Abstract
Dielectric cavity systems, which have been studied extensively so far, have uniform refractive indices of their cavities, and Husimi functions, the most widely used phase space representation of optical modes formed in the cavities, accordingly were derived only for these homogeneous index cavities. For the case of the recently proposed gradient index dielectric cavities (called as transformation cavities) designed by optical conformal mapping, we show that the phase space structure of resonant modes can be revealed through the conventional Husimi functions by constructing a reciprocal virtual space. As examples, the Husimi plots were obtained for an anisotropic whispering gallery mode (WGM) and a short-lived mode supported in a limaçon-shaped transformation cavity. The phase space description of the corresponding modes in the reciprocal virtual space is compatible with the far-field directionality of the resonant modes in the physical space.
Highlights
Maxwell's equations, the governing equations for electromagnetic fields in space-time including media, have fundamental symmetries under various transformations such as wellknown Lorentz transformation and gauge transformation
Exploiting the form invariance of Maxwell's equations for general coordinate transformations, Pendry et al theoretically presented transformation optics (TO) [4] which is a general methodology for designing electromagnetic materials including optical invisibility cloaks, and various photonic devices which manipulate the path of light waves [5, 6]
In order to analyze the characteristics of optical modes supported in 2D dielectric cavities, it is often more useful to represent the optical modes in phase space than merely depict the mode intensity patterns in real physical space
Summary
Maxwell's equations, the governing equations for electromagnetic fields in space-time including media, have fundamental symmetries under various transformations such as wellknown Lorentz transformation and gauge transformation. We show that the Husimi functions can still be useful for optical modes supported in transformation cavities with inhomogeneous refractive indices To this end, we construct a virtual space which we call a reciprocal virtual (RV) space, through the inverse of the conformal mapping, both polarized resonant modes supported in the transformation cavity of the physical space and the corresponding virtual modes in the unit disk cavity of the RV space turn out to be identical. The refractive index of the unit disk cavity in the RV space is uniform Based on these facts, one can obtain a phase space description of the internal waves in a transformation cavity through the conventional Poincaré Husimi functions calculated in the RV space.
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