Discretized Coupled-mode Theory for Fiber Grating Analysis

Abstract

Fiber gratings have invoked intense research attention in recent years. The imprinted grating structures in fiber couple a propagating fundamental core-mode to other modes for resonant wavelengths and hence produce excellent wavelength filtering properties. To enhance the properties according to applications, a lot of grating structures are imposed in fibers. For example, phase-shifted ones, apodized ones, chirped ones, and so forth are applied to the grating structures for both reflection and transmission gratings. For the grating structure analysis, conventional coupled-mode theory and fundamental matrix theory [1] have good agreements with experimental results for almost periodic grating structures. However, the theories could not resolve the grating structures into less than grating periods, since they are based on a Fourier series representation for the grating structure. Even more, in the case that the whole grating length is not long enough compared to grating period or in the case that there exists no periodicity in the grating structure, its profile should be represented by not Fourier series but Fourier integral. These problems can be overcome by the discretized coupled-mode theory proposed in this paper. This method represents the grating structure as it is by discretizations along the structure. Hence, it can analyze arbitrary grating structures and resolve them into less than grating period. For example, if the discretization number is one hundred for the 500 nm fundamental grating period, then Nyquist theorem expects that the analysis should have 10 nm resolution for the grating structure. According to the grating structure variations, the discretization can be adjusted. Therefore, a precise analysis can be obtained no matter how complex the grating structure is. In this scheme, the nonlinear property of fiber grating can be considered easily. Together with Fourier transform analysis, the grating nonlinear response [2] including fiber grating soliton problems can be analyzed with high precision.