This paper deals with a study of the analytic structure and properties of a family of "hyperboloidal" beams (HBs) which spans from the standard Gaussian-beam configuration to "flat-top" beam configurations of interest for low-thermal-noise optical cavities for gravitational interferometry. In this connection, starting from a Gauss-Laguerre-type representation, a generalized duality relation (involving fractional Fourier transform operators of complex order) between two generic HBs is derived.
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