavelengths are obtained for an angle of incidence close to the Brewster angle. Furthermore, it is shown that a three-dimensional Gaussian beam with a circular cross section becomes on reflection near the Brewster angle a Gaussian beam with an elliptical cross section. In addition, an evaluation of the accuracy of the paraxial approximation is done by comparison with an exact calculation. Nonspecular reflection effects have been studied for total reflection after the pioneering work of Goos and Hanchen. Experimental evidence of a lateral shift of a finite beam was first reported in the context of acoustic waves. Many theoretical approaches have been developed; among them, the use of an angular spectral approach has proved to be useful in showing that a variety of nonspecular effects may be defined. 3 These effects are easily understood in the context of an angular spectrum approach based on the following expansion of the incident monochro-matic p-polarized beam Ej: Ei(xjL,zi) = Wk^ f exp(W 'k 2Si2)u(vi) X exp(ikisi * x 1 ji + ikicizi)dsi (1) in the (xi, yi, zi) coordinate system (see Fig. 1), where the time dependence exp(-iwst) has been omitted. The beam waist is w, ki = wic, v is the unit vector [s, c = (1-S2)v2] of the wave vector, and u is the unit vector specifying the polarization, given by us zAv/IzAvI, (2a) up vAu, (2b) for s and p polarization, respectively. The reflected field is then expressed as w 2 k. 2 (wki 2 s 2 Er(XllrZr) = 8v! R(sr)exp-r IU(Vr) 47T f 4 / x exp[+iki(sr * Xj 1 r + CrZr)]dSr (3) in the (Xr, Yr, Zr) coordinate system, where R(Sr) is the reflection factor for p polarization. It is seen from Eq. (3) that the dependence of the reflection factor R(Sr) on the direction of the wave vector Sr introduces a distortion of the beam compared with the symmetric beam. If we assume that the reflected beam remains Gaussian, the distortion can be accounted for by introducing variations of the parameters of the beam. These modifications, which appear as nonspecular effects, are lateral and focal shifts 8A and 3, of the p