Analytical Expressions for Radiation Effects on Guided Mode Propagation in Long Period Gratings

Simple analytical expressions for guided power transmission through a periodic long-period grating with consideration of the significant radiation loss are derived from the complex coupled-mode theory. Under nearly phase-matching condition, the full coupled-mode equations are reduced to a set of two-mode equations for which close-form analytical solutions are readily obtained. For the long period grating with a high refractive index outer cladding, the effect of the radiation field on the guided mode can be well represented and explained by a decay factor for the envelope of the power profile along the waveguide. For the long period grating with infinite cladding, an approximate solution is proposed and shown to be in good agreement with the exact numerical solution.