Nonspherical mirrors to reduce thermoelastic noise in advanced gravitational wave interferometers

We introduce and study a non-Gaussian paraxial cavity mode that has a special beam intensity by properly designing the end mirrors of a resonator. The final aim is to reduce the thermoelastic noise generated by the interaction between the field and the mirror by optimizing the shape of both. We present numerical results for the special design of a resonator as long as the Fabry-P\'erot arms of the gravitational wave detector Laser Interferometric Gravitational Wave Observatory (LIGO). We discuss the alignment stability of such a cavity and we present the most important investigations we have done for implementing the reshaped mirrors we propose in the next LIGO configuration. We implement different numerical tools in order to understand the impact of both a small misalignment in the Fabry-P\'erot cavity and imperfections in the radial profile of the mirrors on the power built up inside the resonators and on the signal at the dark port of the beam splitter. The results of all the simulation runs are compared with an analytical model that takes into account mismatch problems and losses.