A New Look at Numerical Analysis of Uniform Fiber Bragg Gratings Using Coupled Mode Theory

Abstract

Abstract-The coupled mode theory (CMT) is used to analyze uniform Fiber Bragg gratings. The multi-mode CMT is expressed as the first-order vector ordinary differential equations (ODEs) with coefficients depending on the propagation distance. We show in this paper that by changing variables, the original couple mode equations (CMEs) can be re-casted as constant coefficient ODEs. The eigenvalue and eigenvector technique (EVVT), the analytic method for solving constant coefficient ODEs, is then applied to solve the coupled mode equations. Furthermore, we also investigate the application of RungeKutta method (RKM) to the calculation of the global transfer-function matrix for CMEs. We compare the transmission and the reflection

Numerical Analysis of Apodized Fiber Bragg Gratings Using Coupled Mode Theory

“…, n. Therefore, Equation (12) is a system of ODEs with z-dependent coefficients. By changing variables [14], Equation (12) of each section is expressed as a system of ODE with constant coefficients and can be solved by EVVT. (2) Since the parameters δ 0 and δ ν are relatively small compared with the propagation constant β ν the exponential term exp(±i2δ ν z) changes only a few percent over the sub grating section.…”

“…Recently, we presented a rigorous numerical analysis of the uniform FBG problems with both the eigenvalue and eigenvector technique (EVVT) and RKMs on CMEs with sufficiently high number of core and cladding modes in each fiber direction [14]. RKM was the method of choice for CMEs with z-dependent coefficients.…”

“…Since the structure of apodized FBGs is more complicated than that of uniform FBGs, there are many methods presented to explore the performance of apodized FBGs [14][15][16][17]. In this paper, we use CMT to analyze apodized FBGs.…”

“…To reduce the complexity of the problem, a simplified CMT considers only the coupling between a forward core mode and a backward core mode [4][5][6]. In this study, the contra-directional coupling between core modes and cladding modes are considered [14].…”

“…This is because within each uniform FBG section, the transfer matrices still need to be solved by either the modified EVVT method [14] or by RKM. To avoid using modified EVVT we propose the second simplified discrete technique of PUA, where the position of each section is directly substituted into the original CMEs.…”