Evaluation and acceleration of Sommerfeld integrals for stratified media Green´s functions

Several efficient methods are proposed in this paper to evaluate and accelerate the Sommerfeld integrals for stratified media Green´s functions, with the emphasis on the case of half-space. For reference, the slowly convergent method of integration along the Sommerfeld integration path (SIP) is first summarily discussed. To speed up the convergence of the SIs, the deformation is made from the SIP to the steepest descent path (SDP). If branch point is surrounded by the contour, the constant phase path passing through the branch point (BSDP) is added to calculate the contribution of the branch point singularity. Along the deformed path, the integrands have a characteristic of constant phase and fast decaying, which results in the contribution of the path integration mainly from the vicinity of the saddle point and the computation time reduced. Further reducing computation time is realized through the asymptotic approximation based stationary phase method or saddle point method (SPA) along the SDP or the BSDP and physical explanation is enlightened. Numerical results show the validity of the proposed methods.