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.

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