volume | PIER Journals

Abstract

The translational addition theorems for the spherical scalar and vector wave functions are derived in a novel, unified way based on the simple and well-known concepts of the radiation and incoming wave patterns. This approach makes the derivation simpler and more transparent compared to the previous approaches. As a result, we also obtain alternative and partly simpler expressions for the translation coefficients in the vector case.

1. Bruning, J. H. and Y. T. Lo, "Multiple scattering of EM waves by spheres, Part I — Multipole expansion and ray-optical solutions," IEEE Trans. Antennas Propagat., Vol. 19, No. 3, 378-390, May 1971.
doi:10.1109/TAP.1971.1139944 Google Scholar

2. Kokkorakis, G. C., J. G. Fikioris, and G. Fikioris, "EM field induced in inhomogeneous dielectric spheres by external sources," Progress In Electromagnetics Research Symposium, 275-278, Cambridge, USA, March 26-29 2006.

3. Hansen, J. E. (ed.) "Spherical Near-field Antenna Measurements," Peter Peregrinus Ltd., 1988.

4. Greengard, L. and V. Rokhlin, "A new version of the fast multipole method for the Laplace equation in three dimensions," Acta Numerica, 229-269, 1997. Google Scholar

5. Chew, W. C. , J. M. Jin, E. Michielssen, J. M. Song (eds.) Fast and Efficient Algorithms in Computational Electromagnetics, Artech House, 2001.

6. Gumerov, N. A. and R. Duraiswami, Fast Multipole Methods for the Helmholtz Equation in Three Dimensions, Elsevier, 2004.

7. Cheng, H., W. Y. Crutchfield, Z. Gimbutas, L. F. Greengard, J.F.Ethridge, J. Huang, V. Rokhlin, N. Yarvin, and J. Zhao, "A wideband fast multipole method for the Helmholtz equation in three dimensions," J. Comput. Phys., Vol. 216, 300-325, 2006.
doi:10.1016/j.jcp.2005.12.001 Google Scholar

8. Friedman, B. and J. Russek, "Addition theorems for spherical waves," Quart. Appl. Math., Vol. 12, 13-23, 1954. Google Scholar

9. Danos, M. and L. Maximon, "Multipole matrix elements of the translation operator," J. Math. Phys., Vol. 6, No. 5, 766-778, May 1965.
doi:10.1063/1.1704333 Google Scholar

10. Stein, S., "Addition theorems for spherical wave functions," Quart. Appl. Math., Vol. 19, No. 1, 15-24, 1961. Google Scholar

11. Cruzan, O. R., "Translational addition theorems for spherical vector wave functions," Quart. Appl. Math., Vol. 20, No. 1, 33-40, 1962. Google Scholar

12. Borghese, F., P. Denti, G. Toscano, and O. I. Sindoni, "An addition theorem for vector Helmholtz harmonics," J. Math. Phys., Vol. 21, No. 12, 2754-2755, December 1980.
doi:10.1063/1.527572 Google Scholar

13. Felderhof, B. U. and R. B. Jones, "Addition theorems for spherical wave solutions of the vector Helmholtz equation," J. Math. Phys., Vol. 28, No. 4, 836-839, April 1987.
doi:10.1109/8.7220 Google Scholar

14. Wittmann, R. C., "Spherical wave operators and the translation formulas," IEEE Trans. Antennas Propagat., Vol. 36, No. 8, 1078-1087, August 1988. Google Scholar

15. Chew, W. C., Waves and Fields in Inhomogeneuos Media, IEEE Press, 1995.
doi:10.1163/156939393X00787

16. Chew, W. C. and Y. M. Wang, "Efficient ways to compute the vector addition theorem," Journal of Electromagnetic Waves and Applications, Vol. 7, No. 5, 651-665, 1993.
doi:10.1109/8.542073 Google Scholar

17. Kim, K. T., "The translation formula for vector multipole fields and the recurrence relations of the translation coefficients of scalar and vector multipole fields," IEEE Trans. Antennas Propagat., Vol. 44, No. 11, 1482-1487, November 1996.
doi:10.2528/PIER04040601 Google Scholar

18. Kim, K. T., "Symmetry relations of the translation coefficients of the scalar and vector spherical multipole fields," Progress In Electromagnetics Research, Vol. 48, 45-66, 2004.
doi:10.1109/TAP.2007.910358 Google Scholar

19. Kim, K. T., "Efficient recursive generation of the scalar spherical multipole translation matrix," IEEE Trans. Antenna Propagat., Vol. 55, No. 12, 3484-3494, 2007. Google Scholar

20. Chew, W. C., "Vector addition theorem and its diagonalization," Commun. Comput. Phys., Vol. 3, No. 2, 330-341, February 2008.
doi:10.1137/S0036142902405655 Google Scholar

21. Sarvas, J., "Performing interpolation and anterpolation entirely by fast Fourier transform in the 3-D multilevel fast multipole algorithm," SIAM J. Numer. Anal., Vol. 41, No. 6, 2180-2196, 2003.
doi:10.2528/PIER05021001 Google Scholar

22. Wallen, H. and J. Sarvas, "Translation procedures for broadband MLFMA," Progress In Electromagnetics Research, Vol. 55, 47-78, 2005. Google Scholar

23. Wang, P., Y. J. Xie, and R. Yang, "Novel pre-corrected multilevel fast multipole algorithm for electrical large radiation problem," Journal of Electromagnetic Waves and Applications, Vol. 21, No. 13, 1733-1743, 2007.
doi:10.2529/PIERS060907051636 Google Scholar

24. Wallen, H., "Improved interpolation of evanescent plane waves for Fast Multipole Methods," PIERS Online, Vol. 3, No. 6, 764-766, 2007. Google Scholar

25. Jackson, J. D., Classical Electrodynamics, 3rd Ed., Wiley, 1999.

26. Stratton, J. A., Electromagnetic Theory, McGraw-Hill, 1941.

27. Colton, D. and R. Kress, Inverse Acoustic and Elelctromagnetic Scattering Theory, Springler-Verlag, 1998.
doi:10.1063/1.1666629

28. Devaney, A. J. and E. Wolf, "Multipole expansions and plane wave representations of the electromagnetic field," J. Math. Phys., Vol. 15, No. 2, 234-244, February 1974. Google Scholar

29. Abramowitz, M. and I. A. Stegun (eds.), Handbook of Mathematical Functions, Dover Publications, 1970.
doi:10.1002/1098-2760(20000720)26:2<105::AID-MOP11>3.0.CO;2-W

30. Zhao, J.-S. and W. C. Chew, "Applying matrix rotation to the three-dimensional low-frequency multilevel fast multipole algorithm," Microwave Opt. Technol. Lett., Vol. 26, No. 2, July 2000. Google Scholar

Dr. Tommi Dufva

Dr. Jukka Sarvas

Dr. Johan Sten