Coupling Theory of Asymmetric Photonic-crystal Waveguides

Abstract

The physical properties of asymmetric photonic-crystal directional couplers are studied by using the tight-binding theory, which considers that the fleld distributions of photonic- crystal waveguides (PCWs) are localized around the periodic defects. From this model, the analytic formulas which describe the dispersion of the coupler and the eigenmode patterns were derived. By considering coupling up to the third nearest-neighboring between two PCWs, analytic expressions of the dispersion relations of asymmetric photonic crystal coupler are consistent with the results from the plane wave expansion method (PWEM). Due to the broken symmetry, two dispersion curves will never cross. Nevertheless, the eigenmode patterns switch their parity as the symmetric coupler does when wavevector k passes over the decoupling point. Photonic crystal waveguide (PCW) is a structure which introduces considerable quantities of point defects in the photonic crystal (PC). The electromagnetic (EM) wave is strongly conflned in the defect channel, allowing low-loss in Y-blanches and a sharp-bending. Furthermore, creating two line defects to form a directional coupler are used as add/drop fllters, switches and multiplexers. To design photonic crystal devices, the simulation tools such as the PWEM and the flnite difierence time domain method (FDTD) have often been used. However, there is no simple analytic for- mula to analyze physical properties of the coupled photonic crystal waveguides (PCWs), especially to analyze coupling between PCWs (1). Tight-binding theory (TBT) is widely used not only in condensed matter physics but also recently in the EM wave propagation in linear and nonlinear single line defects PCW analytically. They usually considered only coupling between the nearest- neighboring defects that may be good enough in the single PCW or conventional waveguides. Nevertheless, it is insu-cient to only consider the nearest-neighboring defects in the directional coupler made of two identical PCWs (2). The dispersion curves will always split or never cross if only the nearest-neighboring coupling is considered, but the dispersion do cross that was resolved by further considering coupling of the second nearest neighboring defects which causes a sinusoidal modulation to the dispersion curves. Similar to the extended TBT (2) which considers three nearby coupling coe-cients, in this paper, we flrstly derive an analytic solution to describe the dispersion of asymmetric coupled PCWs. This formula provides more generalized discussion and physical insight than what is derived for the case of symmetric coupled PCWs. Especially, the coupled identical PCWs should become asymmetric due to the intensity dependent index of refraction in the nonlinear photonic crystal directional coupler. Secondly, by using the analytic formulas, the phenomena of mode switching and mode patterns in the asymmetric PCWs are extensively analyzed. Finally, the simulation results of considering PCWs of triangular lattices are consistent with our theoretic results. 2. COUPLED EQUATIONS FOR ASYMMSTRIC PHOTONIC CRYSTAL WAVEGUIDES