Abstract
We derive formulas for whispering gallery mode resonances and bending losses in infinite cylindrical dielectric shells and sets of concentric cylindrical shells. The formulas also apply to spherical shells and to sections of bent waveguides. The derivation is based on a Wentzel-Kramers-Brillouin (WKB) treatment of Helmholtz equation and can in principle be extended to any number of concentric shells. A distinctive limit analytically arises in the analysis when two shells are brought at very close distance to one another. In that limit, the two shells act as a slot waveguide. If the two shells are sufficiently apart, we identify a structural resonance between the individual shells, which can either lead to a substantial enhancement or suppression of radiation losses.
Highlights
The quality factor, Q, of whispering gallery mode (WGM) resonators is limited by material purity and radiation losses
The former seems to be nearly attained in large crystalline resonators, where Q > 1011 has been demonstrated [1]
Our study finds application in WGM sensing, where the high Q of micro-rings or micro-spheres promises highly sensitive label-free detection of molecules
Summary
The quality factor, Q, of whispering gallery mode (WGM) resonators is limited by material purity and radiation losses The former seems to be nearly attained in large crystalline resonators, where Q > 1011 has been demonstrated [1]. In the presence of spherical or cylindrical symmetry, the characteristic equation giving the complex WGM frequencies generally involves Bessel function of both large orders and large arguments This double limit is a delicate one to handle numerically, which motivates one to use asymptotic representations. For full and uniform dielectric spheres or cylinders, the positions of the resonances are most accurately given in Schiller’s paper [2], who extended the analysis of Lam, Leung and Young [3] These authors successfully computed the radiative linewidth by substituting their result in the argument of an appropriate spherical Bessel function. One usually keeps solving the WGM characteristic equation numerically to assess the radiation losses [5,6,7,8]
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