By using field expansion in terms of the Legendre polynomials and Schelkunoff functions, Maxwell's equations in the spherical coordinate system are cast into a matrix form which lends itself to the analysis of a multilayer conical waveguide. The matrix formulation is then used to obtain an eigen-value problem whose eigen-values are the allowable wave-numbers for propagation in the radial direction. To verify the proposed numerical approach, it is used to evaluate the resonance frequency of a partially filled spherical resonator. The computed resonance frequencies are then compared with those obtained using commercial software based on the finite-element method. The computation time is enormously reduced using the semianalytical method of this work. Although results are shown for lossless isotropic dielectrics, the method is also applicable to conical waveguides made of lossy dielectrics even with negative permittivity.
1. Sakai, J. I. and E. A. J. Marcatili, "Lossless dielectric tapers with three-dimensional geometry," J. Lightw. Technol., Vol. 9, No. 3, 386-393, 1991. doi:10.1109/50.70017
2. Romanova, E. R., L. A. Melnikov, and E. V. Bekker, "Numerical analysis of total field propagation in linear and nonlinear singlemode tapered fibers," International Conference on Transparent Optical Networks, 161-164, 1999.
3. Paradasi, A., H. E. Hernandez-Figueroa, S. Celaschi, M. L. Brandao, T. Jesus, and M. Mercadante, "Beam propagation method modelling of optical filters based on tapered fibres," IEEE Lasers and Electro-optics Society Annual Meeting, LEOS 96, Vol. 2, 36-37, 1996. doi:10.1109/LEOS.1996.571536
4. Moar, P. N., S. T. Huntingtont, J. Katsifolis, L. W. Cahill, and K. A. N. Roberts, "A near field modelling and measurement of tapered optical fibre devices," IEEE WG5, 54-55, 1997.
6. Srituravanich, W., N. Fang, C. Sun, Q. Luo, and X. Zhang, "Plasmonic nanolithography," Nano Lett., Vol. 4, 1085, 2004. doi:10.1021/nl049573q
7. Liu, Z. W., Q. G. Wei, and X. Zhang, "Surface plasmon interference nanolithography," Nano Lett., Vol. 5, 957, 2005. doi:10.1021/nl0506094
8. Kneipp, K., Y. Wang, H. Kneipp, L. T. Perelman, I. Itzkan, R. R. Dasari, and M. S. Feld, "Single molecule detection using surface-enhanced Raman scattering (SERS)," Phys. Rev. Lett., Vol. 78, 1667, 1997. doi:10.1103/PhysRevLett.78.1667
9. Pettinger, B., B. Ren, G. Picardi, R. Schuster, and G. Ertl, "Nanoscale probing of adsorbed species by tip enhanced Raman spectroscopy," Phys. Rev. Lett., Vol. 92, 096101, 2004. doi:10.1103/PhysRevLett.92.096101
10. Ichimaru, T., N. Hayazawa, M. Hashimoto, Y. Inouye, and S. Kawata, "Tip-enhanced coherent anti-Stokes Raman scattering for vibrational nanoimaging," Phys. Rev. Lett., Vol. 92, 220801, 2004. doi:10.1103/PhysRevLett.92.220801
11. De, A. and G. V. Attimarad, "Numerical analysis of two dimensional tapered dielectric waveguide," Progress In Electromagnetics Research, Vol. 44, 131-142, 2004. doi:10.2528/PIER03062001
12. Siong, C. C. and P. K. Choudhury, "Propagation characteristics of tapered core helical cald dielectric optical fibers," Journal of Electromagnetic Waves and Applications, Vol. 23, 663-674, 2009. doi:10.1163/156939309788019877
13. Jin, X., "Optical fiber index taper-theoretical analysis and experimental demonstration," Int. J. Infrared Millimet. Waves, Vol. 19, No. 6, 875-886, 1998. doi:10.1023/A:1022680608509
15. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite Difference Time-domain Method, 3 Ed., Norwood Artech House Publishers, 2005.
16. Khajehpour, A. and S. A. Mirtaheri, "Analysis of pyramid EM wave absorber by FDTD method and comparing with capacitance and homogenization methods," Progress In Electromagnetics Research Letters, Vol. 3, 123-131, 2008. doi:10.2528/PIERL08021802
17. Kurihara, K., A. Otomo, A. Syouji, J. Takahara, K. Suzuki, and S. Yokoyama, "Superfocusing modes of surface plasmon polaritons in conical geometry based on the quasi-separation of variables approach," J. Phys. A: Math. Theor., Vol. 40, 12479-12503, 2007. doi:10.1088/1751-8113/40/41/015
18. Ruppin, R., "Scattering of electromagnetic radiation by a coated perfect electromagnetic conductor sphere," Progress In Electromagnetics Research Letters, Vol. 8, 53-62, 2009. doi:10.2528/PIERL09041502
19. Issa, N. A. and R. Guckenberger, "Optical nanofocusing on tapered metallic waveguides," Spr. Plas., Vol. 2, 31-37, 2007.
20. Harrington, R. F., Time-harmonic Electromagnetic Fields, IEEE Press Series on Electromagnetic Wave Theory, 2001.
21. Schelkunoff, S. A., Electromagnetic Waves, D. Van Nostrand Company, Inc., Princeton, N.J., 1943.
22. Lindell, I. V. and A. H. Sihvola, "Spherical resonator with db-boundary conditions," Progress In Electromagnetics Research Letters, Vol. 6, 131-137, 2009. doi:10.2528/PIERL09010505
23. Adams, J. C., "On the expression of the product of any two Legendre's coe±cients by means of a series of Legendre's coe±cients," Proc. R. Soc., Vol. 27, 63-71, 1878. doi:10.1098/rspl.1878.0016