STUDY OF EIGENMODE SOLUTION OF MAXWELL'S EQUATIONS

Abstract

The solution of Maxwell's equation is obtained by using eigenmodes and their study is done using three different models of medium halfspace, with horizontal layer and vertical layers for two dimension case. Solutions for three dimensional medium. This can be an alternative approach to model multi-frequency electromagnetic responses with high data density over three-dimensional conductors. The method is based on the fact that each half space conductivity distribution owns a characteristic, continuous spectrum of free-decay eigenfunctions and eigenvalues forming the decay constants. By an expansion in terms of these eigenmodes, any arbitrary frequency-domain electromagnetic response of the conductor can be obtained. The superposition is attractive due to its comparatively low computational cost. However, the crucial factor regarding the .method's efficiency is to find out necessary eigenmodes. The electric eigenmodes are governed by a homogeneous induction equation for the electric field, whose solutions (in the quasi-static approximation which is applicable here) decay exponentially with time. Their determination using finite differences on a staggered grid leads to a large, sparse eigenvalue problem. The superposition is based on the full set of eigenmodes. Since eigenvectors with smallest eigenvalue are important in eigenmode expansion and degeneracy occurs after certain number of eigenmodes an accelerated convergence of the superposition is achieved which allows for a truncation of the eigenmode expansion. The eigenmodes for different models are studied and found most of the eigenmodes can be generated by using shift and compress operator on basic eigenmodes. Correlation analysis is done to find out relation among eigenmodes so that some of the basic eigenmodes can be find out to represent the complete solution of Maxwell's equations. These decreased number of eigenmodes to obtain solution to Maxwell's equation decrease computational cost to electromagnetic modeling over huge data.