Eulerian Geometrical Optics and Fast Huygens Sweeping Methods for Three-Dimensional Time-Harmonic High-Frequency Maxwell's Equations in Inhomogeneous Media

In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that Maxwell’s equations may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, we propose a new Eulerian geometrical-optics method, dubbed the fast Huygens sweeping method, for computing Green’s functions of Maxwell’s equations in inhomogeneous media in the high-frequency regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that a new dyadic-tensor type geometrical-optics ansatz is proposed for Green’s functions which is able to utilize some unique features of Maxwell’s equations. The second novelty is that the Huygens-Kirchhoff secondary source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics associated with the usual geometrical-optics ansatz can be treated automatically. The third novelty is that a butterfly algorithm ∗Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. Email: [email protected] †Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. Email: [email protected] ‡College of Math & Statistics, Chongqing Technology and Business University, Chongqing 400067, PRC. Email: [email protected] §Department of Mathematics, Iowa State University, Ames, IA 50011, USA. Email: [email protected] ¶Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA. Email: [email protected]