Perturbation study of resonances for nearly circular micro-cavities

There is currently a great deal of interest in flat dielectric micro-cavities due to their numerous practical applications and their ability to address fundamental physics issues. But apart from circular cavity, exact expressions of wave-functions and spectrum are still unknown for all other cavity shapes. The difficulty originates from two separated factors: non-linearities induced by the boundary shape and diffraction at the corners. Here we propose an analytical perturbation method for nearly circular cavities and give general formulae up to the second order in the perturbation parameter for the wave-functions, the spectrum, and the far-field pattern. This approach is confirmed with the example of the cut-disk (a chaotic cavity shape with corners) by numerical simulations based on the Boundary Element Method and experiments with organic micro-lasers. This analytical method can be extended to a broad diversity of cavity shapes. We discuss its range of validity in length scale and the limit of the effective refractive index approximation which underlies the two-dimensional approximation.