volume | PIER Journals

Abstract

We investigate the problem of defining propagating constants and modes in metallic waveguides of an arbitrary cross section, filled with a homogeneous bi-isotropic material. The approach follows the guidelines of the classical theory for the isotropic, homogeneous, lossless waveguide: starting with the Maxwell system, we formulate a spectral problem where the square of the propagation constant shows up as the eigenvalue and the corresponding mode as the eigenvector. The difficulty that arises, and this is a feature of chirality, is that the eigenvalue is involved in the boundary conditions. The main result is that the problem is solvable whenever the Dirichlet problem for the Helmholtz equation in the cross section is solvable and a technical hypothesis is fulfilled. Our method, inspired by the null-field method, is quite general and has a good potential to work in various geometries.

1. Lakhtakia, A., Beltrami Fields in Chiral Media, World Scientic, Singapore, 1994.

2. Lindell, I. V., A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media, Artech House, Norwood, 1994.

3. Stratis, I. G., "Electromagnetic scattering problems in chiral media: A review," Electromagnetics, Vol. 19, 547-562, 1999.
doi:10.1080/02726349908908673

4. Olyslager, F., Electromagnetic Waveguides and Transmission Lines, Clarendon Press, Oxford, 1999.

5. Rayleigh, L., "On the passage of electric waves through tubes, or the vibrations of dielectric cylinders," Phil. Mag., Vol. 43, 125-132, 1897.

6. Collin, R. E., Field Theory of Guided Waves, 2nd edition, IEEE Press, New York, 1991.

7. Waldron, R. A., The Theory of Waveguides and Cavities, Maclaren & Sons, London, 1967.

8. Yang , T., S. Song, H. Dong, and R. Ba, "Waveguide structures for generation of terahertz radiation by electro-optical process in GaAs and zugep2 using 1.55 μm fiber laser pulses," Progress In Electromagnetics Research Letters, Vol. 2, 95-102, 2008.
doi:10.2528/PIERL07122806

9. Jarem, J. M., "Propagation, excitation, and orthogonality of modes in a parallel plate, anisotropic waveguide using a modified, coordinate transformation," Progress In Electromagnetics Research B, Vol. 15, 151-173, 2009.
doi:10.2528/PIERB08111005

10. Xu, J., W. X. Wang, L. N. Yue, Y. B. Gong, and Y. Y. Wei, "Electromagnetic wave propagation in an elliptical chiroferrite waveguide," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 14-15, 2021-2030, 2009.
doi:10.1163/156939309789932430

11. Eftimiu, C. and L. W. Pearson, "Guided electromagnetic waves in chiral media," Radio Sci., Vol. 24, No. 3, 351-359, 1989.
doi:10.1029/RS024i003p00351

12. Hollinger, R., V. V. Varadan, and V. K. Varadan, "Eigenmodes in circular waveguide containing an isotropic chiral material," Radio Sci., Vol. 26, No. 5, 1335-1344, 1991.
doi:10.1029/91RS00962

13. Pelet, P. and N. Engheta, "The theory of chirowaveguides," IEEE Trans. Antennas and Propagation, Vol. 38, No. 1, 90-98, 1990.
doi:10.1109/8.43593

14. Svedin, J. A. M., "Propagation analysis of chirowaveguides using the finite element method," IEEE Trans. Microwave Theory Tech., Vol. 38, No. 10, 1488-1496, 1990.
doi:10.1109/22.58690

15. Cessenat, M., Mathematical Methods in Electromagnetism: Linear Theory and Applications, World Scientific, Singapore, 1996.

16. Dautray, R. and J.-L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 3: Spectral Theory and Applications," Springer, 1990.

17. Jackson, J. D., Classical Electrodynamics, 3rd edition, Wiley, New Jersey, 1999.

18. 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.

19. Martin, P. A., "On the null-field for water-wave scattering problems," IMA J. Appl. Math., Vol. 33, 55-69, 1984.
doi:10.1093/imamat/33.1.55

20. Bohren, C. F., "Light scattering by an optically active sphere," Chem. Phys. Letters, Vol. 29, 458-462, 1974.
doi:10.1016/0009-2614(74)85144-4

21. Athanasiadis, C., G. Costakis, and I. G. Stratis, "On some properties of Beltrami fields in chiral media," Rep. Math. Phys., Vol. 45, 257-271, 2000.
doi:10.1016/S0034-4877(00)89036-9

22. Reinert, J., G. Busse, and A. F. Jacob, "Waveguide characterization of chiral material: Theory," IEEE Trans. Microwave Theory Tech., Vol. 43, No. 3, 290-296, 1999.
doi:10.1109/22.750227

23. Sjoberg, D., "Determination of propagation constants and material data from waveguide measurements," Progress In Electromagnetics Research B, Vol. 12, 163-182, 2009.
doi:10.2528/PIERB08121304

24. Ioannidis, A. D., D. Sjoberg, and G. Kristensson, "On the propagation problem in a metallic homogeneous bi-isotropic waveguide,", Department of Electrical and Information Technology-Lund University, TEAT-7178, 1-19, 2009. www.eit.lth.se.

25. Colton, D., L. Paivarinta, and J. Sylvester, "The interior transmission problem," Inverse Probl. Imaging, Vol. 1, 13-28, 2007.

26. Colton, D. and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983.

27. Evans, L. C., "Partial Differential Equations," American Mathematical Society, 1998.

28. Gohberg, I. C. and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, 1969.

29. 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

30. Millar, R. F., "On the completeness of sets of solutions to the Helmholtz equation," IMA J. Appl. Math., Vol. 30, 27-37, 1983.
doi:10.1093/imamat/30.1.27

31. Condon, E. U., "Theories of optical rotatory power," Rev. Mod. Phys., Vol. 9, 432-457, 1937.
doi:10.1103/RevModPhys.9.432

Dr. Andreas D. Ioannidis

Professor Gerhard Kristensson

Dr. Daniel Sjöberg