Predicting electromagnetic fields at high frequencies is of importance to – among others – wireless communication technology and Electromagnetic Compatibility (EMC). Field intensity simulations are difficult and time consuming, hampered by (i) the need to resolve fields at the scale of the wavelength – leading to small mesh sizes and thus large models, (ii) the complexity of the environment and (iii) inherent uncertainty in setting up the models. In the WAMO research group we apply asymptotic methods and Wigner function approaches combined with random matrix theory to simulate electromagnetic wave fields in the presence of intrinsic system complexity. The research is done in close collaboration between the School of Mathematical Sciences (Faculty of Science) and the George Green Institute for Electromagnetics Research (Faculty of Engineering).

**Propagation of Wigner Functions**

Predicting the properties of wave fields in complex environments is an extremely challengingtask of crucial importance to a wide variety of technological and engineering applications, such as vibro-acoustics or electromagnetic (EM) wave modelling. In particular, characterizing the radiation of EM sources reliably, both in free space and within enclosures, is a longstanding research issue. In the context of electromagnetic compatibility (EMC), digital circuits and large printed circuit boards (PCB) embed thousands of electronic devices and metallic tracks and can produce fields reaching dangerous but hard-to-predict levels. We therefore set out an approach for propagating such complex and statistically characterized wave fields exploiting Wigner distribution function (WDF) techniques. The WDF formalism offers a direct route to pure ray-tracing approximations in an operator implementation, while still capturing in its exact implementation the full wave dynamics. The formalism allows one to efficiently treat radiation from complex sources, often having a statistical character. Complexity arises here through the stochastic nature of the radiated field, which may be best described by considering time or frequency averages and thus looking at an ensemble of system realizations. A statistical representation is then appropriate and computationally more efficient than a purely deterministic treatment. Applications of the phase-space propagators include source reconstruction and localization as well as emission source microscopy.

**Related Publications:**

- Propagating wave correlations in complex systems, S. C. Creagh, G. Gradoni, T. Hartmann and G. Tanner, Journal of Physics A 50, 045101 (2016)
- Measurement and Wigner Function Analysis of Field-Field Correlation for Complex PCBs in Near Field, M. Baharuddin, S. Greedy, D. W. P. Thomas, G. Gradoni, S. C. Creagh, G. Tanner, EMC Europe 2016, Wroclaw, Poland
- A Phase-Space Approach for Propagating Field-Field Correlation Functions Near Stochastic Sources, G. Tanner, G. Gradoni, and S. C. Creagh, EMTS2016, Espoo, Finnland
- C. Smartt et al., “Challenges of time domain measurement of field-field correlation for complex PCBs,” 2015 IEEE International Symposium on Electromagnetic Compatibility (EMC), Dresden, 2015, pp. 953-958.
- Radiation of Complex Sources in Reflecting Environments: A Wigner Function Approach, G. Gradoni, S. C. Creagh and G. Tanner, EMC Europe 2014 (Gotheburg, Schweden, 1-4 Sept 2014)
- Wigner Function Approach for Describing the Radiation of Complex Sources, G. Gradoni, S. C. Creagh and G. Tanner, 2014 IEEE International Symposium on Electromagnetic Compatibility, Raleigh, California

**Near Field Wireless Communication**

We study the possibility of substituting hard wiring between chips with wireless communications sustained from edge on-chip antenna elements. Given the scale of integration of modern electronics circuitry, wireless communications take place at small scales, often implying that energy transport is carried out through non-radiating – evanescent – surface waves.Understanding the impact of interference on the performance of a multiple-input-multiple-output (MIMO) based device is of paramount importance in ensuring that a proposed electronic design is both resilient and robust. In this work and as part of the Horizon2020 project *Noisy Electromagnetic Fields for Chip-to-Chip communication* – NEMF21, the effect of element-to-element interference in the creation of multiple channels of a wireless link approaching the near-field regime is studied. Multiple antenna transmit-and receive-arrays are considered for different antenna types and arrangements operating between 5.6 and 60 GHz. In collaboration with TUM, we find that multiple channels can be created even if the antennas interact at sub-wavelength distances, thus confirming previous theoretical predictions [REFERENCE Nossek].

**Wireless communication in complex environments**

At mmWave frequencies, full wave electromagnetic simulations become time consuming, andclassical ray tracing algorithms, e.g., shoot and bounce schemes, converge slowly to the solution. Phase-space methods developed at WAMO are used to perform ray-tracing calculations in large, complex environments and thus provide the right tool for the telecommunication industry to plan network coverage on the vast scale needed. The formulation of a physics-based outdoor-to-indoor transfer function is tackled using efficient methods based on random matrix theory and transfer operator approaches. We are working on predicting the EM energy flow through multiply-connected and complex environments by a fusion of the Random Coupling Model (RCM) with the Dynamical Energy Analysis (DEA) method. Applications are envisaged in the telecommunication industry, naval environments and avionics. This project is funded by Telecom Italia Lab, Italy and the Office of Naval Research (ONR), USA.

**Statistical Electromagnetics**

Electromagnetic propagation within complex, confined systems can be characterized through statistical methods. The assumptions underneath those methods are similar to those adopted in statistical physics for a gas of non-interacting particles confined in a box. In electromagnetics, the energy can be thought as propagated by a “gas’’ of interfering partial waves and thus modelled by a Plane Wave Expansion (PWE). The complexity of the system, e.g., uncertain and/or irregular boundaries of a cavity, is thus captured in the statistics of partial wave phase, amplitude, correlation and direction of propagation, from which total field statistics can be derived through entropy maximisation methods. An alternative is modelling cavity fields as a superposition of ergodic eigen-modes with random modal coefficients and resonant frequencies. The former can be tackled again with a random uncorrelated PWE, also known as Berry’s hypothesis. The latter offers an interesting connection with universal laws predicted by the RMT for the eigen-energies of chaotic systems. Researchers of the WAMO have contributed to establish those approaches and explain numerical and experimental observations on higher order energy statistics, quality factor and coherence bandwidth probability distributions, as well as extreme value field statistics of electromagnetic reverberation chambers. Electromagnetic reverberation, born for performing robust and extreme tests in electromagnetic compatibility and recently used to emulate indoor and outdoor wireless environments, is now transitioning to become part of statistical physics. Current research efforts are dedicated to:

- understanding the number of uncorrelated chamber realizations,
- creating a physical model for the chamber lowest usable frequency,
- studying dynamical effects associated with electromagnetic reverberation, including Fermi acceleration, Doppler shift, transient eigen-modes and diffusion.

**Related Publications:**

- G. Gradoni and L. R. Arnaut, “Higher Order Statistical Characterization of Received Power Fluctuations for Partially Coherent Random Fields,” in IEEE Transactions on Electromagnetic Compatibility, vol. 51, no. 3, pp. 583-591, Aug. 2009.
- G. Gradoni and L. R. Arnaut, “Generalized Extreme-Value Distributions of Power Near a Boundary Inside Electromagnetic Reverberation Chambers,” in IEEE Transactions on Electromagnetic Compatibility, vol. 52, no. 3, pp. 506-515, Aug. 2010.
- Gradoni, G. and Arnaut, L.R., “Minimum-value distribution of random electromagnetic fields for modeling deep fading in wireless communications,” Ann. Telecommun. (2011) 66: 465.
- L. R. Arnaut and G. Gradoni, “Probability Distribution of the Quality Factor of a Mode-Stirred Reverberation Chamber,” in IEEE Transactions on Electromagnetic Compatibility, vol. 55, no. 1, pp. 35-44, Feb. 2013.
- G. Gradoni, V. Mariani Primiani, and F. Moglie, “Reverberation chamber as a multivariate process: FDTD evaluation of correlation matrix and independent positions,” Progress In Electromagnetics Research, Vol. 133, 217-234, 2013.
- G. Gradoni and L. R. Arnaut, “Transient evolution of eigenmodes in dynamic cavities and time-varying media,” in Radio Science, vol. 50, no. 12, pp. 1256-1270, Dec. 2015.
- L. R. Arnaut and G. Gradoni, “Probability Distribution of the Coherence Bandwidth of a Reverberation Chamber,” in IEEE Transactions on Antennas and Propagation, vol. 63, no. 5, pp. 2286-2290, May 2015.
- M. Migliaccio, G. Gradoni and L. R. Arnaut, “Electromagnetic Reverberation: The Legacy of Paolo Corona,” in IEEE Transactions on Electromagnetic Compatibility, vol. 58, no. 3, pp. 643-652, June 2016.