Application of Complex Vectors and Complex Transformations in Solving Maxwell’s Equations

Abstract

Application and implication of using complex vectors and complex transformations in solutions of Maxwell’s equations is investigated. Complex vectors are used in complex plane waves and help to represent this type of waves geometrically. It is shown that they are also useful in representing inhomogeneous plane waves in plasma, single-negative and double-negative metamaterials. In specific I will investigate the Otto configuration and Kretschmann configuration and I will show that in order to observe the minimum in reflection coefficient it is necessary for the metal to be lossy. We will compare this to the case of plasmon-like resonance when a PEC periodic structure is illuminated by a plane wave. Complex transformations are crucial in deriving Gaussian beam solutions of paraxial Helmholtz equation from spherical wave solution of Helmholtz equation. Vector Gaussian beams also will be discussed shortly.