PhD studentship: Phase-space methods for smart electromagnetic environments

Closing Date
Friday, 25th June 2021
Mathematical Sciences

Supervised by Dr Gabriele Gradoni

This project will be based at the University of Nottingham in the School of Mathematical Sciences, with frequent (funded) travels to the University of Cambridge in the Maxwell Centre, Cavendish Laboratory.

Wireless infrastructures for beyond 5G/6G generation of mobile communication systems are required to guarantee unprecedented link performance levels, while minimizing the complexity, the power consumption, and the cost of the architecture. Therefore, alternative solutions to the approach “more information and data through more power and more emissions of electromagnetic waves” are mandatory because of the existing electromagnetic congestion. The solution proposed by the EU Horizon2020 ICT52 RISE-6G project ( is to consider a propagation environment that can be reconfigured via large wall-mounted reconfigurable intelligent surfaces (RISs): A smart electromagnetic environment (SME). Together with wireless infrastructure and users, the SME are treated as a whole system to be optimised for improving the performance of the wireless communication channel.

To enable the design of smart electromagnetic environments and their integration in wireless signal coverage planning tools for telecommunication operators, advanced electromagnetic modelling is required. 

The wave modelling research Group at the University of Nottingham ( have developed innovative high frequency propagation methods inspired by quantum mechanics and based on the phase space representation of waves.

A phase space transformation of electromagnetic fields represents the flow of wave energy in both position and direction of propagation, and it is achieved by calculation of the Wigner function. In the high frequency regime, the Wigner function can be approximated to provide a ray-based picture of the propagated wave fields, including higher order effects such as edge diffraction and diffusion from rough surfaces. Since the Wigner function transports bundles of rays, as opposed to individual rays as in shoot-and-bounce methods, efficient ray tracing methods can be developed in phase space to predict the distribution of wave energy within large and complex environments. The method has been recently applied to predict the spatial correlation function in confined domains with chaotic dynamics.  

 The project will extend the Wigner function approach to include variable and inhomogeneous boundary conditions of confined domains, including RISs. The main objectives will be: - to develop an average phase space representation of ray densities scattered by a reconfigurable surface in both the antenna array and the metamaterial regime; - to integrate the average phase space representation into the dynamical energy analysis code; - to devise a large scale collocation method for multiple RISs to achieve prescribed enhanced/suppressed coverage areas in large reconfigurable environments (train station and smart factory). 

Relevant publications:  [1] D.J. Chappell, D. Löchel, N. Søndergaard, G. Tanner, Dynamical energy analysis on mesh grids: A new tool for describing the vibro-acoustic response of complex mechanical structures, Wave Motion, Volume 51, Issue 4, 2014, Pages 589-597.

[2] Gabriele Gradoni, Stephen C Creagh, Gregor Tanner, Christopher Smartt and David W P Thomas. A phase-space approach for propagating field–field correlation functions (2015) New J. Phys. 17 093027

[3] Adnan F, Blakaj V, Phang S, Antonsen TM, Creagh SC, Gradoni G, Tanner G. (2021) Wireless power distributions in multi-cavity systems at high frequencies. Proc. R. Soc. A 477: 20200228. 

Subject Areas; Applied Mathematics, Wave Modelling

Funding Notes: Summary: UK/EU students - Tuition Fees will be paid, and a full stipend provided at the RCUK rate (£15,285 per annum for 2020/21). There will also be some funds available to support conference attendance. The scholarship length will be 3.5 years. 

Eligibility/Entry Requirements: We require an enthusiastic graduate with a 1st class degree in Mathematics, preferably at MMath/MSc level, or an equivalent overseas qualification (in exceptional circumstances a 2:1 class degree, or equivalent, can be considered). 

It is highly desirable that the successful applicant is familiar with one or several of the following topics: wave asymptotics, electromagnetism, metamaterials, numerical methods. While experience with more than one of the aforementioned mathematical fields would be beneficial, the willingness to learn and engage with all of them is an absolute necessity. The project requires the student to have experience with a scientific computing software package or programming language such as MATLAB, C/C++ and/or Python.

For any enquiries please email

Applications: To apply please send your CV and a Cover letter to

This studentship is open until filled. Early application is strongly encouraged.