Real spectra, complex spectra, compact spectra

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Abstract
Plane-wave spectral and induced source representations of directly excited and (or) scattered fields constitute alternative approaches for analyzing wave propagation. Although the plane-wave spectra, on the one hand, and the source distributions, on the other, generally require continuous superpositions that lead to integral formulations in the spatial-spectral and the physical configuration domains, respectively, phenomena of constructive and destructive interference at high frequencies permit contraction of these distributed constituents around interference maxima represented by stationary points, end points, or other critical points in the integration interval. This leads to a representation of the high-frequency field in terms of physically meaningful compact spectral objects generated by local portions of the source distribution that radiate energy from the (actual or induced) source region to the observer by local plane waves traversing the ray trajectories of the geometrical theory of diffraction. If the critical points in the integrals are real, the compact representation identifies nonevanescent wave bundles emitted by source patches at a real physical location. However, for many wave phenomena involving beam-type initial source fields, concave and convex boundaries, leaky waveguides, etc., as well as damped resonances in the time domain, the spectral contraction occurs around damped complex constituents identifying bundles of evanescent plane waves that travel along complex ray trajectories. Thus the initial source configuration and propagation space must be extended by analytic continuation to complex values. Insisting on real spectral and configurational domains expresses in a smeared-out unnatural manner what is compact and natural in the complex domain. These concepts are illustrated here in various examples, with emphasis on the physical importance of compact representations.

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