Evanescent and real waves in quantum billiards and Gaussian beams

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Abstract
A mode in a confined planar region can contain evanescent waves in its plane-wave superposition. It would seem impossible to construct such a mode by continuation of an external scattering superposition. Which must contain only real plane waves. However, evanescent waves can be expressed as the singular limit of an angular superposition of real plane waves. This is surprising because. In the direction perpendicular to that in which it decays. An evanescent wave oscillates faster than the free-space wavenumber; thus the singular superpositions lie in the class of 'superoscillatory' functions, which vary faster than any of their Fourier components. The superposition is the limit of an exact (i.e. non-paraxial and non-singular) Gaussian beam on its evanescent side slopes. Far from the axis, the beam possesses, on each side, a line of phase singularities (nodal points) which organize the global energy current. The three-dimensional generalization provides an explicit elementary construction of a superoscillatory function.

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