Numerical stable determination of Floquet-modes and the application to the computation of band structures

Abstract

Periodic structures like Bragg-gratings are important components of optical circuits. The analysis of these devices can be done very efficiently by combining eigenmode propagation methods with Floquet's theorem. A particular problem is the determination of the Floquet modes. Transfer matrix formulas are stable only in case of low losses and when the length of the periods is not too big. A stable method, also in the mentioned cases, is presented in this paper. Reflection coefficients are transformed from the output of a periodic segment to its input and the fields are computed in opposite direction. By this, the exponential increasing terms, which lead to the numerical problems are avoided. The formulas are applied to determine eigenmodes in various waveguide structures. Particular periodic structures are photonic crystals (PC), who have very promising features. For tailoring these PCs the knowledge of the band structure is required. With the Floquet modes that have been determined before this band structure has been calculated. A comparison with the literature showed a very good agreement.