Iterative approach as alternative to S-matrix in modal methods

Abstract

The continuously increasing complexity of opto-electronic devices and the rising demands of simulation accuracy lead to the need of solving very large systems of linear equations making iterative methods promising and attractive from the computational point of view with respect to direct methods. In particular, iterative approach potentially enables the reduction of required computational time to solve Maxwell's equations by Eigenmode Expansion algorithms. Regardless of the particular eigenmodes finding method used, the expansion coefficients are computed as a rule by scattering matrix (S-matrix) approach or similar techniques requiring order of 3 M operations. In this work we consider alternatives to the S-matrix technique which are based on pure iterative or mixed direct-iterative approaches. The possibility to diminish the impact of 3 M -order calculations to overall time and in some cases even to reduce the number of arithmetic operations to 2 M by applying iterative techniques are discussed. Numerical results are illustrated to discuss validity and potentiality of the proposed approaches.

Application of the iterative approach to modal methods for the solution of Maxwell's equations

“…Therefore, by using this approach in case of dielectric gratings direct methods such as FMM consume only of an order of 2 M numerical operations to find the expansion coefficients [33].…”

“…Such factor depends on the value of the permittivity and on the geometry of the layers which define the magnitude of rejected submatrices with decreasing exponential multiplier. The overall number of operations required to solve system ( 24) is about of Therefore, by using this approach in case of dielectric gratings direct methods such as FMM consume only of an order of 2 M numerical operations to find the expansion coefficients [33].…”