  We report the surprising observation of directional tunneling escape from
nearly spherical fused-silica optical resonators, in which most of the phase
space is filled with nonchaotic regular trajectories. Experimental and
theoretical studies of the dependence of the far-field emission pattern on both
the degree of deformation and the excitation condition show that
nonperturbative phase-space structures in the internal ray dynamics profoundly
affect tunneling leakage of the whispering-gallery modes.
