JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 18, NO. 12, DECEMBER 2000 2217 Application of Multiple Scales Analysis and the Fundamental Matrix Method to Rugate Filters: Initial-Value and Two-Point Boundary Problem Formulations Mohammed Bataineh, Member, IEEE, and Omar Rafik Asfar, Senior Member, IEEE 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 pertur- bation expansion is more accurate, and permits the use of larger amplitudes of the periodic index variation of the rugate. The cou- pled-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. Index Terms—Apodization techniques, coupled-mode theory, multiple scales analysis, numerical solutions, optical waveguide filters, periodic structures. I. INTRODUCTION W HEN the refractive index varies sinusoidally or, in gen- eral, periodically in the direction perpendicular to the film plane of an optical thin film, the resulting structure is re- ferred to as a rugate filter. This inhomogeneous thin-film optical coating is assumed to be nonabsorbent, nonscattering, linear, isotropic, and nonmagnetic. The characteristic matrix method [1], used in the analysis of stack filters, also could be used to assess the spectral response of these periodic coatings. This re- quires a huge number of layers to represent faithfully the si- nusoidal profile of the rugate filter, resulting in a prohibitively large number of matrix multiplications and, consequently, large computer times. This was realized by Southwell [2], who used Manuscript received October 15, 1999; revised June 8, 2000. M. Bataineh is with the Hijjawi Faculty for Engineering Technology, Yarmouk University, Irbid, Jordan. O. R. Asfar is with the Hijjawi Faculty for Engineering Technology, Yarmouk University, Irbid, Jordan, on leave from Jordan University of Science and Tech- nology, Irbid, Jordan. Publisher Item Identifier S 0733-8724(00)09102-7. coupled-wave theory to reduce the computational time for the case of a constant small amplitude sine-wave profile. Bovard [3] developed an admittance matrix approach for the treatment of rugate filters. Villa et al. [4] used the same approach to calcu- late thin-film thickness of the rugate filter to obtain a specified reflectance. This paper uses the perturbation method of multiple scales to derive coupled-mode equations for rugate filters. This method has been used successfully to treat propagation of waves in cor- rugated open and closed waveguides, optical fibers, and acoustic waveguides [5]–[7]. The method leads to a system of first-order coupled ordinary differential equations for the incident and re- flected fields, which are usually referred to as the coupled-mode equations. The filter response is obtained by considering two different formulations for an essentially two-point boundary- value problem. Formulation, as an initial-value problem, is pos- sible, because the reflection coefficient at the substrate interface is known. However, the incident mode field is unknown, but the problem can be circumvented by normalizing its amplitude, thus enhancing an initial-value formulation. On the other hand, the reflection coefficient at the end of a periodic waveguide sec- tion is zero, if the waveguide is matched or assumed to be in- finite in length. A two-point boundary-value problem formula- tion is necessary in the latter case. Both the initial-value problem and the two-point problem formulations are considered in this paper, and found in agreement with each other, as well as with the characteristic matrix method for small amplitude of the pe- riodic index perturbation. The two-point problem is solved nu- merically by using the fundamental matrix method [8], which has a distinct advantage in terms of speed over the character- istic matrix and the initial-value methods. A new dimension is added to the filtering problem when the perturbation method of multiple scales is carried to higher order in , where is the amplitude of the uniform periodic index variation. In this paper, both first-order and second-order expan- sions are derived and the range of validity of these expansions is investigated. The first-order expansion accurately determines the spectral response for small , such as considered by South- well [2]. However, as increases, a second-order expansion is required, in order to overcome the phase error in the response of the first-order expansion. Apodization techniques, to modulate the refractive index sine-wave profile, were used by various investigators (e.g. 0733–8724/00$10.00 © 2000 IEEE