Designing arbitrary-shaped whispering-gallery cavities based on transformation optics

A novel approach to designing anisotropic whispering gallery modes in gradient index cavities has been reported recently. These cavities, called transformation cavities, can support high-Q whispering gallery modes with directional emission. However, it is usually difficult to find the desired conformal mapping, and it may contain unwanted singularities inside. We show that arbitrary-shaped transformation cavities can be designed by virtue of a quasi-conformal mapping method without confronting such problems. Even though the quasi-conformal mapping method is exploited, we verify that the resulting mappings in our case are strictly conformal. As a demonstration, Q-factor, near field intensity, far field pattern, and phase space description of resonant modes formed in so-designed quadrupole-shaped transformation cavities are presented.