A contribution to the mathematical theory of diffraction. Part I: A note on double Fourier integrals

TL;DR

This paper develops a mathematical framework for deriving explicit far-field asymptotic expansions of physical fields represented by double Fourier integrals, crucial for diffraction theory, by extending complex analysis techniques to two variables.

Contribution

It introduces the bridge and arrow notation to generalize contour indentation for double integrals and establishes the locality principle for identifying contributing points.

Findings

Derived explicit asymptotic components for contributing points.

Validated the theory with numerical examples.

Provided criteria to identify contributing points.

Abstract

We consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to provide a closed-form far-field asymptotic expansion of $u$. In order to do so, we need to generalise the well-established complex analysis notion of contour indentation to integrals of functions of two complex variables. It is done by introducing the so-called bridge and arrow notation. Thanks to another integration surface deformation, we show that, to achieve our aim, we only need to study a finite number of real points in the Fourier space: the contributing points. This result is called the locality principle. We provide an extensive set of results allowing one to decide whether a point is contributing or not. Moreover, to each contributing point, we associate an explicit closed-form far-field asymptotic component of $u$. We conclude the article by validating this theory against full numerical computations for two specific examples.