1. Advancements on Theory and Simulations of FELs
B.W.J. McNeil
University of Strathclyde

2. Much of the material was supplied by colleagues worldwide – thank you!
Outline
Unaveraged simulation codes
Spontaneous Noise & QFEL
Amplifiers
Oscillators
Plasma Accelerator Driven FELs?
Not really FELs, but….
Much of the material was supplied by colleagues worldwide – thank you!

3. Unaveraged simulation codes

4. Motivation
Cannot model broad bandwidth radiation with SVEA – at best, can only model about the frequency range:
ωr/2 < ωr < 3ωr/2
New methods may violate SVEA – e.g. few cycle pulses
Electron evolution constrained to be localised in averaged models while methods such as e.g. EEHG involves large changes in electron position to prepare the beam
Plasma based accelerators can produce relatively short electron pulses with significant energy chirps and spread – large electron pulse shape (current) changes during emission and coherent spontaneous interactions need to be modelled – cannot do with averaged models

5. Every undulator period
FEL Averaged Code
Radiation envelope discretised at radiation period (slice)
Electrons are confined to each slice, with periodic BC’s in phase space each slice: λr
Every undulator period
Current assumed constant within each ‘slice’

6. FEL Unaveraged Code
Radiation field is discretised below wavelength scale – no SVEA
No electron averaging - a Particle In Cell (PIC) model
Electrons free to propagate – no periodic boundary conditions

7. AURORA* (The Roman Goddess of dawn)
*SI Bajlekov, SM Hooker & R Bartolini, Proc. FEL 2009 Liverpool, MOPC42 (2009)

8. This is a 1D code, i.e. no transverse diffractive effects, which solves the coupled radiation-electron equations in frequency space. This has many similarities with MUFFIN [1]. It is written in MATLAB and C. Used to study broad-band seeding of FEL using HHG in a more rigorous way than previously (see its refs [11-13] and [2] below) – Result is that the modelling of the full HHG bandwidth is unnecessary.
HHG seed spectrum
FEL gain spectrum
[1] N. Piovella, Phys. Plasmas 6, 3358 (1999)
[2] BWJ McNeil et al, NJP 9, 82 (2007)

10. This 1D model is interesting in the derivation of its working equations which can model both forward and backward propagation of the radiation field - the latter only of interest for low energy electron beams.

11. Puffin: A three dimensional, unaveraged free electron laser simulation code*
*N. Piovella, Phys. Plasmas 6, 3358 (1999)

12. Can also run in fast 1D mode enabling parameter scans

13. Numerical Solution – Electrons to Macroparticles
Electron beam is discretised into electron macroparticles by method of [1]
Sampling an electron beam with a gaussian current distribution
Create a grid...
[1] – BWJ McNeil, MW Poole & GRM Robb, Phys Rev ST-AB, 6, (2003)

14. Numerical Solution – Electrons to Macroparticles
Electron beam is discretised into electron macroparticles by method of [1]
Sampling an electron beam with a gaussian current distribution
Create a grid, and into each element we assign a macroparticle with charge weight determined by the current distribution
To simulate noise assign random Poisson deviate to each macroparticle charge weight & position
[1] – BWJ McNeil, MW Poole & GRM Robb, Phys Rev ST-AB, 6, (2003)

15. Transverse space
Radiation field is modelled with a grid of nodes with linear interpolant and macroparticles move freely throughout field elements
Electron beam is matched to a uniform focussing channel - in transverse space beam has a constant radius
Electrons therefore remain confined to an inner set of nodes. Outer field nodes are then used for radiation field diffraction
Low frequency components diffract quickly. Periodic boundary conditions cause a low frequency background signal to build up which may interfere with the FEL interaction. Currently, this is filtered out during propagation – other options are available.
Periodic boundary conditions
Periodic boundary conditions

16. Numerical Solution - Field
Field solver uses split-step Fourier method [1]
1st half step: Field diffraction solved with Fourier Transforms – essentially an analytic solution using FFT and iFFT
2nd half step: Self consistent field driving and electron propagation.
Field driven by electron sources via the Finite Element Method (Galerkin) [2] and requires solution of sparse linear system
Electrons driven by field self-consistently using Runge-Kutte 4th order
[1] - R.H. Hardin and F.D. Tappert, SIAM Rev. 15, 453 (1973)
[2] - K.H. Huebner, E.A. Thornton and T.G. Byrom, The Finite Element Method for Engineers

17. Parallelism Uses Fortran 95 with MPI
Parallel Fourier Transforms - FFTW 2.1.5
Parallel Linear Solver for Sparse Systems – uses MUMPS with possible change to Hiper

18. Memory Distribution (ct-z)
Not immediately intuitive how to maintain a consistent memory distribution with minimum processor communication:
- field
- electron beam PN
- Nth Processor P0 P1 P2 P3 P4 P5 P6 P7
(ct-z)

19. Memory Distribution (ct-z)
Not immediately intuitive how to maintain a consistent memory distribution with minimum processor communication:
- field
- electron beam P0 P1 P2 P3 P4 P5 P6 P7
(ct-z)

20. Memory Distribution Compromise
As the # macroparticles >> # field nodes
(The electron beam is in 6D phase space and the field is in 3D real space):
Electrons are distributed uniformly amongst the processors
Complete field can be stored on each processor as:
memory(field) << memory(beam)

21. Code parallel algorithm

22. Machines used so far In house Sun x4600 server
16 cores
DL1 on NW grid at Daresbury Labs, UK
30-40 cores
ARCHIE at University of Strathclyde
cores

23. Current development on UK Hartree HPC Centre
Processor number scaling performed by L. Anton at Hartree

24. Broadband Field - variation with observation angle
- on-axis spectra, then in order of increasing arbitrary angle:
Avg. code range
ω/ωr

25. Broad bandwidth 3D radiation

26. Non-localised electron evolution & very short pulses*
*Campbell & McNeil, ‘A simple model for the generation of ultra-short radiation pulses’, To be published in Proc. FEL 2012, Nara, Japan.

27. Transverse intensity about peak power
Temporal Power
Spectral Power

28. 2-colour operation* *In collaboration with Sven Reiche
(per undulator module)
13 undulator modules in total, laid out as:
Tune modules to one of 2 frequencies, by varying magnetic field only:
*In collaboration with Sven Reiche

29. Variable polarisation
Model variably polarized undulator
Radiation field polarization solved self-consistently – electrons only interact with correct polarization
May be useful investigating FEL concepts with variable polarization and multi-frequency model

30. Shorter pulses & energy modulation e.g.:
DJ Dunning, BWJ McNeil & NR Thompson, PRL 110, (2013) & this workshop

31. EEHG – another look at what can be done in non-averaged limit

32. EEHG mechanism* Periodic BCs
*Stupakov, Phys. Rev. Lett. 102, (2009)

33. EEHG without periodic BC’s or averaging*
*JR Henderson and BWJ McNeil, EPL, 100, (2012)

34. Normal radiator undulator

35. MLOK* radiator undulator
*Thompson, McNeil, PRL 100, (2008)

36. Spontaneous Noise & QFEL

37. The Quantum High-Gain FEL (QFEL)
Courtesy of Gordon Robb
In classical FEL theory, electron-light momentum exchange
is continuous and the photon recoil momentum is neglected
Classical induced
momentum spread (gRmcr)
i.e.
where
one-photon
recoil momentum(ħk)
>>
is the “quantum FEL parameter”
Consider the opposite case where
Classical induced
momentum spread (gmcr)
one-photon
recoil momentum(ħk)
<
i.e.
Electron-radiation momentum exchange is now discrete i.e.
so a quantum model of the electron-radiation interaction is required.

38. The Quantum High-Gain FEL (QFEL)
Courtesy of Gordon Robb
Classical regime (r>>1) _
Quantum regime (r<1) _
Field evolves as in classical FEL
particle model – continuous spectrum
Discrete spectrum in quantum regime
due to photon recoil.
improved temporal coherence but
lower power
See e.g.
G. Preparata, Phys. Rev. A 38, 233(1988)
R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB 9, (2006)
R.Bonifacio, N.Piovella, M.Cola, L.Volpe, NIMA 577, 745 (2007)

39. The Quantum High-Gain FEL (QFEL) – recent work
Courtesy of Gordon Robb
Generation of Harmonics in QFEL
G. R. M. Robb , NIM A593 , 87 (2008)
R. Bonifacio,  G. R. M. Robb and N. Piovella, Optics Comm. 284 , 1004 (2011)
Modelling Spontaneous Emission in QFEL
G. R. M. Robb and R. Bonifacio, Europhys. Lett. 94, (2011).
G. Geloni, V. Kocharyan, and E. Saldin, Europhys. Lett. 98, (2012).
A. Potylitsyn and A. Kol’chuzhkin, Europhys. Lett. 100, (2012).
Competition Between Spontaneous and Stimulated Emission in QFEL
G. R. M. Robb and R. Bonifacio, Phys. Plasmas 19, (2012).
G. R. M. Robb and R. Bonifacio, Phys. Plasmas 20, (2013).

40. - Theory
Experiment

41. The shot-noise ‘relaxes out’ in the drift section
6.5m Drift

42. Before drift After drift Other relevant works:
Note: Effects are presently relevant below x-ray photon energies in visible and perhaps UV.
Daniel Ratner, Shot Noise Suppression in
Linac Beams, This workshop.
Other relevant works:

43. It may be interesting to introduce the space charge effects identified by Gover et al. into the quantum model of Robb & Bonifacio….

44. Amplifiers

46. Harmonic Lasing

47. Harmonic Amplifier FEL*
fundamental
3rd harmonic
A relative phase change between electrons and fundamental radiation of n2π/3 (n - integer) will disrupt the fundamental-electron coupling and so the fundamental’s growth.
2π/3
However, a n2π/3 phase change for the fundamental is a n2π phase change for the 3rd harmonic – The 3rd harmonic interaction therefore suffers no disruption.
*McNeil, Robb & Poole, PAC 2005, Knoxville, Tennessee,
McNeil, Robb, Poole & Thompson, PRL 96, , 2006

48. Using a seeded steady-state model* (i.e. no pulses effects):
Seeded at fundamental
*McNeil, Robb, Poole & Thompson, PRL 96, (2006)

49.
Harmonic lasing:

50. Harmonic amplifier SASE optimisation simulations at LCLS and XFEL*
Fundamental filtered
SASE
SASE
Could be combined with a self-seeding method at fundamental to improve coherence at harmonic.
*Schneidmiller & Yurkov, PRST-AB 15, (2012)

51. Also works in FEL cavity oscillators for low gain*
Such solutions are ‘generic’ and apply across all wavelength ranges from IR to X-ray (XFELO)
*Gavin Cheung (supervisor McNeil), Modelling 4th Generation Light Sources, Undergraduate Project Report,
University of Strathclyde (April 2012).

52. ...and in FEL cavity oscillators in the high gain (RAFEL)*
Such solutions are ‘generic’ and apply across all wavelength ranges from IR to X-ray (XFELO)
*Gavin Cheung (supervisor McNeil), Modelling 4th Generation Light Sources, Undergraduate Project Report,
University of Strathclyde (April 2012).

53. High-Brightness SASE – Stage I*
Mean spike spacing
Bandwidth reduction

54. High-Brightness SASE – Stage II*
*BWJ McNeil, NR Thompson & DJ Dunning, Transform-Limited X-Ray Pulse Generation from a High-Brightness Self-Amplified Spontaneous-Emission Free-Electron Laser, PRL – In Press

55. To appear in Proc. FEL 2012, Nara, Japan
“How to optimize electron delays in a SASE FEL to reach the diffraction limit and operate a high power tapered FEL.”

56. Self-seeded
iSASE
SASE

57. See also:

58. Study of the second saturation in tapered FEL, which limits the amount of energy that can be extracted from an electron beam in an FEL.

59. Oscillators

60. Low Gain, High-Q To appear in Proc. FEL 2012, TUOC05
Provides a useful guide of how to go about modelling FEL oscillators using 1D to full simulation models using a number of codes. Importantly, direct comparison is made with three different operating Jefferson Lab FELs from the IR-UV. Interesting differences in ability of codes to predict correctly the efficiency and the gain.

61. See also:
S.V. Benson & M.D. Shinn, “Simulations of a Free-Electron Laser Oscillator at Jefferson Lab Lasing in the Vacuum Ultraviolet”: to be published in the next edition of the ICFA Newsletter

62. And:
Proc. of FEL 2011 (Shanghai, China), WEOCI1

63. High Gain, Low-Q RAFEL
SSSFEL12, December 2012

64. Regenerative Amplifier FEL

65. WHY A RAFEL?*
Robust FEL cavity design able to generate close to Fourier Transform limited tuneable output from feedback factors F : a self-seeding high gain FEL ideal for short wavelength generation
*Dunning, McNeil & Thompson, NIM A 593, (2008)

66. Example short-pulse RAFEL simulation*
Parameters are typical for a soft x-ray FEL:
Gaussian current electron pulse:
FEL parameter:
Undulator length:
Cavity feedback factor:
Cavity detuning:
*McNeil & Thompson, EPL, 96, 5400 (2011) & 98, (2012)

67. Short pulse RAFEL – stable, coherent output*
Start of undulator
End of undulator
Figure 1: Short pulse RAFEL simulation - showing saturated evolution at the undulator entrance. From top: scaled power at the beginning of the interaction at saturation; the current-weighted bunching; the scaled spectral power as a function of scaled frequency.
Figure 2: Short pulse RAFEL simulation - showing saturated evolution at the undulator exit.
Short ( ), high power, FT limited pulses with none of the jitter associated with SASE.
*McNeil & Thompson, EPL, 96, 5400 (2011) & 98, (2012)

68. RAFEL in MLOK configuration*
Start of undulator
End of undulator
*McNeil & Thompson, EPL, 96, 5400 (2011) & 98, (2012)

70. Plasma Accelerator Driven FELs?

71. And yesterday:

72. 1D theory and 3D (modified GENESIS) simulations show advantages of a transverse gradient FEL undulator for a beam with large energy spread, such as generated by laser-plasma accelerators. The idea for the transverse gradient undulator to compensate for beams with energy spread was originally developed by [1]:
[1] T. Smith, J. M. J. Madey, L. R. Elias and D. A. G. Deacon, J. Appl. Phys. 50, 4580 (1979).

73. For canted poles with undulator parameter of:
then an electron dispersion function of:
such that: ,
ensures that all electrons have the same resonant wavelength and results in a strong high-gain FEL instability.

74. Not really FELs, but….

75. Form factor
Single electron

77. Thank you …and again to those that supplied information of their recent works.