A method of determining narrow energy spread electron beams from a laser plasma wakefield accelerator using undulator radiation

Abstract

In this paper a new method of determining the energy spread of a relativistic electron beam from a laser-driven plasma wakefield accelerator by measuring radiation from an undulator is presented. This could be used to determine the beam characteristics of multi-GeV accelerators where conventional spectrometers are very large and cumbersome. Simultaneous measurement of the energy spectra of electrons from the wakefield accelerator in the 55-70 MeV range and the radiation spectra in the wavelength range of 700-900 nm of synchrotron radiation emitted from a 50 period undulator confirm a narrow energy spread for electrons accelerated over the dephasing distance where beam loading leads to energy compression. Measured energy spreads of less than 1% indicates the potential of using a wakefield accelerator as a driver of future compact and brilliant ultrashort pulse synchrotron sources and free-electron lasers that require high peak brightness beams. (C) 2009 American Institute of Physics. [doi: 10.1063/1.321

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A method of determining narrow energy spread electron beams
from a laser plasma wakefield accelerator using undulator radiation
J. G. Gallacher,1M. P. Anania,1E. Brunetti,1F. Budde,2,6A. Debus,2,5B. Ersfeld,1
K. Haupt,2,3M. R. Islam,1O. Jäckel,2S. Pfotenhauer,2,7A. J. W. Reitsma,1E. Rohwer,3
H.-P. Schlenvoigt,2,4H. Schwoerer,2,3R. P. Shanks,1S. M. Wiggins,1
and D. A. Jaroszynski1,a?
1Department of Physics, Scottish Universities Physics Alliance, University of Strathclyde,
Glasgow G4 0NG, United Kingdom
2Institut für Optik und Quantenelektronik, Friedrich-Schiller-Universität, 07743 Jena, Germany
3Laser Research Institute, University of Stellenbosch, 7602 Matieland, South Africa
4LULI-École Polytechnique, 91128 Palaiseau, France
5Forschungszentrum Dresden-Rossendorf, 01314 Dresden, Germany
6Institut für Laser-und Plasmaphysik, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany
7Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
?Received 8 June 2009; accepted 11 August 2009; published online 2 September 2009?
In this paper a new method of determining the energy spread of a relativistic electron beam from a
laser-driven plasma wakefield accelerator by measuring radiation from an undulator is presented.
This could be used to determine the beam characteristics of multi-GeV accelerators where
conventional spectrometers are very large and cumbersome. Simultaneous measurement of the
energy spectra of electrons from the wakefield accelerator in the 55–70 MeV range and the radiation
spectra in the wavelength range of 700–900 nm of synchrotron radiation emitted from a 50 period
undulator confirm a narrow energy spread for electrons accelerated over the dephasing distance
where beam loading leads to energy compression. Measured energy spreads of less than 1%
indicates the potential of using a wakefield accelerator as a driver of future compact and brilliant
ultrashort pulse synchrotron sources and free-electron lasers that require high peak brightness
beams. © 2009 American Institute of Physics. ?doi:10.1063/1.3216549?
I. INTRODUCTION
Synchrotron sources have become ubiquitous tools for
studying the structure of matter at the atomic and molecular
length scale. These sources are among the largest scientific
instruments in existence and are used by extensive industrial
and academics communities from a wide range of research
fields from physics, chemistry, engineering to the health
sciences.1Conventional synchrotron sources are based on
relativistic electron beams and deliver very high brightness
pulses of electromagnetic radiation over a wide spectral
range stretching from terahertz frequencies to hard x rays.
The next “fourth generation light sources” are based on free-
electron lasers ?FELs? and provide coherent radiation with a
peak brilliances which are many orders of magnitude higher
than available from a synchrotron. Their potential as brilliant
time resolved x-ray probes for taking femtosecond resolution
“snapshots” of molecular and solid-state dynamics1,2makes
them particularly attractive as large national and interna-
tional facilities, such as those being constructed at the Linac
Coherent Light Source ?LCLS? in the United States,3the
X-ray Free Electron Laser ?XFEL? in Europe,4and the
SPring-8 Compact Self-amplified spontaneous emission
Source ?SCSS? in Asia.5
Current light sources utilize rf accelerating structures to
accelerate bunches of electrons to relativistic energies, which
are then injected into magnetic structures, or insertion de-
vices, to produce periodic transverse acceleration and elec-
tromagnetic radiation that can be tuned over a wide spectral
range. Electrical breakdown in rf cavities currently limits the
maximum sustainable electric field of conventional accelera-
tors to less than 100 MV m−1, which results in very large
and expensive devices requiring large infrastructure and
teams of engineers to service them. In addition, the pulse
durations of synchrotrons are usually relatively long, of the
order of 10 ps, which limits their usefulness as an ultrafast
probe. The bunch duration can be reduced to less than 100 fs,
but with a great deal of effort using magnetic compressors
and/or electron bunch slicing techniques.6
A very attractive and compact alternative to rf accelera-
tion is acceleration by plasma waves, a method that was first
proposed in 1979 by Tajima and Dawson.7In these devices,
a plasma density wake is driven either by the Coulomb force
of a charged particle beam8or by the ponderomotive force of
an intense ultrashort laser pulse, as in the laser wakefield
accelerator ?LWFA?.9The wake forms a series of charge den-
sity structures with dimensions of the order of the plasma
wavelength, ?p=2?c/?p, which trail the laser pulse at ap-
proximately the group velocity, ?vg?c?1−ne/nc?1/2?, where
?p is the plasma frequency, ne is the plasma density,
nc=me?0?0
free space, meis the electron mass, ?0is the laser frequency,
and c is the speed of light in vacuum. These “accelerating
cavity” structures due to charge separation produce an elec-
tric field that can be more than three orders of magnitude
2/e2is the critical density, ?0is the permittivity of
a?Electronic mail: d.a.jaroszynski@strath.ac.uk.
PHYSICS OF PLASMAS 16, 093102 ?2009?
1070-664X/2009/16?9?/093102/8/$25.00© 2009 American Institute of Physics
16, 093102-1
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greater than that found in rf accelerators, i.e., in excess of
100 GV m−1.10The first demonstrations of controlled accel-
eration of quasimonoenergetic electron bunches from a
LWFA were reported in 2004.11–13Since then, 1 GeV elec-
tron bunches have been produced in centimeter-long plasma
capillary waveguide accelerating structures.14,15
With the availability of commercial multiterawatt laser
systems based on chirped pulse amplification16there is a real
prospect of accelerators based on plasma becoming widely
used in university-sized establishments as drivers of radia-
tion sources. Advantages of a wakefield based synchrotron
light source include its high peak brilliance and ultrashort
pulse duration, which is a fraction of a plasma oscillation
period, 2?/?p, i.e., of the order of 10 fs for plasma densities
np?1018–1019cm−3.17Such a source would be very attrac-
tive for ultrafast x-ray probing studies.1,2,18Commercial tera-
watt lasers are currently limited to 10 Hz and there is excel-
lent prospect for further development of lasers to increase
their repetition rates to a kilohertz and above. This would
increase the average brilliance. However, the real potential of
a wakefield accelerator is as a compact driver of a FEL,
because their electron beams have very high peak currents,
small emittances and narrow energy spreads.
As a first step toward developing a LWFA based radia-
tion source, narrow-bandwidth synchrotron radiation has
been produced using an electron beam from a LWFA. In
these experiments the wavelength scaling with beam
energy19
and theproduction
demonstrated.20Here undulator radiation was produced by
an electron beam from a LWFAand measurements confirmed
the wavelength scaling with energy and thus the potential for
a sub-10 fs duration x-ray synchrotron source.19,20However,
to exceed the threshold for FEL action at x-ray photon ener-
gies, the major challenge is to produce a very high quality
electron beam with a sufficiently high charge density, low
emittance and small energy spread2to give net gain. rf ac-
celerators can deliver such beams and are currently providing
the basis for developing several large scale x-ray FEL
facilities.1,2LWFAs are currently delivering electron beams
that have promising characteristics, which are deduced from
both measurements and simulations. The current state-of-the-
art wakefield accelerator produces electron beams with a
peak current of between 1 and 10 kA ?deduced from the
measured charge in the range of 10–100 pC and the predicted
bunch duration of 10 fs?11–14,21? and an emittance of the order
of 1? mm mrad.22However, the measured energy spread is
currently a few percent because of the limited resolution of
the electron energy spectrometers used in the studies.11–14,21
In this paper, we present measurements of the electron
energy spectrum of a beam from a LWFA deduced from the
spectrum of synchrotron radiation emitted from an undulator.
We show that the total rms relative energy spread of a LWFA
beam can be close to 1% and demonstrate a new method of
determining the electron energy spread from the measured
synchrotron radiation spectrum, which can be used at arbi-
trarily high electron beam energies. We also provide a theo-
retical explanation for the small observed energy spreads,
whichare smallerthan
simulations.23,24We compare the undulator spectrum de-
of harmonicswas
predictedby previous
duced from the deconvoluted electron energy spectrum, tak-
ing into account the electron beam divergence and detector
acceptance angle, and show that the optical and electron
spectra are self-consistent. Our study demonstrates the vi-
ability of using an undulator as an electron energy spectrom-
eter, which may be particularly useful at high energies
??1 GeV? where conventional high resolution magnetic di-
pole deflection spectrometers become very large and lose
resolution. Furthermore, these measurements highlight the
growing potential of a LWFA as an ultracompact source of
intense, tunable electromagnetic radiation, which is readily
scalable to x-ray wavelengths.
II. WAKEFIELD ACCELERATOR
The experiment to produce undulator synchrotron radia-
tion using laser accelerated electrons has been carried out
using the Jena Ti:sapphire ?JETI? laser.25This laser delivers
85 fs duration laser pulses centered at a wavelength of
795 nm and with an energy of up to 430 mJ on target. The
experimental setup is displayed in Fig. 1 and described in
detail in Ref. 20. The laser pulses are focused by an f/6, 30°
off-axis parabolic ?OAP? mirror to a spot diameter of 11 ?m
at full-width at half-maximum ?FWHM?, yielding a peak in-
tensity of 5?1018W cm−2and a normalized vector poten-
tial a0=1.5. A pulsed supersonic helium gas jet, placed at the
laser beam focus, produces a 2 mm diameter gas plume with
a peak density of 2?1019cm−3. Nonlinear Thomson scatter-
ing from the plasma, at the second harmonic of the laser, is
observed in the direction perpendicular to the laser polariza-
tion and is used to measure the length and position of the
channel, as shown in Fig. 2. The electron beam divergence
and pointing are observed on a removable Lanex scintillating
screen26,27?Fig. 3? and optimized by moving the gas jet po-
sition, direction of the laser beam and tuning the gas density.
This optimization procedure is also an essential step to
steer the electron beam down the undulator axis. An example
of an optimized transverse electron beam profile with a
rms beam divergence of 2 mrad is shown in Fig. 3. This
divergence is consistent with a normalized emittance of
?n=1? mm mrad.
OAP
gas jet
undulator
electron
spectrometer
a
b
c
d
optical
spectrometer
laser
FIG. 1. Setup of the experiment. The laser pulse is focused by an OAP
mirror into a supersonic helium gas jet where electrons are accelerated to
between 50 and 70 MeV. The transverse electron beam profile is monitored
on a removable scintillating screen. ?a? A 1 cm thick, 1 cm inside diameter
lead aperture ?b? protects the undulator magnets from off-axis electrons. The
spectrometer is shielded from direct laser and transition radiation exposure
by a 15 ?m thick aluminum foil ?c? placed at the entrance of the undulator.
The electrons traverse the undulator, produce synchrotron radiation, and are
then dispersed in the magnetic electron spectrometer. Undulator radiation is
collected by a lens ?d? and analyzed by an optical spectrometer.
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Page 3

Following optimization of the electron beam properties,
the scintillating screen is removed and the electrons are
allowed to traverse the undulator, which is placed 40 cm
downstream of the gas jet. A 50 period fixed-gap per-
manent magnet undulator, with wavelength ?u=2 cm and
pole gap of 10 mm, produces an on-axis peak magnetic
field of Bu=0.33 T resulting in a deflection parameter
au=eBu?u/2?m0c=0.6. The undulator is placed in a vacuum
chamber and has the fields of the initial and ultimate three
periods carefully trimmed to ensure on-axis injection and
exit of electrons. This shortens the undulator slightly, which
marginally increases the homogeneous ?1/Nu? component of
the undulator spectrum. A 1 cm thick, 1 cm diameter lead
aperture ?Fig. 1?b?? protects the undulator magnets from off-
axis electron beams.
After traversing the undulator electrons are deflected by
the magnetic field of an electron spectrometer ?constructed
from 200?100 mm2permanent magnets, separated by a
20 mm gap, giving a central magnetic field strength 0.72 T?
placed 185 cm downstream of the gas jet. A scintillating
screen ?Lanex, Konica KR? and a charge-coupled device
?CCD? camera detect the deflected electrons over an energy
range of 14–85 MeV. The scintillation screen intensity has
been calibrated using an imaging plate ?Fuji BAS—MS2025?
to give an absolute measure of the charge per unit energy.27
This takes into account both the response of the image plate
and the scintillating screen.
Undulator radiation is collected within a collection angle
of 3 mrad and focused onto the entrance slit plane of a sym-
metric 200 mm Czerny–Turner spectrometer with a 105 mm
focal length fused silica lens. A thermoelectrically cooled
CCD camera ?Andor DO-420 BN? is used to measure the
spectrum of the undulator radiation. The CCD chip ?1024
?256 pixels? is operated in a hardware binning mode, by
merging 8?12 pixels arrays into superpixels. The spectral
range is set to 540–990 nm and the spectrometer and detector
efficiency carefully calibrated using a standard visible-light
source. The optical spectrometer is shielded against direct
exposure from the laser and plasma light by a 15 ?m thick
aluminum foil placed in front of the undulator.
The maximum electron energy from a wakefield accel-
eration is fixed by the plasma density, which sets the length
over which the accelerating electron beam outruns the
plasma density wake. Because the plasma wake bubblelike
structure travels at a velocity close to the group velocity of
the laser pulse vgand electrons in the wake are quickly ac-
celerated to a velocity very close to the speed of light v?c,
then the difference between the wake and electron velocities
implies that electrons traverse half a relativistic plasma wake
wave ?pover a length Ld=c?0
the experiments described here, electrons driven by the laser
become relativistic and the bubble structure elongates and
widens. Thus the dephasing length increases to Ld=?4/3?
???0
jet where electrons are accelerated is ne?1?1019cm−3,
which gives a relativistic plasma wavelength of 12.8 ?m for
a0=1.5, thus Ld?0.48 mm, which is closely matched to the
length of the gas jet used in the experiment. The maximum
energy an electron can gain can be estimated by integrating
the electric field over the dephasing length. The electric field
amplitude inside the bubble is E0=?a0mc?p/e,28thus the
maximum energy an electron can gain from a plasma wave is
3??0
?p
2/?p
3. When a0?1, as it is in
2/?p
3?c?a0.28The plasma density in the region of the gas
?max=2
2
2?a0.
?1?
Thus
?1?1019cm−3?, we can estimate an upper limit on the maxi-
mum energy an electron can gain to be ?178 MeV.
takingthe plasmadensityinour experiment
FIG. 3. ?Color online? The transverse electron beam profile detected on the
scintillating Lanex screen at the entrance of the undulator. Electron diver-
gences in the range 2–10 mrad are observed. The shot displayed has a rms
divergence of 2 mrad.
FIG. 2. ?Color online? The relativistic plasma channel, self-illuminated at
2? and used to characterize the length and position of the plasma channel.
093102-3A method of determining narrow energy spread…
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Page 4

III. UNDULATOR RADIATION MEASUREMENTS
Linearly polarized undulator radiation is produced by the
periodic transverse acceleration imparted to the electrons by
the Lorentz force of the magnetic field of the undulator. The
main spectral characteristics of the undulator radiation can be
modeled using the well known undulator equation
2h?2?1 +au
? =
?u
2h?z
2=
?u
2
2+ ?2?2?.
?2?
From this it can be seen that the peak wavelength emission
depends on the undulator period ?uand the electron energy
Ee=?mec2, where ? is the Lorentz factor, h is the harmonic
order, and ? is the angle with respect to the electron beam
axis. The small reduction in the longitudinal velocity due to
periodic deflection results in a slight increase in the wave-
length of the emitted radiation by a factor ?1+au
wavelength also exhibits an angular dependence through the
?2?2term. The central radiation cone angular half width
can be approximated as ?cen=1/??Nu, which is approxi-
mately 1–1.3 mrad for the electron energies of the wakefield
accelerator.
Given the known trajectory of an electron in the undu-
lator, the radiation fields can be calculated by directly evalu-
ating the retarded field of the Liénard–Wiechert potential.29
To help characterize the beam we consider the total
brightness of the beam of radiation impinging on a surface as
the power per unit area of the source from a unit solid angle.
The brilliance, which is defined by S?=2?0c?E?2?area, is the
power radiated ?per unit frequency, beam area and solid
angle? by a collimated electron beam of current density jb.
This is given by30
32c?0?2?2?Au????2?sin???
2/2?. The
S???,?? ?
k2Lu
2ejb
??
2
,
?3?
where ? is the angle between observation and motion direc-
tions and the normalized detuning parameter ? is given by
? = ?Nu?
?u
2??2?1 +au
2
2− ?2?2? − 1?,
?4?
and the coefficient Auis given by Ref. 30. Radiation over
a spectral width ??FWHMis emitted within an angle of ap-
proximately ??FWHM. For angles less than Nu
Au????Au?0? where
Au?0? ? au?J0?
4 + 2au
1/2??FWHM/2,
au
2
2?− J1?
au
2
4 + 2au
2??,
?5?
for the fundamental frequency. The spectrum consists of
equally spaced odd harmonics on axis and both even and odd
harmonics off axis, for a planar undulator. The spectrum of
each harmonic is a sinc function ?sin ?2/?2? giving a FWHM
spectral width of
??FWHM
?
?0.9
Nuh.
?6?
For a particular harmonic, the solid angle of radiation emit-
ted into the first lobe is
??FWHM?
0.9
?Nuh?1/2·1 + au
2/2
?
,
?7?
with all harmonics emitted into the solid angle ?FWHM
?1/?z=?1+au
2/2?1/2/? for an on-axis electron beam.
IV. UNDULATOR AS AN ELECTRON SPECTROMETER
To investigate the correspondence between the electron
energy spectra and the undulator radiation spectra, several
electron and optical spectra have been measured simulta-
neously. Figure 4 shows three representative pairs of mea-
surements indicating the peak energy E, the FWHM spectral
width, ?E/E, the peak wavelength, ?, and the FWHM spec-
tral bandwidth, ??/?. Consistent with Eq. ?2?, the wave-
length decreases with increasing electron energy.19This char-
acteristicscalingisclearly
fundamental ?h=1? and second harmonic ?h=2? emission.20
Due to the limited spectral range of the optical spectrometer
system, the fundamental and second harmonic signals could
not be observed simultaneously. The measured radiation
spectrum agrees well with predictions using Eq. ?2? and nu-
merical simulations31shown in Fig. 4?b? for wavelengths
between 740 and 920 nm.
Undulator radiation provides a “signature” or “finger-
print” of the transverse and longitudinal electron beam char-
acteristics. Thus, analysis of the undulator radiation spectra
provides an attractive alternative method of characterizing
observedfor boththe
FIG. 4. ?Color online? ?a? The electron spectra and ?b? the corresponding
optical undulator radiation spectra for three shots. The blue ?solid? and green
?dotted? lines have been multiplied by 5 and 10, respectively for clarity. The
black ?solid? lines show the predicted spectra.
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Page 5

the electron beam properties with high resolution, compared
with using a conventional magnetic spectrometer, particu-
larly at high electron energies. To demonstrate this tech-
nique, an analysis of the experimental spectra shown in
Fig. 4, corresponding to E=64 MeV, ?E=3.4 MeV,
?=740 nm, and ??=55 nm, will be presented. These are
replotted in Fig. 5 ?red lines? for clarity. The corresponding
measured FWHM widths of the electron spectra vary be-
tween 5.3% and 15% ???/? in the range of 2.2%–6.2%?.
The finite divergence and energy spread of the electron
beam reduces the brilliance of the beam and smears out the
spectrum as
S???,?? =?S0??,?,? − ?e?F??,?e?d?d?e,
?8?
where ?eis the direction of electrons and S0is the distribu-
tion for a perfect beam with ??=0 and ??=0.32F??,?e? is
the energy and angular probability distribution of the beam.
The total spectral width of the radiation is the sum of con-
tributions from broadening due to the energy, angular, and
natural spread components
???
??
2
=?2??
??
2
+ ??2?2?2+
1
Nu
2.
?9?
To minimize the spectral width and thus maximize the bril-
liance, the second term, ??2?2?2, can be reduced by matching
the electron beam to the undulator, which minimizes the
beam divergence ?. To minimize the divergence the betatron
wavelength, ??, which is analogous to the Rayleigh range of
a laser beam, should be made approximately equal to the
undulator length.33The rms emittance is defined as the trans-
verse momentum/position phase-space area of the beam33
?rms= ?x2????
px
mec?
2?−?
xpx
mec?
2?
1/2
,
?10?
where pxand x are the transverse momenta and coordinates
of an electron, respectively. The edge or envelope emittance
is four times as large. The un-normalized rms emittance ?rms
of an electron beam of radius reis analogous to the wave-
length of a laser beam ?i.e., has units of length? and is given
by
?rms= ????re= k?re
2,
?11?
where ???is the variance of the ?normalized? transverse ve-
locities and k?=2?/??=auku/2? is the betatron wave num-
ber. The normalized emittance, ?n=?z??rms governs the
broadening through ??. The minimum beam divergence that
is consistent with the smallest average beam radius gives
???Lu, which occurs when the Fresnel number F=re
?1 and the electron beam divergence ?given by ???rms/re?
matches the diffraction angle of radiation emitted by the un-
dulator ????/re?, i.e., when
?2= 2k??rms,
2/?Lu
?12?
and the brilliance is maximized, which is approximately 0.4
mrad for our parameters and a matched beam. However, in
this case the electron beam was not focused into the undula-
tor. Thus the dominant broadening contribution comes from
the divergence of the beam, ?2?2?6% corresponding to the
beam divergence of approximately 2 mrad. The natural width
is 1/Nu=2.3%, taking into account the alteration of the first
and last three periods for electron injection and exit. For the
spectrum of Fig. 5?b?, ???/??FWHM measured=7.4%, which
gives an initial upper limit of 7.0% on the combined contri-
butions from energy and angular spreads. The electron en-
ergy spectral width ??/? is limited to below 1.8%, which is
smaller than the value of 2.4% obtained directly from the
measured electron energy spectrum ?red line in Fig. 5?a??.
To evaluate the actual electron beam energy spread, the
measured spectra can be deconvoluted from the measured
spectrum using the respective instrument response functions.
This has been carried out for both the electron energy spec-
trum ?Fig. 5?a?? and the undulator radiation spectrum ?Fig.
5?b??. In each case, the deconvoluted spectrum STis given by
ST= F−1?
F?CT?
F?RT??,
?13?
where F denotes the Fourier transform, CTis the convoluted
measured spectrum, and RTis the instrument response func-
FIG. 5. ?Color online? ?a? The electron spectra data points ?red solid line?.
The simulated instrument response function of the electron spectrometer,
??/?=1.6% ?blue dashed line?. ?b? The undulator radiation spectra data
points ?black solid line?. A best-fit curve to the data points, ??/?=1.7% ?red
dashed line?. The simulated instrument response function of the magnetic
undulator, ??/?=1.2% ?green dotted line?. The deconvoluted undulator
spectrum, ??/?=1.1% ?blue dash-dotted line?. Note: The upper and lower
axes in these plots give the respective wavelength and energy ?Eq. ?2??.
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Page 6

tion. The instrument response function for the electron spec-
trometer has been simulated using the general particle tracer
?GPT? code34and is illustrated in Fig. 5?a? by the blue line.
The equivalent ??/? for the spectrometer is 1.6%. This
simulation takes into account all the relevant electron beam
parameters and space charge effects.
For the corresponding radiation spectrum, the instrument
response function of the undulator has been estimated using
the three contributions to the measured spectral width of Eq.
?9? such that
RT= F−1?F?rN? ? F?r?? ? F?r???,
?14?
where rN, r?, and r?are the response functions due to natural
broadening, angular spread and energy spread, respectively,
which means that this response function refers back to the
beam energy spread. However, as discussed above, the domi-
nating term is due to the angular spread and so the overall
undulator response function is presented in Fig. 5?b? by the
green line ?the corresponding ??/? is 1.2%?. The deconvo-
luted spectrum ?blue line in Fig. 5?b??; therefore provides a
final spectral width of ??/?=1.2%. As a final check on the
validity of this technique, this deconvoluted undulator spec-
trum has been convoluted with the electron spectrometer re-
sponse function which gives a spectral width of 2.3% which
is very close to the measured width of 2.2%. This confirms
that the actual electron energy spread is close to 1%. The
analysis has also been carried out for other experimental data
shots with good agreement obtained between the measured
and reconstructed spectra.
This method demonstrates the use of an undulator as a
high resolution compact electron spectrometer of arbitrary
resolution where the resolution depends on the number of
undulator periods. At high electron energies an undulator is a
very compact non-intercepting on-line alternative to very
large magnetic dipole electron spectrometers.
The energy spread is reduced when the electrons are
close to the dephasing length Ld??p??
electron beam reaches its maximum energy ??2??p/?0?2,
where ??=?p/?0is the Lorentz factor associated with the
group velocity of the laser. Insight into the generation of
narrow energy spread beams is gained from laser wakefield
simulations. These have been carried out using a self-
consistent reduced model that includes modifications of the
laser pulse due to the local spatiotemporal refractive index
modifications due to the plasma density variations of the
wake.23To simulate self-injection, the electron bunch is in-
jected at an optimal position into the wake behind the laser
pulse. The bunch has a random initial phase-space distribu-
tion, determined by the energy spread and emittance. The
model is implemented in a two-dimensional slab geometry,
where both the laser pulse and the electron bunch are treated
as collections of macroparticles of finite size. The evolution
of the laser pulse a?r,t? is calculated on a spatial grid, on
which the macroparticles are treated classically by solving
the classical equations of motion for the coupled dynamics of
laser pulse, wakefield, and electron bunch. This correctly
models “beam loading” which modifies the wakefield created
by the laser pulse. As an initial condition we have considered
a bunch occupying a volume of about 1 ?m3, and a laser
potential of a0=2 with a spot size of 10 ?m, plasma density
n0=1.2?1019cm−3, and an electron bunch charge 10 and
18 pC, as shown in Fig. 6.
We observe lower energy spread for the beam loaded
case ??2%? because the electron bunch is accelerated by a
uniform electrostatic field due to flattening of the potential.
Without beam loading, the leading part of the electron bunch
experiences a weaker acceleration field than the trailing part,
which leads to a larger energy spread as observed in the
simulations.
2/?, i.e., when the
FIG. 6. ?Color online? ?a? Shows the spatial distribution of Lorentz factor of
electrons at dephasing length ?triangles for no beam loading, red ?lower line?
and green ?upper line? for beam loading with 18 and 10 pC of charge,
respectively?, and ?b? spectrum. Here ?c? shows the energy compression as a
function of propagation distance; triangle for no beam loading, square for
beam loading when electron bunch charge is 18 pC ?circle and triangle for
beam loading when the bunch charge is 10 and 18 pC, respectively?.
093102-6Gallacher et al.Phys. Plasmas 16, 093102 ?2009?
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Page 7

V. APPLICATION OF THE LWFA AS A DRIVER
OF A RADIATION SOURCE
FELs require high brightness electron beams for high
gain operation. This condition is met when the relative en-
ergy spread is less than the gain parameter, i.e., ??/???,35
where ? is the Pierce or FEL gain parameter, which also
determines the efficiency,
2??Ip
? =
1
IA??uau
2??x?
2?
1/3
,
?15?
where Ipis the peak electron beam current, IA=4??0mec3/e
?17 kA, the Alfven-current and ?xis the beam diameter.
The ideal gain length35,36is
Lg=
?u
4??3?.
?16?
In the presence of energy spread, emittance and diffraction,
the Lgis increased by ??+1? to Lg=Lgain??1+??.36To reduce
the gain length and undulator length, ? should be maximized.
This is usually achieved by reducing the beam size, which
requires extremely low emittances. However, as can be seen
from Eqs. ?15? and ?16?, the gain length may also be reduced
by increasing the peak current. As electron bunches from
wakefield accelerators are of the order of 10 fs, much shorter
than the plasma wavelength, and the charge of the order of
100 pC, and the peak current can be very high ??10 kA?,
which is much higher than in rf accelerators. Hence, laser
plasma-based electron accelerators may be ideal drivers for
FELs. The next generation x-ray FELs are designed for rela-
tive energy spreads of 0.1% or less and low emittances
?n?1? mm mrad.3,5Up till now, the smallest energy
spreads of LWFA electron bunches has been of the order of
2.5%.11–15,21,37Our measurement of energy spread ?1% is
comparable to recent reports38–40and represents a very im-
portant step.
Ultrahigh beam currents are subject to strong space
charge forces. When the electron bunch leaves the plasma
and into vacuum, it begins to expand due to space charge
forces. This can result in transverse space-charge expansion
and longitudinal debunching.41However, space charge ex-
plosion may contribute to the energy spread for lower ener-
gies. Simulations that have been carried out using GPT
?Ref. 34? show that space charge only has a very minor im-
pact on the transverse and longitudinal emittances. This im-
proves at higher electron energies, where space charge ef-
fects are much weaker; hence higher quality electron beams
are produced.
As an example, consider a 1 GeV laser-produced elec-
tron beam14as a driver of the advanced laser-plasma high-
energy accelerators toward x rays ?ALPHA-X? undulator42
??u=1.5 cm, Nu=200, au=0.7?. This would lead to 2.5 nm
wavelength, ultrashort, incoherent radiation pulses with a
peak brilliance of B?1023photons/s mrad2mm20.1%
bandwidth. Direct generation of coherent undulator radiation
is more challenging as it requires electron bunches shorter
than the emitted wavelength. Coherent emission from a pre-
bunched beam will lead to infrared and far infrared radiation.
However, to reach the ultraviolet or x-ray spectral region one
must rely on the self-amplified spontaneous emission
?SASE? mechanism. This involves microbunching of the
electron beam with a wavelength periodicity due to the com-
bined action of the laser beam with the undulator field. The
initial field in SASE is synchrotron radiation, which results
in amplified noise. However, an external source such as high
order harmonics from a laser driven rare gas is also used as a
seed.
VI. CONCLUSION
Using the demonstration of the first all optically driven
undulator radiation source, we have illustrated how the un-
dulator radiation spectrum is a signature of the electron beam
energy distribution. This technique will apply to electrons of
arbitrarily high energy and therefore will be of crucial im-
portance in the diagnosis of future multi-GeV electron beams
that cannot be precisely interrogated by magnetic dipole
spectrometers. Analysis of the radiation spectrum allows the
actual electron beam energy spread to be deduced which, in
this case, was close to 1% accounting for the spectrometer
response. Production of such a high quality beam is an im-
portant step toward the use of laser wakefield accelerators as
drivers of synchrotron and FEL sources which require high
electron peak current, low energy spread and low emittance.
Extension of this experiment into the ultraviolet and x-ray
wavelength regime and eventual coherent emission ?SASE?
of light will be an invaluable tool. The use of plasma undu-
lators are also being investigated to shorten the undulator
period.43,44This not only allows the use of lower energy
electrons to reach vacuum ultraviolet wavelengths, but be-
cause of the high peak current, also shows promise as future
compact coherent x-ray sources.
ACKNOWLEDGMENTS
This work was supported by the Deutsche Forschungs-
gemeinschaft under Contract No. TR18. Financial support by
the Access to Research Infrastructures activity in the Sixth
Framework Programme of the EU ?Contract No. RII3-CT-
2003-506350, Laserlab Europe? for conducting the research
is gratefully acknowledged. We thank B. Beleites, F.
Ronneberger, and W. Ziegler for their technical support.
S.M.W. acknowledges the support of the Department of
Physics, Lancaster University, and the Cockcroft Institute,
Daresbury Laboratory, Daresbury, United Kingdom. The UK
team also acknowledges the support of the Research Coun-
cils UK and the EU EuroLEAP NEST Contract No. 028514.
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