Circle-shaped Transformation Cavity via Center-Shift M\"obius Transfomation

TL;DR

This paper investigates circle-shaped transformation cavities with shifted centers, analyzing conformal whispering gallery modes and demonstrating their directional emission and mode persistence without boundary shape deformation.

Contribution

It introduces an alternative design scheme for transformation cavities that avoids using a shrinking parameter and characterizes mode properties in shifted center cavities.

Findings

Shifted center cavities exhibit bi-directional emission of whispering gallery modes.

Conformal whispering gallery modes maintain high mode purity and directionality.

The new design scheme simplifies the creation of transformation cavities without boundary deformation.

Abstract

Circle-shaped transformation cavities of shifted center are the most simple transformation cavities in the sense that they have only spatial index variation without boundary shape deformation. For these cavities, we analyze the characteristics of conformal whispering gallery modes according to center-shift coordinate transformation. For this study, we use an alternative scheme for designing a transformation cavity without using a shrinking parameter that is needed to satisfy total internal reflection condition. It is observed that the isotropic emission of whispering gallery modes in the uniform index circular cavity changes to the bi-directional emission of conformal whispering gallery modes as the center-shift increases. Also, using a purity factor that quantitatively measures in RV space the degree of intactness for the internal wave pattern of resonant modes in transformation cavities compared with corresponding whispering gallery mode in uniform disk cavity, we show that the conformal whispering gallery modes in the circle-shaped transformation cavity has not only directionality but also high persistence for characteristics of whispering gallery modes.