The reflection of a Maxwell-Gaussian beam by a planar surface

The reflection of a three-dimensional vectorial Maxwell-Gaussian beam by a planar surface is studied. The surface is characterized by its complex reflection coefficients $r_s(\bk)$ and $r_p(\bk)$ for TE and TM electromagnetic plane waves of wavevector $\bk$, respectively. The field impinging upon the reflecting surface is modeled as a quasi-monochromatic fundamental Gaussian beam suitably modified in order to satisfy Maxwell equations (Maxwell-Gaussian beam). Analytical expressions, correct up to the second order in a perturbation expansion, are given for the reflected electric and magnetic field, respectively. We found that first order terms in the perturbation expansion account for a longitudinal shift (Goos-H\"{a}nchen effect) of the whole reflected beam, while second order terms modifies the transverse shape of the beam which is, at this order, no longer cylindrically symmetric.