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

摘要

In this work we discuss the possibility to reduce the computationalcomplexity of modal methods, i.e. methods based on eigenmodes expansion, fromthe third power to the second power of the number of eigenmodes. The proposedapproach is based on the calculation of the eigenmodes part by part by usingshift-and-invert iterative technique and by applying the iterative approach tosolve linear equations to compute eigenmodes expansion coefficients. Aspractical implementation, the iterative modal methods based on polynomials andtrigonometric functions as well as on finite-difference scheme are developed.Alternatives to the scattering matrix (S-matrix) technique which are based onpure iterative or mixed direct-iteractive approaches allowing to markedlyreduce the number of required numerical operations are discussed. Additionally,the possibility of diminishing the memory demand of the whole algorithm fromsecond to first power of the number of modes by implementing the iterativeapproach is demonstrated. This allows to carry out calculations up to hundredsof thousands eigenmodes without using a supercomputer.