CEM group concentrates on theoretical research in classical electromagnetic theory with an emphasis on the computational aspects. Currently developed topics include radiation properties of electrically small radiators and scatterers, fundamental bounds, and inverse design. Members of the group have for many years been also involved in classical antenna theory and field propagation in artificial materials.

Current research activities are strongly entangled with a “source concept” paradigm, in which field observables operate over electromagnetic sources, which are the system variables often represented in finite-dimensional bases. The emphasis is placed on the development of original computational codes based on this paradigm.

CEM group members offer variety of topics for final projects and maintain extensive library of books related to classical and quantum electromagnetism, antenna theory, and numerical modeling. The team periodically participates in workshops. We like MATLAB, LaTeX, Beamer, and Ti*k*Z.

### Miloslav Čapek

Associate professor

- Antenna theory
- Numerical methods
- Optimization
- MATLAB, LaTeX, Ti
*k*Z

### Lukáš Jelínek

Associate professor

- Electromagnetic field theory
- Antenna theory
- Artificial electromagnetic materials

### Martin Štrambach

Student

- Machine learning
- GPU computing

Lamye Akrou

- Visit 2019
- Precise characteristic mode decomposition.

- Fundamental Bounds in EM
- Optimal Inverse Design
- Small Antennas
- Modal Decompositions
- Method of Moments

Fundamental bounds determine the best attainable values of physical metrics and are typically evaluated using tools of convex optimization and matrix operators. The fundamental bounds delimit performance of a hypothetical device. In line with the “source concept”, fundamental bounds result in optimal current densities defined in a prescribed region. Formulation of fundamental bounds commonly include physically motivated constraints such as enforcement of self-resonance or complex power balance. Among others, bounds are for example known on minimum antenna Q-factor, maximum antenna gain, minimum dissipation factor, minimum or maximum scattering cross sections.

Quite a few constraints can be used, *e.g.*, a constraint on self-resonance, on complex power balance, or on only partly controllable region. All bounds can be transformed into multi-criteria form describing the mutual trade-offs.

*Figure: Optimal current densities for an L-shape plate (electric size ka = 1/2). The sequence corresponds to a specific trade-off point between dissipation factor and Q-factor, maximal radiation efficiency (externally tuned and self-resonant), and minimal Q-factor, respectively.*

In the current state of the art, a direct shape synthesis is not possible, the main obstacle being the combinatorial nature of an associated optimization problem which is non-polynomial in time. On the other hand, there is a strong evidence that a skilled designer can provide designs with performance close to fundamental bounds...

How close can we go with clever, albeit brute-force techniques? Can we go closer? Is it possible to improve the design performance with the knowledge of the first differences or utilization of machine learning? To address these questions, a novel memetic framework combining local and global optimization routines is developed, combining the advantages of an adjoint formulation of topology optimization and of an evolutional algorithm. Various geometry- and topology-based metrics like shape regularity are being incorporated as well.

*Figure: One run of topology optimization based on the exact re-analysis.*

Electrically small antennas are ubiquitous, yet their true importance is still about to be revealed with the upcoming Internet of Things (IoT). Other technologies relying on small radiators such as implantable antennas, directional nanoscatterers, or field concentrators and absorbers are developed too. For all of them, the performance nontrivially depends on shape of the structure, material distribution, and excitation.

Various metrics relevant mainly in the electrically small regime are studied and formulated in terms of method of moments operators. A few examples are stored energy, ohmic losses, electric and magnetic moments, total active reflection coefficient and others.

*Figure: A meanderline situated in a radiansphere of radius a as a typical representant of planar electrically small antenna.*

The most salient features of many complicated EM phenomena can be revealed by proper modal decomposition, which also reduces the complexity of problem and often offers additional physical insight. Most commonly, the modal decomposition is achieved via eigenvalue problem the modes of which are used as a new basis for the engineering problem at hand.

For example, characteristic modes diagonalize the impedance matrix, result in orthogonal far-fields, and are thus excellent for a design of electrically small MIMO antennas. Other bases, like radiation modes, are perfect for reducing the numerical complexity of fundamental bounds evaluation via convex optimization routines.

*Figure: First two dominant modes on a rectangular plate.*

Method of moments is used to convert linear operator equations into linear algebraic equation systems by representing the solution in a suitable basis of expansion functions. The crucial step of identifying associated Green’s function makes this method ideal for open (radiating) problems since only the region containing sources is discretized.

Application of method of moments to integral equations leads to dense and relatively small (as compared the application to differential equations also known as finite element method) system matrix. Direct solvers are used to invert these matrices up to dimensions of several thousands. For bigger matrices, the efficient indirect solvers are available.

Conventionally, the inversion of a system matrix yields the contrast current flowing in the solution domain. This is, however, only one of many possibilities how to employ this method. Other techniques utilize the system matrix and its derivatives directly.

*Figure: Surface current density (3rd characteristic mode) on a triangularized obstacle.*

- Virtual Prototyping of EM Systems
- Fundamental Bounds
- AToM
- Source Concept

Fast and precise virtual prototyping of EM systems is a strong prerequisite for massive expansion of IoT devices and is one of the final goals of Industry 4.0. This project aims at the development of numerical tools capable to design and simulate wireless systems, to expand existing optimization routines, and to characterize the connectors and calibration kits.

This project has been supported by Technology Agency of the Czech Republic within the frame of project TH04010373, 2018-2021 ("Virtual Prototyping and Validation of Electromagnetic Systems").

*Figure: Parametric sweep of a double cylindrical helix.*

The optimality of passive electromagnetic structures is studied within this project. The main goal is to investigate the existence of and build up an understanding of the physical bounds on the primary physical quantities as well as system metrics related to radiation and scattering phenomena arising in wireless communication and power transfer. This project strives to fill a gap in the knowledge of classical electromagnetism as the investigation of the optimal source current distributions is, in many cases, absent.

This project has been supported by Czech Science Foundation within the frame of project GACR 19-06049S, 2019-2021 ("Fundamental Bounds on Electromagnetic Radiation and Scattering Phenomena and Associated Realizable Subforms").

*Figure: Pareto-optimal solutions with respect to three antenna parameters for three arrangements of dipole arrays.*

AToM (Antenna Toolbox for MATLAB) originated from project TA04010457 (2014-2017) of Technology Agency of the Czech Republic and is written in MATLAB as a semi-open code offering user-friendly operation through GUI or direct access to low level functions. The simulation core of AToM is based on Method of moments solution to field integral equations on wire, planar, and volumetric structures. Together with modal decomposition (characteristic modes), the source concept, feeding synthesis and powerful optimization package (FOPS optimizer), AToM presents unique tool for analysis and synthesis of antennas.

AToM also contains several add-ons, notable examples being the determination of fundamental bounds on antenna metrics, the design of optimal shapes via topology (shape) optimization and parametric strip generator. An extensive list of integrodifferential operators is available both in RWG MoM (2D surface code) and piecewise constant MoM (3D volumetric code).

Visit AToM webpage for more details.

Evaluating antenna and scattering characteristics solely by means of contrast currents flowing in a region occupied by the radiator opens a new paradigm called “source concept”, which also involves diverse methods of calculation, decomposition, and electric and magnetic currents modification. Within the method of moments, the source concept typically leads to observables, the current-quadratic forms of which provide the performance metrics and allows for their efficient analysis or for setting up their fundamental bounds. Automated search for sub-optimal designs via topology optimization is the latest application of the source concept.

A precise and efficient numerical implementation of the source concept is an inevitable part of its practically-oriented development. With respect to this, the MATLAB toolbox AToM is currently being developed at CTU in Prague.

- Internal seminars
- Posters
- Other materials

Author: J. Tucek

Presented on: January 06, 2020

Author: M. Strambach

Presented on: March 03, 2020

Author: V. Neuman

Presented on: April 20, 2020

AToM2TikZ is an extension for AToM package, which allows conversion of AToM plots into LaTeX/Ti*k*Z macros. Package offers additional functionality such as animations, shading, opacity settings and possibility to easily integrate another graphical objects. The presentation contains basic description of plot conversion, examples, how to use package and restrictions of current version of AToM2TikZ. The direct visibility of triangles algorithm is presented at the end of seminar.

Author: M. Masek

Presented on: May 04, 2020

Symmetries are an integral part of our everyday lives. The presentation describes the utilization of symmetries and application of point group theory into MoM framework. Their detailed implementation into Antenna Toolbox for Matlab (AToM) package is explained and thoroughly illustrated. Later on, application examples are presented. Namely: the solution to the modal tracking, block-diagonalizing of any arbitrary MoM operator, or design of orthogonal feeding schemes.

Presented on: February 25, 2020

Presented on: February 25, 2020

Presented on: February 26, 2019

Presented on: February 26, 2019

Presented on: February 26, 2019

Presented on: February 27, 2018

## Templates & LaTeX packages

Authors: M. Masek, M. Capek, and P. Chvojka

A departmental beamer template to create a presentations in LaTeX.

The template is available in the free use at Overleaf.

Authors: M. Masek and M. Capek

A departmental template to create a poster in LaTeX.

The template is available in the free use at Overleaf.

Author: M. Masek

The matbpars package provides a parsing algorithm which is able to print MATLAB commands in LaTeX with the same style as they are written in the original software. Direct inline commands and loading the code from the .m files are included.

The package is still under development for some corner-cases, however, it works correctly for standard usage. Is in the free use at Overleaf.

Casimir Ehrenborg

Guy Vandenbosch

Raúl Rodríguez Berral

- Fundamental Bounds
- Optimal Design
- Modal Decomposition
- Others

## Submitted:

A problem of the erroneous duality gap caused by the presence of symmetries is solved in this paper utilizing point group theory. The optimization problems are first divided into two classes based on their predisposition to suffer from this deficiency. Then, the classical problem of Q-factor minimization is shown in an example where the erroneous duality gap is eliminated by combining solutions from orthogonal subspaces. Validity of this treatment is demonstrated in a series of subsequent examples of increasing complexity spanning the wide variety of optimization problems, namely minimum Qfactor, maximum antenna gain, minimum total active reflection coefficient, or maximum radiation efficiency with self-resonant constraint. They involve problems with algebraic and geometric multiplicities of the eigenmodes, and are completed by an example introducing the selective modification of modal currents falling into one of the symmetry conformal sub-spaces. The entire treatment is accompanied with a discussion of finite numerical precision, and mesh grid imperfections and their influence on the results. Finally, the robust and unified algorithm is proposed and discussed, including advanced topics such as the uniqueness of the optimal solutions, dependence on the number of constraints, or an interpretation of the qualitative difference between the two classes of the optimization problems.

Available online at arXiv.

The optimal spectral efficiency of MIMO antennas in an ideal line-of-sight channel is investigated when bandwidth requirements are placed on the antenna. By posing the problem as a convex optimization problem restricted by the input port Q-factor a semi-analytical expression is formed for its solution. It is shown that this solution is solely dependent on energy modes of the antenna. These modes are compared to the characteristic modes and the ability to induce them through sub-regions of a plate is investigated. The position of these regions is also investigated when they are raised above the ground plane. Their performance is illustrated through spectral efficiency over Q, a quantity that is connected to the true capacity. It is demonstrated that the spatial diversity of the controlled regions correlates with the number of significant energy modes.

Available online at arXiv.

## Published:

*Optics Express, vol. 28, pp. 36584 - 36599, Nov. 2020*

Trade-offs between feasible absorption and scattering cross sections of obstacles confined to an arbitrarily shaped volume are formulated as a multi-objective optimization problem solvable by Lagrangian-dual methods. Solutions to this optimization problem yield a Pareto-optimal set, the shape of which reveals the feasibility of achieving simultaneously extremal absorption and scattering. Two forms of the trade-off problems are considered involving both loss and reactive material parameters. Numerical comparisons between the derived multi-objective bounds and several classes of realized structures are made. Additionally, low-frequency (electrically small, long wavelength) limits are examined for certain special cases.

*IEEE Transactions on Antennas and Propagation*, Oct. 2020

A numerically effective description of the total active reflection coefficient and realized gain are studied for multi-port antennas. Material losses are fully considered. The description is based on operators represented in an entire-domain port-mode basis, *i.e.*, on matrices with favorably small dimensions. Optimal performance is investigated and conditions on optimal excitation and matching are derived. The solution to the combinatorial problem of optimal ports' placement and optimal feeding synthesis is also accomplished. Four examples of various complexity are numerically studied, demonstrating the advantages of the proposed method. The final formulas can easily be implemented in existing electromagnetic simulators using integral equation solver.

*New Journal of Physics*, vol. 22, 073013, Sept. 2020.

A general framework for determining fundamental bounds in nanophotonics is introduced in this paper. The theory is based on convex optimization of dual problems constructed from operators generated by electromagnetic integral equations. The optimized variable is a contrast current defined within a prescribed region of a given material constitutive relations. Two power conservation constraints analogous to optical theorem are utilized to tighten the bounds and to prescribe either losses or material properties. Thanks to the utilization of matrix rank-1 updates, modal decompositions, and model order reduction techniques, the optimization procedure is computationally efficient even for complicated scenarios. No dual gaps are observed. The method is well-suited to accommodate material anisotropy and inhomogeneity. To demonstrate the validity of the method, bounds on scattering, absorption, and extinction cross sections are derived first and evaluated for several canonical regions. The tightness of the bounds is verified by comparison to optimized spherical nanoparticles and shells. The next metric investigated is bi-directional scattering studied closely on a particular example of an electrically thin slab. Finally, the bounds are established for Purcell's factor and local field enhancement where a dimer is used as a practical example.

*IEEE Transactions on Antennas and Propagation*, vol. 67, no. 8, pp. 5282 – 5293, Aug. 2019.

Fundamental bounds on antenna gain are found via convex optimization of the current density in a prescribed region. Various constraints are considered, including self-resonance and only partial control of the current distribution. Derived formulas are valid for arbitrarily shaped radiators of a given conductivity. All the optimization tasks are reduced to eigenvalue problems, which are solved efficiently. The second part of the paper deals with superdirectivity and its associated minimal costs in efficiency and Q-factor. The paper is accompanied by a series of examples practically demonstrating the relevance of the theoretical framework and entirely spanning a wide range of material parameters and electrical sizes used in antenna technology. Presented results are analyzed from the perspective of effectively radiating modes. In contrast to a common approach utilizing modes, the radiating modes of a given body are directly evaluated and analyzed here. All crucial mathematical steps are reviewed in the appendices, including a series of important subroutines to be considered making it possible to reduce the computational burden associated with the evaluation of electrically large structures and structures of high conductivity.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 67, no. 4, pp. 2482-2493, Apr. 2019.

The tradeoff between radiation efficiency and antenna bandwidth, expressed in terms of Q-factor for small antennas, is formulated as a multiobjective optimization problem in current distributions of predefined support. Variants on the problem are constructed to demonstrate the consequences of requiring a self-resonant current as opposed to the one tuned by an external reactance. The tradeoffs are evaluated for sample problems and the resulting Pareto-optimal sets reveal the relative cost of valuing low radiation Q-factor over high efficiency, the cost in efficiency to require a self-resonant current, the effects of lossy parasitic loading, and other insights. Observations are drawn from the sample problems selected, though the tradeoff evaluation method is valid for studying arbitrary antenna geometries. In the examples considered here, we observe that small increases in Q-factor away from its lower bound allow for dramatic increases in efficiency toward its upper bound.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 66, no. 12, pp. 6716-6723, Dec. 2018.

Existing optimization methods are used to calculate the upper bounds on radiation efficiency with and without the constraint on self-resonance. These bounds are used for the design and assessment of small electric-dipole-type antennas. We demonstrate that the assumption of lossless, lumped, and external tuning skews the true nature of radiation efficiency bounds when practical material characteristics are used in the tuning network. A major result is that, when realistic (*e.g.*, finite conductivity) materials are used, small antenna systems exhibit dissipation factors which scale as (ka)^{-4} , rather than (ka)^{-2} as previously predicted under the assumption of lossless external tuning.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 66, no. 9, pp. 4948-4952, Sept. 2018.

Radiation efficiencies of modal current densities distributed on a spherical shell are evaluated in terms of dissipation factor. The presented approach is rigorous, yet simple and straightforward, leading to closed-form expressions. The same approach is utilized for a two-layered shell and the results are compared with other models existing in the literature. Discrepancies in this comparison are reported and reasons are analyzed. Finally, it is demonstrated that radiation efficiency potentially benefits from the use of internal volume, which contrasts with the case of the radiation Q-factor.

Published version available online at IEEEXplore.

*IEEE Transactions on Antennas and Propagation*, vol. 65, no. 8, pp. 4115-4123, Aug. 2017.

Optimal currents on arbitrarily shaped radiators with respect to the minimum quality factor are found using a simple and efficient procedure. The solution starts with a reformulation of the problem of minimizing quality factor Q as an alternative, so-called dual, problem. Taking advantage of modal decomposition and group theory, it is shown that the dual problem can easily be solved and always results in minimal quality factor Q. Moreover, the optimization procedure is generalized to minimize quality factor Q for embedded antennas, with respect to the arbitrarily weighted radiation patterns, or with prescribed magnitude of the electric and magnetic near fields. The obtained numerical results are compatible with previous results based on the composition of modal currents, convex optimization, and quasi-static approximations; however, using the methodology in this paper, the class of solvable problems is significantly extended.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 65, no. 1, pp. 329-341, Jan. 2017.

An optimization problem has been formulated to find a resonant current extremizing various antenna parameters. The method is presented on, but not limited to, particular cases of gain G, quality factor Q, gain to quality factor ratio G/Q, and radiation efficiency η of canonical shapes with conduction losses explicitly included. The Rao-Wilton-Glisson basis representation is used to simplify the underlying algebra while still allowing surface current regions of arbitrary shape to be treated. By switching to another basis generated by a specific eigenvalue problem, it is finally shown that the optimal current can, in principle, be found as a combination of a few eigenmodes. The presented method constitutes a general framework in which the antenna parameters, expressed as bilinear forms, can automatically be extremized.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

_{z}for Dominant Spherical Modes,”

*IET Microwaves, Antennas and Propagation*, vol. 9, no. 10, pp. 1096-1103, Feb. 2015.

This study describes an analytical evaluation of the quality factor Q_{z} in a separable system in which the vector potential is known. The proposed method uses a potential definition of active and reactive power, implicitly avoiding infinite entire space integration and extraction of radiation energy. As a result, all the used quantities are finite, and the calculated Q_{z} is always non-negative function of frequency. The theory is presented on the canonical example of the currents flowing on a spherical shell. The Q_{z} for the dominant spherical transverse magnetic and transverse electric modes and their linear combination are found in closed forms, including both internal and external energies. The proposed analytical method and its results are compared with previously published limits of the quality factor Q.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

## Published:

*International Journal of RF and Microwave Computer-Aided Engineering*, pp. 1-8, May 2020.

A single‐ and multi‐objective optimization package is presented and described in detail. It contains an ensemble of local and global optimization routines. Procedures controlling variable number of dimensions are implemented as well, which is a rare feature among optimization oriented packages. The package is provided as a MATLAB toolbox. It excels in versatility and extensibility, which is demonstrated on a series of examples covering classical electromagnetism and antenna design. It is taken for granted that defining parameters of the optimization method can be set prior to the simulation run. However, its effective performance can be changed during the optimization run thanks to the full control feature. Moreover, it opens new possibilities in merging various algorithms into hybrids, performing complex dynamic programming tasks, or exploiting third party software. These advantages render the package as a perfect tool to deal with nowadays challenging engineering tasks.

*IEEE Transactions on Antennas and Propagation*, vol. 18, no. 11, pp. 2311-2315, Nov. 2019.

A recently introduced technique of topology sensitivity in method of moments is extended by the possibility of adding degrees of freedom (reconstruct) into the underlying structure. The algebraic formulation is inversion-free, suitable for parallelization, and scales favorably with the number of unknowns. The reconstruction completes the nearest neighbors procedure for an evaluation of the smallest shape perturbation. The performance of the method is studied with a greedy search over a Hamming graph representing the structure in which initial positions are chosen from a random set. The method is shown to be an effective data mining tool for machine learning-related applications.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 61, no. 4, pp. 19-29, Aug. 2019.

Considerable time is often spent optimizing antennas to meet specific design metrics. Rarely, however, are the resulting antenna designs compared to rigorous physical bounds on those metrics. Here, we study the performance of optimized planar meander line antennas with respect to such bounds. Results show that these simple structures meet the lower bound on the radiation quality factor (Q-factor) (maximizing single-resonance fractional bandwidth) but are far from reaching the associated physical bounds for efficiency. The relative performance of other canonical antenna designs is comparable in similar ways, and the quantitative results are connected to intuitions from small antenna design, physical bounds, and matching network design.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 67, no. 6, pp. 3889-3901, Jun. 2019.

A method evaluating the sensitivity of a given parameter to topological changes is proposed within the method of moments paradigm. The basis functions are used as degrees of freedom which, when compared to the classical pixeling technique, provide important advantages, one of them being impedance matrix inversion free evaluation of the sensitivity. The devised procedure utilizes port modes and their superposition which, together with only a single evaluation of all matrix operators, leads to a computationally effective procedure. The proposed method is approximately 100 times faster than the contemporary approaches, which allows the investigation of the sensitivity and the modification of shapes in real time. The method is compared with the known approaches and its validity and effectiveness are verified using a series of examples. The procedure can be implemented in up-to-date electromagnetic (EM) simulators in a straightforward manner. It is shown that the iterative repetition of the topology sensitivity evaluation can be used for gradient-based topology synthesis. This technique can also be employed as a local step in global optimizers.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

## Submitted:

A technique of designing antenna excitation realizing orthogonal free-space channels is presented. It is shown that a symmetric antenna geometry is required in order to achieve orthogonality with respect to all physical quantities. A maximal number of reachable uncorrelated channels and a minimal number of feeders required to excite them are rigorously determined from the knowledge of an antenna's symmetries. The number of channels and number of feeders are summarized for the commonly used point groups (a rectangle, a square, etc.). The theory is applied to an example of a rectangular rim where the positions of feeders providing the best total active reflection coefficient, an important metric in multi-port systems, are determined. The described technique can easily be implemented in existing integral solvers.

Available online at arXiv.

## Published:

*IEEE Transactions on Antennas and Propagation*, vol. 68, no. 2, pp. 927-937, Feb. 2020.

Issues in modal tracking in the presence of crossings and crossing avoidances between eigenvalue traces are solved via the theory of point groups. The von Neumann-Wigner theorem is used as a key factor in predictively determining mode behavior over arbitrary frequency ranges. The implementation and capabilities of the proposed procedure are demonstrated using characteristic mode decomposition as a motivating example. The procedure is, nevertheless, general and can be applied to an arbitrarily parametrized eigenvalue problem. A treatment of modal degeneracies is included and several examples are presented to illustrate modal tracking improvements and the immediate consequences of improper modal tracking. An approach leveraging a symmetry-adapted basis to accelerate computation is also discussed. A relationship between geometrical and physical symmetries is demonstrated on a practical example.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 68, no. 2, pp. 927-937, Dec. 2018.

A new method to improve the accuracy and efficiency of characteristic mode (CM) decomposition for perfectly conducting bodies is presented. The method uses the expansion of the Green dyadic in spherical vector waves. This expansion is utilized in the method-of-moment (MoM) solution of the electric-field integral equation to factorize the real part of the impedance matrix. The factorization is then employed in the computation of CMs, which improves the accuracy as well as the computational speed. An additional benefit is a rapid computation of far fields. The method can easily be integrated into existing MoM solvers. Several structures are investigated, illustrating the improved accuracy and performance of the new method.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 65, no. 8, pp. 4134-4145, Aug. 2017.

Characteristic modes (CMs) of a spherical shell are found analytically as spherical harmonics normalized to radiate unitary power and to fulfill specific boundary conditions. The presented closed-form formulas lead to a proposal of precise synthetic benchmarks that can be utilized to validate the method-of-moments matrix or performance of CM decomposition. Dependence on the mesh size, electrical size, and other parameters can systematically be studied, including the performance of various mode tracking algorithms. A notable advantage is the independence on feeding models. Both theoretical and numerical aspects of CM decomposition are discussed and illustrated by examples. The performance of state-of-the-art commercial simulators and academic packages having been investigated, we can conclude that all contemporary implementations are capable of identifying the first dominant modes while having severe difficulties with higher order modes. Surprisingly poor performance of the tracking routines is observed notwithstanding the recent ambitious development.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 65, no. 2, pp. 713-720, Feb. 2017.

Aspects of the theory of characteristic modes (CMs), based on their variational formulation, are presented and an explicit form of a related functional, involving only currents in a spatial domain, is derived. The new formulation leads to deeper insight into the modal behavior of radiating structures as demonstrated by a detailed analysis of three canonical structures: a dipole, an array of two dipoles and a loop, cylinder, and a sphere. It is demonstrated that knowledge of the analytical functional can be utilized to solve important problems related to the theory of CM decomposition such as the resonance of inductive modes or the benchmarking of method of moments code.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 64, no. 12, pp. 5230-5242, Dec. 2016.

This paper describes a powerful, yet simple, procedure how to acquire a current approaching the lower bound of quality factor Q. This optimal current can be determined for an arbitrarily shaped electrically small radiator made of a perfect conductor. Quality factor Q is evaluated by Vandenbosch's relations yielding stored electromagnetic energy as a function of the source current density. All calculations are based on a matrix representation of the integro-differential operators. This approach simplifies the entire development and results in a straightforward numerical evaluation. The optimal current is represented in a basis of modal currents suitable for solving the optimization problem so that the minimum is approached by either one mode tuned to the resonance, or, by two properly combined modes. An overview of which modes should be selected and how they should be combined is provided and results concerning rectangular plate, spherical shell, capped dipole antenna, and fractal shapes of varying geometrical complexity are presented. The reduction of quality factor Q and the G/ Q ratio are studied and, thanks to the modal decomposition, the physical interpretation of the results is discussed in conjunction with the limitations of the proposed procedure.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IET Microwaves, Antennas and Propagation,*vol. 9, no. 1, pp. 10-15, Jul. 2014.

This study describes an effective technique for calculating modal radiation efficiency calculation based on decomposition into characteristic modes. The key assumption is that the current distribution on the perfect electric conductor is almost the same as in the case of a very good conductor, for example, metals such as copper, aluminium and silver. This assumption is verified against the conventional technique, the impedance boundary condition (IBC). The proposed approach does not require any modification of the formulation of method of moments for perfectly conducting surfaces, which is assumed for the modal decomposition. Modal efficiencies provide an additional insight that is useful especially for the design of small antennas. Taking the feeding into account, the modal losses can be summed up to obtain the total efficiency. The technique works perfectly for common metals, is fully comparable with the IBC, and can easily be incorporated into any present-day in-house solver. A numerical analysis of three antennas is presented to demonstrate the merits of the approach. Radiation efficiency of coupled dipoles, an electrically small meandered dipole, and PIFA were investigated by the presented method. The results are in perfect agreement with the reference commercial package.

Published version available online at IEEEXplore.

*IEEE Antennas and Propagation Magazine*, vol. 55, no. 2, pp. 176-189, Apr. 2013.

This paper describes the implementation of a complex MATLAB tool to calculate the characteristic modes and associated antenna parameters. The first code, written in FORTRAN, was presented in the early seventieths by Harrington and Mautz. Here, we utilize MATLAB, which is widely known and used in the antenna community these days. Because eigen-decomposition is time consuming, parallel and distributed computing is used. Thanks to the hundreds of built-in functions in MATLAB, computation of the surface currents from the eigenvectors obtained, as well as other important characteristics, are very easy and effective. The practical features are discussed with two examples.

Published version available online at IEEEXplore.

*IEEE Transactions on Antennas and Propagation*, vol. 60, no. 10, pp. 4556-4567, Oct. 2012.

A new formula for the evaluation of the modal radiation Q factor is derived. The total Q of selected structures is to be calculated from the set of eigenmodes with associated eigen-energies and eigen-powers. Thanks to the analytical expression of these quantities, the procedure is highly accurate, respecting arbitrary current densities flowing along the radiating device. The electric field integral equation, Delaunay triangulation, method of moments, Rao-Wilton-Glisson basis function and the theory of characteristic modes constitute the underlying theoretical background. In terms of the modal radiation Q, all necessary relations are presented and the essential points of implementation are discussed. Calculation of the modal energies and Q factors enable us to study the effect of the radiating shape separately to the feeding. This approach can be very helpful in antenna design. A few examples are given, including a thin-strip dipole, two coupled dipoles a bowtie antenna and an electrically small meander folded dipole. Results are compared with prior estimates and some observations are discussed. Good agreement is observed for different methods.

Published version available online at IEEEXplore.

## Pre-prints:

Hybrid computational schemes combining the advantages of a method of moments formulation of a field integral equation and T-matrix method are developed in this paper. The hybrid methods are particularly efficient when describing the interaction of electrically small complex objects and electrically large objects of canonical shapes such as, spherical multi-layered bodies where the T-matrix method is reduced to the Mie series making the method an interesting alternative in the design of implantable antennas or exposure evaluations. Method performance is tested on a spherical multi-layer model of the human head. Along with the hybrid method, an evaluation of the transition matrix of an arbitrarily shaped object is presented and the characteristic mode decomposition is performed, exhibiting fourfold numerical precision as compared to conventional approaches.

Available online at arXiv.

Three contradictory but state-of-the-art concepts for defining and evaluating stored electromagnetic energy are treated in this communication, and are collated with the widely accepted definition of stored energy, which is the total energy minus the radiated energy. All three concepts are compared, and the results are discussed on an example of a dominant spherical mode, which is known to yield dissimilar results for the concepts dealt with here. It is shown that various definitions of stored energy density immanently imply diverse meanings of the term "radiation".

Available online at arXiv.

Ambiguities in the definition of stored energy within distributed or radiating electromagnetic systems motivate the discussion of the well-defined concept of recoverable energy. This concept is commonly overlooked by the community and the purpose of this communication is to recall its existence and to discuss its relationship to fractional bandwidth. Using a rational function approximation of a system's input impedance, the recoverable energy of lumped and radiating systems is calculated in closed form and is related to stored energy and fractional bandwidth. Lumped circuits are also used to demonstrate the relationship between recoverable energy and the energy stored within equivalent circuits produced by the minimum phase-shift Darlington's synthesis procedure.

Available online at arXiv.

## Published:

*IEEE Access*, vol. 6, pp. 10553-10568, Feb. 2018.

Though commonly used to calculate Q-factor and fractional bandwidth, the energy stored by radiating systems (antennas) is a subtle and challenging concept that has perplexed researchers for over half a century. Here, the obstacles in defining and calculating stored energy in general electromagnetic systems are presented from first principles as well as using demonstrative examples from electrostatics, circuits, and radiating systems. Along the way, the concept of unobservable energy is introduced to formalize such challenges. Existing methods of defining stored energy in radiating systems are then reviewed in a framework based on technical commonalities rather than chronological order. Equivalences between some methods under common assumptions are highlighted, along with the strengths, weaknesses, and unique applications of certain techniques. Numerical examples are provided to compare the relative margin between methods on several radiating structures.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IET Microwaves, Antennas and Propagation*, vol. 11, no. 6, pp. 852-858, May 2017.

A full-wave numerical scheme of polarisability (polarisability) tensors evaluation is presented. The method accepts highly conducting bodies of arbitrary shape and explicitly accounts for the radiation as well as ohmic losses. The method is verified on canonical bodies with known polarisability tensors, such as a sphere and a cube, as well as on realistic scatterers. The theoretical developments are followed by a freely available code whose sole user input is the triangular mesh covering the surface of the body under consideration.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Transactions on Antennas and Propagation*, vol. 64, no. 10, pp. 4575-4576, Oct. 2016.

This paper discusses the methods for evaluating the stored electromagnetic energies and the radiation Q for an arbitrary lossless antenna. New expressions for the stored electromagnetic energies are derived by using the Poynting theorem in the complex frequency domain, and they are compared with previous theory and are validated by numerical examples. The minimization of radiation Q for small antenna is also investigated. There exists an optimal current distribution that minimizes the radiation Q for specified small antenna geometry. The optimized Q and the optimal current distribution for small antenna may be determined by solving a generalized eigenvalue equation obtained from the Rayleigh quotient for the radiation Q.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*Proceedings of Royal Society A*, vol. 472, no. 2188, pp. 1-16, Apr. 2016.

This paper deals with the old yet unsolved problem of defining and evaluating the stored electromagnetic energy—a quantity essential for calculating the quality factor, which reflects the intrinsic bandwidth of the considered electromagnetic system. A novel paradigm is proposed to determine the stored energy in the time domain leading to the method, which exhibits positive semi-definiteness and coordinate independence, i.e. two key properties actually not met by the contemporary approaches. The proposed technique is compared with an up-to-date frequency domain method that is extensively used in practice. Both concepts are discussed and compared on the basis of examples of varying complexity.

Published version available online at Proceedings of Royal Society A.

Final manuscript available online at arXiv.

*Radioengineering*(Feature Article), vol. 2016, no. 1, pp. 12-17, Apr. 2016.

Antenna parameters particularly relevant to electrically small antenna design are reviewed in this paper. Source current definitions are accentuated leading to the introduction of the source concept which advantageously utilize only spatially bounded quantities. The framework of the source concept incorporates powerful techniques such as structural and modal decomposition, operator’s inversion and current optimization, thus opening new, challenging possibilities for antenna design, analysis and synthesis.

Published version available online at Radioengineering.

*IEEE Transactions on Antennas and Propagation*, vol. 63, no. 6, pp. 2787-2790, Jun. 2015.

The functional relation between the fractional bandwidth and the quality factor of a radiating system is investigated in this communication. Several widely used definitions of the quality factor are compared with two examples of RLC circuits that serve as a simplified model of a single-resonant antenna tuned to its resonance. It is demonstrated that for a first-order system, only the quality factor based on differentiation of the input impedance has unique proportionality to the fractional bandwidth, whereas, e.g., the classical definition of the quality factor, i.e., the ratio of the stored energy to the lost energy per one cycle, is not uniquely proportional to the fractional bandwidth. In addition, it is shown that for higher order systems, the quality factor based on differentiation of the input impedance ceases to be uniquely related to the fractional bandwidth.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.

*IEEE Antennas and Propagation Magazine*, vol. 56, no. 6, pp. 172-183, Jun. 2014.

This paper deals with practical aspects of mesh generation for the theory of characteristic modes. First, we describe a tool for surface-mesh generation in MATLAB. The tool is afterwards used for an analysis of relative convergence of modal results computed by an in-house modal analyzer in MATLAB. Different meshing scenarios are selected for a dipole, a rectangular patch, and a rectangular patch with a slot, and a recommendation for a mesh-refinement strategy is given. The study is supported by a simple error analysis, considering the approximation error in the evaluation of moment-matrix elements. It is shown that the results are also applicable for the commercial implementation in FEKO software.

Published version available online at IEEEXplore.

*IEEE Transactions on Antennas and Propagation*, vol. 62, no. 1, pp. 311-318, Jan. 2014.

New expressions are derived to calculate the Q factor of a radiating device. The resulting relations link Q based on the frequency change of the input impedance at the input port (Q_{X} , Q_{Z} ) with expressions based solely on the current distribution on an radiating device. The question of which energies of a radiating system are observable is reviewed, and then the proposed Q factor as defined in this paper is physical. The derivation is based on potential theory rather than fields. This approach hence automatically eliminates all divergent integrals associated with electromagnetic energies in infinite space. The new formulas allow us to study the radiation Q factor for antennas without feeding (through *e.g.*, characteristic modes) as well as fed by an arbitrary number of ports. The new technique can easily be implemented in any numerical software dealing with current densities. To present the merits of proposed technique, three canonical antennas are studied. Numerical examples show excellent agreement between the measurable Q_{Z} derived from input impedance and the new expressions.

Published version available online at IEEEXplore.

Final manuscript available online at arXiv.