ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

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Rostuntsova A. А., Ryskin N. M. Study of character of modulation instability in cyclotron resonance interaction of an electromagnetic wave with a counterpropagating rectilinear electron beam. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 5, pp. 597-609. DOI: 10.18500/0869-6632-003067, EDN: ZKVTFL

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Study of character of modulation instability in cyclotron resonance interaction of an electromagnetic wave with a counterpropagating rectilinear electron beam

Rostuntsova Alena Александровна, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences
Ryskin Nikita Mikhailovich, Saratov Branch of Kotel`nikov Institute of Radiophysics and Electronics of Russian Academy of Sciences

In this paper, the interaction of a monochromatic electromagnetic wave with a counterpropagating electron beam moving in an axial magnetic field is considered. The purpose of this study is to investigate the conditions for occurrence of modulation instability (MI) in such a system and to determine at which parameters of the incident wave the MI is absolute or convective.

Methods. Theoretical analysis of the MI character is carried out by studying the asymptotic form of unstable perturbations using the saddle-point analysis. The analytical results are verified by numerical simulations.

Results. Theoretically, the boundary of change in the character of MI on the plane of input signal parameters (amplitude and detuning of the frequency from the cyclotron resonance) is determined. Numerical simulations confirm that as the signal frequency increases, the regime of self-modulation, which corresponds to the absolute MI, is replaced by the stationary single-frequency transmission corresponding to the convective MI. The numerical results coincide with the analytical ones for the system, which is matched at the end. The matching is implemented by smooth increasing of the guiding magnetic field in the region of electron beam injection.

Conclusion. Determining the analytical conditions for the implementation of the absolute MI is of practical interest, since the emerging self-modulation can lead to the generation of trains of pulses with the spectrum in the form of frequency combs.

This work was supported by Russian Science Foundation under Grant No. 23-12-00291
  1. Benjamin TB. Instability of periodic wavetrains in nonlinear dispersive systems. Proc. R. Soc. Lond. A. 1967;299(1456):59–76. DOI: 10.1098/rspa.1967.0123.
  2. Dodd RK, Eilbeck JC, Gibbon JD, Morris HS. Solitons and Nonlinear Wave Equations. London: Academic Press; 1982. 630 p.
  3. Newell AC. Solitons in Mathematics and Physics. Philadelphia: SIAM; 1985. 260 p. DOI: 10.1137/1.9781611970227.
  4. Ostrovsky LA, Potapov AI. Modulated Waves: Theory and Applications. Baltimore, MD, USA: The Johns Hopkins University Press; 1999. 369 p.
  5. Zakharov VE, Ostrovsky LA. Modulation instability: The beginning. Physica D. 2009;238(5): 540–548. DOI: 10.1016/j.physd.2008.12.002.
  6. Ryskin NM, Trubetskov DI. Nonlinear Waves. Moscow: URSS; 2021. 312 p. (in Russian).
  7. Ryskin NM. Oscillations and Waves in Nonlinear Active Media. Saratov: Saratov University Publishing; 2017. 102 p. (in Russian).
  8. Balyakin AA, Ryskin NM. A change in the character of modulation instability in the vicinity of a critical frequency. Tech. Phys. Lett. 2004;30(3):175–177. DOI: 10.1134/1.1707158.
  9. Balyakin AA, Ryskin NM. Modulation instability in a nonlinear dispersive medium near cut-off frequency. Nonlinear Phenomena in Complex Systems. 2004;7(1):34–42.
  10. Rostuntsova AA, Ryskin NM, Zotova IV, Ginzburg NS. Modulation instability of an electromagnetic wave interacting with a counterpropagating electron beam under condition of cyclotron resonance absorption. Phys. Rev. E. 2022;106(1):014214.
  11. Newell AC. Nonlinear tunnelling. J. Math. Phys. 1978;19(5):1126–1133. DOI: 10.1063/1.523759.
  12. Zotova IV, Ginzburg NS, Zheleznov IV, Sergeev AS. Modulation of high-intensity microwave radiation during its resonant interaction with counterflow of nonexcited cyclotron oscillators. Tech. Phys. Lett. 2014;40(6):495–498. DOI: 10.1134/S1063785014060285.
  13. Zotova IV, Ginzburg NS, Sergeev AS, Kocharovskaya ER, Zaslavsky VY. Conversion of an electromagnetic wave into a periodic train of solitons under cyclotron resonance interaction with a backward beam of unexcited electron-oscillators. Phys. Rev. Lett. 2014;113(14):143901. DOI: 10.1103/PhysRevLett.113.143901.
  14. Ginzburg NS, Zotova IV, Kocharovskaya ER, Sergeev AS, Zheleznov IV, Zaslavsky VY. Self-induced transparency solitons and dissipative solitons in microwave electronic systems. Radiophysics and Quantum Electronics. 2021;63(9–10):716–741. DOI: 10.1007/s11141-021-10092-w.
  15. Benirschke DJ, Han N, Burghoff D. Frequency comb ptychoscopy. Nat. Commun. 2021;12(1):4244. DOI: 10.1038/s41467-021-24471-4.
  16. Hagmann MJ. Scanning frequency comb microscopy–A new method in scanning probe microscopy. AIP Advances. 2018;8(12):125203. DOI: 10.1063/1.5047440.
  17. Gaponov AV, Petelin MI, Yulpatov VK. The induced radiation of excited classical oscillators and its use in high-frequency electronics. Radiophysics and Quantum Electronics. 1967;10(9–10): 794–813. DOI: 10.1007/BF01031607.
  18. Kuzelev MV, Rukhadze AA. Methods of Wave Theory in Dispersive Media. Singapore: World Scientific; 2009. 272 p. DOI: 10.1142/7231.
  19. Barletta A, Celli M. Convective to absolute instability transition in a horizontal porous channel with open upper boundary. Fluids. 2017;2(2):33. DOI: 10.3390/fluids2020033.
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