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We investigate the optical modes of hollow ring resonator with omnidirectional reflector cladding by numerical method. We demonstrate that each $\mathrm{TM}_{0\rm ml}$ mode (or $\mathrm{TE}_{0\rm ml}$ mode) can split into two modes whose wavelengths are very close. The doublet effect is caused by the symmetry breaking at circular cross section when a straight waveguide is curled into a ring shape...
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