Correcting the formalism governing Bloch Surface Waves excited by 3D Gaussian beams

Due to the growing number of publications and applications based on the exploitation of Bloch surface waves and the gross errors and approximations that are regularly used to evaluate the properties of this type of wave, we judge seriously important for successful interpretation and understanding of experiments to implement adapted formalism allowing to extract the relevant information. Through a comprehensive calculation supported by an analytical development, we establish a generalized formula for the propagation length which is different from what is usually employed in the literature. We also demonstrate that the Goos-H\"anchen shift becomes an extrinsic property that depends on the beam dimension with an asymptotic behavior limiting its value to that of the propagation length. The proposed theoretical scheme allows predicting some new and unforeseen results such as the effect due to a slight deviation of the angle of incidence or of the beam-waist position with respect to the structure. This formalism can be used to describe any polarization-dependent resonant structure illuminated by a polarized Gaussian beam.