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

Abstract

Abstract. In this work we discuss the possibility to reduce the computational complexity of modal methods, i.e. methods based on eigenmodes expansion, from the third power to the second power of the number of eigenmodes. The proposed approach is based on the calculation of the eigenmodes part by part by using shift-and-invert iterative technique and by applying the iterative approach to solve linear equations to compute eigenmodes expansion coefficients. As practical implementation, the iterative modal methods based on polynomials and trigonometric functions as well as on finitedifference scheme are developed. Alternatives to the scattering matrix (S-matrix) technique which are based on pure iterative or mixed direct-iterative approaches allowing to markedly reduce the number of required numerical operations are discussed. Additionally, the possibility of diminishing the memory demand of the whole algorithm from second to first power of the number of modes by implementing the iterative approach is demonstrated. This allows to carry out calculations up to hundreds of thousands eigenmodes without using a supercomputer.

Efficient method for the solution of Maxwell’s equations for nanostructured materials

“…The spectral element method can be also used for iterative calculation of eigenmodes, for example, in resonators, or as an integral part of the modal methods. In paper [9] such iterative approach was used to calculate hundreds of thousands of eigenmodes within modal method based on 1-D spectral elements.…”

Improving accuracy of the numerical solution of Maxwell's equations by processing edge singularities of the electromagnetic field

“…The sign of square root in (19) is chosen so that its imaginary part would be positive, whereas for real positive 2 , l n κ the positive square root should be used. Thus, the directions of propagation and decaying of modes coincide.…”

“…Various methods for solving Eq. ( 20) can be found in [4,5,18,19] and the same methods are applicable for solving Eq. ( 6).…”

“…The equations for the coefficients c in Eq. (19) for each layer and the coefficients h in Eq. (15) for each region near the singularity are obtained by equating the values of the field and its normal derivative multiplied by 1 ε − on both sides of the boundary Γ indicated in Fig.…”

“…In numerical calculations we restrict ourselves in expansion (19) to the first M eigenfunctions, and in expansions (3) near each w-th of W angular singularities, situated at the boundary with the coordinate l y between layers l-1 and l, to the first w N functions. Substituting Eq.…”