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

• Modal methods computational complexity reduced to 2nd power of eigenmodes number.
• Memory demand reduced to the 1st power of the number of eigenmodes.
• Eigenmodes are found by batches using Arnoldi method and shift-and-invert technique.
• An iterative procedure is used instead of S- or R-matrix formalism.
• Calculations up to hundreds of thousands eigenmodes without using supercomputers.

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 by applying the iterative technique. The proposed approach is based on the calculation of the eigenmodes part by part by using shift-and-invert iterative procedure and by utilizing 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 finite-difference 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.