Nonlinear mode coupling in whispering-gallery-mode resonators

Whispering-gallery modes (WGMs) have been a source of fascination to physicists since at least the work of Lord Rayleigh in 1910 [1], when he explained the phenomenon that a whisper in one end of the gallery of St. Paul’s Cathedral could be heard at the other end. This phenomenon has practical implications as well, since WGMs in resonators with cylindrical or spherical symmetry can have very high quality (Q) factors. In the past two decades, microcavity resonators have been increasingly used to generate and filter narrowband light [2,3]. The quest to obtain pure optical frequency sources was revolutionized in 2000 with the invention of locked frequency comb lasers [4,5]. This revolution was enabled by technology that allowed the inventors of the comb laser to achieve a factor of 2 (an octave) of bandwidth. However, these sources have the drawback that they are typically bulky and expensive. Today, we may be on the verge of a second revolution in frequency generation. In fact, in the past eight years it has been demonstrated that it is possible to use WGM microresonators to generate solitons and hence broadband combs [6,7], and, within the past year, they have been used to generate nearly an octave of bandwidth and to lock the carrier-envelope phase [8]. However, the process by which these modes are generated remains poorly understood. It has been demonstrated that single modes are governed by the Lugiato-Lefever equation (LLE) [9–11]; however, there is recent experimental evidence that mode coupling can play a critical role in obtaining a broad bandwidth comb [12,13]. That is particularly the case at optical and near-ultraviolet (UV) wavelengths at which the modes typically have normal dispersion and solitons cannot be obtained from a single mode. In this article, we present coupled LLEs that describe the coupling of two WGMs in a resonator in which chromatic dispersion, the Kerr nonlinearity, and an external pump are all present. These equations resemble the coupled nonlinear Schrödinger equations (NLSEs) that describe mode coupling in optical fibers and waveguides [14] and Bose-Einstein condensates [15]. However, the presence of the pump and the periodicity of the resonator change the equations in a fundamental way. It is no longer possible to remove the phase velocity difference from the equations by separately shifting the central frequencies of each of the modes because that