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Transient response of relativistic electron bunches to wave-number selected perturbations
near the micro-bunching instability threshold

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Transient response of relativistic electron bunches towave-number selected perturbations near the microbunching instability threshold
E Roussel1, C Evain1, M Le Parquier1, C Szwaj1, S Bielawski1, M Hosaka2,
N Yamamoto2, Y Takashima2, M Shimada3, M Adachi4, H Zen4,
S Kimura4 and M Katoh4
1

Laboratoire de Physique des Lasers, Atomes et Molécules, UMR CNRS 8523, and Centred’Études et de Recherches Lasers et Applications, Université Lille 1, Sciences et Technologies,
F-59655 Villeneuve dʼAscq Cedex, France
2
Graduate School of Engineering, Nagoya University, Nagoya, 464-8603, Japan
3
High Energy Accelerator Research Organization, KEK, Tsukuba, 305-0801, Japan
4
UVSOR Facility, Institute for Molecular Science, National Institutes of Natural Sciences,
Okazaki 444-8585, Japan
E-mail: serge.bielawski@univ-lille1.fr
Received 29 December 2013, revised 17 April 2014
Accepted for publication 1 May 2014
Published 13 June 2014
New Journal of Physics 16 (2014) 063027
doi:10.1088/1367-2630/16/6/063027

Abstract

Many spatio-temporal systems can undergo instabilities, leading to the spontaneous formation of spatial structures (patterns). However, a range of cases exist for
which the pattern itself is not directly visible because of technical or fundamental
reasons. This is the case for the spontaneous formation of millimeter-scale patterns
appearing inside relativistic electron bunches of accelerators. We demonstrate inthis case how the study of responses to sine external perturbations can be used as a‘probe’ to deduce the characteristic wavenumber of the pattern formation process.
Experiments are performed in the UVSOR-II electron storage ring when theelectron bunch is subjected to so-called microbunching instability, and the sine
perturbations are provided by an external laser. The response is constituted ofpulses of coherent synchrotron radiation, whose amplitude depends on the perturbation wavenumber. Experimental results on the dynamics are compared tonumerical calculations obtained using a Vlasov–Fokker–Planck model.

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citation and DOI.
New Journal of Physics 16 (2014) 063027
1367-2630/14/063027+11$33.00© 2014 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft

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New J. Phys. 16 (2014) 063027

S Online supplementary data available from stacks.iop.org/njp/16/063027/
mmediaKeywords: accelerators, pattern formation, Vlasov, synchrotron radiation,
instabilities
In a relativistic electron bunch, interactions between electrons through electric forces(wakeﬁelds) can be the cause of so-called microbunching instability [1–9], which ischaracterized by the spontaneous formation of patterns in the electron bunch phase space[10–12]. The understanding of this dynamics is crucial, as electron storage rings are extremely
important sources for users of radiation in the VUV, x-ray, and THz ranges. However, incontrast to more classical pattern-forming systems, as in hydrodynamics, chemistry, biology,
optics, etc [13], the study of pattern formation in relativistic electron bunches presents
additional difﬁculties because of the very indirect nature of the possible measurements. Thissituation is illustrated in ﬁgure 1, where we have represented a typical phase space computed
numerically using the parameters of the UVSOR-II storage ring. Above a threshold current, a
two-dimensional pattern is spontaneously formed (ﬁgures 1(b), (c)) and its evolution in time t isusually extremely rich and complex, as the structures continuously rotate in phase space, andappear in a bursting manner. In contrast, the experimental data contain much less information
because they usually consist of the recording of the coherent submillimeter/millimeter radiation(CSR) emitted by the structures using bolometers (ﬁgures 1(d), (e)), or diode detectors. In otherwords, the detected signal D(t) can be viewed mathematically as a projection of the twodimensional pattern f ( z, δe, t ) onto a scalar variable D(t):f ( z , δe, t ) → D ( t ) =
=

∫
∫

+∞
−∞
+∞
−∞

H (k)

+∞

+∞

−∞

−∞

∫ ∫

2

f ( z , δe, t ) exp ( − ikz ) dzdδe dk

H ( k ) FF ( k, t ) dk,(1)

with H(k) a transfer function, and FF ( k , t ) the so-called form factor [14] of the electron bunchat time t. The details of this projection are not important at this stage. However, this alreadyshows that most of the information on the two-dimensional pattern f ( z, δe, t ) is lost, and thateven crucial features such as the wavenumber selected at the instability threshold are difﬁcult toaccess5 from the detected signal D(t).
In this article, we present an alternate approach that consists of systematically examining
the systemʼs transient response to initial perturbations with selected wavenumbers for various
values of the main operating parameter. From common knowledge in nonlinear dynamics, one
may indeed expect a huge increase in the systemʼs response when the system is near a phasetransition point [13] (the threshold of the microbunching instability here), and the perturbation
5

In principle, selected wavenumbers may be identiﬁed from the spectrum of CSR emission. In this millimeter
range spectra are obtained from Fourier Transform spectrometers, which require stable signals. This has beenpossible in few cases, for particular values of parameters. This has not been possible for instance at UVSOR, wherebursting occurs very near threshold.
2

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New J. Phys. 16 (2014) 063027

Figure 1. Illustration (theory versus experiment) of the microbunching instability issue
in storage rings. (a) A relativistic electron bunch (600 MeV, with the length in thecentimeter range for UVSOR-II) experiences round-trips in a vacuum chamber. If theelectron bunch charge exceeds a given threshold, a spatiotemporal instability leads tothe spontaneous formation of patterns inside the electron bunch density, which is
responsible for the emission of strong millimeter and terahertz radiation in the bending
magnets. (b), (c) A typical pattern in phase space computed from the classical
Fokker–Planck–Vlasov model (the integration of equation (2) for currents of 4.5 mA (b)and 12 mA (c), and the parameters from table (1)). The pattern rotates in phase spacewith a frequency of the order of 10 kHz, and its evolution is usually irregular. In
contrast, the data typically available experimentally (d), (e), contain much lessinformation, as they consist of single bolometer signals versus time. Most of theinformation on the two-dimensional structure, as its characteristic wavenumber, are thus
not accessible. The currents for (d) and (e) are 11.1 mA and 12.9 mA, respectively.
Movies of (b) and (c) can be found in the supplementary material, available at stacks.
iop.org/njp/16/036027/mmedia.

has a wavenumber close to the characteristic wavenumber of the instability. From a conceptual
point of view, this perturbation strategy is close to the experiments performed in theRayleigh–Bénard experiments [15], although our goal is different here.
Technically, this attempt to use laser perturbations to probe the pattern formation is also
motivated by the pioneering work of Byrd, Sannibale et al [16]. They demonstrated that a huge
response (a burst of intense coherent synchrotron radiation) can be obtained using short laser
pulses. Here, instead of using short pulses (and thus a broadband initial perturbation), a key
point is of course to shape the laser pulses with a sine proﬁle [17] (at millimeter scale). The
sinusoidally shaped laser pulses that are chosen are moreover slightly longer than the electron
bunch (several centimeters) (ﬁgure 2(a)).
The experiments were conducted in the UVSOR-II storage ring, operating at an energy of
E0 = 600 MeV, in single-bunch low-alpha mode [18] (the machine parameters are displayed intable 1). In these conditions, the electron bunch has a duration of 37 ps RMS at low current, a
relative energy spread σE E0 = 3.4 × 10−4 RMS, and the threshold for the microbunching3

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New J. Phys. 16 (2014) 063027

Figure 2. The experimental ‘seeding’ of patterns using sinusoidally modulated laser

perturbations. (a) The setup for the sine perturbations of the electron bunch. (b)Bolometer signal versus time for two modulation wavenumbers: ν̄m = 1.7 cm −1 (strong
pattern seeding is visible as a delayed response around t = 60 μs) and ν̄m = 3 cm −1 (nonoticeable seeding occurs). (c) Bolometer signal versus time and modulation
wavenumbers, showing that seeding is effective only for wavenumbers around 1.7
cm −1. The current is I = 7 mA. (d) The evolution of ﬁgure (c) with the current.

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New J. Phys. 16 (2014) 063027

Table 1. The parameters corresponding to the experiment and used in the numericalsimulations. Note that the natural bunch length is the value at zero current (the actual
length is longer at ﬁnite current).
Nominal energy
Energy spread
Relative energy spread
Natural bunch length
Laser pulse duration
Revolution period
Dipole radius of curvature:
Vacuum chamber height:
Synchrotron frequency
Synchrotron damping time−1

( 2πf τ )s s

E0 = 600 MeV
σE = 0.204 MeV RMSσδ = σE E0 = 3.4 × 10−4 RMS
36.6 ps RMS (σz = 10.98 mm RMS)
300 ps FWHM (38.2 mm RMS)
T0 = 177 ns
R = 2.2 m
2 h = 3.8 × 10−2 m
fs = 9.09 kHz
τs = 19 ms
ϵ = 0.9215 × 10−3

instability is typically of the order of 7 mA (i.e. an electron bunch charge of ≈1.2 nC). The
bursts of millimeter radiation are monitored by an InSb bolometer (QMC QFI/2) with a 2 μs
response time, placed on the BL6B beamline [19]. The laser pulses interact with the electron
bunch in an undulator (ﬁgure 2(a), and the optical setup is similar to the one described in [17]).
We used an ampliﬁed laser (Coherent Mira and Legend) delivering 3 mJ, 300 ps FWHM long
chirped pulses at 800 nm, and with a 1 kHz repetition rate. These pulses are sinusoidally
modulated using the chirped pulse beating technique [20], which allows the modulation period
to be chosen simply by adjusting the delay of a Michelson interferometer. The spectroscopic
wavenumber ν̄m of the perturbation can be typically chosen from the sub-ps range to tens ofpicoseconds.
The data are recorded as follows. For each chosen sine perturbation wavenumber ν̄m, werecord the transient bolometer temporal trace (which roughly corresponds to the induced patternamplitude versus time). Furthermore, the responses versus ν̄m are systematically recorded versus
current I because the effect is expected just below the instability threshold. The data acquisition
is automated, using a Lecroy oscilloscope (WR104MXI) in sequence mode, and automatic
scanning of the Michelson interferometer. At each current value, we record the responses to 200
values of ν̄m from 0.48 to 4.18 cm −1 (this takes 8 s). The electron beam current is decreased at a
slower rate, using the natural decay of the electron bunch charge, or additional losses whenneeded.
Schematically, three current zones are observed. At high current (ﬁgure 2(d)), the laser has
no clear effect, and we simply observe the usual strong bursts of millimeter CSR radiation that
already occur spontaneously without the laser (and for which a pattern like the ones inﬁgure 1(b), (c) is expected theoretically). The expected ‘pattern seeding’ effect is observed for
currents that are near the microbunching instability threshold (I ≈ 7 mA). The signature
consists of the delayed pulse [ﬁgure 2(b), red curve] at a time of ≈60 μs. The pattern seeding
effect then decreases when the current is further decreased to very low values (I ≈ 4 − 5 mA andbelow), where the quasi-instantaneous response to the laser perturbation eventually remains

5

E Roussel et al

New J. Phys. 16 (2014) 063027(this peak near t = 0 does not involve the nonlinear dynamics of the system and was studied in[14, 17, 21]).
Furthermore, systematic analysis of this bolometer signal versus modulation wavenumber ν̄m reveals that the seeding is effective only in a precise domain of ν̄m , in the 1.5-2
cm −1range. This effect is clearly visible in ﬁgure 2(b), where the green and red curves
represent the responses to wavenumber outside and inside the adequate wavenumber range
for seeding. This resonance effect appears as a characteristic feature in the colorscale
representation of bolometer signals versus ν̄m [ﬁgure 2(c)], and this feature appears essentiallynear the instability threshold [ﬁgure 2(d)].
We have veriﬁed that this peak at t ≈ 60 μs is unique, using scans of ν̄m up to 32 cm −1.
Moreover, the relative sizes of the delayed and quasi-immediate responses depend on current,
showing that the delayed peak is affected by the effect of wakeﬁelds. This will be a main
comparison point with theoretical predictions.
A classical framework for theoretical studies is the one-dimensional Vlasov–Fokker–
Planck approach. The temporal evolution of the electron density f ( q, p, θ ) reads [11]:

∂f
∂f
∂f
∂f ⎞
∂ ⎛
−p
+ q − IcEwf= 2ϵ ⎜ pf +
⎟.
∂p ⎠
∂θ
∂q
∂p
∂p ⎝

(

)

(2)

In this equation, θ is a continuous and dimensionless variable associated with the number ofturns in the storage ring: θ = 2πfs t , where t is the time (in seconds) and fs is the synchrotron
frequency (in the tens of kilohertz range here). The longitudinal position q and relative
momentum p are the deviation from the so-called synchronous electron (with position z 0 andenergy E0 ). q and p are expressed in units of the equilibrium bunch length σz and energy spread()σδ at zero current: q = ( z − z 0 ) σz , and p = ( E − E0 ) σE . ϵ = 1 2πfs τs , where τs is thesynchrotron damping time. In usual cases, ϵ ≪ 1, meaning that synchrotron oscillations are
weakly damped. Ic = I 2πef2πσRT , with R the dipole radius of curvature, T0 the revolution period, andE 0s

I the average beam current (in Amperes). All parameters are in MKS units and f ( q, p) is+∞+∞∫−∞ f ( q, p) dqdp = 1.
−∞

normalized so that ∫Ewf characterizes the interaction between electrons, and is the term responsible for the
microbunching instability. It represents the average longitudinal electric ﬁeld created by thewhole bunch, per Ampere of currentwakeﬁeld created by one electron Ewf1e ( q ), and theelectron bunch charge density [22].
The precise expression of Ewf1e is a main ingredient of the modeling, and depends on theelectron bunch path and detailed boundary conditions in the accelerator (e.g. the vacuum
chamber). For the present test, we use the wakeﬁeld produced by an electron on a curved
trajectory, between two parallel plates of height 2 h , which is one of the simplest models
reproducing the experimentally observed dynamics in storage rings [22]. In MKS units, thewakeﬁeld Ewf ( q ) can be expressed as [23]:6E Roussel et al

New J. Phys. 16 (2014) 063027⎛ 3γ 3 ⎞σzζ ⎟dζρ ( q − ζ ) w ⎜−∞⎝ 2R ⎠
⎛
⎞
+∞
σz
⎜
ζ ⎟,
dζρ ( q − ζ ) G2 ⎜
32 ⎟
−∞
2
R
h
R
) ⎠
⎝ (

γ 4T0
4
×
3
4πϵ0R 2Ewf ( q ) =−T0
8πϵ0h 2

∫

∫

+∞

(3)

where the functions w and G2 are given by equations (3.12,5.22b) of [22], and the otherparameters are given in table 1. The convolution is computed in Fourier space using anapproach similar to the one proposed by Qiang [24] (see [23]). Calculation of the perturbationinduced by the laser is detailed in the supplementary data, available at stacks.iop.org/njp/16/
063027/mmedia, andρ ( q) =

∫

+∞

−∞

dpf ( q , p)

(4)

The integration of equation (2) is performed using the semi-Lagrangian approach. The
scheme of [25, 26] is implemented using a parallel MPI strategy. The longitudinal q
coordinate is divided into slices of equal size, each being associated with a computer core.
The computations were performed using a mesh of 400×400 points, with a physical size (in qand p) of 20 × 20 centered on ( q = 0, p = 0), and a time step h = 2π 2000. The laser
perturbation is always applied after the transients have been damped out (typically 1000
synchrotron periods in situations far from the threshold). Sixteen CPU cores are typicallyused. For comparison with the experiments, the equivalent of the bolometer signal B ( θ ) is
computed from the electron bunch density distribution ρ ( q ). The CSR emitted over a 1 − 9.1
k2

cm −1 bandwidth is calculated: B ( θ ) = ∫ ZR ( k ) ρ˜ ( k ) 2dk , with ZR the real part of the CSR
k1

impedance [27], k1 = 6.9 and k 2 = 63, and ρ̃ ( k ) the Fourier transform of ρ ( q ). The bolometersignal B ( θ ) is ﬁnally processed by a ﬁrst-order ﬁlter, with 2μs response time. We
systematically integrated equation (2) for various values of the modulation wavenumber beam
current. In addition, since we are interested in the effect of the wakeﬁeld on the electron
bunch dynamics, we also integrated, in each case, the corresponding equation ‘without
wakeﬁeld’ (i.e. taking I = 0). This will serve as a reference case, to which departures will be
attributed to wakeﬁeld effects.
The dynamical behavior is similar to the experimental one, and the simulated bolometer
response is composed of two successive peaks (ﬁgure 3(e)). As in the experiment, the delayed
response around 55-60 μs is strongly enhanced when the modulation wavenumber is around
1.5 cm −1 (red curve), and is extremely weak otherwise (green curve). Moreover, we can verify
that the delayed peak size is dependent on the collective effects (wakeﬁeld), by comparing thedata to the reference results ‘without wakeﬁeld’ (i.e. I = 0 and all other parameters unchanged,
black curve). The phase space evolution is represented in ﬁgures 3(a)–(d). Analyses versus
the excitation wavenumber (ﬁgure 3(f)) conﬁrms that the maximum delayed response occurs
at ν̄m ≈ 1.5 cm −1.
Further study of wavenumber selection can be performed by plotting the height of thedelayed peak versus modulation wavenumber. As expected (ﬁgure 4(a)), a good agreement
between theory and experiment is found. Furthermore, it is instructive to compare these7

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New J. Phys. 16 (2014) 063027

Figure 3. Numerical calculations of the response to a modulated laser perturbation (withν̄m =1.5 cm −1 and I = 3.5 mA). (a) The phase space just before laser modulation, (b)–(d)the phase space at different times after the laser perturbation: θ = 0, π 2, π, respectively(for clarity, the initial phase space distribution (a) has been subtracted), (e) the deduced
bolometer signals, and (f) the bolometer signal versus excitation wavenumber. Note thatθ = 2π corresponds to one synchrotron period (110 μs).

8

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New J. Phys. 16 (2014) 063027

Figure 4. The characteristic wavenumbers of the instability. (a) Laser seeding strategy:

height of the delayed peak versus ν̄m (red: experiment, black: numerical simulation). (b)CSR emission spectrum above threshold without laser perturbation (simulation, I = 4.5
mA).

curves with the CSR spectrum of the ‘free-running system’ (i.e. without laser seeding) above
the instability threshold. The emitted spectrum (ﬁgure 4(b)) is found to coincide with thetuning curve of ﬁgure 4(a), and this agreement was found for all currents in this zone. Thisshows that the seeding strategy is a promising technique for examining the characteristic
wavenumbers of the instability.
As this technique provides new possibilities for testing theories, we also tried to ﬁnd
possible differences between the theoretical and experimental ﬁndings. One difference is theratio between the delayed response and the immediate one (the ratio is larger experimentally,
as can be seen in the red curves of ﬁgures 2(b) and 3(e)). Further reﬁnement of the theory isbeyond the scope of this paper. However, we believe that this type of experiment may also
point out differences between the observations and existing models in other storage rings, andhelp progress in reﬁning existing theories (e.g. wakeﬁelds).
In conclusion, the manipulation of a relativistic electron bunch using specially shaped
laser pulses allows us to obtain dynamical information on the spatio-temporal instability
occurring in relativistic electron bunches circulating in a storage ring. This information
includes the characteristic wavenumber and the shape of the transients occurring after
perturbation. The present test at UVSOR-II reveals a satisfying agreement with the
Vlasov–Fokker–Planck equation with a parallel-plates wakeﬁeld. Tests in other accelerators
are possible a priori when a laser perturbation is achievable, in particular in storage rings
equipped with a laser slicing setup.

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New J. Phys. 16 (2014) 063027

Acknowledgments

We would like to thank LeCroy for providing the oscilloscope in the ﬁrst experiments. This
work was supported by the Joint Studies Program of the IMS, the JSPS fellowship program forresearch in Japan (S-09171), the PICS project from CNRS, the Grant-in-aid for scientiﬁc
researches (B20360041) of JSPS, and the ANR (Blanc 2010-042301), and used HPC resourcesfrom GENCI TGCC/IDRIS (2013-x2013057057). CERLA is supported by the French
Ministère chargé de la Recherche, the Région Nord-Pas de Calais, and the FEDER.

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