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Progress In Electromagnetics Research
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SURFACE GREEN'S FUNCTION OF THE HELMHOLTZ EQUATION IN SPHERICAL COORDINATES

By T. Rother, M. Kahnert, A. Doicu, and J. Wauer

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Abstract:
The surface Green's function belonging to the non-spherical exterior boundary value problem of Helmholtz's equation in spherical coordinates is derived. This is performed in two ways, first by applying the Separation of Variables method, and, second, by using the Method of Lines as a special Finite-Difference technique. With this Green's function we are able to resolve some contradictions concerning conceptual aspects of the Separation of Variables method, the Finite-Difference methods, and the Boundary Integral Equation methods which have been developed for rigorously solving non-separable boundary value problems. The necessary mathematical background, the relation to Waterman's T matrix, and simplifications due to certain symmetry properties of the boundary surface will be discussed. In this paper we focus on the scalar problem. The extension to the vector case for electromagnetic wave scattering is in preparation and will be published later.

Citation:
T. Rother, M. Kahnert, A. Doicu, and J. Wauer, "Surface Green's Function of the Helmholtz Equation in Spherical Coordinates," Progress In Electromagnetics Research, Vol. 38, 47-95, 2002.
doi:10.2528/PIER02091902
http://www.jpier.org/PIER/pier.php?paper=0209192

References:
1. Rother, T. and K. Schmidt, "The discretized Mie-formalism for electromagnetic scattering," Progress in Electromagnetic Research, J. A. Kong (ed.), 91–183, EMW Publishing, Cambridge, MA, 1997.

2. Taflove, A., Computational Electrodynamics — The Finite-Difference Time-Domain Method, Artech House, Boston, 1995.

3. Tsang, L., J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing, John Wiley & Songs, New York, 1995.

4. Rother, T., "General aspects of solving Helmholtz’s equation underlying eigenvalue and scattering problems in electromagnetic wave theory," J. Electromagn. Waves Appl., Vol. 13, 867-888, 1999.
doi:10.1163/156939399X00330

5. Lippmann, B. A., "Note on the theory of gratings," J. Opt. Soc. Amer., Vol. 43, 408, 1953.
doi:10.1364/JOSA.43.000408

6. Millar, R. F., "The Rayleigh hypothesis and a related least-squares solution to scattering problems for periodic surfaces and other scatterers," Radio Sci., Vol. 8, 785-796, 1973.
doi:10.1029/RS008i008p00785

7. Morse, P. M. and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill Book Company, Inc., 1953.

8. Schulz, F. M., K. Stamnes, and J. J. Stamnes, "Point group symmetries in electromagnetic scattering," J. Opt. Soc. Am. A, Vol. 16, 853-865, 1999.
doi:10.1364/JOSAA.16.000853

9. Zagorodnov, I. A. and R. P. Tarasov, "Finite groups in numerical solution of electromagnetic scattering problems on non-spherical particles," Light Scattering by Nonspherical Particles: Halifax Contributions, 99-102, Army Research Laboratory, Adelphi, MD, 2000.

10. Kahnert, F. M., J. J. Stamnes, and K. Stamnes, "Application of the extended boundary condition method to homogeneous particles with point group symmetries," Appl. Opt., Vol. 40, 3110-3123, 2001.
doi:10.1364/AO.40.003110

11. Rother, T., K. Schmidt, and S. Havemann, "Light scattering on hexagonal ice columns," J. Opt. Soc. Am. A, Vol. 18, 2512-2517, 2001.
doi:10.1364/JOSAA.18.002512

12. Waterman, P. C., "Matrix formulation of electromagnetic scattering," Proc. IEEE, Vol. 53, 805-812, 1965.
doi:10.1109/PROC.1965.4058

13. Kleinman, R. E., G. F. Roach, and S. E. G. Strom, "The null field method and modified Green function," Proc. R. Soc. Lond. A, Vol. 394, 121-136, 1984.
doi:10.1098/rspa.1984.0072

14. Ramm, A. G., Scattering by Obstacles, D. Reidel, Dordrecht, 1986.
doi:10.1007/978-94-009-4544-9

15. Tai, C.-T., Dyadic Green Functions in Electromagnetic Theory, IEEE Press, Piscataway, 1993.

16. Sommerfeld, A., Partial Differential Equations in Physics, Academic Press, New York, 1949.

17. Waterman, P. C., "Symmetry, unitarity, and geometry in electromagnetic scattering," Phys. Rev. D, Vol. 3, 825-839, 1971.
doi:10.1103/PhysRevD.3.825

18. Colton, D. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer Verlag, Berlin, 1992.
doi:10.1007/978-3-662-02835-3

19. Doicu, A., Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Analysis Using Discrete Sources, Academic Press, New York, 2000.

20. Dallas, A. G., "On the convergence and numerical stability of the second waterman scheme for approximation of the acoustic field scattered by a hard object,", Technical Report, No. 2000-7, 1–35, Dept. of Mathematical Sciences, Univ. of Deleware, 2000.

21. Schmidt, K., T. Rother, and J. Wauer, "The equivalence of applying the extended boundary condition and the continuity conditions for solving electromagnetic scattering problems," Optics Comm., Vol. 150, 1-4, 1998.
doi:10.1016/S0030-4018(98)00113-8

22. Pregla, R. and W. Pascher, "The methods of lines," Numerical Techniques for Microwave and Millimeter Wave Passive Structures, T. Itoh (ed.), 381–446, Wiley, New York, 1989.

23. Wilkinson, J. H., The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, 1965.

24. Pregla, R., "About the nature of the method of lines," Arch. Elektr. Bertragungstech., Vol. 41, 370-386, 1987.

25. Pregla, R., "Higher order approximations for the difference operator in the method of lines," IEEE Microwave and Guided Wave Letters, Vol. 5, 53-55, 1995.
doi:10.1109/75.342150

26. Dreher, A. and T. Rother, "New aspects of the method of lines," IEEE Microwave and Guided Wave Letters, Vol. 5, 408-410, 1995.
doi:10.1109/75.473526

27. Bishop, D. M., Group Theory and Chemistry, Dover Publications, Mineola, 1993.

28. Zakharov, E. V., S. I. Safronov, and R. P. Tarasov, "Finite-order abelian groups in the numerical analysis of linear boundary-value problems of potential theory," Comput. Maths. Math. Phys., Vol. 32, 34-50, 1992.

29. Zakharov, E. V., S. I. Safronov, and R. P. Tarasov, "Finite group algebras in iterational methods of solving boundary-value problems of potential theory," Comput. Maths. Math. Phys., Vol. 33, 907-917, 1993.

30. Kahnert, F. M., J. J. Stamnes, and K. Stamnes, "Can simple particle shapes be used to model scalar optical properties of an ensemble of wavelength-sized particles with complex shapes?," J. Opt. Soc. Am. A, Vol. 19, 521-531, 2002.
doi:10.1364/JOSAA.19.000521

31. Mishchenko, M. I., "Light scattering by randomly oriented axially symmetric particles," J. Opt. Soc. Am. A., Vol. 8, 871-882, 1991.
doi:10.1364/JOSAA.8.000871

32. Khlebtsov, N. G., "Orientational averaging of light-scattering observables in the T-matrix approach," Appl. Opt., Vol. 31, 5359-5365, 1992.
doi:10.1364/AO.31.005359

33. Mackowski, D. W. and M. I. Mishchenko, "Calculation of the T matrix and the scattering matrix for ensembles of spheres," J. Opt. Soc. Am. A, Vol. 13, 2266-2278, 1996.
doi:10.1364/JOSAA.13.002266

34. Kahnert, F. M., J. J. Stamnes, and K. Stamnes, "Using simple particle shapes to model the Stokes scattering matrix of ensembles of wavelength-sized particles with complex shapes: possibilities and limitations," J. Quant. Spectrosc. Radiat. Transfer, Vol. 74, 167-182, 2002.
doi:10.1016/S0022-4073(01)00194-7

35. Schulz, F. M., K. Stamnes, and J. J. Stamnes, "Scattering of electromagnetic waves by spheroidal particles: A novel approach exploiting the T-matrix computed in spheroidal coordinates," Appl. Opt., Vol. 37, 7875-7896, 1998.
doi:10.1364/AO.37.007875

36. Kahnert, F. M., J. J. Stamnes, and K. Stamnes, "Surface-integral formulation for electromagnetic scattering in spheroidal coordinates," J. Quant. Spectrosc. Radiat. Transfer, in press.


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