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.

I. Introduction

Long Period gratings (LPG) have attracted widespread interest in optical communications and sensing systems for their capabilities of coupling light from the waveguide core to the cladding [1]–[3]. Compared with Bragg gratings, the power exchange in LPG happens between the co-propagating core mode and cladding modes. As a result, the coupling mechanism of LPG makes it highly susceptive to refractive index of the outer cladding (also refers to surrounding media in the text) [4]. There have been many theoretical studies and experimental investigations on the LPG transmission spectra with varying surrounding media [5]–[9]. For a waveguide structure with lower refractive index of the outer cladding than that of the inner cladding, the core modes and cladding modes are both guided modes. Consequently, the power exchange occurs between guided core modes and cladding modes and can be well analyzed by the conventional coupled mode theory [10]. However, if the surrounding material has the equal or higher refractive index than that of the inner cladding material, since the radiation loss becomes significant, one has to resort to the continuous radiation modes or the approximate leaky modes [5], [6]. Under this circumstance, the conventional coupled-mode theory becomes cumbersome. Although there have been several previous works for investigation of guided-radiation mode couplings or guided-leaky mode couplings [6], [7], [9], little attention has been paid to the impact of the power leakage loss which is crucial for understanding the physical insights of power exchanges between the guided mode and radiative/leaky modes.