Nonlinear Theory: Optical Mode Analysis

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


The previous chapter dealt with the nonlinear theory in the steady-state regime based on the slowly varying envelope approximation (SVEA). Most of the time-dependent free-electron laser simulation codes that are in use at the present time deal either with an extension of the SVEA in order to solve the wave equation or a particle-in-cell simulation where Maxwell’s equations are solved using a finite-difference time-domain (FDTD) algorithm. The time-dependent formulation presented in this chapter is an extension of the SVEA, in which the SVEA is extended by allowing the slowly varying amplitude to vary in both axial position and time. A time-dependent formulation is necessary to simulate short-wavelength free-electron lasers employing radio-frequency linear accelerators (RF linacs) or storage rings. RF linacs produce high-energy beams with picosecond pulse times and bunch charges of at most several nano-Coulombs. In X-ray free-electron lasers, the actual bunch charge used is about 250 pC or less. Since the growth rate depends upon the peak current, it is desirable to produce bunches with peak currents of several hundred to several thousand amperes, and this requires compression of the bunch to sub-picosecond pulse times. As a result, the slippage of the optical field relative to the electrons can be significant. In addition to describing the slippage of the optical pulse, time dependence is also needed to study the spectral properties of the optical field such as the temporal coherence, linewidth, sideband production, etc. Furthermore, in contrast to the guided-mode analysis used for the steady-state formulation presented in the preceding chapter, the three-dimensional formulations presented in this chapter make use of superpositions of Gaussian optical modes to represent the radiation fields.


Time dependence Slippage Gaussian optical modes Gauss-Hermite modes Gauss-Laguerre modes Slowly-Varying Envelope Approximation SVEA Particle-in-Cell simulation Finite-difference time domain FDTD Optical guiding Gain guiding Refractive guiding Separable beam limit Lagrangian time coordinate Group velocity Quasi-static assumption Shot-noise Poisson statistics Quiet-start Slices Elliptical wigglers JJ-factor APPLE-II wiggler Generalized Pierce parameter Ming-Xie’s parameterization Quadrupole and dipole fields Self-Amplified Spontaneous Emission SASE Rayleigh range Vacuum diffraction Gouy phase shift Source-Dependent Expansion SDE Quadrupole lattice 


  1. 1.
    S. Reiche, Genesis 1.3: a fully time-dependent FEL simulation code. Nucl. Inst. Methods A429, 243 (1999)CrossRefGoogle Scholar
  2. 2.
    E.L. Saldin, E.A. Schneidmiller, M. Yurkov, FAST: three-dimensional time-dependent FEL simulation code. Nucl. Inst. Methods Phys. Res. A429, 233 (1999)CrossRefGoogle Scholar
  3. 3.
    H.P. Freund, S.G. Biedron, S.V. Milton, Nonlinear harmonic generation in free-electron lasers. IEEE J. Quantum Electron. 36, 275 (2000)CrossRefGoogle Scholar
  4. 4.
    L. Giannessi, in Proceedings of the 2006 FEL Conference (, 2006), p. MOPPH026
  5. 5.
    W. Fawley, An Informal Manual for GINGER and Its Post-Processor XPLOTGIN. LBID-2141, CBP Tech Note-104, UC-414 (1995)Google Scholar
  6. 6.
    H.P. Freund, Time-dependent simulation of free-electron laser amplifiers and oscillators. Phys. Rev. ST-AB 8, 110701 (2005)Google Scholar
  7. 7.
    L.T. Campbell, B.W.J. McNeil, PUFFIN: a three-dimensional, unaveraged free-electron laser simulation code. Phys. Plasmas 19, 093119 (2012)CrossRefGoogle Scholar
  8. 8.
    C.K. Birdsall, A.B. Langdon, Plasma Physics Via Computer Simulation (McGraw-Hill, New York, 1985)Google Scholar
  9. 9.
    N. Piovella, High gain free electron laser amplifiers starting from coherent and incoherent spontaneous emission. Phys. Plasmas 6, 3358 (1999)CrossRefGoogle Scholar
  10. 10.
    E.T. Scharlemann, A.M. Sessler, J.S. Wurtele, Optical guiding in a free-electron laser. Phys. Rev. Lett. 54, 1925 (1985)CrossRefGoogle Scholar
  11. 11.
    G.T. Moore, High-gain and large-diffraction regimes of the free-electron laser. Nucl. Inst. Methods A250, 381 (1986)CrossRefGoogle Scholar
  12. 12.
    J.E. LaSala, D.A.G. Deacon, E.T. Scharlemann, Optical guiding simulations for high-gain short-wavelength free-electron lasers. Nucl. Inst. Methods A250, 389 (1986)CrossRefGoogle Scholar
  13. 13.
    A. Amir, Y. Greenzweig, Three-dimensional free-electron laser gain and evolution of optical modes. Nucl. Inst. Methods A250, 404 (1986)CrossRefGoogle Scholar
  14. 14.
    P. Luchini, S. Solimeno, Optical guiding in a free-electron laser. Nucl. Inst. Methods A250, 413 (1986)CrossRefGoogle Scholar
  15. 15.
    M. Xie, D.A.G. Deacon, Theoretical study of free-electron laser active guiding in the small signal regime. Nucl. Inst. Methods A250, 426 (1986)CrossRefGoogle Scholar
  16. 16.
    J. Gallardo, L.R. Elias, Multimode dynamics in a free-electron laser with energy shift. Nucl. Inst. Methods A250, 438 (1986)CrossRefGoogle Scholar
  17. 17.
    B.D. McVey, Three-dimensional simulations of free-electron laser physics. Nucl. Inst. Methods A250, 449 (1986)CrossRefGoogle Scholar
  18. 18.
    R.H. Pantell, J. Feinstein, Free-electron laser mode propagation at saturation. IEEE J. Quantum Electron. QE–23, 1534 (1987)CrossRefGoogle Scholar
  19. 19.
    P. Sprangle, A. Ting, C.M. Tang, Analysis of radiation focusing and steering in the free-electron laser by use of a source dependent expansion technique. Phys. Rev. A 36, 2773 (1987)CrossRefGoogle Scholar
  20. 20.
    S.Y. Cai, A. Bhattacharjee, T.C. Marshall, Optical guiding in a Raman free-electron laser. IEEE J. Quantum Electron. QE–23, 1651 (1987)CrossRefGoogle Scholar
  21. 21.
    P. Luchini, More on optical guiding in a free-electron laser. Nucl. Inst. Methods A259, 150 (1987)CrossRefGoogle Scholar
  22. 22.
    R.W. Warren, B.D. McVey, Bending and focusing effects in a free-electron laser oscillator I: simple models. Nucl. Inst. Methods A259, 154 (1987)CrossRefGoogle Scholar
  23. 23.
    B.D. McVey, R.W. Warren, Bending and focusing effects in a free-electron laser oscillator II: numerical simulations. Nucl. Inst. Methods A259, 158 (1987)CrossRefGoogle Scholar
  24. 24.
    A. Bhattacharjee, S.Y. Cai, S.P. Chang, J.W. Dodd, T.C. Marshall, Observations of optical guiding in a Raman free-electron laser. Phys. Rev. Lett. 60, 1254 (1988)CrossRefGoogle Scholar
  25. 25.
    G. Bourianoff, B. Moore, M. Rosenbluth, F. Waelbroeck, H. Waelbroeck, H.V. Wong, Adaptive eigenmode expansion for 3-D free-electron laser simulations. Nucl. Inst. Methods A272, 340 (1988)CrossRefGoogle Scholar
  26. 26.
    T.M. Antonsen, B. Levush, Optical guiding in the separable beam limit. Nucl. Inst. Methods A272, 472 (1988)CrossRefGoogle Scholar
  27. 27.
    A. Fruchtman, Optical guiding in a sheet-beam free-electron laser. Phys. Rev. A 37, 2989 (1988)CrossRefGoogle Scholar
  28. 28.
    M.H. Whang, S.P. Kuo, Self-focusing of laser pulses in magnetized relativistic electron beams. Nucl. Inst. Methods A272, 477 (1988)CrossRefGoogle Scholar
  29. 29.
    Y.J. Chen, S. Solimeno, L. Carlomusto, Optical guiding in a free-electron laser with full account of electron wiggling and 3-D propagation. Nucl. Inst. Methods A272, 490 (1988)CrossRefGoogle Scholar
  30. 30.
    J.C. Gallardo, G. Dattoli, A. Renieri, T. Hermsen, Integral equation for the laser field: multimode description of a free-electron laser oscillator. Nucl. Inst. Methods A272, 516 (1988)CrossRefGoogle Scholar
  31. 31.
    A. Bhowmik, S. Bitterly, R.A. Cover, P. Kennedy, R.H. Labbe, Transverse mode control in high gain free-electron lasers with grazing incidence, unstable ring resonators. Nucl. Inst. Methods A272, 524 (1988)CrossRefGoogle Scholar
  32. 32.
    M. Xie, D.A.G. Deacon, J.M.J. Madey, The guided mode expansion in free-electron lasers. Nucl. Inst. Methods A272, 528 (1988)CrossRefGoogle Scholar
  33. 33.
    T.M. Antonsen, G. Laval, Suppression of sidebands by diffraction in a free-electron laser. Phys. Fluids B 1, 1721 (1989)CrossRefGoogle Scholar
  34. 34.
    M. Lontano, A.M. Sergeev, A. Cardinali, Dynamical self-focusing of high-power free-electron laser radiation in a magnetized plasma. Phys. Fluids B 1, 901 (1989)CrossRefGoogle Scholar
  35. 35.
    A. Bhattacharjee, S.Y. Cai, S.P. Chang, J.W. Dodd, A. Fruchtman, T.C. Marshall, Theory and observation of optical guiding in a free-electron laser. Phys. Rev. A 40, 5081 (1989)CrossRefGoogle Scholar
  36. 36.
    N. Metzler, T.M. Antonsen, B. Levush, Nonlinear optical guiding in the separable beam limit. Phys. Fluids B 2, 1038 (1990)CrossRefGoogle Scholar
  37. 37.
    B. Hafizi, A. Ting, P. Sprangle, C.M. Tang, Effect of tapering on optical guiding and sideband growth in a finite-pulse free-electron laser. Nucl. Inst. Methods A296, 442 (1990)CrossRefGoogle Scholar
  38. 38.
    H.P. Freund, C.L. Chang, Effect of the lower beat wave on optical guiding in planar wiggler free-electron lasers. Phys. Rev. A 42, 6737 (1990)CrossRefGoogle Scholar
  39. 39.
    H.P. Freund, T.M. Antonsen Jr., The relationship between optical guiding and the relative phase in free-electron lasers. IEEE J. Quantum Electron. QE–27, 2539 (1991)CrossRefGoogle Scholar
  40. 40.
    T.J. Orzechowski, E.T. Scharlemann, D.B. Hopkins, Measurement of the phase of the electromagnetic wave in a free-electron laser amplifier. Phys. Rev. A 35, 2184 (1987)CrossRefGoogle Scholar
  41. 41.
    H.P. Freund, Multimode nonlinear analysis of free-electron laser amplifiers in three dimensions. Phys. Rev. A 37, 3371 (1988)CrossRefGoogle Scholar
  42. 42.
    C. Penman, B.W.J. McNeil, Simulation of input electron noise in the free-electron laser. Opt. Commun. 90, 82 (1992)CrossRefGoogle Scholar
  43. 43.
    B.W.J. McNeil, M.W. Poole, G.R.M. Robb, Unified model of electron beam shot noise and coherent spontaneous emission in the helical wiggler free-electron laser. Phys. Rev. ST-AB 6, 070701 (2003)Google Scholar
  44. 44.
    L. Giannessi, Harmonic generation and linewidth narrowing in seeded free-electron lasers, in Proceedings of the 26th International Conference on Free-Electron Lasers (, 2004), p. 37
  45. 45.
    H.P. Freund, L. Giannessi, W.H. Miner Jr., The effect of shot noise on the start up of the fundamental and harmonics in free-electron lasers. J. Appl. Phys. 104, 123114 (2008)CrossRefGoogle Scholar
  46. 46.
    W.M. Fawley, Algorithm for loading shot noise microbunching in multidimensional free-electron laser simulation codes. Phys. Rev. ST-AB 5, 070701 (2002)Google Scholar
  47. 47.
    J. Bahrdt, W. Frentrup, A. Gaupp, M. Scheer, W. Gudat, G. Ingold, S. Sasaki, Elliptically polarized insertion devices at BESSY II. Nucl. Inst. Methods A467-468, 21 (2001)CrossRefGoogle Scholar
  48. 48.
    A.A. Lutman et al., Polarization control in an x-ray free-electron laser. Nat. Photonics 10, 468 (2016)CrossRefGoogle Scholar
  49. 49.
    H.P. Freund, P.J.M. van der Slot, D.L.A.G. Grimminck, I.D. Setya, P. Falgari, Three-dimensional, time-dependent simulation of free-electron lasers with planar, helical, and elliptical undulators. New J. Phys. 19, 023020 (2017)CrossRefGoogle Scholar
  50. 50.
    J.R. Henderson, L.T. Campbell, H.P. Freund, B.W.J. McNeil, Modelling elliptically polarized free electron lasers. New J. Phys. 18, 062003 (2016)CrossRefGoogle Scholar
  51. 51.
    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
  52. 52.
    A. Yariv, Quantum Electronics, 2nd edn. (John Wiley & Sons, New York, 1967)Google Scholar
  53. 53.
    P.A. Sprangle, A. Ting, C.M. Tang, Analysis of radiation focusing and steering in the free-electron laser by use of a source-dependent expansion. Phys. Rev. A 36, 2773 (1987)CrossRefGoogle Scholar
  54. 54.
    X.J. Wang, T. Watanabe, Y. Shen, R. Li, J.B. Murphy, T. Tsang, H.P. Freund, Efficiency enhancement using electron energy detuning in a laser seeded free electron laser amplifier. Appl. Phys. Lett. 91, 181115 (2007)CrossRefGoogle Scholar
  55. 55.
    X.J. Wang, H.P. Freund, D. Harder, W.H. Miner Jr., J.B. Murphy, H. Qian, Y. Shen, X. Yang, Efficiency and spectrum enhancement in a tapered free-electron laser amplifier. Phys. Rev. Lett. 103, 154801 (2009)CrossRefGoogle Scholar
  56. 56.
    D.C. Quimby, S.C. Gottschalk, F.E. James, K.E. Robinson, J.M. Slater, A.S. Valla, Development of a 10-meter, wedged-pole undulator. Nucl. Inst. Methods A285, 281 (1989)CrossRefGoogle 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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