Nonlinear propagation of leaky TE-polarized electromagnetic waves in a metamaterial Goubau line
Abstract
Propagation of leaky TE-polarized electromagnetic waves in the Goubau line (a perfectly conducting cylinder covered by a concentric dielectric layer) filled with nonlinear metamaterial medium is studied. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered.
Keyword : leaky TE waves, metamaterial Goubau Line, nonlinear permittivity, nonlinear eigenvalue problem, numerical method
How to Cite
Smolkin, E., & Smirnov, Y. (2021). Nonlinear propagation of leaky TE-polarized electromagnetic waves in a metamaterial Goubau line. Mathematical Modelling and Analysis, 26(3), 372-382. https://doi.org/10.3846/mma.2021.13077
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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Yu. Smirnov, E. Smolkin and D. Valovik. Nonlinear double-layer bragg waveguide: Analytical and numerical approaches to investigate waveguiding problem. Advances in Numerical Analysis, 2014:1–11, 2014. https://doi.org/10.1155/2014/231498
E. Smolkin. Propagation of TE waves in a double-layer nonlinear inhomogeneous cylindrical waveguide. In Days on Diffraction’2014 Proceedings, pp. 204–209, Saint Petersburg, Russia, 2014. IEEE. https://doi.org/10.1109/DD.2014.7036451
E. Smolkin. Goubau line filled with nonlinear medium: Numerical study of TMpolarized waves. In Proceedings of the 2015 17th International Conference on Electromagnetics in Advanced Applications, pp. 1572–1575, Turin, Italy, 2015. IEEE. https://doi.org/10.1109/ICEAA.2015.7297390
E. Smolkin. The azimuthal symmetric hybrid waves in nonlinear cylindrical waveguide. In Proceedings of the 2017 Progress in Electromagnetics Research Symposium, PIERS 2017, pp. 348–353, Saint Petersburg, Russia, 2017. IEEE. https://doi.org/10.1109/PIERS.2017.8261763
E. Smolkin and Yu. Shestopalov. Nonlinear Goubau line: analyticalnumerical approaches and new propagation regimes. Journal of Electromagnetic Waves and Applications, 31(8):781–797, 2017. https://doi.org/10.1080/09205071.2017.1317036
E. Smolkin, Yu. Shestopalov and M. Snegur. Diffraction of TM polarized electromagnetic waves by a nonlinear inhomogeneous metal-dielectric waveguide. In Proceedings of the 2017 19th International Conference on Electromagnetics in Advanced Applications, ICEAA 2017, pp. 1288–1291, Verona Italy, 2017. IEEE.
E. Smolkin and D. Valovik. Numerical solution of the problem of propagation of TM-polarized electromagnetic waves in a nonlinear two-layered dielectric cylindrical waveguide. In Mathematical Methods in Electromagnetic Theory (MMET), 2012 International Conference on, pp. 68–71, Kharkiv, Ukraine, 2012. IEEE. https://doi.org/10.1109/MMET.2012.6331288
E. Smolkin and D. Valovik. Calculation of the propagation constants of inhomogeneous nonlinear double-layer circular cylindrical waveguide by means of the cauchy problem method. Journal of Communications Technology and Electronics, 58:762–769, 2013. https://doi.org/10.1134/S1064226913060132
E. Smolkin and D. Valovik. Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity. Advances in Mathematical Physics, 2015, 2015. https://doi.org/10.1155/2015/614976
E. Smolkin and D. Valovik. Nonlinear propagation of coupled electromagnetic waves in a circular cylindrical waveguide. Computational Mathematics and Mathematical Physics, 57(8):1294–1309, 2017. https://doi.org/10.1134/S0965542517080127
A.W. Snyder and J. Love. Optical waveguide theory. Springer, 1983.
A. Sommerfeld. Uber die fortpflanzung elektrodynamischer wellen la¨ngs eines drahtes. Annals of Physics, 303(2):233–290, 1899. https://doi.org/10.1002/andp.18993030202
L.A. Vainstein. Electromagnetic Waves. Radio i svyaz, Moscow, 1988.
V.G. Veselago. The electrodynamics of substances with simultaneously negative values of ε and µ. Soviet Physics Uspekhi, 10(4):509–514, 1968. https://doi.org/10.1070/PU1968v010n04ABEH003699
R.W. Ziolkowski and E.Heyman. Wave propagation in media having negative permittivity and permeability. PHYSICAL REVIEW E, 64(5), 2001. https://doi.org/10.1103/PhysRevE.64.056625
M.J. Adams. An Introduction to Optical Waveguides. New York: Wiley, 1951.
N.N. Akhmediev and A.Ankevich. Solitons, Nonlinear Pulses and Beams. Chapman and Hall, London, 1997.
G. Goubau. Surface waves and their application to transmission lines. Journal of Applied Physics, 21(11):1119–1128, 1950. https://doi.org/10.1063/1.1699553
G. Goubau. Open wire lines. IRE Trans. Microwave Theory and Technique, 4(4):197–200, 1956. https://doi.org/10.1109/TMTT.1956.1125062
F. Harms. Elektromagnetische wellen an einem draht mit isolierender zylindrischer hu¨lle. Annals of Physics, 328(6):44–60, 1907. https://doi.org/10.1002/andp.19073280603
L.D. Landau, E.M. Lifshitz and L.P. Pitaevskii. Course of Theoretical Physics. Electrodynamics of Continuous Media. Butterworth-Heinemann, Oxford, 1993.
N. Marcuvitz. On field representations in terms of leaky modes or eigenmodes. IRE Transactions on Antennas and Propagation, 4(3):192–194, 1956. https://doi.org/10.1109/TAP.1956.1144410
F. Monticone and A. Alu. Leaky-wave theory, techniques, and applications: From microwaves to visible frequencies. In Proceedings of the IEEE, volume 103(5), pp. 793–821. IEEE, 2015. https://doi.org/10.1109/JPROC.2015.2399419
A.A. Oliner. Leaky waves: Basic properties and applications. In Proceedings of 1997 Asia-Pacific Microwave Conference, Navier-Stokes Equations and Related Nonlinear Problems, pp. 397–400, Hong Kong/China, 1997. IEEE. https://doi.org/10.1109/APMC.1997.659407
V.P. Silin P.N. Eleonskii, L.G. Oganes’yants. Cylindrical nonlinear waveguides. Soviet Physics Jetp, 35:44–47, 1972.
R.A. Shelby, D.R.Smith and S.Schultz. Experimental verification of a negative index of refraction. Science, 292(5514):77–79, 2001. https://doi.org/10.1126/science.1058847
Y.R. Shen. The Principles of Nonlinear Optics. John Wiley and Sons, New York, 1984.
Yu. Smirnov and E. Smolkin. On the existence of non-polarized azimuthalsymmetric electromagnetic waves in circular dielectric waveguide filled with nonlinear isotropic homogeneous medium. Wave Motion, 77:77–90, 2018. https://doi.org/10.1016/j.wavemoti.2017.11.001
Yu. Smirnov, E. Smolkin and V. Kurseeva. The new type of non-polarized symmetric electromagnetic waves in planar nonlinear waveguide. Applicable Analysis, 98(3):483–498, 2019. https://doi.org/10.1080/00036811.2017.1395865
Yu. Smirnov, E. Smolkin and D. Valovik. Nonlinear double-layer bragg waveguide: Analytical and numerical approaches to investigate waveguiding problem. Advances in Numerical Analysis, 2014:1–11, 2014. https://doi.org/10.1155/2014/231498
E. Smolkin. Propagation of TE waves in a double-layer nonlinear inhomogeneous cylindrical waveguide. In Days on Diffraction’2014 Proceedings, pp. 204–209, Saint Petersburg, Russia, 2014. IEEE. https://doi.org/10.1109/DD.2014.7036451
E. Smolkin. Goubau line filled with nonlinear medium: Numerical study of TMpolarized waves. In Proceedings of the 2015 17th International Conference on Electromagnetics in Advanced Applications, pp. 1572–1575, Turin, Italy, 2015. IEEE. https://doi.org/10.1109/ICEAA.2015.7297390
E. Smolkin. The azimuthal symmetric hybrid waves in nonlinear cylindrical waveguide. In Proceedings of the 2017 Progress in Electromagnetics Research Symposium, PIERS 2017, pp. 348–353, Saint Petersburg, Russia, 2017. IEEE. https://doi.org/10.1109/PIERS.2017.8261763
E. Smolkin and Yu. Shestopalov. Nonlinear Goubau line: analyticalnumerical approaches and new propagation regimes. Journal of Electromagnetic Waves and Applications, 31(8):781–797, 2017. https://doi.org/10.1080/09205071.2017.1317036
E. Smolkin, Yu. Shestopalov and M. Snegur. Diffraction of TM polarized electromagnetic waves by a nonlinear inhomogeneous metal-dielectric waveguide. In Proceedings of the 2017 19th International Conference on Electromagnetics in Advanced Applications, ICEAA 2017, pp. 1288–1291, Verona Italy, 2017. IEEE.
E. Smolkin and D. Valovik. Numerical solution of the problem of propagation of TM-polarized electromagnetic waves in a nonlinear two-layered dielectric cylindrical waveguide. In Mathematical Methods in Electromagnetic Theory (MMET), 2012 International Conference on, pp. 68–71, Kharkiv, Ukraine, 2012. IEEE. https://doi.org/10.1109/MMET.2012.6331288
E. Smolkin and D. Valovik. Calculation of the propagation constants of inhomogeneous nonlinear double-layer circular cylindrical waveguide by means of the cauchy problem method. Journal of Communications Technology and Electronics, 58:762–769, 2013. https://doi.org/10.1134/S1064226913060132
E. Smolkin and D. Valovik. Guided electromagnetic waves propagating in a two-layer cylindrical dielectric waveguide with inhomogeneous nonlinear permittivity. Advances in Mathematical Physics, 2015, 2015. https://doi.org/10.1155/2015/614976
E. Smolkin and D. Valovik. Nonlinear propagation of coupled electromagnetic waves in a circular cylindrical waveguide. Computational Mathematics and Mathematical Physics, 57(8):1294–1309, 2017. https://doi.org/10.1134/S0965542517080127
A.W. Snyder and J. Love. Optical waveguide theory. Springer, 1983.
A. Sommerfeld. Uber die fortpflanzung elektrodynamischer wellen la¨ngs eines drahtes. Annals of Physics, 303(2):233–290, 1899. https://doi.org/10.1002/andp.18993030202
L.A. Vainstein. Electromagnetic Waves. Radio i svyaz, Moscow, 1988.
V.G. Veselago. The electrodynamics of substances with simultaneously negative values of ε and µ. Soviet Physics Uspekhi, 10(4):509–514, 1968. https://doi.org/10.1070/PU1968v010n04ABEH003699
R.W. Ziolkowski and E.Heyman. Wave propagation in media having negative permittivity and permeability. PHYSICAL REVIEW E, 64(5), 2001. https://doi.org/10.1103/PhysRevE.64.056625