7 September 2006 Analytic structure and generalized duality relations for a family of hyperboloidal beams and supporting mirrors of potential interest for future gravitational wave detection interferometers
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Abstract
For the baseline design of future gravitational wave detection interferometers, use of optical cavities with nonsphericalmirrors 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 Laser Interferometer Gravitational-wave Observatory (LIGO) optical cavity design are discussed.
© (2006) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Vincenzo Galdi, Vincenzo Galdi, Giuseppe Castaldi, Giuseppe Castaldi, Vincenzo Pierro, Vincenzo Pierro, Innocenzo M. Pinto, Innocenzo M. Pinto, Juri Agresti, Juri Agresti, Erika D'Ambrosio, Erika D'Ambrosio, Riccardo DeSalvo, Riccardo DeSalvo, } "Analytic structure and generalized duality relations for a family of hyperboloidal beams and supporting mirrors of potential interest for future gravitational wave detection interferometers", Proc. SPIE 6290, Laser Beam Shaping VII, 629004 (7 September 2006); doi: 10.1117/12.679412; https://doi.org/10.1117/12.679412
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