Differential form representation of stochastic electromagnetic fields

In this work, we revisit the theory of stochastic elec tromagnetic fields using exterior differential forms. We present a short overview as well as a brief introdu ction to the application of differential forms in electromagnetic theory. A differential form is in principle a quantity, which can be integrated. For the case of a three-dimensional space, the domains of integration could either be lines (1D), areas (2D) or volumes (3D). The corresponding differential forms are then given as one-forms, two-forms and three-forms. We describe electric-, and magnetic fields by differential one-forms while electric-, and magnetic displacement fields are given by two-forms. Since charge density is a q uantity that is given as a density in space, we use pseudo scalars, or differential three-forms to des cribe it. A well-known result from the field of mathematics, the lemma of Poincaré, can then be used to ensure the existence of a field, just by the presence of some charge density. Probably the most important advantage, which makes exterior calculus superior to vectors, when considering electromagneti c fields, is that the formalism becomes completely independent of the choice of a specific coordinate system. Within the framework of exterior calculus, we deri ve equations for the second order moments, describing stochastic electromagnetic fields. Also the equations for the double one-forms, or two-forms respectively, relating the second order moments are independent fo r the choice of a particular basis. Since the resulting objects are continuous quantities in sp ace, a discretization sc heme based on the Method of Moments (MoM) is introduced for numerical treatment. The MoM is applied in such a way, that the notation of exterior calculus is maintained while we still arrive at the same set of algebraic equat ions in the end, we would have obtained when formulating the theory using the traditional notation of vector calculus. Fo r properly setting up the Method of Moments, we introduce an inner product for differential forms. We show that our inner product is valid by exploring some important properties, like sesquilinearity and positive semi-definiteness. We conclude our work with an analytic calculation of the propagation of correlations of two Hertzian dipoles using the formalism of exterior calculus. We see that even though we consider uncorrelated sources, the cross-correlati on spectra of the excited elec tric field are non-vanishing.      «

In this work, we revisit the theory of stochastic elec tromagnetic fields using exterior differential forms. We present a short overview as well as a brief introdu ction to the application of differential forms in electromagnetic theory. A differential form is in principle a quantity, which can be integrated. For the case of a three-dimensional space, the domains of integration could either be lines (1D), areas (2D) or volumes (3D). The correspondin...     »