Energy Recovered Linacs and Ponderomotive Spectral Broadening

1
Energy Recovered Linacs andPonderomotive
Spectral Broadening

• G. A. Krafft
• Jefferson Lab

2
Outline

• Recirculating linacs defined and described
• Review of recirculating SRF linacs
• Energy recovered linacs
• Energy recovered linacs as light sources
• Ancient history lasers and electrons
• Dipole emission from a free electron
• Thomson scattering
• Motion of an electron in a plane wave
• Equations of motion
• Exact solution for classical electron in a plane
  wave
• Applications to scattered spectrum
• General solution for small a
• Finite a effects
• Ponderomotive broadening
• Sum Rules
• Conclusions

3
Schematic Representation of Accelerator Types
RF Installation
Beam injector and dump
Beamline
Ring
Recirculating Linac
Linac
4
Why Recirculate?

• Performance upgrade of a previously installed
  linac
• Stanford Superconducting Accelerator and MIT
  Bates doubled their energy this way
• Cheaper design to get a given performance
• Microtrons, by many passes, reuse expensive RF
  many times to get energy up. Penalty is that the
  average current has to be reduced proportional to
  1/number passes, for the same installed RF.
• Jefferson Lab CEBAF type machines add passes
  until the decremental gain in RF system and
  operating costs no longer pays for an additional
  recirculating loop
• Jefferson Lab FEL and other Energy Recovered
  Linacs (ERLs) save the cost of higher average
  power RF equipment (and much higher operating
  costs) at higher CW operating currents by
  reusing beam energy through beam recirculation.

5
Beam Energy Recovery
Recirculation path length in standard
configuration recirculated linac. For energy
recovery choose it to be (n 1/2)?RF. Then
6
Beam Energy Recovery
Recirculation path length in herring-bone
configuration recirculated linac. For energy
recovery choose it to be n?RF. Note additional
complication path length has to be an integer at
each and every different accelerating cavity
location in the linac.
7
Comparison between Linacs and Storage Rings

• Advantage Linacs
• Emittance dominated by source emittance and
  emittance growth down linac
• Beam polarization easily produced at the
  source, switched, and preserved
• Total transit time is quite short
• Beam is easily extracted. Utilizing source laser
  control, flexible bunch patterns possible
• Long undulaters are a natural addition
• Bunch durations can be SMALL (10-100 fsec)

8
Comparison Linacs and Storage Rings

• Advantage Storage Rings
• Up to now, the stored average current is much
  larger
• Very efficient use of accelerating voltage
• Technology well developed and mature
• Disadvantage Storage Rings
• Technology well developed and mature (maybe!)
• The synchrotron radiation damping equilibrium,
  and the emittance and bunch length it generates,
  must be accepted

9
Power Multiplication Factor

• Energy recovered beam recirculation is nicely
  quantified by the notion of a power
  multiplication factor
• where Prf is the RF power needed to
  accelerate the beam
• By the first law of thermodynamics (energy
  conservation!) k lt 1
• in any linac not recirculated. Beam
  recirculation with beam deceleration somewhere is
  necessary to achieve k gt 1
• If energy IS very efficiently recycled from the
  accelerating to the decelerating beam

10
High Multiplication Factor Linacs
Recirculated Linacs
Normal Conducting Recirculators kltlt1
LBNL Short Pulse X-ray Facility (proposed) k0.1
CEBAF (matched beam load) k0.99 (typical) k0.8
High Multiplication Factor Superconducting
Linacs
JLAB IR DEMO k16 JLAB 10 kW Upgrade
k33 Cornell/JLAB ERL k200 (proposed) BNL PERL
k500 (proposed)
Will use the words High Multiplication Factor
Linac for those designs that feature high k.
11
Comparison Accelerator Types
Typical results by accelerator type
12
More Modern Reason to Recirculate!

• A renewed general interest in beam recirculation
  has arisen due to the success of Jefferson Labs
  high average current energy recovered Free
  Electron Lasers (FELs), and the broader
  realization that it may be possible to achieve
  beam parameters Unachievable in storage rings
  or linacs without recirculation.
• ERL synchrotron source Beam power in a typical
  synchrotron source is (100 mA)(5 GeV)500 MW.
  Realistically, even the federal govt. will be
  unable to provide a third of a nuclear plant to
  run a synchrotron source. Idea is to use the high
  multiplication factor possible in energy
  recovered designs to reduce the power load. Pulse
  lengths of order 100 fsec or smaller may in be
  possible in an ERL source impossible at a
  storage ring. Better emittance may be possible
  too.
• The limits, in particular the average current
  carrying capacity of possible recirculated linac
  designs, are not yet determined and may be far in
  excess of what the FELs can do!

13
Challenges for Beam Recirculation

• Additional Linac Instability
• Multipass Beam Breakup (BBU)
• Observed first at Illinois Superconducting
  Racetrack Microtron
• Limits the average current at a given
  installation
• Made better by damping non-fundamental
  electromagnetic High Order Modes (HOMs) in the
  cavities
• Best we can tell at CEBAF, threshold current is
  around 20 mA, measured to be several mA in the
  FEL
• Changes based on beam recirculation optics
• Turn around optics tends to be a bit different
  than in storage rings or more conventional
  linacs. Longitudinal beam dynamics gets coupled
  more strongly to the transverse dynamics and
  nonlinear corrections different
• HOM cooling will perhaps limit the average
  current in such devices.

14
Challenges for Beam Recirculation

• High average current sources needed to provide
  beam
• Right now, looks like a good way to get there is
  with DC photocathode sources as we have in the
  Jefferson Lab FEL.
• Need higher fields in the acceleration gap in the
  gun.
• Need better vacuum performance in the beam
  creation region to increase the photocathode
  lifetimes.
• Goal is to get the photocathode decay times above
  the present storage ring Toushek lifetimes
• Beam dumping of the recirculated beam can be a
  challenge.

15
Recirculating SRF Linacs
16
The CEBAF at Jefferson Lab

• Most radical innovations (had not been done
  before on the scale of CEBAF)
• choice of Superconducting Radio Frequency (SRF)
  technology
• use of multipass beam recirculation
• Until LEP II came into operation, CEBAF was the
  worlds largest implementation of SRF technology.

17
CEBAF Accelerator Layout
C. W. Leemann, D. R. Douglas, G. A. Krafft, The
Continuous Electron Beam Accelerator Facility
CEBAF at the Jefferson Laboratory, Annual
Reviews of Nuclear and Particle Science, 51,
413-50 (2001) has a long reference list on the
CEBAF accelerator. Many references on Energy
Recovered Linacs may be found in a recent ICFA
Beam Dynamics Newsletter, 26, Dec. 2001
http//icfa-usa/archive/newsletter/icfa_bd_nl_26.p
df
18
CEBAF Beam Parameters
19
Short Bunches in CEBAF
Wang, Krafft, and Sinclair, Phys. Rev. E, 2283
(1998)
20
Short Bunch Configuration
Kazimi, Sinclair, and Krafft, Proc. 2000 LINAC
Conf., 125 (2000)
21
dp/p data 2-Week Sample Record
Energy Spread less than 50 ppm in Hall C, 100 ppm
in Hall A
Secondary Hall (Hall A)
Primary Hall (Hall C)
Energy drift
X and sigma X in mm
Energy drift
Energy spread
1E-4
Energy spread
23-Mar
27-Mar
31-Mar
4-Apr
23-Mar
27-Mar
31-Mar
4-Apr
Date
Time
Courtesy Jean-Claude Denard
22
Energy Recovered Linacs

• The concept of energy recovery first appears in
  literature by Maury Tigner, as a suggestion for
  alternate HEP colliders
• There have been several energy recovery
  experiments to date, the first one in a
  superconducting linac at the Stanford SCA/FEL
• Same-cell energy recovery with cw beam current up
  to 10 mA and energy up to 150 MeV has been
  demonstrated at the Jefferson Lab 10 kW FEL.
  Energy recovery is used routinely for the
  operation of the FEL as a user facility
• Maury Tigner, Nuovo Cimento 37 (1965)
• T.I. Smith, et al., Development of the
  SCA/FEL for use in Biomedical and Materials
  Science Experiments, NIMA 259 (1987)

23
The SCA/FEL Energy Recovery Experiment

• The former Recyclotron beam recirculation system
  could not be used to obtain the peak current
  required for FEL lasing and was replaced by a
  doubly achromatic single-turn recirculation line.
  
• Same-cell energy recovery was first demonstrated
  in an SRF linac at the SCA/FEL in July 1986
• Beam was injected at 5 MeV into a 50 MeV linac
  (up to 95 MeV in 2 passes)
• Nearly all the imparted energy was recovered. No
  FEL inside the recirculation loop.

T. I. Smith, et al., NIM A259, 1 (1987)
24
CEBAF Injector Energy Recovery Experiment

• N. R. Sereno, Experimental Studies of
  Multipass Beam Breakup and Energy Recovery using
  the CEBAF Injector Linac, Ph.D. Thesis,
  University of Illinois (1994)

25
Instability Mechanism
Courtesy N. Sereno, Ph.D. Thesis (1994)
26
Threshold Current
Growth Rate
where
If the average current exceeds the threshold
current
have instability (exponentially growing cavity
amplitude!)
Krafft, Bisognano, and Laubach, unpublished (1988)
27
Jefferson Lab IR DEMO FEL
Wiggler assembly
Neil, G. R., et. al, Physical Review Letters, 84,
622 (2000)
28
FEL Accelerator Parameters
29
ENERGY RECOVERY WORKS
Gradient modulator drive signal in a linac cavity
measured without energy recovery (signal level
around 2 V) and with energy recovery (signal
level around 0).
Courtesy Lia Merminga
30
Longitudinal Phase Space Manipulations
Simulation calculations of longitudinal dynamics
of JLAB FEL
Piot, Douglas, and Krafft, Phys. Rev. ST-AB, 6,
0030702 (2003)
31
Phase Transfer Function Measurements
Krafft, G. A., et. al, ERL2005 Workshop Proc. in
NIMA
32
Longitudinal Nonlinearities Corrected by
Sextupoles
Nominal Settings
Sextupoles Off
Basic Idea is to use sextupoles to get T566 in
the bending arc to compensate any curvature in
the phase space.
33
IR FEL Upgrade
34
IR FEL 10 kW Upgrade Parameters
35
ERL X-ray Source Conceptual Layout
36
Why ERLs for X-rays?
ESRF 6 GeV _at_ 200 mA
ERL 5 GeV _at_ 10-100 mA
ex ey ? 0.01 nm mrad B 1023
ph/s/mm2/mrad2/0.1BW LID 25 m
ex 4 nm mrad ey 0.02 nm mrad B 1020
ph/s/mm2/mrad2/0.1BW LID 5 m
ERL (no compression)
ESRF
ERL (w/ compression)
t
37
Brilliance Scaling and Optimization

• For 8 keV photons, 25 m undulator, and 1 micron
  normalized emittance, X-ray source brilliance
• For any power law dependence on charge-per-bunch,
  Q, the optimum is
• If the space charge/wake generated emittance
  exceeds the thermal emittance eth from whatever
  source, youve already lost the game!
• BEST BRILLIANCE AT LOW CHARGES, once a given
  design and bunch length is chosen! Therefore,
  higher RF frequencies preferred
• Unfortunately, best flux at high charge

38
ERL Phase II Sample Parameters

rms values
39
ERL X-ray Source Average Brilliance and Flux
Courtesy Qun Shen, CHESS Technical Memo 01-002,
Cornell University
40
ERL Peak Brilliance and Ultra-Short Pulses
Courtesy Q. Shen, I. Bazarov
41
Cornell ERL Phase I Injector
Injector Parameters
Beam Energy Range 5 15a MeV Max Average Beam
Current 100 mA Max Bunch Rep. Rate _at_ 77 pC 1.3
GHz Transverse Emittance, rms (norm.) lt 1b
mm Bunch Length, rms 2.1 ps Energy Spread,
rms 0.2
a at reduced average current b corresponds to 77
pC/bunch
42
Beyond the space charge limit
Cornell ERL Prototype Injector Layout
Solenoids
2-cell SRF cavities
500-750 kV DC Photoemission Gun
Merger dipoles into ERL linac
Buncher
Injector optimization
0.1 mm-mrad, 80 pC, 3ps
Courtesy of I. Bazarov
43
Nonlinear Thomson Scattering

• Many of the the newer Thomson Sources are based
  on a PULSED laser (e.g. all of the high-energy
  lasers are pulsed by their very nature)
• Have developed a general theory to cover
  radiation calculations in the general case of a
  pulsed, high field strength laser interacting
  with electrons in a Thomson scattering
  arrangement.
• The new theory shows that in many situations the
  estimates people do to calculate flux and
  brilliance, based on a constant amplitude models,
  need to be modified.
• The new theory is general enough to cover all
  1-D undulator calculations and all pulsed laser
  Thomson scattering calculations.
• The main new physics that the new calculations
  include properly is the fact that the electron
  motion changes based on the local value of the
  field strength squared. Such ponderomotive forces
  (i.e., forces proportional to the field strength
  squared), lead to a red-shift detuning of the
  emission, angle dependent Doppler shifts of the
  emitted scattered radiation, and additional
  transverse dipole emission that this theory can
  calculate.

44
Ancient History

• Early 1960s Laser Invented
• Brown and Kibble (1964) Earliest definition of
  the field strength parameters K and/or a in the
  literature that Im aware of
• Interpreted frequency shifts that occur at high
  fields as a relativistic mass shift.
• Sarachik and Schappert (1970) Power into
  harmonics at high K and/or a . Full calculation
  for CW (monochromatic) laser. Later referenced,
  corrected, and extended by workers in fusion
  plasma diagnostics.
• Alferov, Bashmakov, and Bessonov (1974)
  Undulator/Insertion Device theories developed
  under the assumption of constant field strength.
  Numerical codes developed to calculate real
  fields in undulators.
• Coisson (1979) Simplified undulator theory,
  which works at low K and/or a, developed to
  understand the frequency distribution of edge
  emission, or emission from short magnets, i.e.,
  including pulse effects

45
Coissons Spectrum from a Short Magnet
Coisson low-field strength undulator spectrum
R. Coisson, Phys. Rev. A 20, 524 (1979)
46
Dipole Radiation
Assume a single charge moves in the x direction
Introduce scalar and vector potential for fields.
Retarded solution to wave equation (Lorenz gauge),
47
Dipole Radiation
Polarized in the plane containing and

48
Dipole Radiation
Define the Fourier Transform
With these conventions Parsevals Theorem is
Blue Sky!
This equation does not follow the typical (see
Jackson) convention that combines both positive
and negative frequencies together in a single
positive frequency integral. The reason is that
we would like to apply Parsevals Theorem easily.
By symmetry, the difference is a factor of two.
49
Dipole Radiation
For a motion in three dimensions
Vector inside absolute value along the magnetic
field
Vector inside absolute value along the electric
field. To get energy into specific polarization,
take scaler product with the polarization vector
50
Co-moving Coordinates

• Assume radiating charge is moving with a velocity
  close to light in a direction taken to be the z
  axis, and the charge is on average at rest in
  this coordinate system
• For the remainder of the presentation, quantities
  referred to the moving coordinates will have
  primes unprimed quantities refer to the lab
  system
• In the co-moving system the dipole radiation
  pattern applies

51
New Coordinates
Resolve the polarization of scattered energy into
that perpendicular (s) and that parallel (p) to
the scattering plane
52
Polarization
It follows that
So the energy into the two polarizations in the
beam frame is
53
Comments/Sum Rule

• There is no radiation parallel or anti-parallel
  to the x-axis for x-dipole motion
• In the forward direction ?'? 0, the radiation
  polarization is parallel to the x-axis for an
  x-dipole motion
• One may integrate over all angles to obtain a
  result for the total energy radiated

Generalized Larmor
54
Sum Rule
Total energy sum rule
Parsevals Theorem again gives standard Larmor
formula
55
Energy Distribution in Lab Frame
By placing the expression for the Doppler shifted
frequency and angles inside the transformed beam
frame distribution. Total energy radiated from
d'z is the same as d'x and d'y for same dipole
strength.
56
Bend
Undulator
Wiggler
e
e
e
57
Weak Field Undulator Spectrum
Generalizes Coisson to arbitrary observation
angles
58
Strong Field Case
Why is the FEL resonance condition?
59
High K
Inside the insertion device the average (z)
velocity is
and the radiation emission frequency redshifts by
the 1K2/2 factor
60
Thomson Scattering

• Purely classical scattering of photons by
  electrons
• Thomson regime defined by the photon energy in
  the electron rest frame being small compared to
  the rest energy of the electron, allowing one to
  neglect the quantum mechanical Dirac recoil on
  the electron
• In this case electron radiates at the same
  frequency as incident photon for low enough field
  strengths
• Classical dipole radiation pattern is generated
  in beam frame
• Therefore radiation patterns, at low field
  strength, can be largely copied from textbooks
• Note on terminology Some authors call any
  scattering of photons by free electrons Compton
  Scattering. Compton observed (the so-called
  Compton effect) frequency shifts in X-ray
  scattering off (resting!) electrons that depended
  on scattering angle. Such frequency shifts arise
  only when the energy of the photon in the rest
  frame becomes comparable with 0.511 MeV.

61
Simple Kinematics
e-
Beam Frame
Lab Frame
62
In beam frame scattered photon radiated with wave
vector
Back in the lab frame, the scattered photon
energy Es is
63
Electron in a Plane Wave
Assume linearly-polarized pulsed laser beam
moving in the direction (electron charge is e)
Polarization 4-vector
Light-like incident propagation 4-vector
Krafft, G. A., Physical Review Letters, 92,
204802 (2004), Krafft, Doyuran, and Rosenzweig ,
Physical Review E, 72, 056502 (2005)
64
Electromagnetic Field
Our goal is to find xµ(t)(ct(t),x(t),y(t),z(t))
when the 4-velocity uµ(t)(cdt/dt,dx/dt,dy/dt,dz/d
t)(t) satisfies duµ/dt eFµ?u?/mc where t is
proper time. For any solution to the equations of
motion.
Proportional to amount frequencies up-shifted
going to beam frame
65
? is exactly proportional to the proper time
On the orbit
Integrate with respect to ? instead of t. Now
where the unitless vector potential is f(?)-eA(?
)/mc2.
66
Electron Orbit
Direct Force from Electric Field
Ponderomotive Force
67
Energy Distribution
68
Effective Dipole Motions Lab Frame
And the (Lorentz invariant!) phase is
69
Summary

• Overall structure of the distributions is very
  like that from the general dipole motion, only
  the effective dipole motion, including physical
  effects such as the relativistic motion of the
  electrons and retardation, must be generalized
  beyond the straight Fourier transform of the
  field
• At low field strengths (f ltlt1), the distributions
  reduce directly to the classical Fourier
  transform dipole distributions
• The effective dipole motion from the
  ponderomotive force involves a simple projection
  of the incident wave vector in the beam frame
  onto the axis of interest, multiplied by the
  general ponderomotive dipole motion integral
• The radiation from the two transverse dipole
  motions are compressed by the same angular
  factors going from beam to lab frame as appears
  in the simple dipole case. The longitudinal
  dipole radiation is also transformed between beam
  and lab frame by the same fraction as in the
  simple longitudinal dipole motion. Thus the usual
  compression into a 1/? cone applies

70
Weak Field Thomson Backscatter
With F p and f ltlt1 the result is identical to
the weak field undulator result with the
replacement of the magnetic field Fourier
transform by the electric field Fourier transform
Undulator
Thomson Backscatter
Driving Field
Forward Frequency
Lorentz contract Doppler
Double Doppler
71
High Field Strength Thomson Backscatter
For a flat incident laser pulse the main results
are very similar to those from undulaters with
the following correspondences
Undulator
Thomson Backscatter
Field Strength
Forward Frequency
Transverse Pattern
NB, be careful with the radiation pattern, it is
the same at small angles, but quite a bit
different at large angles
72
Forward Direction Flat Laser Pulse
20-period equivalent undulator
73
(No Transcript)
74
Realistic Pulse Distribution at High a
In general, its easiest to just numerically
integrate the lab-frame expression for the
spectrum in terms of Dt and Dp. A 105 to 106
point Simpson integration is adequate for most
purposes. Flat pulses reproduce previously known
results and to evaluate numerical error, and
Gaussian amplitude modulated pulses. One may
utilize a two-timing approximation (i.e., the
laser pulse is a slowly varying sinusoid with
amplitude a(?)), and the fundamental expressions,
to write the energy distribution at any angle in
terms of Bessel function expansions and a ?
integral over the modulation amplitude. This
approach actually has a limited domain of
applicability (K,alt0.1)
75
Forward Direction Gaussian Pulse
Apeak and ?0 chosen for same intensity and same
rms pulse length as previously
76
Radiation Distributions Backscatter
Gaussian Pulse s at first harmonic peak
Courtesy Adnan Doyuran (UCLA)
77
Radiation Distributions Backscatter
Gaussian p at first harmonic peak
Courtesy Adnan Doyuran (UCLA)
78
Radiation Distributions Backscatter
Gaussian s at second harmonic peak
Courtesy Adnan Doyuran (UCLA)
79
90 Degree Scattering
80
90 Degree Scattering
And the phase is
81
Radiation Distribution 90 Degree
Gaussian Pulse s at first harmonic peak
Courtesy Adnan Doyuran (UCLA)
82
Radiation Distributions 90 Degree
Gaussian Pulse p at first harmonic peak
Courtesy Adnan Doyuran (UCLA)
83
Polarization Sum Gaussian 90 Degree
Courtesy Adnan Doyuran (UCLA)
84
Radiation Distributions 90 Degree
Gaussian Pulse second harmonic peak
Second harmonic emission on axis from
ponderomotive dipole!
Courtesy Adnan Doyuran (UCLA)
85
THz Source
86
Radiation Distributions for Short High-Field
Magnets
And the phase is
Krafft, G. A., to be published Phys. Rev. ST-AB
(2006)
87
Wideband THz Undulater
Primary requirements wide bandwidth and no
motion and deflection. Implies generate A and B
by simple motion. One half an oscillation is
highest bandwidth!
88
THz Undulator Radiation Spectrum
89
Total Energy Radiated
Lienards Generalization of Larmor Formula (1898!)
Baruts Version
From ponderomotive dipole
Usual Larmor term
90
Some Cases
Total radiation from electron initially at rest
For a flat pulse exactly (Sarachik and Schappert)
91
For Circular Polarization
Only other specific case I can find in literature
completely calculated has usual circular
polarization and flat pulses. The orbits are then
pure circles
Sokolov and Ternov, in Radiation from
Relativistic Electrons, give
(which goes back to Schott and the turn of the
20th century!) and the general formula checks out
92
Conclusions

• Recent development of superconducting cavities
  has enabled CW operation at energy gains in
  excess of 20 MV/m, and acceleration of average
  beam currents of 10s of mA.
• The ideas of Beam Recirculation and Energy
  Recovery have been introduced. How these concepts
  may be combined to yield a new class of
  accelerators that can be used in many interesting
  applications has been discussed. Ive given you
  some indication about the historical development
  of recirculating SRF linacs.
• The present knowledge on beam recirculation and
  its limitations in a superconducting environment,
  leads us to think that recirculating accelerators
  of several GeV energy, and with beam currents
  approaching those in storage ring light sources,
  are possible.

93
Conclusions

• Ive shown how dipole solutions to the Maxwell
  equations can be used to obtain and understand
  very general expressions for the spectral angular
  energy distributions for weak field undulators
  and general weak field Thomson Scattering photon
  sources
• A new calculation scheme for high intensity
  pulsed laser Thomson Scattering has been
  developed. This same scheme can be applied to
  calculate spectral properties of short, high-K
  wigglers.
• Due to ponderomotive broadening, it is simply
  wrong to use single-frequency estimates of flux
  and brilliance in situations where the square of
  the field strength parameter becomes comparable
  to or exceeds the (1/N) spectral width of the
  induced electron wiggle
• The new theory is especially useful when
  considering Thomson scattering of Table Top
  TeraWatt lasers, which have exceedingly high
  field and short pulses. Any calculation that does
  not include ponderomotive broadening is
  incorrect.

94
Conclusions

• Because the laser beam in a Thomson scatter
  source can interact with the electron beam
  non-colinearly with the beam motion (a piece of
  physics that cannot happen in an undulator),
  ponderomotively driven transverse dipole motion
  is now possible
• This motion can generate radiation at the second
  harmonic of the up-shifted incident frequency on
  axis. The dipole direction is in the direction of
  laser incidence.
• Because of Doppler shifts generated by the
  ponderomotive displacement velocity induced in
  the electron by the intense laser, the frequency
  of the emitted radiation has an angular
  asymmetry.
• Sum rules for the total energy radiated, which
  generalize the usual Larmor/Lenard sum rule, have
  been obtained.