Abstract
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.
© 2000 IEEE
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