PIER
 
Progress In Electromagnetics Research
ISSN: 1070-4698, E-ISSN: 1559-8985
Home | Search | Notification | Authors | Submission | PIERS Home | EM Academy
Home > Vol. 127 > pp. 445-459

COMPLEX POINT SOURCE FOR THE 3D LAPLACE OPERATOR

By M.-J. Gonzalez-Morales, R. Mahillo-Isla, C. Dehesa-Martinez, and E. Gago-Ribas

Full Article PDF (258 KB)

Abstract:
The research about the so-called \emph{complex beams}, localized solutions of the Helmholtz wave equation, lead to the problem of finding the sources of such solutions, which may be formally expressed as a Dirac delta function of a complex argument. To investigate about the meaning of the Dirac delta distribution of complex argument, the Green's function of the 3D Poisson problem with a point source localized at an imaginary position in free space is considered. The main physical features of the potential created by that source are described. The inverse problem consists in looking for the real source distribution which causes that potential. The sources appear on a disk in the real space. Their physical interpretation requires a regularization process based on including the border of the disk.

Citation:
M.-J. Gonzalez-Morales, R. Mahillo-Isla, C. Dehesa-Martinez, and E. Gago-Ribas, "Complex point source for the 3D laplace operator," Progress In Electromagnetics Research, Vol. 127, 445-459, 2012.
doi:10.2528/PIER12032305
http://www.jpier.org/pier/pier.php?paper=12032305

References:
1. Appell, P. E., "Quelques remarques sur la théorie des potentiels multiformes," Mathematische Annalen, Vol. 30, 155-156, 1887.
doi:10.1007/BF01564536

2. Arnaud, J. A. and H. Kogelnik, "Gaussian light beams with general astigmatism," Applied Optics, Vol. 8, No. 8, 1687-1694, 1969.
doi:10.1364/AO.8.001687

3. Izmest'ev, A. A., "Wave fields of beam type and spatial quantization of the angular momentum," Theoretical and Mathematical Physics, Vol. 7, No. 3, 591-599, 1971.
doi:10.1007/BF01032079

4. Deschamps, G. A., "Gaussian beam as a bundle of complex rays," Electron. Lett., Vol. 7, No. 23, 684-685, 1971.
doi:10.1049/el:19710467

5. Felsen, L. B., "Complex-source-point solutions of the field equations and their relations to the propagation and scattering of Gaussian beams," Symp. Math., Vol. 18, 39-56, 1976.

6. Lumori, M. L., "Gaussian beam modeling of SAR enhancement in paraxial and non-paraxial regions of biological tissues," Progress In Electromagnetics Research M, Vol. 11, No. 1-12, 2010.

7. Varaulta, S., S. Boliolib, and J. Sokoloffc, "Scattering by an array of rods using the gaussian beam formalism coupled to the scattering matrix method," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 8-9, 1131-1145, 2011.
doi:10.1163/156939311795762169

8. Menga, H. F., W. B. Doub, and J. L. Zhangc, "Generation of mathieu beams in millimeter wave band using diffractive elements," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 16, 2296-2307, 2011.
doi:10.1163/156939311798146999

9. Zhang, T.-L., Z.-H. Yan, F.-F. Fan, and B. Li, "Design of a ku-band compact corrugated horn with high gaussian beam effciency," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 1, 123-129, 2011.
doi:10.1163/156939311793898297

10. Lim, S.-H., J.-H. Han, S.-Y. Kim, and N.-H. Myung, "Azimuth beam pattern synthesis for airborne SAR system optimization," Progress In Electromagnetics Research, Vol. 106, 295-309, 2010.
doi:10.2528/PIER10061901

11. Mokhtari, A. and A. A. Shishegar, "Rigorous 3D vectorial gaussian beam modeling of demultiplexing performance of virtually-imaged-phased-arrays," Progress In Electromagnetics Research M, Vol. 13, 1-6, 2010.
doi:10.2528/PIERM10041604

12. Gago-Ribas, E., M. J. González-Morales, and C. Dehesa-Martínez, "Analytical parametrization of a 2D real propagation space in terms of complex electromagnetic beams," IEICE Trans. on Electronics, Vol. E80-C, No. 11, 1434-1439, 1997.

13. González-Morales, M. J., C. Dehesa-Martínez, and E. GagoRibas, "About complex extensions and their application in electromagnetics," Springer Proceedings in Physics, Vol. 104, 81-86, Springer,2006.
doi:10.1007/3-540-30636-6_8

14. Mahillo-Isla, R. and M. J. González-Morales, "Plane wave spectrum of 2D complex beams," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 8-9, 1123-1131, 2009.

15. González-Morales, M. J., R. Mahillo-Isla, E. Gago-Ribas, and C. Dehesa-Martínez, "3D complex beams in the spatial and the spectral domains," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 8-9, 1103-1112, 2010.
doi:10.1163/156939310791586124

16. González-Morales, M. J., R. Mahillo-Isla, E. Gago-Ribas, and C. Dehesa-Martínez, "Complex polar coordinates in electromagnetics," Journal of Electromagnetic Waves and Applications, Vol. 25, No. 2-3, 389-398, 2011.
doi:10.1163/156939311794362795

17. Kaiser, G., "Complex-distance potential theory, wave equations,and physical wavelets," Mathematical Methods in the Applied Sciences, Special Issue on Clifford Analysis in Applications, Vol. 25, 1577-1588, F. Sommen and W. Sproessig, Editors, 2002.

18. Kaiser, G., "Physical wavelets and their sources: Real physics in complex spacetime," Topical Review, Journal of Physics A:Mathematical and General, Vol. 36, No. 30, 291-338, 2003.
doi:10.1088/0305-4470/36/30/201

19. Mahillo-Isla, R., M. J. González-Morales, and C. DehesaMartínez, "Regularization of complex beams," 12th International Conference on Mathematical Methods in EM Theory, 242-244, 2008.

20. Tagirdzhanov, A. M., A. S. Blagovestchenskii, and A. P. Kiselev, "Complex source: Singularities in real space," Progress In Electromagnetics Research Symposium Proceedings, 1527-1530, Moscow,2009.

21. Tagirdzhanov, A. M., A. S. Blagovestchenskii, and A. P. Kiselev, "Complex source wavefields: Sources in real space," J. Phys. A:Math. Theor., Vol. 44, 2011.

22. Gleiser, R. J. and J. A. Pullin, "Appell rings in general relativity," Class. Quantum Grav., 977-985, 1988.

23. Gel'fand, I. B. and G. E. Shilov, Generalized Functions, Academic Press, Inc., 1964.


© Copyright 2014 EMW Publishing. All Rights Reserved