It is well known from transformation optics that a light pathway can be designed with artificial materials. When a coordinate transform technique is applied to optically resonating dielectric structures, interesting phenomena can be observed as well. Generally, a long-lived whispering gallery mode (WGM) has no preferential direction of radiation because of its rotationally symmetric structure. However, if the space inside the resonator is transformed so that the discontinuity of coordinates exists, it becomes possible to reconcile directional emission with WGMs. Here, we transform only the inner space of a deformed optical cavity, e.g., the Limacon cavity into a virtual perfect disk via a conformal mapping and show these two seemingly incompatible behaviors can be observed simultaneously. The refractive index profiles that realize the transformed space can be obtained from the conformal space transformation. The resonant mode calculated with a transformed boundary element method shows that the WGMs can restore for the deformed cavity. The Husimi function calculated for this transformed cavity shows a weighted band-like profile, which implies that the optical rays inside the cavity is maintaining its reflecting angle as is for the original cavity. However, the far-field pattern shows anisotropic emission of radiation because it is determined by tunneling through the rotationally asymmetric boundary. For example, the conformal WGMs in Limacon and center-shifted triangular cavities exhibit bidirectional and uni-directional emission patterns in the far-field, respectively. These conformal WGM cavities with both the ultra-high quality factor and the directional light emission may be used in the realization of efficient directional light sources.
