At the present time, LIGO uses gaussian beams supported by spherical mirrors in its arm cavities. These beams have large intensity gradients all-across the mirror and they average poorly over the bumps and valleys that arise in mirrors due to random thermal fluctuations.
Flat-topped beams are being considered for advanced LIGO. This kind of beams have constant intensity on a large central region and therefore average better over thermal fluctuations. So far two configurations for flat-topped beams have been proposed. I will discuss a family of flat-topped beams that range in-between the ones studyed so far. Although the end-states are likely to do better when it comes to thermal fluctuations, they are only marginally stable from the optical point of view and therefore it may make sense to use one of the beams discussed here.