It is described how strongly directional emission may be achieved from whispering gallery modes in optical
resonators which are only very slightly deformed from spherical or circular geometry. A theoretical description
is offered which uses an extension of eikonal approximation to complex ray families. Natural boundaries arise in
the complexified ray families which prevents direct application of ray-theoretical methods. By treating the ray
dynamics perturbatively, however, it may be possible to obtain explicit approximation of the exterior field for
sufficiently weakly deformed resonators.