Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 096, 21 pages      arXiv:1212.1971      https://doi.org/10.3842/SIGMA.2012.096
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

Time-Frequency Integrals and the Stationary Phase Method in Problems of Waves Propagation from Moving Sources

Gennadiy Burlak a and Vladimir Rabinovich b
a) Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, Mor. México
b) National Polytechnic Institute, ESIME Zacatenco, D.F. México

Received July 29, 2012, in final form December 02, 2012; Published online December 10, 2012

Abstract
The time-frequency integrals and the two-dimensional stationary phase method are applied to study the electromagnetic waves radiated by moving modulated sources in dispersive media. We show that such unified approach leads to explicit expressions for the field amplitudes and simple relations for the field eigenfrequencies and the retardation time that become the coupled variables. The main features of the technique are illustrated by examples of the moving source fields in the plasma and the Cherenkov radiation. It is emphasized that the deeper insight to the wave effects in dispersive case already requires the explicit formulation of the dispersive material model. As the advanced application we have considered the Doppler frequency shift in a complex single-resonant dispersive metamaterial (Lorenz) model where in some frequency ranges the negativity of the real part of the refraction index can be reached. We have demonstrated that in dispersive case the Doppler frequency shift acquires a nonlinear dependence on the modulating frequency of the radiated particle. The detailed frequency dependence of such a shift and spectral behavior of phase and group velocities (that have the opposite directions) are studied numerically.

Key words: dispersive media; two-dimensional stationary phase method; electromagnetic wave; moving modulated source.

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References

  1. Afanasiev G.N., Vavilov-Cherenkov and synchrotron radiation. Foundations and applications, Fundamental Theories of Physics, Vol. 142, Kluwer Academic Publishers, New York, 2005.
  2. Afanasiev G.N., Kartavenko V.G., Radiation of a point charge uniformly moving in a dielectric medium, J. Phys. D: Appl. Phys. 31 (1998), 2760-2776.
  3. Afanasiev G.N., Kartavenko V.G., Magar E.N., Vavilov-Cherenkov radiation in dispersive medium, Phys. B 269 (1999), 95-113.
  4. Babich V.M., Buldyrev V.S., Molotkov I.A., The space-time ray method, Leningrad. Univ., Leningrad, 1985.
  5. Belonogaya E.S., Galyamin S.N., Tyukhtin A.V., Properties of Vavilov-Cherenkov radiation in an anisotropic medium with a resonant dispersion of permittivity, J. Opt. Soc. Amer. B. 28 (2011), 2871-2878.
  6. Blestein N., Handelsman R.A., Asymptotic extention of integrals, Dover Publication, New York, 1975.
  7. Bolotovskii B.M., Vavilov-Cherenkov radiation: its discovery and applications, Phys. Usp. 52 (2009), 1099-1110.
  8. Brillouin L., Wave propagation and group velocity, Pure and Applied Physics, Vol. 8, Academic Press, New York, 1960.
  9. Carusotto I., Artoni M., La Rocca G.C., Bassani F., Slow group velocity and Cherenkov radiation, Phys. Rev. Lett. 87 (2001), 064801, 4 pages.
  10. Chen J., Wang Y., Jia B., Geng T., Li X., Feng L., Qian W., Liang B., Zhang X., Gu M., Zhuang S., Observation of the inverse Doppler effect in negative-index materials at optical frequencies, Nature Photonics 5 (2011), 239-242.
  11. Choi M., Lee, Seung H., Kim Y., Kang Seung B., Shin J., Kwak M.H., Kang K.Y., Lee Y.H., Park N., Min B., A terahertz metamaterial with unnaturally high refractive index, Nature 470 (2011), 369-373.
  12. Cohen L., Time-frequency analysis, Prentice-Hall Inc., Englewood Cliffs, N.J., 1995.
  13. Doil'nitsina E.G., Tyukhtin A.V., Radiated power of oscillators traveling in a moving medium, Radiophys. Quantum Electron. 50 (2007), 287-298.
  14. Duan Z., Wu B.I., Xi S., Chen H., Chen M., Research progress in reversed Cherenkov radiation in double-negative metamaterials, Progr. Electromag. Res. 90 (2009), 75-87.
  15. Dung H.T., Buhmann S.Y., Knöll L., Welsch D.G., Scheel S., Kästel J., Electromagnetic-field quantization and spontaneous decay in left-handed media, Phys. Rev. A 68 (2003), 066606, 15 pages, quant-ph/0306028.
  16. Fedoryuk M.V., Method of the steepest descent, Nauka, Moscow, 1977.
  17. Felsen L.B., Marcuvitz N., Radiation and scattering of waves, Prentice-Hall Microwaves and Fields Series, Prentice-Hall Inc., Englewood Cliffs, N.J., 1973.
  18. Galyamin S.N., Tyukhtin A.V., Electromagnetic field of a moving charge in the presence of a left-handed medium, Phys. Rev. B 81 (2010), 235134, 14 pages.
  19. Ginzburg V.L., Propagation of electromagnetic waves in plasma, Nauka, Moscow, 1960.
  20. Ginzburg V.L., Radiation by uniformly moving sources (Vavilov-Cherenkov effect, transition radiation, and other phenomena), Phys. Usp. 39 (1996), 973-982.
  21. Ginzburg V.L., Theoretical physics and astrophysics, 2nd ed., Nauka, Moscow, 1981.
  22. Grudsky S.M., Obrezanova O.A., Rabinovich V.S., Sound propagation from a moving air source in the ocean covered by ice, Acoustical Phys. 41 (1995), 1-6.
  23. Il'ichev V.I., Rabinovich V.S., Rivelis E.A., Hoha U.V., Acoustic field of a moving narrow-band source in oceanic waveguides, Dokl. Phys. 304 (1989), 1123-1127.
  24. Jackson J.D., Classical electrodynamics, 2nd ed., John Wiley & Sons Inc., New York, 1975.
  25. Jelley J.V., Cherenkov radiation and its applications, Pergamon Press, London, 1958.
  26. Landau L.D., Lifshits E.M., Course of theoretical physics, Vol. 2, The classical theory of fields, 4th ed., Pergamon Press, Oxford, 1975.
  27. Landau L.D., Lifshits E.M., Course of theoretical physics, Vol. 9, Statistical physics, Part 2, Theory of the condensed state, Pergamon Press, Oxford, 1981.
  28. Landau L.D., Lifshits E.M., Theoretical physics, Vol. 2, Field theory, Nauka, Moscow, 1988.
  29. Landau L.D., Lifshits E.M., Theoretical physics, Vol. 8, Electrodynamics of continuous media, Nauka, Moscow, 1982.
  30. Lewis R.M., Asymptotic methods for the solution of dispersive hyperbolic equations, in Asymptotic Solutions of Differential Equations and Their Applications, Wiley, New York, 1964, 53-107.
  31. Lu J., Grzegorczyk T., Zhang Y., Pacheco J., Wu B.I., Kong J., Chen M., Cherenkov radiation in materials with negative permittivity and permeability, Opt. Express 11 (2003), 723-734.
  32. Luo C., Ibanescu M., Johnson S.G., Joannopoulos J.D., Cherenkov radiation in photonic crystals, Science 299 (2003), 368-371.
  33. Melamed T., Felsen L.B., Pulsed-beam propagation in lossless dispersive media. I. Theory, J. Opt. Soc. Amer. A 15 (1993), 1268-1276.
  34. Obrezanova O.A., Rabinovich V.S., Acoustic field generated by moving sources in stratified waveguides, Wave Motion 27 (1998), 155-167.
  35. Obrezanova O.A., Rabinovich V.S., Acoustic field of a source moving along stratified waveguide surface, Acoustical Phys. 39 (1993), 517-521.
  36. Pendry J.B., Holden A.J., Robbins D.J., Stewart W.J., Magnetism from conductors and enhanced nonlinear phenomena, IEEE Trans. Microwave Theory Tech. 47 (1999), 2075-2084.
  37. Pendry J.B., Schurig D., Smith D.R., Controlling electromagnetic fields, Science 312 (2006), 1780-1782.
  38. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P., Numerical recipes. The art of scientific computing, 3rd ed., Cambridge University Press, Cambridge, 2007.
  39. Shalaev V.M., Optical negative-index metamaterials, Nature Photonics 1 (2007), 41-48, arXiv:1109.0084.
  40. Shin J., Shen J.T., Fan S., Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad bandwidth, Phys. Rev. Lett. 102 (2009), 093903, 4 pages.
  41. Shubin M.A., Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001.
  42. Smith D.R., Padilla J.W., Vier D.C., Nemat-Nasser S.C., Schultz S., Composite medium with simultaneously negative permeability and permittivity, Phys. Rev. Lett. 84 (2000), 4184-4187.
  43. Smolyaninov I.I., Hung Y.J., Davis C.C., Magnifying superlens in the visible frequency range, Science 315 (2007), 1699-1701, physics/0610230.
  44. Sommerfeld A., Mechanics of deformable bodies. Lectures on theoretical physics, Vol. II, Academic Press Inc., New York, 1950.
  45. Soukoulis C.M., Wegener M., Past achievements and future challenges in the development of three-dimensional photonic metamaterials, Nature Photonics 5 (2011), 523-530, arXiv:1109.0084.
  46. Stevens T.E., Wahlstrand J.K., Kuhl J., Merlin R., Cherenkov radiation at speeds below the light threshold: phonon-assisted phase matching, Science 291 (2001), 627-630.
  47. Stix T.H., Waves in plasmas, Springer-Verlag, New York, 1992.
  48. Swanson D.G., Plasma waves, 2nd ed., IOP Publishing, Bristol, 2003.
  49. Taflove A., Hagness S.C., Computational electrodynamics: the finite-difference time-domain method, 2nd ed., Artech House Inc., Boston, MA, 2000.
  50. Tyukhtin A.V., Energy characteristics of radiation from an oscillating dipole moving in a dielectric medium with resonant dispersion, Tech. Phys. 50 (2005), 1084-1088.
  51. Tyukhtin A.V., Galyamin S.N., Vavilov-Cherenkov radiation in passive and active media with complex resonant dispersion, Phys. Rev. E 77 (2008), 066606, 8 pages.
  52. Veselago V.G., The electrodynamics of substances with simultaneously negative values of ε and μ, Sov. Phys. Usp. 10 (1968), 509-514.
  53. Vorobev V.V., Tyukhtin A.V., Cherenkov radiation in a wire metamaterial, Phys. Rev. Lett. 108 (2012), 184801, 4 pages.
  54. Waters K.R., Hughes M.S., Brandenburger G.H., Miller J.G., On a time-domain representation of the Kramers-Krönig dispersion relations, J. Acoust. Soc. Amer. 108 (2000), 2114-2119.
  55. Xi S., Chen H., Jiang T., Ran L., Huangfu J., Wu B.I., Kong J.A., Chen M., Experimental verification of reversed Cherenkov radiation in left-handed metamaterial, Phys. Rev. Lett. 103 (2009), 093903, 4 pages.
  56. Xiao S., Drachev V.P., Kildishev A.V., Ni X., Chettiar U.K., Hsiao-Kuan Y., Shalaev V.M., Loss-free and active optical negative-index metamaterials, Nature 466 (2010), 735-738.
  57. Ziolkowski R.W., Heyman E., Wave propagation in media having negative permittivity and permeability, Phys. Rev. E 64 (2001), 056625, 15 pages.

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