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10.1117/2.1200905.1566
The quantum free-electron laser
Rodolfo Bonifacio, Nicola Piovella, Gordon Robb, and
Dino Jaroszynski
An ultracompact brilliant coherent x-ray source, where both the accelerator and the wiggler are provided by intense laser pulses, promises
unsurpassed spectral and temporal qualities.
Ultrashort pulses of x-ray radiation from synchrotron sources
have become ubiquitous tools for investigating the structure
of matter. Their immense usefulness has led to the development of large international facilities. These are based on radiofrequency accelerating cavities and magnetic undulators, and
provide brief radiation pulses capable of probing and taking
‘snapshots’ of molecules and solid-state matter. However, synchrotron sources produce pulses of incoherent radiation that are
limited to relatively low peak brilliance and durations of order a
picosecond and longer. As a next significant step in advancing xray sources, the free-electron laser (FEL) produces femtosecondduration pulses with a peak brilliance seven orders of magnitude higher than synchrotrons.
Several large international teams are constructing FELs to produce x-ray radiation through self-amplified spontaneous emission (SASE): the Linac Coherent Light Source1 in Stanford, CA,
the European XFEL2 in Hamburg, Germany, and the SPring8 Compact SASE Source3 in Hyōgo prefecture, Japan. One
drawback of such sources is that they produce pulses composed of many random superradiant spikes with a broad noise
spectrum.4 In the classical picture of the FEL, this spiky x-ray
pulse results from the random initial phases of electrons entering the amplifier. However, it is clear from quantum theory that
the emission process is discrete. Moreover, it must include quantization of the electron motion, which completely changes both
the properties of the emitted radiation and the resulting momentum distribution of the electrons. Accordingly, an FEL operating
in the quantum regime should offer improved performance over
its classical counterpart, in particular, enhanced spectral brightness and degree of coherence.
When an electron emits a photon, the momentum recoil is h̄k. This is naturally quantized and can assume only
the discrete values n(h̄k). In classical FEL theory, the initial
spontaneous-radiation field is amplified through the ‘ponderomotive’ force resulting from the interference of the radiation
Figure 1. Numerical solutions for Lb = 40 Lc (Lb : Electron-bunch
length. Lc : Cooperation length.) and δ = 0 (δ: Frequency detuning),
in (a, c) the classical regime (ρ̄ = 5 and z̄ = 40) and (b, d) the quantum regime (ρ̄ = 0.1 and z̄ = 40). Graphs (a) and (b) show the scaled
intensity, and graphs (c) and (d) the corresponding scaled power spectra as a function of scaled frequency ω̄ = (ω0 − ω)/2ρω, where ω is
the resonance frequency and ω0 the relative frequency with respect to
the ωs and divided by ρ, the free-electron-laser parameter. The dotted
line in (a) marks the front edge of the electron pulse. Ā: Normalized
vector-field potential of the amplified free-electron-laser radiation. z̄:
Scaled wiggler length. z̄1 = ( z − vt)/ Lc , where v is the velocity of the
electrons and t a time interval.
and undulator fields. This leads to electron bunching on a
wavelength scale and exponential amplification with a rate
governed by ρ, the FEL parameter.5 ρ depends on the undulator period, and magnetic-field strength and electron-beam
parameters such as, e.g., the Lorentz factor at resonance for a
particular wavelength of the amplified light, γr , peak current,
and emittance. The number of photons emitted depends on ρ,
and is given by the quantum-FEL (QFEL) parameter6
mcγr
,
(1)
h̄k
which is the ratio of the maximum classical momentum spread
(of order mcγr ρ) to h̄k. When ρ̄ 1, many momentum levels are
involved since the momentum spread is much larger than the
ρ̄ = ρ
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10.1117/2.1200905.1566 Page 2/3
level spacing. The discreteness of the momentum becomes irrelevant, and one recovers the classical behavior, characterized by a
random series of superradiant spikes. The spectrum of the emitted field is broad and chaotic. Conversely, when ρ̄ ≤ 1, an electron emits a single photon and makes a single momentum transition. The result is a single narrow-line spectrum that is Fourierlimited by the electron-beam duration, i.e., ∆ω/ω ' λ / Lb .6, 7
This means that a QFEL operating in the Ångstrom region with
an electron-bunch length Lb = 1mm could generate radiation
with a relative linewidth of 10−7 , much smaller than the envelope linewidth 2ρ of the classical SASE spectrum (typically of
order 10−3 ).
Hence, the QFEL could be a very promising x-ray source
generating quasi-monochromatic radiation (although at a lower
power than in a classical SASE FEL) and a formidable tool for
ultra-high-resolution process studies. The ‘quantum purification’ of the SASE spectrum can be interpreted by the following simple argument. The maximum induced energy spread in
an FEL is δγ /γ ∼ ρ, which in terms of momentum spread is
δp = mc δγ ∼ ρ̄ (h̄k). The QFEL parameter ρ̄ yields the ratio between the maximum momentum spread (induced in the classical
regime) and the photon recoil momentum h̄k. Quantum effects
become important when ρ̄ < 1, since then the discreteness of
momentum exchange is relevant. This provides a simple explanation of the origin of the broad and spiky classical spectrum
and its reduction to a single line in the quantum regime (see
Figure 1 and videos8, 9 ). Experimental realization of a QFEL
requires a laser wiggler instead of the magnetic wiggler usually used in classical SASE experiments.1–3 In a laser-wiggler
configuration, a low-energy electron beam backscatters the photons of a counterpropagating high-power laser into a photon frequency upshifted by a factor 4γ 2 . However, such a
choice sets stringent conditions on the electron- and laser-beam
parameters.10
We propose to exploit the new generation of laser-driven
wakefield accelerators,11 where electrons are accelerated to high
energies by the electrostatic forces of a laser-driven plasma
wave. The advantage is that both the electron beam and the laser
beam acting as a wiggler are contained in a guiding structure.
The electrons are continuously focused by the transverse fields
of the ion ‘bubble,’ while a preformed plasma acts as a waveguide to lead the wiggler laser in maintaining perfect overlap
over many Rayleigh lengths. Furthermore, because the accelerator and the FEL are ‘all-optical’ (they both use lasers to provide
accelerating and wiggler fields, respectively), they can be placed
on a very compact footprint, or perhaps one should even say fingerprint. It should be possible to construct a QFEL driven by a
wakefield accelerator that is only a few centimeters long. This
presents several significant challenges. The first and most stringent is to produce an electron beam with a sufficiently small
energy spread, which must be less than the recoil momentum.
This sets a limit of σγ /γ < 10−4 , which can be alleviated somewhat by going to very short wavelengths, e.g., 0.05Å. However,
the peak current of the electron beam should be greater than
10kA and preferably close to 100kA, which prevailing wisdom
does not rule out.
Our next efforts will focus on operating a QFEL with harmonics to reach even shorter wavelengths, either in the seeded or in
the SASE mode.
Author Information
Rodolfo Bonifacio
Istituto Nazionale di Fisica Nucleare (INFN)
Milan, Italy
and
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro, Brazil
In 1984, Rodolfo Bonifacio laid the foundations for the high-gain
FEL starting from noise, the so-called SASE FEL, which is central to several international programs. He received the Michelson Medal from the Franklin Institute for his studies of optical
bistability, and the Einstein Medal from the Society for Quantum Optics and Quantum Electronics for his pioneering work
on the FEL. Recently, he and colleagues proposed a completely
new QFEL regime.
Nicola Piovella
INFN
Milano, Italy
and
Dipartimento di Fisica
Università degli Studi di Milano
Milan, Italy
Nicola Piovella was born in Milan in 1959. He received a PhD
in physics from the University of Milan in 1990 with a thesis on
superradiance in FELs. Since 1996 he has been with the Department of Physics of the University of Milan, working on collective effects in beam and atomic physics. His research interests
are free-electron lasers, Bose-Einstein condensation, and laser
cooling.
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10.1117/2.1200905.1566 Page 3/3
Gordon Robb and Dino Jaroszynski
Physics Department
University of Strathclyde
Glasgow, Scotland
Gordon Robb is a lecturer. His research interests involve
various collective, nonlinear interactions between light and matter. These include free-electron lasing and collective scattering of
light by cold atomic gases.
Dino Jaroszynski is director of the Electron and Terahertz to Optical Pulse Source (TOPS) and leads the Advanced Laser Plasma
High-energy Accelerators towards X-rays (ALPHA-X) project to
develop radiation sources based on laser-plasma accelerators.
He has made pioneering observations of superradiance in FELs
and has studied short-pulse effects and coherent start-up of FELs
due to prebunching.
References
1. LCLS Design Study Group, LCLS design study, Tech. Rep. SLAC-R521, Stanford
University, 1998. http://www-ssrl.slac.stanford.edu/lcls/CDR
2. R. Brinkmann et al., TESLA XFEL: first stage of the LCLS, Design Study SLACR521, Stanford University, 1998. http://www-ssrl.slac.stanford.edu/lcls/CDR
3. T. Shintake, Status of the SCSS test accelerator and XFEL project in Japan, EPAC’06,
2006. http://www-xfel.spring8.or.jp
4. R. Bonifacio, L. De Salvo, P. Pierini, N. Piovella, and C. Pellegrini, Spectrum,
temporal structure, and fluctuations in a high-gain free electron laser starting from noise,
Phys. Rev. Lett. 73, p. 70, 1994.
5. R. Bonifacio, C. Pellegrini, and L. Narducci, Collective instabilities and high-gain
regime in a free electron laser, Opt. Commun. 50, p. 373, 1984.
6. R. Bonifacio, N. Piovella, G. R. M. Robb, and A. Schiavi, Quantum regime of free
electron lasers starting from noise, Phys. Rev. ST Accel. Beams 9, p. 090701, 2006.
7. R. Bonifacio, N. Piovella, M. M. Cola, L. Volpe, A. Schiavi, and G. R. M. Robb, The
quantum free electron laser, Nucl. Instrum. Methods Phys. Res. A 593, p. 69, 2008.
8. http://spie.org/documents/newsroom/videos/1566/classical.avi Video of the
spectral and temporal evolution in the classical regime of the QFEL. (Credit: Gordon
Robb, University of Strathclyde)
9. http://spie.org/documents/newsroom/videos/1566/quantum.avi Video of
the spectral and temporal evolution in the quantum regime of the QFEL. (Credit:
Gordon Robb, University of Strathclyde)
10. R. Bonifacio, N. Piovella, M. M. Cola, and L. Volpe, Experimental requirements for
X-ray compact free electron lasers with a laser wiggler, Nucl. Instrum. Methods Phys.
Res. A 577, p. 745, 2007.
11. D. A. Jaroszynski et al., Radiation sources based on laser-plasma interactions, Phil.
Trans. R. Soc. A 364, pp. 689–710, 2006.
c 2009 SPIE