Application of multiple scales analysis and the fundamental matrix method to rugate filters: initial-value and two-point boundary problem formulations

In this paper, the filtering problem of apodized rugates is solved by deriving first-order, as well as second-order, coupled-mode equations via the perturbation method of multiple scales. The first-order perturbation equations are the same as those of coupled-mode theory. However, the second-order perturbation expansion is more accurate, and permits the use of larger amplitudes of the periodic index variation of the rugate. The coupled-mode equations are solved numerically by using two different formulations. The first approach is a two-point boundary-value problem formulation, based on the fundamental matrix solution, that is essentially the exact solution for the unapodized rugate. The second approach is an initial-value problem formulation, that uses backward integration of the coupled-mode equations. Comparison with the characteristic matrix method is made for the case of unapodized rugate in terms of speed and accuracy, and it is found that the fundamental matrix solution is the fastest. The accuracy of the multiple scales solution is measured in terms of the amplitude error and the phase error of the filter's spectral response, taking the characteristic matrix solution as a reference for the unapodized rugate. The proposed formulations are utilized to calculate the spectral response of apodized rugates.