Cavity with a deformable mirror for tailoring the shape of the eigenmode

We demonstrate an optical cavity that supports an eigenmode with a flattop spatial profile -- a profile that has been proposed for the cavities in the Advanced Laser Interferometer Gravitational Wave Observatory, the second-generation laser interferometric gravitational wave observatory -- because it provides better averaging of the spatially dependent displacement noise on the surface of the mirror than a Gaussian beam. We describe the deformable mirror that we fabricated to tailor the shape of the eigenmode of the cavity and show that this cavity is a factor of 2 more sensitive to misalignments than a comparable cavity with spherical mirrors supporting an eigenmode with a Gaussian profile.