In this work, we revisit the theory of stochastic electromagnetic fields using exterior differential forms. We present a short overview as well as a brief introduction to the application of differential forms in electromagnetic theory. Within the framework of exterior calculus we derive equations for the second order moments, describing stochastic electromagnetic fields. Since the resulting objects are continuous quantities in space, a discretization scheme 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 equations as obtained for the case of formulating the theory using the traditional notation of vector calculus. We conclude with an analytic calculation of the radiated electric field of two Hertzian dipole, excited by uncorrelated random currents.

The most widely used concept for the formulation of Maxwell's
equations is the vector field approach. Even though vector calculus became a
quasi-standard for engineering applications, other formulations like tensor
calculus, quaternions and differential forms could provide deeper insight
into the underlying physics

Characterization and modeling of stationary stochastic electromagnetic fields
using field correlations has been expanded to the transmission line matrix
method in

As in the case of deterministic fields, it can be expected that differential
forms may lead to a better understanding of stochastic electromagnetic
fields. Within this work, we take advantage of differential form
representation for the modeling of noisy processes. Noise is an inevitable
perturbation in wireless communication scenarios. While noise is a stochastic
process also interfering signals originating from deterministic processes may
have to be treated as quasi-stochastic signals due to lack of knowledge or
the prohibitive complexity to model the deterministic process. Accurate noise
modeling is crucial with respect to electromagnetic compatibility (EMC),
electromagnetic interference (EMI), and signal integrity (SI) considerations
for the design of electronic components and systems. A careful modeling of
noisy processes and stochastic electromagnetic fields shows also potential to
improve wireless- and on-chip communication

The calculus of exterior differential forms was introduced by Élie

Maxwell's equations are often given in terms of differential equations,
incorporating “curl” and “div” operators in traditional vector calculus
notation. This representation can be generalized to differential forms, by
introducing the

With the framework presented so far, we can postulate Maxwell's equations in
a very general and coordinate independent way, starting with Gauss' law,
which relates an electric displacement field

As pointed out by

The correlation double one-forms and two-forms for the electric field and the
source currents, respectively are continuous in space. In order to enable a
numerical treatment of problems related to stochastic electromagnetic fields,
a discretization scheme has to be introduced. We use the method of moments
(MoM) to transform field problems to network problems following

In the next step, we expand the electric field

We can now calculate correlation matrices for the generalized voltages

In order to show, that exterior calculus could improve our understanding of electromagnetic fields in general, and stochastic electromagnetic fields in particular, we perform analytic calculations on the stochastic emission of two Hertzian dipoles.

The Green's double one-form

A sketch of the setup we use for our analytic considerations is given by
Fig.

Setup for analytic calculations.

The distance between the dipoles is

Using the shorthand functions

The locations of our observation points are given by the coordinates,

The electric fields at locations

We have revisited the the theoretic description of stochastic
electromagnetic fields and introduced exterior calculus for their
description. The method of moments was applied within the framework of
differential forms in order to obtain equations which can be treated
numerically. These equations are equivalent to those obtained from
traditional vector calculus. The main advantage, however, of the formulation
presented over the traditional formulation, given in terms of the vector
calculus notation, is the formulation's independence of any choice of a
particular coordinate system. Other benefits, like the graphical
representation of stochastic field forms, may be worth to be further
explored. We derived the correlation matrix of stochastic generalized
voltages from the correlation matrix of generalized currents. The involved
linear operators, i.e. the impedance matrices

The authors declare that they have no conflict of interest.

This work was supported by the European Union's Horizon 2020 research and innovation programme under grant no. 664828 (NEMF21). Edited by: R. Schuhmann Reviewed by: two anonymous referees