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.
The limit sensitivity of interferometric gravitational wave antennas is set by the thermal noise in the dielectric mirror
coatings. These are currently made of alternating quarter-wavelength high/low index material layers with low
mechanical losses. The quarter-wavelength design yields the maximum reflectivity for a fixed number of layers, but not
the lowest noise for a prescribed reflectivity. This motivated our recent investigation of optimal thickness
configurations, which guarantee the lowest thermal noise for a targeted reflectivity. This communication provides a
compact overview of our results, involving nonperiodic genetically-engineered and truncated periodically-layered
configurations. Possible implications for the advanced Laser Interferometer Gravitational wave Observatory (LIGO) are
We highlight the self-similar properties of the reflection coefficient of optical-Cantor prefractals multilayers. These properties are first illustrated by means of numerical results obtained using the characteristic matrix method and then are analytically investigated by resorting to the small reflection approximation.
In this paper, we propose a new synthesis method which allows the design of equi-ripple dual-band impedance transformers and is based on a straightforward generalization of the well-known Chebyshev synthesis of single band transformer. As compared to a single-band Chebyshev transformer encompassing both required passbands, the proposed design yields significantly better performance.