volume | PIER Journals

Abstract

The di®raction of a plane electromagnetic wave by an ideal metallic sphere (Mie's theory) is investigated by a new method. The method represents the charge disturbances (polarization) by a displacement field in the positions of the mobile charges (electrons) and uses the equation of motion for the polarization together with the electromagnetic potentials. We employ a special set of orthogonal functions, which are combinations of spherical Bessel functions and vector spherical harmonics. This way, we obtain coupled integral equations for the displacement field, which we solve. In the non-retarded limit (Coulomb interaction) we get the branch of "spherical" (surface) plasmons at frequencies ω = ω_psqrt(l/(2(l/ + 1)), where ω_p is the (bulk) plasma frequency and l = 1, 2,.... When retardation is included, for an incident plane wave, we compute the field inside and outside the sphere (the scattered field), the corresponding energy stored by these fields, Poynting vector and scattering cross-section. The results agree with the so-called theory of "effective medium permittivity", although we do not start the calculations with the dielectric function. In turn, we recover in our results the well-known dielectric function of metals. We have checked the continuity of the tangential components of the electric field and continuity of the normal component of the electric displacement at the sphere surface, as well as the conservation of the energy flow and re-derived the "optical theorem". In the limit of small radii (in comparison with the electromagnetic wavelength) the sphere exhibits a series of resonant absorptions at frequencies close to the plasmon frequencies given above. For large radii these resonances disappear.

1. Mie, G., "Beitrage zur optik truber medien, speziell kolloidaler metallosungen," Ann. Physik, Vol. 25, 377-445, 1908.
doi:10.1002/andp.19083300302 Google Scholar

2. Van De Hulst, H. C., Light Scattering by Small Particles, Wiley, 1957.

3. Doyle, W. T. and A. Agarwal, "Optical extinction of metal spheres," J. Opt. Soc. Am., No. 55, 305-309, 1965.
doi:10.1364/JOSA.55.000305 Google Scholar

4. Crowell, J. and R. H. Ritchie, "Radiation decay of Coulomb-stimulated plasmons in spheres," Phys. Rev., Vol. 172, 436-440, 1968.
doi:10.1103/PhysRev.172.436 Google Scholar

5. Ashkin, A. and J. M. Dziedzic, "Observation of resonances in the radiation pressure on dielectric spheres," Phy. Rev. Lett., Vol. 38, 1351-1355, 1977.
doi:10.1103/PhysRevLett.38.1351 Google Scholar

6. Chylek, P., J. T. Kiehl, and M. K. W. Ko, "Narrow resonance structures in the Mie scattering characteristics," Appl. Optics, Vol. 17, 3019-3021, 1978.
doi:10.1364/AO.17.003019 Google Scholar

7. Conwell, P. R., P. W. Barber, and C. K. Rushforth, "Resonant spectra of dielectric spheres," J. Opt. Soc. Am., Vol. A1, 62-67, 1984.
doi:10.1364/JOSAA.1.000062 Google Scholar

8. Marston, P. L. and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev., Vol. A30, 2508-2516, 1984. Google Scholar

9. Chang, S., J. T. Kim, J. H. Jo, and S. S. Lee, "Optical force on a sphere caused by the evanescent field of a Gaussian beam; Effects of multiple scattering," Optics Commun., Vol. 139, 252-261, 1997.
doi:10.1016/S0030-4018(97)00144-2 Google Scholar

10. Ruppin, R., "Optical properties of small metallic spheres," Phys. Rev., Vol. B11, 2871-2876, 1975. Google Scholar

11. Messinger, B. J., K. U. Von Raben, R. K. Chang, and P. W. Barber, "Local fields at the surface of noble-metal microspheres," Phys. Rev., Vol. B24, 649-657, 1981. Google Scholar

12. Arnold, S., A. B. Pluchino, and K. M. Leung, "Influence of surface-mode-enhanced local fields on photophoresis," Phys. Rev., Vol. A29, 654-660, 1984. Google Scholar

13. Brechignac, C., P. Cahuzac, J. Leygnier, and A Sarfati, "Optical response of large lithium clusters: Evolution toward the bulk," Phys. Rev. Lett., Vol. 70, 2036-2039, 1993.
doi:10.1103/PhysRevLett.70.2036 Google Scholar

14. Markowicz, P., K. Kolwas, and M. Kolwas, "Experimental determination of free-electron plasma damping rate in large sodium clusters," Phys. Lett., Vol. A236, 543-547, 1997. Google Scholar

15. Brechignac, C., P. Cahuzac, N. Kebaili, J. Leygnier, and H. Yoshida, "Interband effect in the optical response of strontium clusters," Phys. Rev., Vol. B61, 7280-7283, 2000. Google Scholar

16. Apostol, M. and G. Vaman, "Electromagnetic field interacting with a semi-infinite plasma," J. Opt. Soc. Am., Vol. A26, 1747-1753, 2009.
doi:10.1364/JOSAA.26.001747 Google Scholar

17. Born, M. and E. Wolf, Principles of Optics, Pergamon, 1959.

18. Lorentz, H. A., The Theory of Electrons, Leipzig, 1916.

19. Apostol, M. and G. Vaman, "Plasmons and polaritons in a semi-infinite plasma and a plasma slab," Physica B, in print. Google Scholar

20. Apostol, M. and G. Vaman, "Electromagnetic eigenmodes in matter. van der Waals-London and Casimir forces," J. Theor. Phys., Vol. 177, 1-12, 2009. Google Scholar

21. Edmonds, A. R., Angular Momentum in Quantum Mechanics, Princeton, 1957.

22. Blatt, J. M. and V. E. Weisskopf, Theoretical Nuclear Physics, Dover, 1991.

23. Erdelyi, A., Higher Transcendental Functions, Vol. 2, Bateman Project, McGraw-Hill, 1957.

24. Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1972.

25. Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series and Products, 930, 8.533, Academic Press, New York, 2000.

Professor Marian Apostol

Dr. Georgeta Vaman