Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals

Optimizing diffraction catastrophe evaluation

A theoretical analysis is proposed, aimed at investigating the character of those power series expansions recently considered for the evaluation of several types of diffraction catastrophes. A hyperlinear convergence is found to be the signature for such expansions, so that the results of the numerical experiments recently carried out find a meaningful interpretation in terms of the accelerating action operated by the Weniger transformation. As an important by-product of our analysis, simple criteria, aimed at numerically optimizing the diffraction catastrophe evaluations, are provided through analytical expressions.

Evaluation of cuspoid and umbilic diffraction catastrophes of codimension four

The evaluation of the two diffraction catastrophes of codimension four, namely, the butterfly and the parabolic umbilic, is here proposed by means of a simple computational approach developed in the past to characterize the whole hierarchy of the structurally stable diffraction patterns produced by optical diffraction in three-dimensional space. In particular, after expanding the phase integral representations of butterfly and parabolic umbilic in terms of (slowly) convergent power series, the retrieving action of the Weniger transformation on them is investigated through several numerical experiments. We believe that the methodology and the results presented here could also be of help for the dissemination of catastrophe optics to the widest scientific audience.

On the numerical evaluation of umbilic diffraction catastrophes

A simple computational approach is proposed for the evaluation of umbilic diffraction catastrophes which, together with cuspoids, describe the whole hierarchy of the structurally stable diffraction patterns that can be produced by optical diffraction. In this paper, after expanding the double integral representations of hyperbolic and elliptic umbilics as convergent power series, the action of the Weniger transformation on them is studied. Exact expressions for the "on-axis" umbilic field have also been found, which extend previously published results to complex values of the control parameter. Numerical experiments aimed at giving evidence of the effectiveness and implementative ease of the approach are eventually presented.

Free-space asymptotic far-field series

A systematic procedure for deriving the complete asymptotic series in inverse powers of the distance from the origin of free-space 2D and 3D, scalar and vectorial, monochromatic electromagnetic fields is derived here. Each term of the series is expressed in closed form through the use of a differential operator acting on the angular spectrum. A simple recursive routine for computing the derivatives is provided. Examples of application are also given.

Joint use of the Weniger transformation and hyperasymptotics for accurate asymptotic evaluations of a class of saddle-point integrals. II. Higher-order transformations

The use of hyperasymptotics (H) and the Weniger transformation (WT) has been proposed, in a joint fashion, for decoding the divergent asymptotic series generated by the steepest descent on a wide class of saddle-point integrals evaluated across Stokes sets [R. Borghi, Phys. Rev. E 78, 026703 (2008)]. In the present sequel, the full development of the hyperasymptotic-Weniger transformation (H-WT) up to the second order in H is derived. Numerical experiments, carried out on several classes of saddle-point integrals, including the swallowtail diffraction catastrophe, show the effectiveness of the second-level H-WT, in particular when the integrals are evaluated beyond the asymptotic realm.