Analysis of ElectromagneticWave Propagation in Frequency Bands of Nonlinear Metamaterials

Authors

  • Nikolaos L. Tsitsas Department of Informatics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Keywords:

Analysis of ElectromagneticWave Propagation in Frequency Bands of Nonlinear Metamaterials

Abstract

Electromagnetic wave propagation phenomena in nonlinear metamaterials are investigated for waves propagating either in the left-handed frequency band or in the frequency band gaps. In the left-handed band, we implement directly the reductive perturbation method to Faraday’s and Amp´ere’s laws and derive a second-and a thirdorder nonlinear Schr¨odinger (NLS) equation, describing solitons of moderate and ultra-short pulse widths, respectively. Then, we find necessary conditions and derive exact bright and dark soliton solutions of these equations. On the other hand, in the frequency band gaps with negative linear effective permittivity and positive permeability (where linear electromagnetic waves are evanescent), we derive two short-pulse equations (SPEs) for the high- and low-frequency band gaps. The structure of the SPEs solutions is discussed, and connections with the NLS soliton solutions are presented. Numerical simulations of the SPEs solutions are included and compared with those of the reduced wave equations. Directions towards the modelling of wave propagation in nonlinear chiral metamaterials are pointed out.

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Author Biography

Nikolaos L. Tsitsas, Department of Informatics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece

Nikolaos L. Tsitsas was born
in Athens, Greece, in 1979. He
received the Diploma and Ph.D.
Degree in Electrical Engineer-
ing from the National Technical
University of Athens (NTUA) in
2002 and 2006 respectively, and
the M.Sc. Degree in Applied Mathematics from the
National and Kapodistrian University of Athens in
2005.
From 2008 to 2011, he was an Adjunct Lecturer
at the School of Applied Mathematical and Phys-
ical Sciences of the NTUA. From 2009 to 2011,
he was an Adjunct Lecturer at the Hellenic Army
Academy. Since 2012, he has been an Assistant
Professor at the Department of Informatics of the
Aristotle University of Thessaloniki. He is the
author or coauthor of 30 papers in scientific jour-
nals and 36 papers in conference proceedings. His
research interests include analytical and numerical
methods in direct and inverse wave scattering and
propagation theory as well as numerical solution
techniques of integral and partial differential equa-
tions.
Dr. Tsitsas is a member of the IEEE, the Optical
Society of America, the American Mathematical
Society, and the Technical Chamber of Greece.

References

G. V. Eleftheriades and K. G. Balmain, Negative-

Refraction Metamaterials. Fundamental Principles

and Applications, John Wiley, New Jersey, 2005.

C. Caloz and T. Itoh, Electromagnetic Metama-

terials. Transmission Line Theory and Microwave

Applications, John Wiley, New Jersey, 2006.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-

Nasser, and S. Schultz, “Composite Medium with

Simultaneously Negative Permeability and Permit-

tivity,” Phys. Rev. Lett., vol. 84, pp. 4184-4187,

May 2000.

A. Shelby, D. R. Smith, and S. Schultz, “Experi-

mental Verification of a Negative Index of Refrac-

tion,” Science, vol. 292, pp. 77-79, Apr. 2001.

M. C. Tang, S. Q. Xiao, D. Wang, J. Xiong,

K. Chen, and B. Z. Wang, “Negative Index of

Reflection in Planar Metamaterial Composed of

Single Split-Ring Resonators,” ACES J., vol. 26,

pp. 250-258, Mar. 2011.

V. Shalaev, “Optical Negative-Index Metamateri-

als,” Nature Photonics, vol. 1, pp. 41-48, Jan. 2007.

V. G. Veselago, “Electrodynamics of Substances

with Simultaneously Negative Values of Sigma and

Mu,” Sov. Phys. Usp., vol. 10, pp. 509, 1968.

J. B. Pendry, “Negative Refraction Makes a Perfect

Lens,” Phys. Rev. Lett., vol. 85, pp. 3966-3969,

Oct. 2000.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire,

“Metamaterials and Negative Refractive Index,”

Science, vol. 305, pp. 788-792, Aug. 2004.

J. B. Pendry and D. R. Smith, “Reversing Light

with Negative Refraction,” Phys. Today, vol. 57,

pp. 37-43, Jun. 2004.

S. A. Ramakrishna, “Physics of Negative Refrac-

tive Index Materials,” Rep. Prog. Phys., vol. 68,

pp. 449-521, Feb. 2005.

C. M. Soukoulis, M. Kafesaki, and

E. N. Economou, “Negative-Index Materials:

New Frontiers in Optics,” Adv. Materials, vol. 18,

pp. 1941-1952, Aug. 2006.

J. C. Liu, W. Shao, and B. Z. Wang, “A Dual-Band

Metamaterial Design using Double SRR Struc-

tures,” ACES J., vol. 26, pp. 459-463, Jun. 2011.

Y. Huang, G. Wen, T. Li, and K. Xie, “Positive-

Negative-Positive Metamaterial Consisting of Fer-

rimagnetic Host and Wire Array,” ACES J., vol. 25,

pp. 696-702, Aug. 2010.

G. Antonini, “Reduced Order Models for Meta-

material Transmission Lines,” ACES Newsletter,

vol. 21, pp. 78-103, 2006.

A. A. Zharov, I. V. Shadrivov, and Yu. S. Kivshar,

“Nonlinear Properties of Left-Handed Metamateri-

als,” Phys. Rev. Lett., vol. 91, 037401, Jul. 2003.

V. M. Agranovich, Y. R. Shen, R. H. Baughman,

and A. A. Zakhidov, “Linear and Nonlinear Wave

Propagation in Negative Refraction Metamateri-

als,” Phys. Rev. B, vol. 69, 165112, Apr. 2004.

N. Lazarides and G. P. Tsironis, “Coupled Non-

linear Schr ̈odinger Field Equations for Electro-

magnetic Wave Propagation in Nonlinear Left-

Handed Materials,” Phys. Rev. E, vol. 71, 036614,

Mar. 2005.

I. V. Shadrivov, A. A. Zharov, N. A. Zharova, and

Yu. S. Kivshar, “Nonlinear Left-Handed Metamate-

rials,” Radio Science, vol. 40, RS3S90, May 2005.

M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Po-

liakov, G. D’Aguanno, N. Mattiucci, M. J. Bloe-

mer, and A. M. Zheltikov, “Generalized Nonlinear

Schrodinger Equation for Dispersive Susceptibil-

ity and Permeability: Application to Negative In-

dex Materials,” Phys. Rev. Lett., vol. 95, 013902,

Jul. 2005.

S. C. Wen, Y. J. Xiang, X. Y. Dai, Z. X. Tang,

W. H. Su, and D. Y. Fan, “Theoretical Models

for Ultrashort Electromagnetic Pulse Propagation

in Nonlinear Metamaterials,” Phys. Rev. A, vol. 75,

, Mar. 2007.

I. V. Shadrivov and Yu. S. Kivshar, “Spatial Soli-

tons in Nonlinear Left-Handed Metamaterials,” J.

Opt. A: Pure Appl. Opt., vol. 7, pp. S68-S72,

Feb. 2005.

A. Aceves, “Optical Gap Solitons: Past, Present,

and Future; Theory and Experiments,” Chaos,

vol. 10, pp. 584-589, Sep. 2000.

S. Longhi, “Gap Solitons in Metamaterials,” Waves

in Random and Complex Media, vol. 15, pp. 119-

, Feb. 2005.

T. Sch ̈afer and C. E. Wayne, “Propagation of Ultra-

Short Optical Pulses in Cubic Nonlinear Media,”

Physica D, vol. 196, pp. 90-105, Sep. 2004.

N. Costanzino, V. Manukian, and C. K. R. T. Jones,

“Solitary Waves of the Regularized Short Pulse

and Ostrovsky Equations,” SIAM J. Math. Anal.,

vol. 41, pp. 2088-2106, 2009.

N. L. Tsitsas, N. Rompotis, I. Kourakis,

P. G. Kevrekidis, and D. J. Frantzeskakis,

“Higher-Order Effects and Ultrashort Solitons

in Left-Handed Metamaterials,” Phys. Rev. E,

vol. 79, 037601, Mar. 2009.

N. L. Tsitsas, T. P. Horikis, Y. Shen,

P. G. Kevrekidis, N. Whitaker, and

D. J. Frantzeskakis, “Short Pulse Equations

and Localized Structures in Frequency Band

Gaps of Nonlinear Metamaterials,” Phys. Let. A,

vol. 374, pp. 1384-1388, Mar. 2010.

I. Kourakis and P. K. Shukla, “Nonlinear Prop-

agation of Electromagnetic Waves in Negative-

Refraction-Index Composite Materials,” Phys. Rev.

E, vol. 72, 016626, Jul. 2005.

T. Taniuti, “Reductive Perturbation Method and

Far Fields of Wave-Equations,” Prog. Theor. Phys.

Suppl., vol. 55, pp. 1-35, 1974.

Y. Kodama and A. Hasegawa, “Nonlinear Pulse-

Propagation in a Monomode Dielectric Guide,”

IEEE J. Quantum Electron., vol. 23, pp. 510-524,

May 1987.

A. Hasegawa and Y. Kodama, Solitons in Optical

Communications, Clarendon Press, Oxford, 1995.

J. V. Moloney and A. C. Newell, Nonlinear Optics,

Westview Press, Oxford, 2004.

G. P. Agrawal, Nonlinear Fiber Optics, Academic

Press, London, 2007.

K. Hizanidis, D. J. Frantzeskakis, and C. Polymilis,

“Exact Travelling Wave Solutions for a General-

ized Nonlinear Schrodinger Equation,” J. Phys. A:

Math. Gen., vol. 29, pp. 7687-7703, Dec. 1996.

L. Kang, Q. Zhao, H. Zhao, and J. Zhou, “Mag-

netically Tunable Negative Permeability Metama-

terial Composed by Split Ring Resonators and

Ferrite Rods,” Opt. Express, vol. 16, pp. 8825-

, Jun. 2008.

S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck,

N. C. Panoiu, and R. M. Osgood, “Near-Infrared

Double Negative Metamaterials,” Opt. Express,

vol. 13, pp. 4922-4930, Jun. 2005.

R. Camassa and D. D. Holm, “An Integrable

Shallow-Water Equation with Peaked Solitons,”

Phys. Rev. Lett., vol. 71, pp. 1661-1664, Sep. 1993.

A. Sakovich and S. Sakovich, “Solitary Wave So-

lutions of the Short Pulse Equation,” J. Phys. A:

Math. Gen., vol. 39, pp. L361-L367, Jun. 2006.

T. G. Mackay and A. Lakhtakia, “Plane Waves with

Negative Phase Velocity in Faraday Chiral Medi-

ums,” Phys. Rev. E, vol. 69, 026602, Feb. 2004.

J. B. Pendry, “A Chiral Route to Negative Refrac-

tion,” Science, vol. 306, pp. 1353-1355, Nov. 2004.

V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD

Formulation for Dispersive Chiral Media using

the Z Transform Method,” IEEE Trans. Antennas

Propagat., vol. 53, pp. 3374-3384, Oct. 2005.

N. Wongkasem, A. Akyurtlu, K. A. Marx,

Q. Dong, J. Li, and W .D. Goodhue, “Development

of Chiral Negative Refractive Index Metamateri-

als for the Terahertz Frequency Regime,” IEEE

Trans. Antennas Propagat., vol. 55, pp. 3052-3062,

Nov. 2007.

N. L. Tsitsas, A. Lakhtakia, and

D. J. Frantzeskakis, “Vector Solitons in Nonlinear

Isotropic Chiral Metamaterials,” J. Phys. A: Math.

Theor., vol. 44, 435203, Oct. 2011.

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Published

2022-05-02

How to Cite

[1]
N. L. . Tsitsas, “Analysis of ElectromagneticWave Propagation in Frequency Bands of Nonlinear Metamaterials”, ACES Journal, vol. 27, no. 2, pp. 169–180, May 2022.

Issue

2012: Vol 27 Iss 2

Section

General Submission