# Derivation of Extended Parabolic Theories for Vector Electromagnetic Wave Propagation

## Abstract

The recent application of pseudo-differential operator and functional integral methods to the factored scalar Helmholtz equation has yielded extended parabolic wave theories and corresponding path integral solutions for a large variety of acoustic wave propagation problems. These known techniques are applied here to the full vector electromagnetic problem resulting in a new first-order Weyl pseudo-differential equation, which is recognized as an extended parabolic wave equation. Perturbation treatments of the Weyl composition equation for the operator symbol matrix yield high-frequency and other asymptotic wave theories. Unlike the scalar Helmholtz equation case, the one-way vector EM equation (and a scalar analogue provided by the Klein-Gordon equation of relativistic physics) require the solution of a generalized quadratic operator equation. While these operator solutions do not have a simple formal representation as in the straightforward square root case, they are conveniently constructed in the Weyl pseudo-differential operator calculus.

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