Matrix analysis

Abstract

Hooke postulated, in the seventeenth century, that light waves must be transverse but his idea was forgotten. Young and Fresnel put forward the same idea in the nineteenth century and accompanied their postulation with a theoretical description of light based on transverse waves. Forty years later, Maxwell proved that light must be a transverse wave and that E and H, for a plane wave in an isotropic medium with no free charge and no currents, are mutually perpendicular and lie in a plane normal to the direction of propagation, k. Convention requires that we use the electric vector to label the direction of the electromagnetic wave’s polarization. The direction of the displacement vector is called the direction of polarization and the plane containing the direction of polarization and the propagation vector is called the plane of polarization. The selection of the electric field is not completely arbitrary, except in relativistic situations, when v≈c, the interaction of the electromagnetic wave with matter will be dominated by the electric field. Assume that a plane wave is propagating in the z-direction. In complex notation, the plane wave is given by (Formula Presented) The procedure used to decompose an arbitrary polarization into components parallel to two axes of a Cartesian coordinate system, is a technique used extensively in vector algebra to simplify mathematical calculations. According to the mathematical formalism associated with this technique, the polarization is described in terms of a set of basis vectors, ei. An arbitrary polarization would be expressed as 2....