Attosecond betatron radiation pulse train
aa r X i v : . [ phy s i c s . p l a s m - ph ] A ug Attosecond betatron radiation pulse train
Vojtˇech Horn´y ∗ Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden andInstitute of Plasma Physics, Czech Academy of Sciences,Za Slovankou 1782/3, 182 00 Praha 8, Czech Republic
Miroslav Kr˚us
Institute of Plasma Physics, Czech Academy of Sciences,Za Slovankou 1782/3, 182 00 Praha 8, Czech Republic
Wenchao Yan
Institute of Physics, Czech Academy of Sciences,ELI BEAMLINES, Na Slovance 1999/2,182 21 Praha 8, Czech Republic andKey Laboratory for Laser Plasmas (MOE), School of Physics and Astronomy,Shanghai Jiao Tong University, Shanghai 200240, China
T¨unde F¨ul¨op
Department of Physics, Chalmers University of Technology, 412 96 Gothenburg, Sweden (Dated: August 13, 2020) bstract High-intensity X-ray sources are essential diagnostic tools for science, technology and medicine.Such X-ray sources can be produced in laser-plasma accelerators, where electrons emit short-wavelength radiation due to their betatron oscillations in the plasma wake of a laser pulse. Contem-porary available betatron radiation X-ray sources can deliver a collimated X-ray pulse of durationon the order of several femtoseconds from a source size of the order of several micrometres. In thispaper we demonstrate, through particle-in-cell simulations, that the temporal resolution of such asource can be enhanced by an order of magnitude by a spatial modulation of the emitting relativis-tic electron bunch. The modulation is achieved by the interaction of the that electron bunch with aco-propagating laser beam which results in the generation of a train of equidistant sub-femtosecondX-ray pulses. The distance between the single pulses of a train is tuned by the wavelength of themodulation laser pulse. The modelled experimental setup is achievable with current technologies.Potential applications include stroboscopic sampling of ultrafast fundamental processes. ∗ [email protected] NTRODUCTION
Sub-femtosecond high brightness X-ray pulses are in high demand by research commu-nities in the fields of biology, material science or femtochemistry [1], as well as by industryand medicine[2]. Such pulses can be used as a diagnostic tool to resolve the structure anddynamics of dense matter, proteins, and study fundamental physical phenomena such aschemical reactions, lattice vibrations or phase transitions. Currently, high brightness X-raysources are produced by large scale facilities based on radiation emission by relativistic elec-tron bunches, e.g. synchrotron light sources [3] and X-ray free electron lasers [4]. This limitstheir general availability for many of the potential users. Here, we propose a new method toproduce a train of equidistant sub-femtosecond X-ray pulses with a currently available lasersystems.Acceleration of electron bunches by the plasma wakefield driven by laser [5, 6], electron [7],or proton [8] beams provides a promising alternative to the aforementioned concepts. Themajor advantage of plasma based accelerators is their ability to sustain acceleration gradientsof the order of hundreds of GeV/m, which is approximately three orders of magnitude higherthan is attainable with standard radiofrequency accelerators. Thus, the electrons can beaccelerated to energies of the order of hundreds of MeV in a few millimeters. During theacceleration process, the electron bunch undergoes transverse betatron oscillations due tothe presence of the transverse electric field. As a result, betatron radiation [9–11] with asynchrotron-like [12] spectrum, typically in the X-ray range, is emitted.The betatron radiation characteristics depend on the electron Lorentz factor γ , plasmaelectron density n e , betatron oscillation amplitude r β , and number of oscillation periods N . The radiation spectrum is characterized by a critical energy, close to the peak of thesynchrotron spectrum, given in practical units ~ ω c [eV] = 5 . × − γ n e [cm -3 ] r β [ µ m].The average photon number with energy ~ ω c emitted by an electron is N X = 5 . × − N K ,where K = 1 . × − γ / n / e [cm -3 ] r β [ µ m] is the strength parameter [13, 14]. Severalapplications of such betatron sources have been demonstrated, e.g. diagnosing biologicalsamples[15] and probing extreme states of matter[16], but others would require higher photonnumber and benefit from increased energy efficiency and better tunability.Several recent studies suggest methods for enhancing betatron radiation emission, mostlybased on the increase of the betatron oscillation amplitude. This can be achieved by an axial3agnetic field, either self-generated or external [17, 18]; by a delayed modulation laser pulse[19]; by the interaction of the electron beam with a high intensity optical lattice formed bythe superposition of two transverse laser pulses [20]; by using structured laser pulses [21]; orby the interaction of electrons with the tail of the plasma wave drive pulse[22–25].The betatron oscillation can also be tuned by manipulation of the plasma density. Thiscan be done in several ways, e.g. by using a tilted shock front in the acceleration phase[26], an axially modulated plasma density[27], off-axis laser alignment to a capillary plasmawaveguide [28], transverse density gradient [29, 30], or tailoring the dynamics of the nonlinearplasma wave in a way that electrons find themselves behind its first period (the bubble) fora certain period of time, where their oscillations are amplified due to the opposite polarityof transverse fields [31]. Also, injection of matter by irradiating solid micro-droplets [32] ornanoparticles [33] may provide enhancement of the generated betatron X-ray intensity.The conversion efficiency from laser-light to X-ray can be increased by using a hybridscheme, which combines a low-density laser-driven plasma accelerator with a high-densitybeam-driven plasma radiator[34]. Increase of betatron light by localized injection of a groupof electrons in the shape of an annulus was also reported [35]. The X-ray flux can alsobe increased due to shortening of the betatron oscillation wavelength during the naturallongitudinal expansion of bubble [36].In this paper, we propose an experimental setup where, in addition to an enhancementof the betatron radiation flux, a train of sub-femtosecond X-ray pulses is generated. It isachieved by separation of the electron bunch accelerated in the laser wakefield into a train ofequidistant sub-bunches by a delayed modulation laser pulse, see Figure 1a) for a schematicof the proposed setup. The separation interval between the pulses corresponds to half of themodulation pulse wavelength and each pulse in the train is even shorter.Generation of electron bunch trains has been studied previously. They originate eitherfrom conventional radiofrequency accelerators [37–39], from laser wakefield accelerators em-ploying self-injection controlled by driver pulse shaping [40] or optical injection by crossingtwo wakefields[41], or from plasma wakefield accelerator injected due to the bubble lengthoscillation on the density downramp [42]. The advantage of the scheme described in thispaper over the aforementioned ones is that the electron bunching is well controlled by themodulator on the sub-micron scale. Thus, the emitted signal comprises of the train of X-raypulses with an unprecedented repetition rate.4ulse-trains composed of sub-femtosecond X-ray pulses can enhance the temporal res-olution of sampling of ultrafast fundamental physical processes by an order of magnitude,whilst maintaining its other advantageous features such as a small source size of severalmicrons enabling high-resolution images and a relatively small cost of the required lasersystems compared to the large scale facilities such as synchrotrons or free electron lasers.A broadband X-ray pulse-train could sample physical processes occurring on femtosecondtime-scales by e.g. X-ray absorption spectroscopy (XAS) or polychromatic (Laue) X-raydiffraction. In all cases, the image observed at the detector (typically a CCD camera) wouldbe composed of a series of sharp and fuzzy regions. As the time-delay between the X-raypulses in a train is set by the wavelength of the modulation pulse, the dynamics of thesampled process can be extracted from the configuration of the sharp region on the detectedimage. This approach is analogous to stroboscopic measurement of fast processes, see Fig-ure 1b) for a schematic illustration. In attosecond science, stroboscopic images have beenalready recorded[43] with high harmonics emission[44]. Our source, despite being incoherenton its wavelength, provides higher photon energy which results in the increased penetrabilitythrough the investigated sample. drive pulse modulation pulse bubble FIG. 1.
Schematics of the proposed setup and the application configuration . (a) Amoderately high-intensity laser pulse creates a plasma cavity free of electrons (bubble). An electronbunch is injected in the rear part of the bubble, along with a weaker modulation pulse, with adelay that is such that it propagates with the electron bunch. (b) Illustration of stroboscropicmeasurement of fast processes using a modulated X-ray probe. ESULTS
A driving laser-pulse of moderate intensity ( I . W · cm -2 ), linearly polarized inthe y -direction, propagates in the longitudingal ( x ) direction in an underdense plasma (inpractice, n e is in the order of 10 cm -3 ) and creates a moderately nonlinear plasma wave.Its first period, the so-called “bubble”, is an ion cavity free of electrons which are expelledby the strong ponderomotive force of the driving pulse. The electron bunch is located inthe rear part of the bubble. It is injected transversely ( y -direction), either by self-injection,or as is the case in this paper, by controlled injection on the density downramp. A weakermodulation pulse ( I . W · cm -2 ) with wavelength λ m is injected to follow the drivingpulse. Its electric field, polarized in the y -direction, still dominates over the electrostatictransverse field of the bubble. The delay between the pulses is chosen in a way that itshigh-intensity part co-propagates with the electron bunch.As the modulation pulse propagates within the bubble, its group velocity is approximatelyequal to the speed of light in vacuum v g,m / c . The average longitudinal velocity of anelectron in the bunch is lower, due to the relativistic limitation caused by transverse betatronoscillations. The accelerated electrons oscillate transversely on a sine-like trajectory becausethey gained a considerable transverse momentum dominantly by the fields of the modulationpulse, but also by the injection process and by the electrostatic transverse fields of thebubble. Every periodic increase of their transverse velocity leads to a decrease of theirlongitudinal velocity. As a result, the modulation pulse steadily overtakes the electronbunch. Consequently, an electron from the bunch experiences the action of a periodicallyvarying transverse component of the Lorentz force as it propagates backward with respectto the modulation pulse.The transverse electron motion can be described by the equation of motion d p y / d t ≈ q e (1 − β x ) E ,y,m cos( k m ξ ), where q e is electron charge, E ,y,m is the electric field amplitudeof the modulation pulse, k m ξ is the phase of the modulation pulse, with k m = 2 π/λ m beingthe modulation pulse wavenumber and ξ = x − x − v g,m t the coordinate co-moving with themodulation pulse. Here, we assumed | p x | ≫ | p y | , p x ≫ m e c , and considered the modulationpulse as a plane wave, which is applicable in regions around the propagation axis, whereits magnetic field is proportional to its electric field B z ≈ E y /c . Thus, the electrons flowbackward with respect to the modulation pulse and due to the phase dependence of the6ransverse force, they are periodically pushed in the ± y − direction. This effect itself leads toenhancement of the betatron radiation emission in comparison with a standard case withoutthe modulation pulse.From the positions where cos( k m ξ ) = 0, the absolute value transverse momentum of theelectrons decreases and the longitudinal momentum grows; the latter one is largest at theturning points of their trajectory where p y = 0. Thus, the turning points related to themodulation pulse phase are the same for all electrons of the bunch. Large longitudinalmomenta together with low transverse momenta result in a clustering of the bunch electronsin the nests co-moving with the modulation pulse. Alternatively stated: the original electronbunch is microbunched. As the betatron radiation is mainly emitted at the turning pointsof the electron trajectories, its temporal profile is composed of intensity peaks separated by λ m / c , i.e. a train of X-ray pulses is emitted and the delay between the pulses is adjustableby choosing λ m .The effect of microbunching can be understood as a forced betatron resonance. Contraryto previous cases with the modulation by the tail of the plasma wave drive pulse[23, 25],where the electron beam experiences a long acceleration period before it catches the laserpulse which resulting in limited controllability of the X-ray source, we reach the betatronresonance immediately from the moment of injection. Numerical simulation
The process of michrobunching and its fingerprint on the betatron radiation signal isstudied by means of 2D particle-in-cell (PIC) simulations and their post–processing. Abubble regime configuration with modest laser parameters is chosen for the demonstrationof the process. The parameters used in the simulation are the following: plasma electrondensity n = 2 . × cm -3 , driver laser wavelength λ d = 0 . µ m, waist size (radius at1/e of maximum intensity) w = 10 µ m, pulse length (FWHM of intensity) τ = 20 fs, andnormalized driver laser intensity a ,d = eE ,d /m e cω = 1 . I = 6 . × W cm − . Its focal spot is located at x f,m = 110 µ m. The modulation pulsehas the same fundamental parameters with the exception of normalized intensity, which is a ,m = 0 .
2, and wavelength λ m = λ d / . × W cm − . It isdelayed by 58 fs and its focal spot is located at x f,m = 410 µ m. Both pulses are linearly7
54 656 658-4-2024384 386 388-4-2024115 116 117 118 119-4-2024
115 116 117 118 11901234
384 386 38801234
654 656 65801234
FIG. 2.
Plasma bubble evolution and electron microbunching.
Snapshots of the electrondensity at the injection time (0 . . . x − axis. polarized in the y − direction.Self-injection of electrons in the plasma wakefield does not occur with these parameters ifthe plasma density is constant. Instead, a plasma density profile is chosen so that controlledinjection occurs. In the simulations, the density profile is set in the following way. A 10 µ mlong vacuum is located at the left edge of the simulation box, then a 50 µ m linear densityup-ramp follows until the electron density reaches 2 n e . Nevertheless, the nature of thepresented injection scheme does not depend on the plasma-edge density ramp. Afterwards,a 35 µ m long density plateau follows; then the density linearly drops to n e over a distance of25 µ m. On this down-ramp, the controlled injection occurs[45]. The PIC simulations wereperformed with the epoch code, see the Methods section for details.The snapshots of the electron density during the injection and acceleration process areshown in Figure 2. The density profile in the panel corresponding to the injection time( t = 0 .
505 051015
Time: 2.3 ps Time: 2.3 ps E y [TV/m] n e [ n ]
655 656 657 658-4-2024 -505 a =0a =0.2 a) b) c) (cid:1) m FIG. 3.
Electron bunch structure and energy spectrum a) Electron density of the trappedelectrons (plotted the simulation cells where average kinetic energy of electrons is higher than10 MeV) and the transverse electric field at t = 2 . t = 4 . together with the transverse electric field. Apparently, the electric field of the modulationpulse dominates over the electrostatic field of the bubble in the region around the axis wherethe electron bunch is located. The bunch itself has a sawtooth-shape. The distance betweenthe x − coordinates of the turning points is λ m /
2. The peak values of the electron densityare located in these turning points.Figure 3b) shows the positions and transverse momenta of the accelerated electrons. Thepositions between the peaks of the density bunch profile and the dominant direction of thetransverse component of the electron momentum confirm that the electrons propagate back-wards in the frame co-moving with the modulation pulse. These findings can be interpretedas the electron bunch as a whole performs snake-like motion in the direction of − ξ . Thismeans that the modulation pulse effectively induces the microbunching of injected electronsand the distance between single microbunches is λ m / λ m / c , as will beshown later. 9he electron energy spectrum in time of 4.0 ps just before the structure begins to dephaseis shown in Figure 3c); blue and red lines show the cases without and with the modulationpulse, respectively. The spectra comprise a clear peak which corresponds to the electronsaccelerated in the first period of the plasma wave due to the controlled injection. Although,the relative energy spread is rather high. However, for the purpose of betatron radiationgeneration the energy spread is not a determining factor. The presence of the modulatorleads to further electron energy gain compared to the reference case: the electrons receivethe energy stored in the modulator by direct laser acceleration [46, 47]. The estimatedaccelerated charge (electron energy higher than 25 MeV) is about 4 to 8 pC in both cases.There are about 1.3% less electrons trapped when the modulator is present. Betatron radiation spectrogram
Figure 4 shows the spectrograms, i.e. both temporal and energy profiles of the betatronradiation, with and without the modulation pulse; for details see the Methods section. Fourdifferent cases are presented: a) the case when the modulator is not present, (b) with λ m = λ d and a ,m = 0 .
6, (c) with λ m = λ d / a ,m = 0 .
1, and (d) with λ m = λ d / a ,m = 0 . ≈ µ mshown in Figure 3. Nevertheless, while the signal is continuous in the case without themodulator (Fig. 4a), the modulated signals (Fig. 4b-d) exhibit trains of ultrashort pulses.Moreover, the spectrograms show that the betatron radiation critical energy is also modu-lated in time. In average, the energy of radiation is considerably higher when the modulatoris present. The inset in panel (d) confirms the correlation between the energy distributionof electrons within the bunch and the temporal and energy profile of emitted X-rays.Figure 5 shows the temporal profiles of betatron radiation. Whereas the blue curvebelonging to reference case (a) does not vary significantly, the other three curves (b-d)show several clear peaks. The red curve represents the case (b); three dominant peaks arepresent. The peak-to-peak distances is between the first and the second and the second andthe third dominant peaks are 1.35 fs and 1.29 fs, respectively. This is in good agreementwith the theoretically expected value λ m / c = 1 . (cid:0)
23 02468 (cid:2) -(cid:3)9d(cid:4) λ m = λ d /(cid:5)(cid:6) a =0.20 5 (cid:7) (cid:8)(cid:9) .5 0 (cid:10) (cid:11) (cid:12)(cid:13)(cid:14) c) (cid:15) (cid:16) (cid:17)(cid:18) .522.5 (cid:19) (cid:20)(cid:21)(cid:22) b) (cid:23) (cid:24) (cid:25)(cid:26) (cid:27)(cid:28)(cid:29) a) FIG. 4.
Spectrograms of the betatron radiation emitted by the electrons.
Temporal andenergy profiles are shown for a reference case without a modulator and for three different modulatorpulse cases. The signal close to t = 0 corresponds to the front of the bunch and arrives first atthe detector. The inset in the panel d) shows the electron energy distribution within the bunch.It displays a matrix of the average electron energy in cell; only the cells with average energy over10 MeV are shown. Both temporal and energy profile of emitted X-rays are correlated with theinner structure of the bunch. Note that the x -axis is reversed. (0.46 ± λ m / c = 0 . ± (cid:30) (cid:31) !" $ % & λ m = λ d ’() a =0.2 *+a, .e0234568 c :;< λ m = λ d , a =0.6 λ m = λ d =>? a @ABC b)c) D E F G HI J KL .522.5 M NOP Q R S E cr T UV WX E cr Y Z[\ ]^ E cr _ ‘bc fg E cr h ijk lm E cr a) b) FIG. 5.
Energy spectra for the emitted betatron radiation a) Temporal profile of thebetatron radiation for a reference case without a modulator and for three different modulator pulsecases. The inset, corresponding to the case λ m = λ d / a ,m = 0 . λ m = λ d and a ,m = 0 .
6, (c) λ m = λ d / a ,m = 0 .
1, and (d) λ m = λ d / a ,m = 0 . The number of electrons within the bunch differs by less than 3.5% between all fourcompared cases.The estimated total energy within the pulse train is 0.10 nJ in case (a). Itincreases greatly when the modulator in present: it is 0.45 nJ, 0.65 nJ, and 2.2 nJ in cases(b-d), respectively. The increase is caused partly by the higher energy of the electrons andpartly by the higher amplitude of betatron oscillations.Finally, the time-integrated energy spectra on axis for all the cases (a-d) are shown inFigure 5b, including information about the critical energy of the emitted signal in all cases.The critical energy of the case (d) is 5.3 × higher than in the reference case (a). DISCUSSION
We propose a method for producing a train of ultrashort X-ray pulses by modifying thestandard laser wakefield accelerator setup delivering betatron radiation. This is accom-plished by adding a delayed modulation laser pulse to follow the plasma wave in the region12here the electron bunch is injected. As a result, the betatron oscillations of the acceleratedelectrons are driven dominantly by the fields of the modulation pulse and not by electro-static fields of the bubble. The turning points of the betatron trajectories are the same forall accelerated electrons and the electrons cluster there.In other words, the electron bunch is microbunched and the longitudinal distance betweenthe single bunches is half of the modulation pulse wavelength λ m . This property is imprintedon the temporal profile of the emitted X-rays. Thus the betatron radiation signal is composedof a train of pulses separated by a factor of λ m / c , which is 440 as when third harmonicsof a standard Ti:sapphire laser pulse is used as the modulator. Moreover, the energy andintensity of the emitted X-rays are also enhanced. The resulting X-ray source could enableobservation of temporal evolution of ultrafast phenomena on the time scale of hundreds ofattoseconds.The process of electron microbunching was further tested in a relatively broad parameterspace. The scheme works in the densities 1 . × cm -3 – 6 × cm -3 . The sharpestmicrobunching occurs in lower densities, as higher density leads to the lower plasma wavephase velocity causing the structure decay due to dephasing. The results that are presentedthroughout the paper are given after 3.5 ps of acceleration time ( t = 4 . a ,m ∈ [0 . , . METHODS
2D PIC simulations were performed with the EPOCH [53] code. The simulations were runin the moving simulation box with dimensions 80 µ m × µ m. The grid resolution was 90and 12 cells per λ d in the longitudinal and transverse directions, respectively. Initially, twoelectron macroparticles were placed in every cell. The plasma is represented as an electrongas; the ions were considered as a homogeneous static background. In total, approximately2 . × macroparticles were simulated.The temporal profile of betatron radiation was calculated using the method based onthe Fourier transform of the emitted signal which can be determined by using trajectoriesof the trapped electrons[54]. It takes advantage of the fact that each electron performsbetatron motion in the wiggler regime and the emitted signal is composed of a series ofsharp peaks radiated at the turning points of the electron trajectories separated by relativelylong intervals of silence. Thus, it is possible to store the times when the single peaks ofall the tracked electrons were emitted and construct the betatron radiation spectrogramfrom that. This method is applicable even for the discussed case of X-ray emission bymicrobunched electrons, because the level of microbunching does not suffice to emit coherentelectromagnetic radiation more energetic than ultraviolet. 20 000 of the tracked electronmacroparticles were processed in each case. ACKNOWLEDGMENTS
The authors thank V´aclav Petrˇz´ılka from IPP CAS, Evangelos Siminos from GothenburgUniversity, and Julien Ferri and Longqing Yi from Chalmers University of Technology, for14heir suggestions and fruitful discussions. This work was supported by the Ministry of Edu-cation, Youth and Sports of the Czech Republic within the project LQ1606, from the HighField Initiative (CZ.02.1.01/0.0/0.0/15 003/0000449) from European Regional DevelopmentFund, and also received funding from the European Research Council (ERC) under the Eu-ropean Union’s Horizon 2020 research and innovation programme under grant agreementNo 647121.Access to computing and storage facilities owned by parties and projects contributingto the National Grid Infrastructure MetaCentrum provided under the programme
Projectsof Large Research, Development, and Innovations Infrastructures (CESNET LM2015042),and to ECLIPSE cluster of ELI-Beamlines project and at Chalmers Centre for Computa-tional Science and Engineering (C3SE) provided by the Swedish National Infrastructure forComputing is greatly appreciated as well.
AUTHOR CONTRIBUTIONS STATEMENT
V.H. and W.Y. conceived the idea. V.H. performed the simulations. V.H. and M.K.developed the theoretical interpretation. All authors discussed the findings and contributedto the writing of the manuscript.
ADDITIONAL INFORMATION
Competing financial interests
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