Decay rates of optical modes unveiling the island structures in mixed phase space

We explore the decay rates of optical modes in asymmetric microcavities with mixed phase space across a wide range of wavelengths that extend deep into the semiclassical, i.e., short-wavelength, limit. Implementing an efficient numerical method, we computed ${10}^{6}$ eigenmodes and discovered that certain decay rates form sequential separate branches with increasing wave number that eventually merge into smooth curves. The analysis of the localization properties and Husimi distributions reveals that each branch corresponds to a periodic orbit in the closed classical system. Our findings show that these decay rates gradually resolve the structure of the islands in mixed phase space as we approach the short-wavelength limit. We present an effective semiclassical model incorporating wave-number-dependent localization, Fresnel reflection, and the Goos-H"anchen shift and demonstrate that these effects are crucial in accounting for the observed branches of decay rate curves.