{'fraction_non_alphanumeric': 0.010113780025284451, 'fraction_numerical': 0.0025284450063211127, 'mean_word_length': 5.285714285714286, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Cylindrical Coordinates Coupled Mode Theory

We present a cylindrical coordinates coupled mode formulation for coupling between two degenerate modes of a traveling wave resonator. The resonator is assumed to have rotational symmetry and therefore two degenerate modes propagating in opposite directions. We analyze coupling of the resonance modes in space due to a perturbation in the resonator and derive first-order coupling equations. We also present an application of the method in modeling the mode coupling in a ring resonator with an integrated distributed Bragg reflector. For the presented structure, the results from coupled mode analysis agree with the finite elements method simulation results, and the method dramatically reduces the simulation time from 2 h to less than 5 min.