Generation of UR Harmonics in Undulators with Multiperiodic Fields

  • K. V. ZhukovskyEmail author

Undulator radiation (UR) of some elliptical undulators with multiperiodic magnetic fields of sin-sin and sincos configuration is investigated, as is also UR of a planar and a spiral undulator in the presence of additional harmonics of the magnetic field. Exact analytical expressions for the UR spectrum and the intensity in terms of generalized special Bessel functions are obtained; the corresponding Bessel coefficients are derived. Comparison of the obtained results with the available experimental data for real devices along with numerical simulations confirms the accuracy of the analytical formulas. The possibility of generation of a strong fifth UR harmonic induced by the third harmonic of the undulator field is demonstrated. In free electron lasers (FELs) this effect leads to the generation of a strong fifth harmonic of coherent radiation against the background of a weak third harmonic. In a spiral undulator with a third harmonic of the field, large values of the Bessel coefficients are obtained for the fifth UR harmonic; this makes it possible to use such an undulator as a prebuncher, i.e., to bunch electrons, at the wavelength of the fifth harmonic in a cascade FEL with amplification of the higher harmonics. The corresponding FEL is modeled.


undulator radiation harmonic generation free electron laser two-frequency undulator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. L. Ginzburg, Izv. Akad. Nauk SSSR. Fiz., 11, 1651 (1947).Google Scholar
  2. 2.
    H. Motz, W. Thon, and R. N. J. Whitehurst, Appl. Phys., 24, 826 (1953).CrossRefGoogle Scholar
  3. 3.
    V. G. Bagrov, G. S. Bisnovatyi-Kogan, V. A. Bordovitsyn, et al., Theory of Radiation of Relativistic Particles [in Russian], Fizmatlit, Moscow (2002).Google Scholar
  4. 4.
    L. A. Artsimovich and I. Ya. Pomeranchuk, Zh. Eksp. Teor. Fiz., 16, 379 (1946).ADSGoogle Scholar
  5. 5.
    I. M. Ternov, V. V. Mikhailin, and V. R. Khalilov, Synchrotron Radiation and Its Applications [in Russian], Publishing House of Moscow State University, Moscow (1980). 1980.Google Scholar
  6. 6.
    D. F Alferov, Yu. A. Bashmakov, and P. A. Cherenkov, Usp. Fiz. Nauk, 157, Nо. 3, 389 (1989).CrossRefGoogle Scholar
  7. 7.
    D. F. Alferov, Yu. A. Bashmakov, and E. G. Bessonov, Zh. Тekh. Fiz., 43, Nо. 10, 2126–2132 (1973).Google Scholar
  8. 8.
    N. M. Kroll and W. A. McMullin, Phys. Rev. A, 17, 300 (1978).ADSCrossRefGoogle Scholar
  9. 9.
    W. B. Colson, Nucl. Instrum. Methods A, 393, 82 (1997).ADSCrossRefGoogle Scholar
  10. 10.
    P. Sprangle and R. A. Smith, Phys. Rev. A, 21, 293 (1980).ADSCrossRefGoogle Scholar
  11. 11.
    G. Margaritondo and P. R. Ribic, J. Synchrotron Rad., 18, 101–108 (2011).CrossRefGoogle Scholar
  12. 12.
    K. Zhukovsky, J. Electromagn. Waves Appl., 29, 132 (2015).CrossRefGoogle Scholar
  13. 13.
    G. Dattoli, V. V. Mikhailin, and K. Zhukovsky, J. Appl. Phys., 104, 124507-1– 124507-8 (2008).ADSGoogle Scholar
  14. 14.
    G. Dattoli, K. V. Zhukovsky, and V. V. Mikhailin, Moscow Univ. Phys. Bull., 64, Nо. 5, 507–512 (2009).ADSCrossRefGoogle Scholar
  15. 15.
    K. Zhukovsky, J. Electromagn. Waves Appl., 28, Nо. 15, 1869–1887 (2014).Google Scholar
  16. 16.
    G. Dattoli, A. Doria, L. Giannessi, and P. L. Ottaviani, Nucl. Instrum. Methods Phys. Res. A, 507, 388–391 (2003).ADSCrossRefGoogle Scholar
  17. 17.
    Qika Jia, Phys. Rev. ST-AB, 14, 060702 (2011).ADSGoogle Scholar
  18. 18.
    K. V. Zhukovsky, Moscow Univ. Phys. Bull., 73, Nо. 4, 364–371 (2018); DOI: Scholar
  19. 19.
    K. Zhukovsky, Laser Part. Beams, 34, 447 (2016).ADSCrossRefGoogle Scholar
  20. 20.
    B. Prakash, V. Huse, M. Gehlot, et al., Optik, 127, 1639–1643 (2016).ADSCrossRefGoogle Scholar
  21. 21.
    Jeevakhan Hussain and G. Mishra, Nucl. Instrum. A, 656, 101–106 (2011).ADSCrossRefGoogle Scholar
  22. 22.
    G. Mishra and Jeevakhan Hussain, Nucl. Instrum. A, 621, 637–642 (2010).ADSCrossRefGoogle Scholar
  23. 23.
    K. Zhukovsky, Nucl. Instrum. B, 369, 9 (2016).ADSCrossRefGoogle Scholar
  24. 24.
    K. Zhukovsky, Opt. Commun., 353, 35 (2015).ADSCrossRefGoogle Scholar
  25. 25.
    K. V. Zhukovsky, Moscow Univ. Phys. Bull., 73, Nо. 5, 462–467 (2018); DOI: Scholar
  26. 26.
    K. Zhukovsky and A. Kalitenko, J. Synchrotron Rad., 26, 159–169 (2019).CrossRefGoogle Scholar
  27. 27.
    K. Zhukovsky and A. Kalitenko, J. Synchrotron Rad., 26, 605–606 (2019).CrossRefGoogle Scholar
  28. 28.
    K. V. Zhukovsky, Zh. Tekh. Fiz., 89, Nо. 3, 426–435 (2019).Google Scholar
  29. 29.
    K. V. Zhukovsky and A. M. Kalitenko, Russ. Phys. J., 62, Nо. 2, 354–362 (2019).Google Scholar
  30. 30.
    G. Dattoli, V. V. Mikhailin, P. L. Ottaviani, and K. Zhukovsky, J. Appl. Phys., 100, 084507 (2006).ADSCrossRefGoogle Scholar
  31. 31.
    K. Zhukovsky and I. Potapov, Laser Part. Beams, 35, 326 (2017).ADSCrossRefGoogle Scholar
  32. 32.
    K. Zhukovsky, EPL, 119, 34002 (2017).ADSCrossRefGoogle Scholar
  33. 33.
    K. Zhukovsky, Russ. Phys. J., 60, Nо. 9, 1630–1637 (2017).Google Scholar
  34. 34.
    K. Zhukovsky, J. Phys. D, 50, 505601 (2017).CrossRefGoogle Scholar
  35. 35.
    K. Zhukovsky, J. Appl. Phys., 122, 233103 (2017).ADSCrossRefGoogle Scholar
  36. 36.
    K. Zhukovsky, Opt. Commun., Nо. 418, 57–64 (2018).Google Scholar
  37. 37.
    K. Zhukovsky, J. Optics., 20, Nо. 9, 095003 (2018).Google Scholar
  38. 38.
    K. V. Zhukovsky, I. A. Potapov, and A. M. Kalitenko, Radiophys. Quant. Electr., 61, Nо. 3, 216–231 (2018).Google Scholar
  39. 39.
    T. Tanaka and H. Kitamura, J. Synchr. Rad., 8, 1221 (2001).CrossRefGoogle Scholar
  40. 40.
    K. Lee, J. Mun, S. Hee Park, et al., Nucl. Instrum. Meth. Phys. Res. A, 776, 27–33 (2015).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.M. V. Lomonosov Moscow State UniversityMoscowRussia

Personalised recommendations