Izvestiya of Saratov University.


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Davidovich M. V. Negative Dispersion, Refraction and Backward Polaritons: Impedance Approach. Izvestiya of Sarat. Univ. Physics. , 2019, vol. 19, iss. 2, pp. 95-112. DOI: 10.18500/1817-3020-2019-19-2-95-112

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Negative Dispersion, Refraction and Backward Polaritons: Impedance Approach

Davidovich Mikhail Vladimirovich, Saratov State University

Background and Objectives: The dispersion equations of surface plasmon-polaritons are derived for the general case of layered dissipative structures. The waves are classified as gliding with energy flow into structure from vacuum and leakage ones. The dispersion equations and conditions for the existence of slow and fast gliding and leaky waves, as well as forward and backward waves are considered. It is shown that for improper gliding and leakage monochromatic waves (in particular, for the Zenneck wave), the group velocity does not match the rate of energy transfer, especially in the bands of resonances, bandgaps and bands of strong spatial dispersion. We demonstrate the convenience of the impedance approach to the tasks. Results: The general form of the dispersion equation for polaritons in the multilayered structures, including thin 2D films, are obtained. The main results of the paper consist in the derived dispersion equations and their numerical solution, the conditions for the existence of forward and backward polaritons and slow or fast polaritons. The type of polariton is determined by the sign of the reactive part of the input impedance for this type of wave. The positive (inductive) one corresponds to a forward polariton and the negative (capacitive) one – to a backward polariton. The slowdown is determined by the ratio of reactive and active parts of the input impedance. A slow surface plasmon occurs when the input impedance is highly reactive. The presence of spot with a backward wave and negative refraction allows us to implement control of plasmons, in particular, to carry out its focusing. The negative refraction does not necessarily occur in the presence of a backward wave.


1. Economou E. N. Surface Plasmons in Thin Films. Phys. Rev., 1969, vol. 182, no. 2, pp. 539‒545. DOI: https://doi.org/10.1103/PhysRev.182.539

2. Tournoisa P., Laude V. Negative group velocities in metalfi lm optical waveguides. Optics Communications, 1997, vol. 137, pp. 41–45.

3. Liu Y. M., Pile D. F. P., Liu Z., Wu D., Sun C., Zhang X. Negative group velocity of surface plasmons on thin metallic fi lms. Proc. SPIE, 2006, vol. 6323, pp. 63231M(1‒9). DOI: https://doi.org/10.1117/12.681492

4. Fedyanin D. Yu., Arsenin A. V., Leiman V. G., Gladun A. D. Surface plasmon-polatritones with negative and zero group velocities propagating in thin metal fi lms. Quantum Electronics, 2009, vol. 39, no. 8, pp. 745–750. DOI: https://doi.org/10.1070/QE2009V039N08ABEH014072

5. Zuev V. S., Zueva G. Ya. Very slow surface plasmons: Theory and practice (Review). Optics and Spectroscopy, 2008, vol. 107, no. 4, pp. 614–628. DOI: https://doi.org/10.1134/S0030400X09100166

6. Fedyanin D. Yu., Arsenin A. V., Leiman V. G., Gladun A. D. Backward waves in planar insulator-metalinsulator waveguide structures. J. Opt., 2010, vol. 12, pp. 015002(1‒7). DOI: https://doi.org/10.1088/2040-8978/12/1/015002

7. Tao J., Wang Q. J., Zhang J., Luo Y. Reverse surfacepolariton cherenkov radiation, Scientifi c Reports, 2016, vol. 6, pp. 30704(1‒6). DOI: https://doi.org/10.1038/srep30704

8. Davidovich M. V. Plasmons in multilayered plan-layered structures. Quantum Electronics, 2017, vol. 47, no. 6, pp. 567‒579. DOI: https://doi.org/10.1070/QEL16272

9. Davidovich M. V. Maximum deceleration and negative dispersion of plasmons along a metal layer. Technical Physics Letters, 2017, vol. 43, no. 11, pp. 1023‒1026. DOI: https://doi.org/10.1134/S1063785017110207

10. Davidovich M. V., Meshchanov V. P. Dispersion of surface plasmons on metasurfaces: the method of tensor Green’s functions. Antenny [Antennas], 2017, no. 8 (240), pp. 3‒16 (in Russian).

11. Davidovich M. V. On the Condition for Transformation of a Fast Surface Wave into a Slow Surface Wave. Journal of Communications Technology and Electronics, 2018, vol. 63, no. 6, pp. 497–504. DOI: https://doi.org/10.1134/S106422691806005

12. Mikhailov S. A., Ziegler K. New electromagnetic mode in graphene, Phys. Rev. Lett., 2007, vol. 99, no. 1, pp. 016803(1‒4). DOI: https://doi.org/10.1103/PhysRevLett.99.016803

13. Hanson G. W. Dyadic Green’s functions and guided surface waves for a surface conductivity model of graphene. J. Appl. Phys., 2008, vol. 103, pp. 064302(8). DOI: https://doi.org/10.1063/1.289145

14. Davidovich M. V., Bushuev N. A. On the possibility of creation of vacuum electronic amplifi ers on the surface plasmons. II vserossijskaya ob”edinennoj nauchnoj konferenciya “Problemy SVCH ehlektroniki” [II All- Russian United scientifi c conference “Problems of microwave electronics”]. Moscow, OOO Media Publ., 2015, pp. 113‒117 (in Russian).

15. Morozov M. Yu., Moiseenko I. M., Popov V. V. Amplifi cation of plasma waves in shielded active graphene. Tech. Phys. Lett., 2016, vol. 42, no. 1, pp. 40–42. DOI: https://doi.org/10.1134/S1063785016010144

16. Popov V. V., Polischuk O. V., Davoyan A. R., Ryzhii V., Otsuji T., Shur M. S. Plasmonic terahertz lasing in an array of graphene nanocavities. Phys. Rev., 2012, vol. B86, pp. 195437(1–6). DOI: https://doi.org/10.1103/PhysRevB.86.195437

17. Anenkov V. V., Shevchenko V. V. Fundamental Modes of a Nonsymmetric Waveguide from Metamaterial. Journal of Communications Technology and Electronics, 2011, vol. 56, no. 2, pp. 115‒124. DOI: https://doi.org/10.1134/S1064226911100020

18. Davidovich M. V. Hyperbolic Medium of Finite-Length Wires. Journal of Experimental and Theoretical Physics, 2018, vol. 127, no. 1, pp. 1–19. DOI: https://doi.org/10.1134/S1063776118070178

19. Davidovich M. V. Diamagnetism and paramagnetism of the metamaterial of the rings with the current. JETP Lett., 2018, vol. 108, no. 5, pp. 228–233. DOI: https://doi.org/10.1134/S0370274X18170010

20. Ahiezer A. I., Ahiezer I. A. Electromagnetics and electromagnetic waves. Moscow, Vysshaya shkola Publ., 1985. 504 p. (in Russian).

21. Davidovich M. V. Propagation of signals through a dissipative fi lter and the negative time delay. Technical Physics, 2012, vol. 57, no. 3, pp. 15‒22. DOI: https://doi.org/10.1134/S1063784212030048

22. Rytov S. M. Some theorems on group velocity electromagnetic waves. J. Exp. Theor. Phys., 1947, vol. 17, pp. 930‒936.

23. Schulz-DuBois E. O. Energy transport velocity of electromagnetic propagation in dispersive media. Proc. IEEE, 1969, vol. 57, no. 10, pp. 1748‒1757.

24. Belov P. A., Simovsky C. R., Tretyakov S. A. Backward waves and negative refraction in photonic (electromagnetic) crystals. Journal of Communications Technology and Electronics, 2004, vol. 49, no. 11, pp. 1285‒1294 (in Russian).

25. Agranovich V. M. Negative refraction in the optical range and nonlinear wave propagation. Physics-Uspekhi, 2004, vol. 174, no. 6, pp. 683–684. DOI: https://doi.org/10.3367/UFNr.0174.200406i.0683

26. Agranovich V. M., Gartstein Yu. N. Spatial dispersion and negative refraction of light. Physics-Uspekhi, 2006, vol. 176, no. 10, pp. 1051–1068. DOI: https://doi.org/10.3367/UFNr.0176.200610c.1051

27. Rautian S. G. Refl ection and refraction at the boundary of a medium with negative group velocity. Physics-Uspekhi, 2008, vol. 178, no. 10, pp. 1017‒1024. DOI: https://doi.org/10.3367/UFNr.0178.200810a.1017

28. Simovsky K. R. On material parameters of metamaterials (a Review). Optics and Spectroscopy, 2009, vol. 107, no. 5, pp. 766–793. DOI: https://doi.org/10.1134/S0030400X09110101

29. Makarov V. P., Rukhadze A. A. Electromagnetic waves with negative group velocity and the tensor of energy-momentum. Physics-Uspekhi, 2011, vol. 181, iss. 12, pp. 1357‒1368. DOI: https://doi.org/10.3367/UFNr.0181.201112n.1357

30. Davidovich M. V. Zakony sohraneniya i plotnosti energii i impul'sa elektromagnitnogo polya v dispergiruyushchej srede [The laws of conservation and density of energy and momentum of the electromagnetic fi eld in a dispersing medium]. Saratov, Izd-vo Sarat. un-ta, 2012. 112 p. (in Russian).

31. Davidovich M. V. Vtekayushchie i vytekayushchie nesobstvennye mody ‒ analiz dissipativnyh dispersionnyh uravnenij i volna Cenneka [Gliding and leakage improper waves ‒ the analysis of dissipative dispersive equations and Zenneck wave]. Saratov, Izd-vo Sarat. un-ta, 2014. 104 p. (in Russian).

32. Vainstein L. A. Elektromagnitnye volny [Electromagnetic waves]. Moscow, Radio i svyaz’ Publ., 1988. 440 p. (in Russian).

33. Davidovich M. V. Why can’t a negative refractive index be used. Izv. Saratov Univ. (N. S.), Ser. Physics, 2011, vol. 11, iss. 1, pp. 42–47 (in Russian).

34. Landau L. D., Lifshits E. M. Course of Theoretical Physics in 10 volumes. Vol. 8. Electrodynamics of continuous media. Pergamon Press, 1960. 460 p.

35. Lagarkov A. N., Kisel V. N., Sarychev A. K., Semenenko V. N. Electrophysics and electrodynamics of metamaterials. Thermophysics of High Temperatures, 2010, vol. 48, no. 6, pp. 983–999. DOI: https://doi.org/10.1134/S0018151X10060258

36. Davidovich M. V. Analysis of Photonics and nanoplasmonics structures by the method of integral equations. Naukooemkie Tekhnologii, 2016, vol. 17, no. 5, pp. 8‒18 (in Russian).

37. Vendik I. B., Vendik O. G., Gashinova M. S. Artifi cial dielectric medium possessing simultaneously negative permittivity and negative magnetic permeability. Tech. Phys. Lett., 2006, vol. 32, iss. 5, pp. 429–433. DOI: https://doi.org/10.1134/S106378500605018X

38. Davidovich M. V. Analysis of plasmons and planehomogenization in photonic crystals and hyperbolic metamaterials. J. Exp. Theor. Phys., 2016, vol. 160, iss. 6, pp. 928–941. DOI: https://doi.org/10.1134/S106377611611025X

39. Lovat G., Hanson G. W., Araneo R., Burghignoli P. Semiclassical spatially dispersive intraband conductivity tensor and quantum capacitance of graphene. Phys. Rev., 2013, vol. B 87, pp. 115429(11). DOI: https://doi.org/10.1103/PhysRevB.87.115429

40. Vaskovsky V. A., Locke E. G. Forward and backward noncollinear waves in magnetic fi lms. Physics-Uspekhi, 2006, vol. 176, iss. 5, pp. 557–562. DOI: https://doi.org/10.3367/UFNr.0176.I200605.0557

41. Keller Yu. I., Makarov P. A., Shatrov V. G., Shcheglov V. I. Surface magnetostatic waves in the ferrite plate with dissipation. Part 1, 2. Journal of Radioelectronics, 2016, no. 2, 3. Available at: http://jre.cplire.ru/mac/feb16/2/text.html; http://jre.cplire.ru/jre/mar16/1/text.html (accessed 14 April 2019) (in Russian).