In this paper, the coupled mode theory is used to analyze apodized fiber Bragg gratings (FBGs). Since the profile of gratings varies with the propagation distance, the coupled mode equations (CMEs) of apodized FBGs are solved by the fourth-order Runge-Kutta method (RKM) and piecewise-uniform approach (PUA). We present two discretization techniques of PUA to analyze the apodization profile of gratings. A uniform profile FBG can be expressed as a system of first-order ordinary differential equations with constant coefficients. The eigenvalue and eigenvector technique as well as the transfer matrix method is applied to analyze apodized FBGs by using PUAs. The transmission and reflection efficiencies calculated by two PUAs are compared with those computed by RKM. The results show that the order of the local truncation error of RKM is h-4, while both PUAs have the same order of the local truncation error of h-2. We find that RKM, capable of providing fast-convergent and accurate numerical results, is a preferred method in solving apodized FBG problems.
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