Matrix Formalism for Optimal Computational Design (IEEE AP-S/URSI 2023)

The 2023 IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, Portland, Oregon, USA, July 23rd, 8:20 – 12:00


The success of computational electromagnetics in recent decades stems from the possibility to numerically approach non-canonical scenarios, i.e., those unsolvable analytically. For this reason, numerical methods such as FEM, MoM, or FDTD are used and also available in powerful commercial simulators. They, however, typically focus on solving a problem in the meaning to compute fields and currents from given excitations. Many other techniques are available, with the system at hand discretized and characterized in terms of matrix operators. Modal decompositions decouple excitation from the properties of an obstacle. Fundamental bounds indicate the best possible performance. Shape and topology optimization schemes can isolate well-performing structures for given objective parameters. Mastering these techniques requires good knowledge of matrix algebra and its application to electromagnetism and optimization.

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 will offer a good balance between math, EM theory, code implementation, and live demos covering diverse applications. The participants will receive the presented codes and worksheets summarizing the theory. The presented materials will be accessible after the conference from a remote repository.


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


  • (08:20 – 08:30) Introduction
  • (08:30 – 09:05) Matrix formalism in integral equation (Part 1)
  • (09:05 – 09:45) Modal analysis (Part 2)
  • (09:45 – 10:25) Fundamental bounds (Part 3)
  • (10:25 – 10:40) Coffee break
  • (10:40 – 11:15) Inverse design based topology sensitivity (Part 4)
  • (11:15 – 11:45) Antenna array port modes (Part 5)
  • (11:45 – 12:00) Discussion and conclusion


Lectures can be downloaded from THIS link with the password received on-site.

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 24/07/23), unzip and add them to MATLAB path.
  • The examples are prepared in MATLAB (version 2021a and newer are recommended) and can be downloaded HERE (updated 24/07/23). 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., Gustafsson, M., Jelinek, L., Kadlec, P.: Memetic Scheme for Topology Optimization Using Exact Reanalysis of Method-of-Moments Models – Part 1: Theory and Implementation, arxiv:, 2021.
  22. Capek, M., Gustafsson, M., Jelinek, L., Kadlec, P.: Memetic Scheme for Topology Optimization Using Exact Reanalysis of Method-of-Moments Models – Part 2: Examples and Properties, arxiv:, 2021.
  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.

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

Last edit: 2023-07-05 | 2:23:32 PM