  For the baseline design of the advanced Laser Interferometer
Gravitational-wave Observatory (LIGO), use of optical cavities with
non-spherical mirrors supporting flat-top ("mesa") beams, potentially capable
of mitigating the thermal noise of the mirrors, has recently drawn a
considerable attention. To reduce the severe tilt-instability problems
affecting the originally conceived nearly-flat, "Mexican-hat-shaped" mirror
configuration, K. S. Thorne proposed a nearly-concentric mirror configuration
capable of producing the same mesa beam profile on the mirror surfaces.
Subsequently, Bondarescu and Thorne introduced a generalized construction that
leads to a one-parameter family of "hyperboloidal" beams which allows
continuous spanning from the nearly-flat to the nearly-concentric mesa beam
configurations. This paper is concerned with a study of the analytic structure
of the above family of hyperboloidal beams. Capitalizing on certain results
from the applied optics literature on flat-top beams, a physically-insightful
and computationally-effective representation is derived in terms of
rapidly-converging Gauss-Laguerre expansions. Moreover, the functional relation
between two generic hyperboloidal beams is investigated. This leads to a
generalization (involving fractional Fourier transform operators of complex
order) of some recently discovered duality relations between the nearly-flat
and nearly-concentric mesa configurations. Possible implications and
perspectives for the advanced LIGO optical cavity design are discussed.
