Optimal Antennas: Operators, Limits, and Design (EuCAP 2024)

18th European Conference on Antennas and Propagation, 2024, Glasgow, Scotland, UK, March 22nd, 14:00 – 17:00


The success of computational electromagnetics in recent decades stems from the possibility to numerically approach non-canonical scenarios, i.e., those that cannot be solved analytically. This short course summarizes recent advancements in evaluating fundamental bounds, inverse design, and modal decomposition. An efficient framework utilizing matrix formalism is necessary for all these diverse techniques. This formalism, including various techniques, tricks, and identities, will be presented. The course aims to provide a good balance between mathematics, electromagnetic theory, code implementation, and live demonstrations that cover diverse applications.

After the course, the participant will be able to:

  • identify and evaluate matrices corresponding to various operators stemming from integral equations,
  • formulate antenna problems and solve them using convex optimization,
  • quantify the cost of extra design constraints (self-resonance, restricted controllable region, etc.),
  • use software to compute bounds for practical design cases and – if needed – to implement his/her own codes,
  • use topology optimization to improve antenna designs and to automate antenna design,
  • apply the presented methodology to optimization problems involving antenna arrays and multi-port antennas,
  • understand unsolved problems in the field and start his/her own research in that area.

The course will offer a good balance between mathematics, EM theory, code implementation, and live demos covering diverse applications. The participants will be provided with codes determining the bounds on the antenna metrics (gain, radiation efficiency, Q-factor) and performing topology optimization. A worksheet with analytical solutions will be distributed as well.


Mats Gustafsson (Lund University, Sweden, EU)
Miloslav Capek (Czech Technical University, Czech Republic, EU)


  • (14:00 – 14:15) Introduction
  • (14:15 – 14:40) Integral equations and related operators (Theory: physical background, numerical evaluation of the operators, algebraic properties, modal analysis. Practical part: live demonstrations.)
  • (14:40 – 15:20) Fundamental bounds (Theory: convex optimization, formulation of quadratic problems. Practical part: live demonstrations.)
  • (15:20 – 15:40) Coffee break
  • (15:40 – 16:20) Inverse design (Theory: properties, available techniques. Practical part: live demonstrations.)
  • (16:20 – 16:50) Antenna synthesis and optimal excitation
  • (16:50 – 17:00) Discussion and conclusion


Lectures can be downloaded from THIS link with the password received on-site.
Cheat-sheet with integral-equation-based matrix operators is HERE.

Practical part – Instructions

  • The participants are welcome to bring their laptops to actively participate in the practical part. In that case they are requested to download SW packages from HERE (updated 22/03/24), unzip and add them to MATLAB path.
  • The examples are prepared in MATLAB (version 2023a and newer are recommended) and can be downloaded HERE (updated 22/03/24). To unzip, use the password received via e-mail or on-site.
  • To run the examples, download the software package consisting of AToM (Antenna Toolbox for MATLAB).
  • Run the m-file Ex0_HelloMatrices.m to verify that all packages are available and the hardware is compatible.
  • Should you have any persisting problem with the test example Ex0_HelloMatrices.m, contact us and attach the automatically generated file Ex0_LogFile.out to the e-mail.


  1. Gustafsson, M., Tayli, D., Cismasu,M.: Physical Bounds of Antennas, in Handbook of Antenna Technologies, 2015, DOI 10.1007/978-981-4560-75-7_18-1.
  2. Gustafsson, M., Tayli, D., Ehrenborg, C., Cismasu, M., Norbedo, S.: Antenna Current Optimization using MATLAB and CVX, FERMAT , Vol. 15, No. 5, p. 1-29, 2016.
  3. Jelinek, L., Capek, M.: Optimal Currents on Arbitrarily Shaped Surfaces, IEEE Trans. Antennas Propag., Vol. 65, No. 1, p. 329-341, 2017.
  4. Molesky, S., Lin, Z., Piggott, A.Y. et al.: Inverse Design in Nanophotonics. Nature Photonics, Vol. 12, pp. 659 – 670, 2018.
  5. Chao, P., Strekha, B., Defo, R. K., Molesky, S., Rodriguez, A. W.: Physical Limits in Electromagnetism, Nature Reviews Physics, July 2022.
  6. Christiansen, R. E., Sigmund, O.: Compact 200 Line MATLAB Code for Inverse Design in Photonics by Topology Optimization: Tutorial, Journal of the Optical Society of America B, Vol. 38, No. 2 pp. 510-520, 2021.
  7. Harrington, R. F.: Field Computation by Moment Methods, Wiley – IEEE Press, 1993.
  8. Harrington, R. F.: Antenna Excitation For Maximum Gain, IEEE Trans. Antennas Propag., Vol. 13, No. 6, pp. 896-903, 1965.
  9. Gustafsson, M., Sohl, C., Kristensson, G.: Physical Limitations on Antennas of Arbitrary Shape, Proc. R. Soc. A , Vol. 463, pp. 2589-2607, 2007.
  10. Gustafsson, M., Nordebo, S.: Optimal Antenna Currents for Q, Superdirectivity, and Radiation Patterns Using Convex Optimization, IEEE Trans. Antennas Propag., Vol. 61, No. 3, pp. 1109-1118, 2013.
  11. Capek, M., Jelinek, L.: Optimal Composition of Modal Currents For Minimal Quality Factor Q, IEEE Trans. Antennas Propag., Vol. 64, No. 12, pp. 5230-5242, 2016.
  12. Capek, M., Jelinek, L., Hazdra, P.: On the Functional Relation between Quality Factor and Fractional Bandwidth, IEEE Trans. Antennas Propag., Vol. 63, No. 6, pp. 2787-2790, 2015.
  13. Capek, M., Gustafsson, M., Schab, K.: Minimization of Antenna Quality Factor, IEEE Trans. Antennas Propag., Vol. 65, No. 8, pp. 4115-4123, 2017.
  14. Gustafsson, M., Capek, M.: Maximum Gain, Effective Area, and Directivity, IEEE Trans. Antennas Propag., Vol. 67, No. 8, pp. 5282 – 5293, 2019.
  15. Gustafsson, M., Fridén, J., Colombi, D.: Antenna Current Optimization for Lossy Media With Near-Field Constraints, IEEE Antennas Wireless Propag. Lett., Vol. 14, pp. 1538-1541, 2015.
  16. Gustafsson, M., Capek, M., Schab, K.: Tradeoff Between Antenna Efficiency and Q-Factor, IEEE Trans. Antennas Propag., Vol. 67, No. 4, pp. 2482-2493, 2019.
  17. Gustafsson, M., Schab, K., Jelinek, L., Capek, M.: Upper bounds on absorption and scattering, New Journal of Physics, Vol. 22, No. 7, p. 073013, 2020.
  18. Ehrenborg, C., Gustafsson, M., Capek, M.: Capacity Bounds and Degrees of Freedom for MIMO Antennas Constrained by Q-Factor, IEEE Trans. Antennas Propag., Vol. 69, No. 9, pp. 5388-5400, 2021.
  19. Capek, M., Jelinek, L., Gustafsson, M.: Shape Synthesis Based on Topology Sensitivity, IEEE Trans. Antennas Propag., Vol. 67, No. 6, pp. 3889 – 3901, 2019.
  20. Capek, M., Jelinek, L., Schab, K., Gustafsson, M., Jonsson, B. L. G., Ferrero, F., Ehrenborg, C.: Optimal Planar Electric Dipole Antennas: Searching for Antennas Reaching the Fundamental Bounds on Selected Metrics, IEEE Antennas and Propagation Magazine, Vol. 61, No. 4, pp. 19-29, 2019.
  21. Capek, M., Jelinek, L., Kadlec, P., Gustafsson, M.: Optimal Inverse Design Based on Memetic Algorithms – Part 1: Theory and Implementation, IEEE Trans. Antennas Propag., Vol. 71, No. 11, pp. 8806-8816, Nov. 2023.
  22. Capek, M., Jelinek, L., Kadlec, P., Gustafsson, M.: Optimal Inverse Design Based on Memetic Algorithms – Part 2: Examples and Properties, IEEE Trans. Antennas Propag., Vol. 71, No. 11, pp. 8817-8826, Nov. 2023.
  23. Schab, K., Rothschild, A., Nguyen, K., Capek, M., Jelinek, L., Gustafsson, M.: Trade-Offs in Absorption and Scattering by Nanophotonic Structures, Optics Express, Vol. 28, No. 24, pp. 36584-36599, 2020.
  24. Schab, K., Jelinek, L., Capek, M., Gustafsson, M.: Upper Bounds on Focusing Efficiency, Optics Express, Vol. 20, Issue 25, pp. 45705-45723, 2022.

Contacts: miloslav.capek[at]fel.cvut.cz (practical part) and mats.gustafsson[at]eit.lth.se (theoretical part). Please, add always the second lecturer into a copy as well.

Last edit: 2024-03-22 | 1:23:32 AM