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Coherent Emission: Linear Theory

  • H. P. Freund
  • T. M. AntonsenJr.
Chapter

Abstract

In order to give rise to stimulated emission, it is necessary for the electron beam to respond in a collective manner to the radiation field and to form coherent bunches. This can occur when a light wave traverses an undulatory magnetic field such as a wiggler because the spatial variations of the wiggler and the electromagnetic wave combine to produce a beat wave, which is essentially an interference pattern. It is the interaction between the electrons and this beat wave which gives rise to the stimulated emission in free-electron lasers. In the case of a magnetostatic wiggler, this beat wave has the same frequency as the light wave, but its wavenumber is the sum of the wavenumbers of the electromagnetic and wiggler fields. As a result, the phase velocity of the beat wave is less than that of the electromagnetic wave, and it is called a ponderomotive wave. Since the ponderomotive wave propagates at less than the speed of light in vacuo, it can be in synchronism with electrons that are limited by that velocity. Our purpose in this chapter is to give a detailed discussion of the free-electron laser as a linear gain medium as well as to provide a comprehensive derivation of the relevant formulae for the gain in various configurations in both the idealized one-dimensional and the realistic three-dimensional limits. To this end, we derive the expressions for the gain in both the low- and high-gain regimes. The low-gain regime is relevant to short-wavelength free-electron laser oscillators driven by high-energy but low-current electron beams. In contrast, the results in the high (exponential)-gain regime are usually described in terms of a dispersion equation and are appropriate to free-electron laser amplifiers and SASE driven by intense relativistic electron beams.

Keywords

Ponderomotive wave SASE Vlasov-Maxwell equations Poisson’s equation Space-charge Wiggler strength parameter Pendulum equation Separatrix Linear stability Compton regime Raman regime Negative-mass regime Pierce parameter Low-gain regime Spectral function High-gain regime Dispersion equation Beam-plasma waves Thermal effects Thermal function JJ-factor Axial magnetic field Waveguide mode analysis Floquet’s theorem Optical mode analysis Ming Xie parameterization 

References

  1. 1.
    V.P. Sukhatme, P.A. Wolff, Stimulated Compton scattering as a radiation source – theoretical limitations. J. Appl. Phys. 44, 2331 (1973)CrossRefGoogle Scholar
  2. 2.
    T.J.T. Kwan, J.M. Dawson, A.T. Lin, Free-electron laser. Phys. Fluids 20, 581 (1977)CrossRefGoogle Scholar
  3. 3.
    N.M. Kroll, W.A. McMullin, Stimulated emission from relativistic electrons passing through a spatially periodic transverse magnetic field. Phys. Rev. A 17, 300 (1978)CrossRefGoogle Scholar
  4. 4.
    J.M.J. Madey, Relationship between mean radiated energy, mean squared radiated energy and spontaneous power spectrum in a power series expansion of the equations of motion in a free-electron laser. Nuovo Cimento 50B, 64 (1979)CrossRefGoogle Scholar
  5. 5.
    T.J.T. Kwan, J.M. Dawson, Investigation of the free-electron laser with a guide magnetic field. Phys. Fluids 22, 1089 (1979)CrossRefGoogle Scholar
  6. 6.
    I.B. Bernstein, J.L. Hirshfield, Amplification on a relativistic electron beam in a spatially periodic transverse magnetic field. Phys. Rev. A 20, 1661 (1979)CrossRefGoogle Scholar
  7. 7.
    P. Sprangle, R.A. Smith, V.L. Granatstein, Free-electron lasers and stimulated scattering from relativistic electron beams, in Infrared and Millimeter Waves, vol. 1, ed. by K. J. Button, (Academic, New York, 1979), p. 279CrossRefGoogle Scholar
  8. 8.
    L. Friedland, J.L. Hirshfield, Free-electron laser with a strong axial magnetic field. Phys. Rev. Lett. 44, 1456 (1980)CrossRefGoogle Scholar
  9. 9.
    H.P. Freund, P. Sprangle, D. Dillenburg, E.H. da Jornada, B. Liberman, R.S. Schneider, Coherent and incoherent radiation from free-electron lasers with an axial guide field. Phys. Rev. A 24, 1965 (1981)CrossRefGoogle Scholar
  10. 10.
    B. Bernstein, L. Friedland, Theory of free-electron laser in combined helical pump and axial guide magnetic fields. Phys. Rev. A 23, 816 (1981)CrossRefGoogle Scholar
  11. 11.
    A. Gover, P. Sprangle, A generalized formulation of free-electron lasers in the low-gain regime including transverse velocity spread and wiggler incoherence. J. Appl. Phys. 52, 599 (1981)CrossRefGoogle Scholar
  12. 12.
    S.T. Stenholm, A. Bambini, Single-particle theory of the free-electron laser in a moving frame. IEEE J. Quantum Electron. QE-17, 1363 (1981)CrossRefGoogle Scholar
  13. 13.
    C.C. Shih, A. Yariv, Inclusion of space-chsrge effects with Maxwell's equations in the single-particle analysis of free-electron lasers. IEEE J. Quantum Electron. QE-17, 1387 (1981)CrossRefGoogle Scholar
  14. 14.
    R. Coisson, Energy loss calculation of gain in a plane sinusoidal free-electron laser. IEEE J. Quantum Electron. QE-17, 1409 (1981)CrossRefGoogle Scholar
  15. 15.
    W.B. Colson, The nonlinear wave equation for higher harmonics in free-electron lasers. IEEE J. Quantum Electron. QE-17, 1417 (1981)CrossRefGoogle Scholar
  16. 16.
    N.M. Kroll, P.L. Morton, M.N. Rosenbluth, Free-electron lasers with variable parameter wigglers. IEEE J. Quantum Electron. QE-17, 1436 (1981)CrossRefGoogle Scholar
  17. 17.
    H.S. Uhm, R.C. Davidson, Free-electron laser instability for a relativistic annular electron beam in a helical wiggler field. Phys. Fluids 24, 2348 (1981)CrossRefGoogle Scholar
  18. 18.
    H.P. Freund, P. Sprangle, D. Dillenburg, E.H. da Jornada, R.S. Schneider, B. Liberman, Collective effects on the operation of free-electron lasers with an axial guide field. Phys. Rev. A 26, 2004 (1982)CrossRefGoogle Scholar
  19. 19.
    J.A. Davies, R.C. Davidson, G.L. Johnston, Compton and Raman free-electron laser stability properties for a cold electron beam propagating through a helical magnetic field. Aust. J. Plant Physiol. 33, 387 (1985)Google Scholar
  20. 20.
    L.K. Grover, R.H. Pantell, Simplified analysis of free-electron lasers using Madey’s theorem. IEEE J. Quantum Electron. QE-21, 944 (1985)CrossRefGoogle Scholar
  21. 21.
    L.F. Ibanez, S. Johnston, Finite-temperature effects in free-electron lasers. IEEE J. Quantum Electron. QE-19, 339 (1983)CrossRefGoogle Scholar
  22. 22.
    E. Jerby, A. Gover, Investigation of the gain regimes and gain parameters of the free-electron laser dispersion equation. IEEE J. Quantum Electron. QE-21, 1041 (1985)CrossRefGoogle Scholar
  23. 23.
    H.P. Freund, R.C. Davidson, D.A. Kirkpatrick, Thermal effects on the linear gain in free-electron lasers. IEEE J. Quantum Electron. 27, 2550 (1991)CrossRefGoogle Scholar
  24. 24.
    L. Friedland, A. Fruchtman, Amplification on relativistic electron beams in combined helical and axial magnetic fields. Phys. Rev. A 25, 2693 (1982)CrossRefGoogle Scholar
  25. 25.
    H.S. Uhm, R.C. Davidson, Helically distorted relativistic beam equilibria for free-electron laser applications. J. Appl. Phys. 53, 2910 (1982)CrossRefGoogle Scholar
  26. 26.
    W.A. McMullin, R.C. Davidson, Low-gain free-electron laser near cyclotron resonance. Phys. Rev. A 25, 3130 (1982)CrossRefGoogle Scholar
  27. 27.
    H.S. Uhm, R.C. Davidson, Free-electron laser instability for a relativistic solid electron beam in a helical wiggler field. Phys. Fluids 26, 288 (1983)CrossRefGoogle Scholar
  28. 28.
    H.P. Freund, P. Sprangle, Unstable electrostatic beam modes in free-electron laser systems. Phys. Rev. A 28, 1835 (1983)CrossRefGoogle Scholar
  29. 29.
    C. Grebogi, H.S. Uhm, Vlasov susceptibility of relativistic magnetized plasma and application to free-electron lasers. Phys. Fluids 29, 1748 (1986)CrossRefGoogle Scholar
  30. 30.
    N.S. Ginzburg, Diamagnetic and paramagnetic effects in free-electron lasers. IEEE Trans. Plasma Sci. PS-15, 411 (1987)CrossRefGoogle Scholar
  31. 31.
    H.P. Freund, R.C. Davidson, G.L. Johnston, Linear theory of the collective Raman interaction in a free-electron laser with a planar wiggler and an axial guide field. Phys. Fluids B 2, 427 (1990)CrossRefGoogle Scholar
  32. 32.
    J.R. Cary, T.J.T. Kwan, Theory of off-axis mode production by free-electron lasers. Phys. Fluids 24, 729 (1981)CrossRefGoogle Scholar
  33. 33.
    T.J.T. Kwan, J.R. Cary, Absolute and convective instabilities in two-dimensional free-electron lasers. Phys. Fluids 24, 899 (1981)CrossRefGoogle Scholar
  34. 34.
    H.P. Freund, S. Johnston, P. Sprangle, Three-dimensional theory of free-electron lasers with an axial guide field. IEEE J. Quantum Electron. QE-19, 322 (1983)CrossRefGoogle Scholar
  35. 35.
    H.P. Freund, A.K. Ganguly, Three-dimensional theory of the free-electron laser in the collective regime. Phys. Rev. A 28, 3438 (1983)CrossRefGoogle Scholar
  36. 36.
    P. Luchini, S. Solimeno, Gain and mode-coupling in a three-dimensional free-electron laser: a generalization of Madey's theorem. IEEE J. Quantum Electron. QE-21, 952 (1985)CrossRefGoogle Scholar
  37. 37.
    M.N. Rosenbluth, Two-dimensional effects in free-electron lasers. IEEE J. Quantum Electron. QE-21, 966 (1985)CrossRefGoogle Scholar
  38. 38.
    B.Z. Steinberg, A. Gover, S. Ruschin, Three-dimensional theory of free-electron lasers in the collective regime. Phys. Rev. A 36, 147 (1987)CrossRefGoogle Scholar
  39. 39.
    C.J. Elliot, M.J. Schmitt, Small-signal gain for a planar free-electron laser with a period magnetic field. IEEE Trans. Plasma Sci. PS-15, 319 (1987)CrossRefGoogle Scholar
  40. 40.
    A. Fruchtman, High-density thick beam free-electron laser. Phys. Rev. A 37, 4259 (1988)CrossRefGoogle Scholar
  41. 41.
    T.M. Antonsen, P.E. Latham, Linear theory of a sheet beam free-electron laser. Phys. Fluids 31, 3379 (1988)CrossRefGoogle Scholar
  42. 42.
    V.K. Tripathi, C.S. Liu, A slow wave free-electron laser. IEEE. Trans. Plasma Sci. PS-17, 583 (1989)CrossRefGoogle Scholar
  43. 43.
    A. Fruchtman, H. Weitzner, Raman free-electron laser with transverse density gradients. Phys. Rev. A 39, 658 (1989)CrossRefGoogle Scholar
  44. 44.
    L.H. Yu, S. Krinsky, R.L. Gluckstern, Calculation of universal scaling function for free-electron laser gain. Phys. Rev. Lett. 64, 3011 (1990)CrossRefGoogle Scholar
  45. 45.
    Y.H. Chin, K.-J. Kim, M. Xie, Three-dimensional theory of the small-signal, high-gain free-electron laser including betatron oscillations. Phys. Rev. A 46, 6662 (1992)CrossRefGoogle Scholar
  46. 46.
    Y.H. Chin, K.-J. Kim, M. Xie, Three-dimensional free-electron laser dispersion relation including betatron oscillations. Nucl. Instr. Meth. A318, 481 (1992)CrossRefGoogle Scholar
  47. 47.
    J.R. Pierce, Traveling Wave Tubes (Van Nostrand, New York, 1950)Google Scholar
  48. 48.
    J. Masud, T.C. Marshall, S.P. Schlesinger, F.G. Yee, W.M. Fawley, E.T. Scharlemann, S.S. Yu, A.M. Sessler, E.J. Sternbach, Sideband control in a millimeter-wave free-electron laser. Phys. Rev. Lett. 58, 763 (1987)CrossRefGoogle Scholar
  49. 49.
    J. Fajans, G. Bekefi, Measurements of amplification and phase shift in a free-electron laser. Phys. Fluids 29, 3461 (1986)CrossRefGoogle Scholar
  50. 50.
    M. Xie, Design optimization for an x-ray free electron laser driven by the SLAC linac, in Proc. IEEE 1995 Particle Accelerator Conference, vol. 183, IEEE Cat. No. 95CH35843 (1995)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • H. P. Freund
    • 1
  • T. M. AntonsenJr.
    • 2
  1. 1.University of Maryland, University of New MexicoViennaUSA
  2. 2.University of MarylandPotomacUSA

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