Tailored laser pulse chirp to maintain optimum radiation pressure acceleration of ions
aa r X i v : . [ phy s i c s . p l a s m - p h ] N ov Tailored laser pulse chirp to maintain optimum radiation pressureacceleration of ions
F. Mackenroth a) and S.S. Bulanov Max Planck Institute for the Physics of Complex Systems, Dresden, Germany Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: 21 November 2018)
Ion beams generated with ultra-intense lasers-plasma accelerators hold promises to providecompact and affordable beams of relativistic ions. One of the most efficient accelerationsetups was demonstrated to be direct acceleration by the laser’s radiation pressure. Due toplasma instabilities developing in the ultra-thin foils required for radiation pressure acceler-ation, however, it is challenging to maintain stable acceleration over long distances. Recentstudies demonstrated, on the other hand, that specially tailored laser pulses can shorten therequired acceleration distance suppressing the onset of plasma instabilities. Here we extendthe concept of specific laser pulse shapes to the experimentally accessible parameter of afrequency chirp. We present a novel analysis of how a laser pulse chirp may be used to drivea foil target constantly maintaining optimal radiation pressure acceleration conditions forin dependence on the target’s areal density and the laser’s local field strength. Our resultsindicate that an appropriately frequency chirped laser pulse yields a significantly enhancedacceleration to higher energies and over longer distances suppressing the onset of plasmainstabilities.Keywords: laser-plasma ion acceleration, radiation pressure acceleration, high-power laserapplications
I. INTRODUCTION
Beams of relativistic ions serve a wide range of ap-plications from technical material science, over medicalapplications to even fundamental studies of high energyphysics. Some of these applications particularly benefitfrom short, dense ion beams, not necessarily of ultra-highenergy . Relativistic ion beams with the necessary highfluxes can be accelerated by high power lasers , whichhave undergone considerable development over the pastdecades , with several facilities breaking the Petawatt(PW) barrier already operational , or in planning .Consequently, the acceleration of ions to relativistic en-ergies by high-power lasers is among the most intenselystudied applications of such laser systems .As a result of this deep interest in laser-ion accelera-tion, there were several technical approaches proposed,to overcome the challenges of this application, suchas the experimentally most widely studied target nor-mal sheath acceleration (TNSA) , Coulomb explo-sion (CE) , hole boring (HB) , relativistic trans-parency (RT) , shock wave acceleration (SWA) ,magnetic vortex acceleration (MVA) , standing waveschemes , and several others to the highly efficient ra-diation pressure acceleration (RPA), . In this latterregime, a thin solid density foil target is quickly ionizedby the laser pulse to form a plasma, which reflects the in-coming radiation and is consequently accelerated by thelaser’s radiation pressure. Furthermore, in the foil’s restframe the laser’s frequency will appear down-shifted by afactor 2 γ , where γ = ǫ/m i is the foil’s relativistic factor,with ǫ and m i being the energy and mass of a single ion a) Electronic mail: [email protected] of the foil, respectively. It is thus apparent, that for rela-tivistic foil energies ( γ ≫
1) the laser light’s frequency isstrongly reduced leading to almost complete transfer oflaser energy to the foil. In this regime, due to the rela-tivistic time dilation, the acceleration is maintained overa long time during which the foil almost co-propagateswith the laser and constantly experiences its radiationpressure. It was shown that in an ideal setting this leadsto the foil reaching an energy proportional to that of theaccelerating laser pulse. We note that the development ofinstabilities and the presence of other limiting factors,for example, laser group velocity and transverse targetexpansion , limit the effectiveness of the RPA. How-ever, it was shown recently, that the laser pulse tailoringand special target engineering might compensate theselimiting factors.The RPA was never experimentally tested in the ultra-relativistic regime due to the lack of necessary laser facili-ties, however there are experimental indications that thisscheme also works in the nonrelativistic regime .It is less stable and less effective than in the ultra-relativistic regime, mainly because the foil reflectivityis no longer perfect, but depends complicatedly on thetarget areal density, as well as the laser’s intensity, andfrequency . Including this non-trivial parameter de-pendence of the foil reflectivity in a one-dimensionalmodel of its dynamics, it was demonstrated that theion energies are optimized if for a foil of density andthickness n e and l , respectively, moving with a momen-tum p ( t ) at position x ( t ), corresponding to the single-particle energy ε ( t ), by the radiation pressure of a laserpulse with electric field envelope E ( t, x ) and frequency ω ( t, x ) = 2 π/λ ( t, x ), are related via the following opti-mum condition a ( t, x ( t )) = γ ( t ) ǫ ( t, x ( t )) , (1)where a ( t, x ) = | eE ( t, x ) | / ( m e ω ( t, x )) is the laser’sdimensionless amplitude, the parameter ǫ ( t, x ) = π ( ln e ) / ( λ ( t, x ) n cr ( t, x )), introduced in , is the target’sareal density normalized to the product of the laser’swavelength λ and the critical plasma density n cr ( t, x ) = m e ω ( t, x ) / (4 πe ), where e < m e are the elec-tron charge and mass, respectively, and units with c ≡ .In this paper, we study how the optimum condition(1) can be optimized through a tailored frequency profile,instead. We are going to demonstrate that the changesof the reflectivity can be counteracted by a complicatedfrequency chirp of the driving laser pulse and derive aclosed analytical form of the laser’s required frequencydependence. We note that the problem of the laser chirpinfluence on the ion acceleration was addressed in a num-ber of papers , however a systematic analytical treat-ment of this problem in the case of a thin foil RPA wasmissing. II. THEORY
We begin by reformulating the optimum condition interms of basic quantities as |E ( t, x ) | = γ ( t ) , (2)where we introduced the scaled electric field E ( t, x ) = E ( t, x ) /E foil , where E foil := 2 π | e | ln e is the static, one-dimensional charge separation field of the foil. Next, wenote that eq. (2) is independent of the laser’s frequencyand thus infer that if the laser’s electric field and fre-quency are independent the laser’s frequency cancels outof the optimum condition and, provided the field am-plitude is varied appropriately, eq. (2) is fulfilled for allfrequencies. The same conclusion can be drawn fromeq. (23) of , which is independent of the laser’s fre-quency.On the other hand, the frequency still does impactthe acceleration process heavily, despite the fact that theoptimum condition is independent of it. To demonstratethis, we turn to the foil’s equation of motion dp dt = K | E ( t, x ) | πln e p m i + p ( t ) − p ( t ) p m i + p ( t ) + p ( t ) (3) dx k dt = p ( t ) p m i + p ( t ) K = 2 | ρ | + | α | , where m i is the mass of a single ion in the foil and ρ and α are the foil’s its reflection and absorption coefficient,respectively. We rewrite these equations to be expressedin the Lorentz invariant laser phase η = ω ( t − x k ( t )),where again x k ( t ) is the foil’s position, to read dp dη = dp dt dtdη = | ρ ( η ) eE ( η ) | eE foil p m i + p ( η ) p m i + p ( η ) + p ( η )(4) dtdη = p m i + p ( t ) p m i + p ( t ) − p ( t ) . Separating the variables in this equation its general so-lution was found to be given by p ( η ) = 12 (cid:18) h + D − m i h + D (cid:19) (5) h := p ( η ) + q p ( η ) + m i (6) D := Z ηη dη ′ | ρ ( η ′ ) eE ( η ′ ) | eE foil . (7)We continue by rewriting condition (2) as a function ofthe phase-dependent momentum. We find that maintain-ing the optimum acceleration condition is ensured by themomentum fulfilling the condition p opt0 ( η ) = m i p E ( η ) − . (8)We immediately conclude that this condition can onlybe satisfied for |E ( η ) | ≥
1, whence we have to focus onthis parameter regime in the following. We do so byhaving our analysis only start at the time instant η defined by |E ( η ) | = 1, where we assumed the foil tobe at position x ( η ) = 0. In the following we assumethe laser’s intensity envelope to be given by a Gaus-sian of FWHM τ L , modeled by a field envelope of theform E ( η ) = E max exp (cid:2) − η/τ L ) (cid:3) , independentlyof the pulse chirp. We choose to fix the pulse envelope inaccordance with the above reasoning, rather than, e.g.,assuming a constant value of a or laser intensity, as wewish to model a pulsed laser field of a given intensityprofile, in agreement to experimental setups. Followingthis model, the threshold condition can be analyticallysolved to give the phase at which the laser reaches thethreshold of the foil transparency η = τ L s log [ E max ]2 log(2) , (9)such that the acceleration can only be optimized in theinterval η ∈ [ − η , η ]. For all numerical examples studiedbelow we are going to consider the acceleration only ininterval η ∈ [ − η , ω x ( t )3210 p ( t ) / m i [ × − ] (cid:0) p opt ( t ) , x opt0 ( t ) (cid:1) ( p ( t ) , x ( t )) FIG. 1. Comparison of the foil’s momentum in units of anaccelerated ion’s mass as a function of its position in unitsof wavelength in the general as well as the optimized case forthe parameters given in the text. with τ L = 10 /ω . In order to test the improved ion ac-celeration regime, we numerically integrate eq. (8) andcompare it to a full numerical solution of the system (4).Furthermore, as we additionally wish to highlight theenhancement in the accelerated foil’s position x ( η ), weadditionally solve its equation of motion dx dη = dx dt dtdη = p ( t ) p m i + p ( t ) − p ( t ) , (10)where we made use of the relativistic velocity relation dx /dt = p ( t ) / p m i + p ( t ). In order to analyze the so-lutions of the respective equations of motion for the foil’smomentum and position in the general as well as the op-timized case we study a foil of density n e = 10 cm − ≈ n cr for radiation of 800 nm wavelength, and thickness l = 10 nm accelerated by a moderately relativistic laserwith intensity I = 1 . × W / cm . Furthermore, as wewish to analyze the efficiency enhancement as a functionof space-time, we plot the foil’s momentum in the labo-ratory frame as a function of its position by means of aparametric plot with the coordinates ( p ( η ) /m i , ω x ( η ))(s. fig. 1). We find a significant enhancement of the foil’smomentum, when accelerated according to the optimumcondition p opt0 , derived from eq. (8). We furthermorenote that while we only consider the acceleration up tothe phase value η = 0, the physical time extends to val-ues larger than 0, for the simple reason that the non-trivial foil displacement x ( t ) leads to a non-trivial re-lation between phase η and time t = η + x ( t ). In thisrespect, we also note that since p opt0 ( t ) > p ( t ) it willbe x opt0 ( t ) > x ( t ) as reflected in the fact that η = 0is reached at larger displacements x opt0 ( η ) in the opti-mized regime. We note that this can be interpreted asthe optimized case leading to a reduction of the accel-eration length, i.e., the same ion energy can be reachedover smaller distance. This effect is clearly inferable fromthe fact that in the optimized case the ion momentumat any given foil position is is higher than in the unop-timized case, indicating that the optimized pulse chirpneeds significantly less distance to accelerate ions to acertain energy. Usually the acceleration length is limited by either the Rayleigh length or the transverse expansionof the target, or both. In the case of transversely flat-top(e.g., super-Gaussian) laser pulses, which are often sug-gested to be employed to produce quasi-mono energeticion beams, the acceleration length is limited by the factthat such pulses do not propagate without changing theirtransverse shape. Thus, any technique that allows forreaching some ion energy over shorter distance is boundto optimize the acceleration process.Having established the improved performance of eq. (8)we continue to discuss how a frequency chirp can be usedto maintain the optimum condition eq. (1). To this end,we require the reflection coefficient entering eq. (4). Afoil’s reflection coefficient depends on the laser’s and foil’sparameters in its rest frame ( γ ≈
1) according to ρ ( η ) = ǫ ( η ) a ( η ) (cid:16)(cid:2) ∆ − (cid:3) + 4 a (cid:17) + ∆ − (cid:16) [∆ − + 4 a (cid:17) + ∆ + 1 , (11)where we defined the difference ∆ ( η ) := a ( η ) − ǫ ( η ).Transforming this relation back into the laboratory frameamounts to the replacement ǫ ( η ) → γ ( η ) ǫ ( η ). Hence,from eq. (1) we read off that both in the foil’s rest frameand the laboratory frame the optimum condition can beused to simplify the reflection coefficient. Consequently,in this work we can always use ρ ( η ) = (cid:0) a (cid:1) − a ) + 1 . (12)Assuming then properties of a typical foil of 10 nm thick-ness and a density of n e = 10 / cm , approximately cor-responding to 50 times the critical density for an opticallaser beam of 800 nm wavelength, driven by a laser withthe above assumed Gaussian temporal profile and peakintensity I = 10 W / cm we find a decisive dependencyof the reflection coefficient on the incident frequency (s.fig. 2). We can also read off that this dependency ismore pronounced the in the laser’s rising edge where a islower, indicating that the laser frequency gives the finesttunability during the starting phase of RPA. Based onthese results, it is apparent that even though eq. (8) isformally independent of the laser’s frequency, an appro-priate pulse chirp can still be used to tailor the reflectioncoefficient and hence the overall acceleration. This ap-proach is complementary to the tuning through an op-timized intensity profile , which is aimed at the ultra-relativistic regime. In contrast, the here presented fre-quency optimization is most apt to steer and stabilizethe commonly highly unstable initial phase of radiationpressure acceleration. To find an optimal pulse chirp, adifferential approach is favorable: Provided we can matchthe foil’s momentum to its ideal value at a given time in-stant, from that time on the optimum condition can beenforced on its differential equation of motion (8). Theoptimum momentum, fulfilling eq. (2) perpetually, on the ω/ω ρ η = − η η = − η / η = 0 FIG. 2. Reflection coefficient as a function of the laser’sfrequency at various phases of the driving laser pulse fora foil of 10 nm thickness and a density of n e = 10 / cm ( ≈ n cr ( ω = 1 . other hand, changes as dp opt dη = m i E ′ ( η ) E ( η ) p E ( η ) − . (13)This differential change can be matched to eq. (4) pro-vided the reflection coefficient is given by ρ ( η ) = vuut p E ( η ) − E ( η ) ! m i E ′ ( η ) | eE ( η ) | , (14)where we additionally assumed eq. (8) to be fulfilledand the foil to be initially at rest p ( η ) = 0. Equatingeqs. (12,14) and solving for the laser frequency numeri-cally will be a benchmark for our analytical results. Wenote that provided the optimal reflection coefficient (14)is maintained throughout the acceleration process, thefoil ions’ momentum will develop according to eq. (13)and be given by eq. (8). Additionally, we see that eq. (14)is only meaningfully defined in the regime E ′ ( η ) ≥ γ ( η ) would be re-quired to maintain the optimum condition, which wouldcorrespond to deceleration instead of the desired accel-eration. We thus have to limit our analysis to the firsthalf of the pulse, discriminated by E ′ ( η ) ≥
0. We notethat the results of Ref. indicate that the optimal pulseprofile has only the rising edge.In accordance with the technical development we aregoing to consider high-intensity laser pulses defined bythe condition a ≫ a ( η ) ≈ γ ( η ) ǫ ( η ),where we did not yet assume eq. (1) to hold exactly. Wecan then study the acceleration process for the case of anultra-relativistic foil p i ≫ m i , as is common in investi-gations of the RPA scheme, and in addition in the foil’snot commonly considered beginning, still non-relativisticacceleration phase p i ∼ m i . A. Non-relativistic foil motion
We begin studying the motion of a non-relativistic foil1 − γ ( η ) ≪
1, which, according to eq. (2) translates tothe condition |E ( η ) | ∼
1. From eqs. (1) we deduce that inthis case close to the optimum drive regime the foil’s andlaser’s parameters are linked by ǫ ≈ a . Furthermore,from eq. (9) we see that in the regime E ∼ η ≪ τ L , such thatthe field will not be strongly changing E ′ ( η ) ≪ E (0) /τ L .From eq. (14) we thus see that the optimal reflection coef-ficient has to be rather small, which is achievable for largelaser frequencies (compare fig. 2). This in turn, however,implies small values of a , even for high laser powers. Inorder to corroborate this conjecture, we need to find ananalytic expression for the laser’s frequency structure en-suring that the foil’s reflection coefficient from eq. (12)is matched to its optimal reflection derived in eq. (14).To find such a solution for the optimized laser frequency,we again solve the foil’s equation of motion (4) throughseparation of variables and find that, when neglectingabsorption in the foil, in the non-relativistic regime themomentum of a foil initially at rest is given by p ( η ) = Z ηη dη ′ | ρ ( η ′ ) eE ( η ′ ) | eE foil (15)The same result can be found expanding solution (5) tolowest order in p /m i . From eq. (8) we deduce that thephase dependent reflection coefficient required to meetthe optimum condition is given by ρ ( η ) = s m i E ′ ( η ) | eE ( η ) | p E ( η ) − , (16)which is equivalent to approximating the solution ofeq. (14) in the regime p E ( η ) − ≪ E ( η ). Equatingthis to the approximation (12) we find that an appropri-ate laser frequency chirp can ensure condition (1) to besatisfied in the regime E ∼ ω optNR ( η ) = eE ( η ) m e | eE ( η ) | p E ( η ) − − m i E ′ ( η ) q m i E ′ ( η ) | eE ( η ) | p E ( η ) − . (17)Comparing this analytical expression to the numericallyconsistent solution of eqs. (12,14) for a foil of 10 µ m thick-ness and n e = 9 × cm ≈ n crit accelerated by alaser pulse of intensity I = 10 W / cm and τ L = 10 /ω we find good agreement with the outlined derivation (s.fig. 3). We also find our conjecture confirmed that therequired laser frequencies are several hundreds of eV, asissued above. B. Ultra-relativistic foil motion
We now turn to studying the motion of an ultra-relativistic foil γ ( η ) ≫
1, which, according to eq. (2)translates to the condition |E ( η ) | ≫
1. In this regime, ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ − − η/η ω o p t ( η ) [ e V ] analyticsnumerics FIG. 3. Optimal laser frequency in the non-relativistic modelcase (parameters in the text) according to the analytical so-lution eq. (17) (solid red) in comparison to the numerical so-lution of eqs. (12,14) (blue crosses). as can be deduced from eq. (9) we see that the accel-eration will be occurring over the whole phase interval η ∈ [ − τ L , τ L ]. Consequently, in contrast to the previ-ously studied non-relativistic case, the field’s derivativecan be estimated to be of the order E ′ ( η ) ∼ E ( η ) /τ L .Hence, we can estimate from eq. (14) that in this regimethe optimal reflection coefficient will be of the order ρ . p m i / | eE ( η ) | τ L . This indicates that for ultra-relativistic foil motion in the pulse center, i.e., for largestfield strengths, the optimally matched reflection coeffi-cient again has to be small, which in the present schemeis achievable by large frequencies. On the other hand,from eq. (12) one infers that the maximal reflection coef-ficient ρ → a → ∞ , corresponding to thelow-frequency limit ω →
0. As a result, in the beginningphase of the acceleration the reflection coefficient can stillbe small, facilitating the use of optical laser frequencies.From this result, however, we infer a further restriction:The optimum condition (2) can only be maintained up tothe phase instant, where the value of the required opti-mally matched reflection coefficient from eq. (14) exceedsthe maximum achievable value ρ max . This phase instantis implicitly defined by the condition E ′ ( η ) ≤ | eE foil | m i . (18)Assuming again the Gaussian field shape E ( η ) = E max exp (cid:2) − η/τ L ) (cid:3) , in the pulse’s rising edge thescaled field’s derivative is maximal at the phase instant˜ η = − τ L / p log(2), whence for the specified pulse shapewe can rewrite eq. (18) as a maximal condition for thepulse duration in the form τ L ≥ m i p log(2) | eE foil | E max e − , (19)which ensures that the optimally matched reflection co-efficient can always be matched by eq. (12). Since we fixthe intensity profile, the change in laser frequency leads ⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯⨯ − − η/η ω o p t ( η ) [ e V ] analyticsnumerics FIG. 4. Optimal laser frequency in the ultra-relativistic modelcase (parameters in the text) according to the analytical so-lution eq. (21) (solid red) in comparison to the numerical so-lution of eqs. (12,14) (blue crosses). to a changing its value accordingly by a couple of ordersof magnitude. Consequently, for ultra-relativistic foil mo-tion, the matching between eq. (12) and the foil’s opti-mal reflection coefficient eq. (14) requires to again solveeq. (4) by separating the variables and integration. Theresult is formally the same expression for the foil’s mo-mentum as in the non-relativistic case but, from eq. (8)we deduce that in the case γ ≫ ρ ( η ) = s m i E ′ ( η ) eE ( η ) (20)which is equivalent to approximating eq. (14) in theregime E ( η ) ≫
1. Equating this to the approximation(12) we arrive at an equation which is very similar tothe one obtained in the non-relativistic case. We find theappropriate laser frequency chirp, ensuring condition (1)to be satisfied in the regime γ ≫ ω optUR ( η ) = eE ( η ) m e (cid:12)(cid:12) eE ( η ) (cid:12)(cid:12) − m i E ′ ( η ) p m i E ′ ( η ) | eE ( η ) | ! . (21)By virtue of the condition E ( η ) ≤ m i E ′ ( η ) ≤ (cid:12)(cid:12) eE ( η ) (cid:12)(cid:12) and conclude that latter condition(18) to be required for ω optUR ( η ) to have positive, i.e., phys-ical solutions. We can hence interpret that condition asthe physical prerequisite of the optimally matched reflec-tion coefficient to be reachable through tuning the fre-quency. Comparing the analytical expression (21) to thenumerically consistent solution of eqs. (12,14) for a foilof 1 nm thickness and density n e = 10 cm ≈ n crit accelerated by a laser pulse of intensity I = 10 W / cm and duration τ L = 3 × /ω we find our analytical ap-proximation very well confirmed (s. fig. 4). We also findthe required laser frequencies to lie in significantly lowerenergy ranges as compared to the nonrelativistic case.This behavior can be explained by the observation thatthe equality between eqs. (12,20) in the regime E ( η ) ≫ a than the equality be-tween eqs. (12,16) in the regime E ( η ) ∼
1. Physically, thistranslates to the observation that an ultra-relativistic foilcan withstand stronger laser acceleration, as experiencedin lower frequency fields. Also the divergence of the op-timum frequencies for later phases η → E ′ ( η ) goes to zero in thisregime, indicating that the optimum reflection coefficient(20) vanishes as well, which is achieved for very large fre-quencies, only. We furthermore note that at the timeinstant η = 0 in the above example the foil has alreadybeen accelerated to γ ( η = 0) = E ( η = 0) ∼
10, while con-stantly maintaining the optimum condition (1). This ex-ample indicates that indeed the suggested method is alsoapplicable to ultra-relativistic foil motion, provided onecan supply the required high-frequency photon beams.
C. Conclusion
In summary, we have presented a systematic study ofhow an appropriately chosen frequency chirp serves tomaintain an optimum condition in the radiation pressureacceleration of a thin foil. We presented analytical ex-pressions for the required pulse chirp in two limiting casesof nonrelativistic as well as ultra-relativistic foil motion.Comparing these limiting cases to exact numerical solu-tions of the defining equations of the pulse’s frequencystructure required to maintain optimal acceleration con-ditions we found excellent agreement between the exactand approximate solutions. While we found the requiredfrequencies to be beyond the capabilities of nowadaysavailable technology, the presented conceptual analysismay still prove useful for an improved understanding ofthe overall acceleration process.SSB acknowledges support by Laboratory Directed Re-search and Development (LDRD) funding from LawrenceBerkeley National Laboratory provided by the Director,and the U.S. Department of Energy Office of Science, Of-fices of High Energy Physics and Fusion Energy Sciencesunder contract No. DE-AC 02-05 CH11231. S. Busold, A. Almomani, V. Bagnoud, W. Barth, S. Bedacht,A. Blaevi, O. Boine-Frankenheim, C. Brabetz, T. Burris-Mog,T. Cowan, O. Deppert, M. Droba, H. Eickhoff, U. Eisen-barth, K. Harres, G. Hoffmeister, I. Hofmann, O. Jaeckel,R. Jaeger, M. Joost, S. Kraft, F. Kroll, M. Kaluza, O. Kester,Z. Lecz, T. Merz, F. N¨urnberg, H. Al-Omari, A. Orzhekhovskaya,G. Paulus, J. Polz, U. Ratzinger, M. Roth, G. Schau-mann, P. Schmidt, U. Schramm, G. Schreiber, D. Schumacher,T. Stoehlker, A. Tauschwitz, W. Vinzenz, F. Wagner, S. Yaramy-shev, and B. Zielbauer, Nuclear Instruments and Methods inPhysics Research Section A: Accelerators, Spectrometers, De-tectors and Associated Equipment , 94 (2014), proceedingsof the first European Advanced Accelerator Concepts Workshop2013. F. Mackenroth, A. Gonoskov, and M. Marklund, The EuropeanPhysical Journal D , 204 (2017). C. Danson, D. Hillier, N. Hopps, and D. Neely, High Power LaserScience and Engineering , e3 (2015). C. J. Hooker, J. L. Collier, O. Chekhlov, R. Clarke, E. Divall,K. Ertel, B. Fell, P. Foster, S. Hancock, A. Langley, D. Neely,J. Smith, and B. Wyborn, J. Phys. IV , 673 (2006). K. Nakamura, H. S. Mao, A. J. Gonsalves, H. Vincenti, D. E.Mittelberger, J. Daniels, A. Magana, C. Toth, and W. P. Lee-mans, IEEE Journal of Quantum Electronics , 1 (2017). Z. Major, S. A. Trushin, I. Ahmad, M. Siebold, C. Wandt,S. Klingebiel, T.-J. Wang, J. A. F. Lop, A. Henig, S. Kruber,R. Weingartner, A. Popp, J. Osterhoff, R. H¨orlein, J. Hein,V. Pervak, A. Apolonski, F. Krausz, and S. Karsch, The Re-view of Laser Engineering , 431 (2009). J. Zou, C. Le Blanc, D. Papadopoulos, G. Chriaux, P. Georges,G. Mennerat, F. Druon, L. Lecherbourg, A. Pellegrina,P. Ramirez, and et al., High Power Laser Science and Engi-neering (2015), 10.1017/hpl.2014.41. J. Kawanaka, K. Tsubakimoto, H. Yoshida, K. Fujioka, Y. Fuji-moto, S. Tokita, T. Jitsuno, N. Miyanaga, and G.-E. D. Team,Journal of Physics: Conference Series , 012044 (2016). G.A. Mourou, G. Korn, W. Sandner and J.L. Collier (edi-tors),
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