Theory of Fourier mode coupling for long-period fiber gratings

A novel theory, namely, Fourier mode coupling (FMC) theory for long-period fiber gratings (LPFGs) is proposed in this paper. During analyzing the co-propagating coupling between the core mode and cladding modes in LPFGs, the Fourier transform relations among the amplitude coefficients of co-propagating coupled-modes are found for the first time, to the best of authors' knowledge. The general expressions of the coupling and transmission spectra of LPFGs are also deduced from the combination of Fourier transform with the well-known coupled-mode theory. In the proposed FMC theory, the spectral characteristics of the LPFGs without over-coupling are derived from the calculation of co-propagating mode coupling in the spatial domain spectrum, which is the Fourier transform result of refractive index perturbation in the LPFG. According to the FMC theory, the spectra of the LPFGs in different perturbation amplitudes and lengths are numerically simulated here. A measured transmission spectrum is also compared with the calculated transmission spectra based on the FMC theory and the coupled mode theory, respectively. The comparison shows that the FMC theory and the derived spectra for LPFGs are in consistance with the coupled-mode theory and the practical spectra of LPFGs respectively. The FMC has many features, these being simple, fast and accurate, which could be employed for spectrum analysis of any LPFG with an arbitrary distribution of refractive index perturbation along the fiber axis.