However, the computational cost of determining the impedance matrix varies depending on the chosen Green function and requires numerical integration when all near-field contributions are included.

The mosaic representation described in [ 29 ] is used and the comparison between the two propagated CFs is performed on a selected square of the mosaic. This is shown in figure 2 along with resulting energy densities. There is a good agreement between the two correlation patterns both in terms of height and spreading of the correlation length.

### Introduction to Electromagnetic Fields and Waves

Even better agreement is found for the energy densities reported in figure 2 d , e , both accurately reproducing the measured intensity in figure 2 f. To have a better insight into the accuracy of the approximated WF method, we use the exact MoM method as a reference where an exact WF propagator has been used in previous work [ 28 , 33 ].

In particular, the differences between both the propagated energy densities and the measured data have been calculated for the example in figure 2 , in order to quantify the relative errors of the two methods with respect to measurements. The error plots are shown in figure 3. Both the propagated energy densities have small differences compared to the measured data with the numerically exact MoM method being more accurate than the approximated WF method, as expected.

However, the MoM is computationally more intensive than the approximated WF and does not transparently provide information on space-angular properties of emissions as the WF does. The WF method propagates input data with the level of efficiency of a double Fourier transform. In this example, the computation time of the MoM-based propagation, with full calculation of impedances including near-field contributions, takes hours, while the computation time of the WF-based propagation takes minutes in a standard desktop computer. Since the MoM corresponds to an exact propagator, propagation inaccuracies are only related to discretization and discrepencies are likely due to measurement uncertainty.

Theoretical methods and measurement techniques apply to other field components and therefore the same level of agreement is expected for other components of the CT, whence we envisage a successful reconstruction of the average Poynting vector for partially coherent stochastic vector fields.

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Upper row: comparison between field CFs obtained from a WF-based and b MoM-based propagators and c with measured data. Lower row: energy densities obtained from d WF-based and e MoM-based propagators and f with measured data.

Difference between the intensity from a WF-based propagator and measured data and b MoM-based propagator and measured data. This paper has presented a comparison of approaches to stochastic field measurements and modelling using WFs and MoM. It is shown that by quantifying the coherency tensor of stochastic fields, a wave-dynamical phase-space representation can be devised to extract both the energy flow vector and the local energy density. The spatial and directional properties can be extracted from the CT through the WT and the relation between the WT and Poynting vector is derived.

This is then used to develop a propagation rule for the CT. A comparative study based on experimentally measured stochastic fields and propagated fields using the WF and the MoM technique is provided showing good agreement. It is shown that Heaviside anticipated many of these advanced ideas leading to the new field of statistical electromagnetics. Using Maxwell's equations for a source- and charge-free medium in the p domain,. Now, since averaging and integration commute, and using the property of the electric CT, , along with the Hermiticity of , we may write. Note that the components of this operator parallel to the source plane act simply through the delta kernel.

All authors analysed the comparison results. All authors gave final approval for publication. N, is gratefully acknowledged. National Center for Biotechnology Information , U. Published online Oct Creagh , 1 Gregor Tanner , 1 and David W. Thomas 2. Stephen C. David W. Author information Article notes Copyright and License information Disclaimer.

Accepted Jul Abstract This paper reviews recent progress in the measurement and modelling of stochastic electromagnetic fields, focusing on propagation approaches based on Wigner functions and the method of moments technique. Keywords: electromagnetic fields, stochastic fields, propagation, Heaviside.

Introduction Oliver Heaviside was a gifted engineer and a tenacious supporter of the treatise of J. From deterministic to statistical electromagnetic theory Heaviside's masterpiece Electromagnetic Theory [ 7 , 40 , 41 ] contains a thorough analysis of two important properties of the electromagnetic wave EMW : — The electric and magnetic vector forces are subject to an effect of self-induction. Stress is made on this effect in pure plane EM waves, where electric and magnetic forces have a constant ratio , electric and magnetic energies are equal , they have the property of being perpendicular to one another […] and their plane is in the wavefront, or the direction of motion of waves is perpendicular to E and to H.

It is the direction of the flux of energy. Of particular importance is the introduction of the vector product as a tool for unveiling the principle of the continuity of the energy flux. It is remarked that the principle of the continuity of energy is a special form of that of its conservation , or Newton's principle of the conservation or persistence of energy. However, in the ordinary understanding of the conservation principle, it is the integral amount of energy that is conserved, and nothing is said about its distribution or its motion. This involves continuity of existence in time, but not necessarily in space also.

But if we can localise energy definitely in space, then we are bound to ask how energy gets from place to place. As we will see later in the paper, the concept of lines guiding the energy flux was introduced by Heaviside independently from J. Poynting, and using the vector EMW notation. Open in a separate window. Figure 1. Loop probe moved above the cavity-backed aperture by the scanner system. Figure 2. Figure 3. Conclusion This paper has presented a comparison of approaches to stochastic field measurements and modelling using WFs and MoM.

Appendix Using Maxwell's equations for a source- and charge-free medium in the p domain,. Author's contributions G. Competing interests We declare we have no competing interests. References 1. Maxwell J. A treatise on electricity and magnetism , vol. Dover Publications. Harman PM. Bruce J. Hunt, The Maxwellians. Ithaca and London: Cornell University Press, Mahon B.

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Oliver Heaviside: maverick mastermind of electricity. Nahin P. Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age. Hong S. Wireless: from Marconi's black-box to the audion. Russer P. Ferdinand Braun — a pioneer in wireless technology and electronics. Contributions to the history of the Royal Swedish Academy of Sciences, vol. Firenze, Italy: Firenze University Press. Heaviside O. Electromagnetic theory, vol 1. Darrigol O. Differential forms and electromagnetic field theory invited paper. PIER Progr. Haider M, Russer JA. Differential form representation of stochastic electromagnetic fields.

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## Electromagnetism

The quasiparticle view of wave propagation. IEEE 79 , — Hunt B. The Maxwellians. Yaghjian A.

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An overview of near-field antenna measurements. Antennas Propag. Modeling electromagnetic emissions from printed circuit boards in closed environments using equivalent dipoles. IEEE Trans.