volume | PIER Journals

Abstract

Spherical harmonics are classical analysis tools in many science and engineering domains. For analyzing the electromagnetic fields of antennas in the frequency domain, the mostly used formulation is the one proposed by Hansen. This article proposes an alternative solution, relying on spin spherical harmonics. On a sphere, the tangential components of the electric and magnetic fields are represented by means of harmonics of spin ±1. Then new closed-form relations are established between the spin spherical harmonics and the ones formulated by Hansen. A sampling theorem and fast transforms that are consistent with spin spherical harmonics are used. The radiations of spin spherical harmonics of order 1 are related to elementary dipoles and Huygens sources in circular polarization. Finally, numerical experiments are performed with a horn antenna and a GNSS antenna installed on an aircraft. They show that a very large radiating system with a band-limit of 2048 can be efficiently analyzed by means of fast spin spherical harmonic transforms, with a computation time of 2 minutes, approximately.

1. Stratton, Julius Adams, Electromagnetic Theory, John Wiley & Sons, 2007.
doi:10.1002/9781119134640

2. Felsen, Leopold B. and Nathan Marcuvitz, Radiation and Scattering of Waves, John Wiley & Sons, 1994.
doi:10.1109/9780470546307

3. Van Bladel, Jean G., Electromagnetic Fields, John Wiley & Sons, 2007.
doi:10.1002/047012458x

4. Hansen, Jesper E., Spherical Near-field Antenna Measurements, Vol. 26, IET, 1988.
doi:10.1007/978-1-4615-6459-1_33

5. Ramamoorthi, Ravi and Pat Hanrahan, "A signal-processing framework for reflection," ACM Transactions on Graphics (TOG), Vol. 23, No. 4, 1004-1042, 2004.
doi:10.1145/1027411.1027416 Google Scholar

6. Choi, Cheol Ho, Joseph Ivanic, Mark S. Gordon, and Klaus Ruedenberg, "Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion," The Journal of Chemical Physics, Vol. 111, No. 19, 8825-8831, 1999.
doi:10.1063/1.480229 Google Scholar

7. Ritchie, David W. and Graham J. L. Kemp, "Fast computation, rotation, and comparison of low resolution spherical harmonic molecular surfaces," Journal of Computational Chemistry, Vol. 20, No. 4, 383-395, 1999.
doi:10.1002/(sici)1096-987x(199903)20:4<383::aid-jcc1>3.0.co;2-m Google Scholar

8. Simons, Frederik J., F. A. Dahlen, and Mark A. Wieczorek, "Spatiospectral concentration on a sphere," SIAM Review, Vol. 48, No. 3, 504-536, 2006.
doi:10.1137/s0036144504445765 Google Scholar

9. Swenson, Sean and John Wahr, "Methods for inferring regional surface-mass anomalies from Gravity Recovery and Climate Experiment (GRACE) measurements of time-variable gravity," Journal of Geophysical Research: Solid Earth, Vol. 107, No. B9, ETG 3-1-ETG 3-13, 2002.
doi:10.1029/2001jb000576 Google Scholar

10. Whaler, K. A., "Downward continuation of Magsat lithospheric anomalies to the Earth's surface," Geophysical Journal International, Vol. 116, No. 2, 267-278, 1994.
doi:10.1111/j.1365-246x.1994.tb01797.x Google Scholar

11. Turcotte, D. L., R. J. Willemann, W. F. Haxby, and John Norberry, "Role of membrane stresses in the support of planetary topography," Journal of Geophysical Research: Solid Earth, Vol. 86, No. B5, 3951-3959, 1981.
doi:10.1029/jb086ib05p03951 Google Scholar

12. Wieczorek, Mark A. and Roger J. Phillips, "Potential anomalies on a sphere: Applications to the thickness of the lunar crust," Journal of Geophysical Research: Planets, Vol. 103, No. E1, 1715-1724, 1998.
doi:10.1029/97je03136 Google Scholar

13. Bennett, C. L., A. J. Banday, K. M. Górski, G. Hinshaw, P. Jackson, P. Keegstra, A. Kogut, G. F. Smoot, D. T. Wilkinson, and E. L. Wright, "Four-year COBE* DMR cosmic microwave background observations: Maps and basic results," The Astrophysical Journal, Vol. 464, No. 1, L1, 1996.
doi:10.1086/310075 Google Scholar

14. Jarosik, N., C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, M. Halpern, R. S. Hill, G. Hinshaw, A. Kogut, E. Komatsu, et al. "Seven-year wilkinson microwave anisotropy probe (WMAP*) observations: Sky maps, systematic errors, and basic results," The Astrophysical Journal Supplement Series, Vol. 192, No. 2, 14, 2011.
doi:10.1088/0067-0049/192/2/14 Google Scholar

15. Newman, E. T. and R. Penrose, "Note on the bondi-metzner-sachs group," Journal of Mathematical Physics, Vol. 7, No. 5, 863-870, 1966.
doi:10.1063/1.1931221 Google Scholar

16. Matute, E. A., "On the vector solutions of Maxwell equations with the spin-weighted spherical harmonics," ArXiv Preprint Physics/0604096, 2006.
doi:10.48550/arXiv.physics/0604096 Google Scholar

17. Torres del Castillo, G. F., "Spin-weighted spherical harmonics and their applications," Revista mexicana de física, Vol. 53, 125-134, 2007. Google Scholar

18. Shannon, C. E., "Communication in the presence of noise," Proceedings of the IRE, Vol. 37, No. 1, 10-21, 1949.
doi:10.1109/jrproc.1949.232969 Google Scholar

19. Kowsky, W. S., "A quadrature formula over the sphere with application to high resolution spherical harmonic analysis," Bulletin géodésique, Vol. 60, 1-14, 1986.
doi:10.1007/BF02519350 Google Scholar

20. Driscoll, J. R. and D. M. Healy, "Computing Fourier transforms and convolutions on the 2-sphere," Advances in Applied Mathematics, Vol. 15, No. 2, 202-250, 1994.
doi:10.1006/aama.1994.1008 Google Scholar

21. McEwen, Jason D. and Yves Wiaux, "A novel sampling theorem on the sphere," IEEE Transactions on Signal Processing, Vol. 59, No. 12, 5876-5887, 2011.
doi:10.1109/tsp.2011.2166394 Google Scholar

22. Mézières, Nicolas, Benjamin Fuchs, Laurent Le Coq, Jean-Marie Lerat, Romain Contreres, and Gwenn Le Fur, "On the application of sparse spherical harmonic expansion for fast antenna far-field measurements," IEEE Antennas and Wireless Propagation Letters, Vol. 19, No. 5, 746-750, 2020.
doi:10.1109/lawp.2020.2978170 Google Scholar

23. Price, Matthew A. and Jason D. McEwen, "Differentiable and accelerated spherical harmonic and Wigner transforms," Journal of Computational Physics, Vol. 510, 113109, 2024.
doi:10.1016/j.jcp.2024.113109 Google Scholar

24. Mézières, Nicolas, Laurent Le Coq, and Benjamin Fuchs, "Phaseless spherical near-field antenna measurements with reduced samplings," IEEE Transactions on Antennas and Propagation, Vol. 71, No. 9, 7447-7456, 2023.
doi:10.1109/tap.2023.3298149 Google Scholar

25. McEwen, Jason D., Boris Leistedt, Martin Büttner, Hiranya V. Peiris, and Yves Wiaux, "Directional spin wavelets on the sphere," ArXiv Preprint ArXiv:1509.06749, 2015.
doi:10.48550/arXiv.1509.06749 Google Scholar

26. Morlaas, Christophe, Bernard Souny, and Alexandre Chabory, "Helical-ring antenna for hemispherical radiation in circular polarization," IEEE Transactions on Antennas and Propagation, Vol. 63, No. 11, 4693-4701, 2015.
doi:10.1109/tap.2015.2479640 Google Scholar

27. Macabiau, Christophe, Carl Milner, Alexandre Chabory, Norbert Suard, Catalina Rodriguez, Mikaël Mabilleau, Jonathan Vuillaume, and Sixtine Hegron, "Nominal bias analysis for ARAIM user," Proceedings of the 2015 International Technical Meeting of The Institute of Navigation, 713-732, Dana Point, CA, USA, Jan. 2015.

28. Quennelle, A., Alexandre Chabory, R. Contreres, G. LeFur, and P. Pouliguen, "Formulation of the measurement problem and application to antenna measurement correction," 2023 International Conference on Electromagnetics in Advanced Applications (ICEAA), 572-572, Venice, Italy, 2023.
doi:10.1109/iceaa57318.2023.10297713

29. Djelloul, E. M., A. Chabory, C. Morlaas, and R. Douvenot, "Closed-form expression for the radiation of an arbitrarily oriented stochastic elementary dipole," International Conference on Electromagnetics in Advanced Applications (ICEAA), Palermo, Italy, 2025.

Dr. Alice Quennelle

Prof. Alexandre Chabory

Dr. Romain Contreres