Swift-Hohenberg Equation with Third Order Dispersion for Optical Resonators

Summary form only given. The control of linear and nonlinear properties of Photonic Crystal Fibers (PCF) has led to several applications in optoelectronics, sensing, and laser science [1]. In particular, driven optical microcavities are widely used for the generation of optical frequency combs. They can be modeled by the Lugiato-Lefever equation that possesses solutions in the form of localized structures (LSs) [2]. Optical frequency-combs generated in high-Q Kerr resonators [3] are in fact the spectral content of the stable temporal pattern occurring in the cavity. In this communication, we study the dynamics of an all ring cavity filled with a photonic crystal fiber and driven by a coherent beam operating near to the resonant frequency of the cavity [4]. By using a multiple -scale reduction of the Lugiato-Lefever equation with high order dispersion, we show that the dynamics of this optical device, when operating close to the critical point associated with bistability, is captured by a real order parameter equation in the form of a generalized Swift-Hohenberg equation. A Swift-Hohenberg equation has been derived for several areas of nonlinear science such as chemistry, biology, ecology, optics, and laser physics [5, 6]. However, the peculiarity of the obtained generalized Swift- Hohenberg equation for photonic crystal fiber resonators is that it possesses a third -order dispersion.