Tunable Laser Positron Source
A. Sahai (PI), V. Harid, M. Golkowski, University of Colorado J. Cary, Tech-X, A. Thomas, Michigan, S. Palaniyappan, LANL, H. Chen, LLNL T. Tajima, UCI, V. Shiltsev, Fermilab
22nd Accelerator Test Facility (ATF) Users' Meeting Funding source: DOE / NSF December 3-5, 2019 - Brookhaven National Laboratory Funding status: proposed 2 Key Scientific Questions
§ Can tunable positron beams be produced using laser wakefield accelerators ?
§ Is it possible to control the interaction between ultrashort positron-electron jets / showers and laser wakefield plasma wave ?
§ What are the limits of the range of tunability of laser produced positron beams ?
§ Which applications can benefit from an unprecedented ultrashort positron beam ? 3 BNL-ATF - Tunable Laser Positron Source BNL-ATF BNL-ATF primary beam secondary beam sub-picosecond stable 1-5 Joule synchronized e- beam high-Z CO2 laser pulse e-beam & CO2 laser e+ target plasma plasma stage # 1 stage # 2 e+
BNL-ATF BNL-ATF many Tesla sub-picosecond e+-e- plasma Superconducting Nano-Coulomb PHYSICAL REVIEW ACCELERATORSshower ANDreflector BEAMS 21, 081301 (2018) 75 MeV e-beam PHYSICAL REVIEW ACCELERATORS AND BEAMS 21, 081301 (2018) magnet BNL-ATF laser, plasma and particle diagnostics
PHYSICALQuasimonoenergetic REVIEW ACCELERATORS laser plasma AND BEAMS positron21, 081301 accelerator (2018) Quasimonoenergeticusing particle-shower laser plasma plasma-wave positron interactions accelerator using particle-shower plasma-wave interactions Aakash A. Sahai* Quasimonoenergetic laserAakash plasma A. Sahai positron* accelerator Department ofusing Physics particle-shower and John Adams Institute plasma-wave for Accelerator interactions Science, Blackett Laboratory, Department of PhysicsImperial and John College Adams London, Institute for SW7 Accelerator 2AZ, United Science, Kingdom Blackett Laboratory, Imperial College London, SW7 2AZ, United Kingdom Aakash A. Sahai* Department of Physics and(Received(Received John Adams 15 15 January January Institute 2018; 2018; for publishedAccelerator published 8 Science, August 8 August 2018) Blackett 2018) Laboratory, Imperial College London, SW7 2AZ, United Kingdom AnAn all-optical all-optical centimeter-scale centimeter-scale laser-plasma laser-plasma positron positron accelerator accelerator is modeled is modeled to produce to produce quasimonoe- quasimonoe- nergeticnergetic beams beams with with tunable tunable (Received ultrarelativistic ultrarelativistic 15 January energies. 2018; energies. published A new A new 8principle August principle 2018)elucidated elucidated here describes here describes the the trapping of divergent positrons that are part of a laser-driven electromagnetic particle-shower with a large trappingAn all-optical of divergent centimeter-scale positrons that laser-plasma are part ofpositron a laser-driven accelerator electromagnetic is modeled to produce particle-shower quasimonoe- with a large energynergeticenergy spread spread beams and and with their their tunable acceleration acceleration ultrarelativistic into into a a quasimonoenergetic energies. quasimonoenergetic A new principle positron positron elucidated beam beam in ahere laser-driven in describes a laser-driven plasma the plasma wave.trappingwave. Proof Proof of of divergent of this this principle principle positrons using that are analysis analysis part of and aand laser-driven particle-in-cell particle-in-cell electromagnetic simulations simulationsdemonstrates particle-shower demonstrates that, with under a that, large limits under limits definedenergydefined here, spread here, existing existingand their lasers lasers acceleration can accelerate accelerate into a quasimonoenergetic hundreds hundreds of MeV of MeV pC positron quasi-monoenergetic pC quasi-monoenergetic beam in a laser-driven positron positronplasma bunches. bunches. By providing an affordable alternative to kilometer-scale radio-frequency accelerators, this compact Bywave. providing Proof of an this affordable principle using alternative analysis and to particle-in-cell kilometer-scale simulations radio-frequency demonstrates accelerators, that, under limits this compact positron accelerator opens up new avenues of research. positrondefined accelerator here, existing opens lasers up can new accelerate avenues hundreds of research. of MeV pC quasi-monoenergetic positron bunches. By providing an affordable alternative to kilometer-scale radio-frequency accelerators, this compact DOI: 10.1103/PhysRevAccelBeams.21.081301 DOI:positron10.1103/PhysRevAccelBeams.21.081301 accelerator opens up new avenues of research. MonoenergeticDOI: 10.1103/PhysRevAccelBeams.21.081301 positron accelerators intrinsic to positron- − Monoenergeticelectron (eþ − e positron) colliders accelerators at energy intrinsic frontiers to[1,2] positron-have electronbeenMonoenergetic (e fundamentale−) colliders positron to accelerators many at energy important intrinsic frontiers to discoveries positron-[1,2] have þ − − beenelectron[3–6] fundamentalthat (eþ underpin− e ) collidersto the standard many at energy model. important frontiers Apart[1,2] from discoverieshave high- been fundamental to many important discoveries [3 6]energythat underpin physics (HEP), the standard monoenergetic model.eþ-beams Apart from of mostly high- – [3sub-MeV–6] that energies underpin are the also standard used model. in many Apart areas from of material high- energy physics (HEP), monoenergetic e -beams of mostly energyscience physics[7,8], medicine (HEP), monoenergetic[9] and appliedeþþ-beams antimatter of mostly physics sub-MeVsub-MeV[10]. Applications energies energies are are have also also however used used in in not many many had areas areas ready of of material access material to sciencesciencepositron[7,8][7,8] accelerators, medicine, medicine and[9][9] haveandand applied had to antimatter rely antimatter on alternative physics physics [10]. Applications have however not had ready access to [10].sources Applications such as β haveþ-decay however[11], (p,n) not reaction had ready[12] and access pair- to positronpositronproduction accelerators accelerators[13] of and and MeV-scale have have had photons to to rely rely on from on alternative alternative—fission sourcesreactors such[14] as, neutron-captureβþ-decay [11], (p,n) reactions reaction[15][12]or MeV-scaleand pair- sourcesproduction such as[13]βþ-decayof MeV-scale[11], (p,n) photons reaction from[12]—fissionand pair- e−-beams impinging on a high-Z target [16]. productionreactors [14][13], neutron-captureof MeV-scale reactions photons[15] or from MeV-scale—fission Positron accelerators have evidently been scarce due to reactorse−-beams[14], impinging neutron-capture on a high-Z reactions target [16][15]. or MeV-scale e−-beamscomplexitiesPositron impinging accelerators involved on a inhave high-Z the evidently production target been[16] and scarce. isolation due to of complexitieselusive particles involved like inpositrons the production[2,16] in and addition isolation to of the Positroncosts associated accelerators with the have large evidently size of radio-frequency been scarce due (rf) to complexitieselusive particles involved like in positrons the production[2,16] in addition and isolation to the of costsaccelerators associated[17] with. The the size large of size conventional of radio-frequency rf accelerators (rf) elusiveacceleratorsis dictated particles by[17] like the. The distance positrons size of over conventional[2,16] whichin charged rf addition accelerators particles to the FIG. 1. Schematic of all-optical centimeter-scale schemes of quasimonoenergetic laser-plasma positron accelerator using the costsisgain associated dictated energy by under with the distance the the largeaction over size of which breakdown of radio-frequency charged limited particles[18] (rf)FIG. 1. Schematic of all-optical centimeter-scale schemes of interaction of e e− showers with plasma-waves. acceleratorsgaintens of energy MVm[17] under−.1 Therf fields the size action sustained of conventional of breakdown using metallic rflimited accelerators structures[18] quasimonoenergeticþ − laser-plasma positron accelerator using the − is dictatedtensthat of reconfigure MVm by the−1 rf distance fields transverse sustained over electromagnetic using which metallic charged waves structures particles into interactionFIG. of 1.eþ Schematic− e showers of with all-optical plasma-waves. centimeter-scale schemes of gainthatmodes energy reconfigure with under axial thetransverse fields. action Thiselectromagnetic of breakdown limit also waves complicates limited into[18] thusquasimonoenergetic produced have to be laser-plasma captured in positron a flux concentrator, accelerator using the − tensmodesefficient of MVm with positron−1 rf axial fields production fields. sustained This[2,13] limitusing, which also metallic has complicates required structures a thusturnedinteraction produced around have and of e to transportedþ − bee capturedshowers back in[19] witha fluxfor plasma-waves. concentrator, reinjection into efficientmulti-GeV positrone−-beam production from a[2,13] kilometer-scale, which has rf required accelerator a turnedthe same around rf and accelerator. transported back [19] for reinjection into that reconfigure transverse electromagnetic waves into multi-GeV[17] to interacte−-beam with from a target. a kilometer-scale Furthermore, rf the accelerator positrons the sameAdvancements rf accelerator. in rf technologies have demonstrated modes with axial fields. This limit also complicates thus produced1 have to be captured in a flux concentrator, [17] to interact with a target. Furthermore, the positrons 100AdvancementsMVm− -scale in fields rf technologies[20] but explorations have demonstrated beyond the efficient positron production [2,13], which has required a turned1 around and transported− back [19] for reinjection into * 100standardMVm− model-scale at fields TeV-scale[20] buteþ explorations− e center-of-mass beyond the ener- Corresponding− author. − [email protected]*Correspondinge -beam author. from a kilometer-scale rf acceleratorstandardgiesthe still model same remain atrf TeV-scale accelerator.unviable.e Moreover,þ − e center-of-mass the progress ener- of non- gies still remain unviable. Moreover, the progress of non- [17][email protected] interact with a target. Furthermore, the positrons HEP applicationsAdvancements of eþ-beams in rf technologies has been largely have stagnant. demonstrated Published by the American Physical Society under the terms of HEPRecent100 applicationsMVm efforts of1 on-scaleeþ compact-beams fields hasand[20] been affordablebut largely explorations positron stagnant. accel- beyond the Publishedthe Creative by the Commons American Attribution Physical Society 4.0 International under the termslicense. of Recent efforts on− compact and affordable positron accel- * erator design based on advanced acceleration− techniques CorrespondingtheFurtherCreative distribution Commons author. of this Attribution work must 4.0 maintain International attributionlicense. to eratorstandard design based model on at advanced TeV-scale accelerationeþ − e techniquescenter-of-mass ener- Furtherthe author(s) distribution and the of published this work article must maintain’s title, journal attribution citation, to [21,22]gieshave still unfortunately remain unviable. been unsatisfactory. Moreover, the Production progress of non- [email protected] author(s) and the published article’s title, journal citation, [21,22] have− unfortunately been unsatisfactory. Production and DOI. of eþ −−e showers using high-energy electrons from and DOI. of eþHEP− e applicationsshowers using of e high-energyþ-beams has electrons been largely from stagnant. Published by the American Physical Society under the terms of Recent efforts on compact and affordable positron accel- the Creative2469-9888 Commons=18=21(8)= Attribution081301(7) 4.0 International license. 081301-1erator design Published based by on the advanced American Physical acceleration Society techniques Further2469-9888 distribution=18=21(8) of this=081301(7) work must maintain attribution 081301-1 to Published by the American Physical Society the author(s) and the published article’s title, journal citation, [21,22] have unfortunately been unsatisfactory. Production − and DOI. of eþ − e showers using high-energy electrons from
2469-9888=18=21(8)=081301(7) 081301-1 Published by the American Physical Society letters to nature
regolith build-up. In contrast, on a highly porous asteroid, only ...... small impacts (perhaps D , 1 km for Mathilde) produce ejecta deposits outside the crater rim (like shot 1648), whereas blankets Spatial sampling of crystal electrons would be absent around large craters. Only a small amount of ejecta by in-¯ight annihilation of would escape Mathilde. Such asteroids would liberate signi®cant meteoritic material only from catastrophic impacts that shatter and fast positrons disperse the whole body. High porosity does not guarantee formation of compaction A. W. Hunt*², D. B. Cassidy*², F. A. Selim³, R. Haakenaasen§, craters. For example, dry sand has a porosity of 35%, but sand T. E. Cowan², R. H. Howell², K. G. Lynnk & J. A. Golovchenko*¶# craters form primarily by excavation, with signi®cant ejecta blankets at all sizes. Compaction in sand is minimal because it is already near * Department of Physics, Harvard University, Cambridge, Massachusetts 02138, a `fully dense' state (the most ef®cient packing of particles). In this USA case, compaction cratering could only occur by crushing of the ² Lawrence Livermore National Laboratory, Livermore, California 94550, USA constituent sand grains, which requires stresses much higher than ³ Engineering Physics Department, Alexandria University, Alexandria 21544, those experienced by most of the cratered material. Most granular Egypt silicate materials are at their fully dense state when their bulk density § Norwegian Defense Research Establishment, 2027 Kjeller, Norway is in the range 2±3 g cm-3. Thus, compaction cratering in silicates k Department of Physics, Washington State University, Pullman, can only occur if the bulk density is well below ,2 g cm-3. We note Washington 99164, USA that large craters on the martian moons Phobos and Deimos ¶ Division of Engineering and Applied Sciences, Harvard University, Cambridge, (densities ,1.9 g cm-3) do not show strong evidence of compaction Massachusetts 02138, USA effects12, probably because they are close to the fully dense state. # The Rowland Institute for Science, Cambridge, Massachusetts 02142, USA Furthermore, even initially highly porous asteroids about ten times ...... larger than Mathilde's diameter would have lithostatic stresses Energetic, positively charged particles travelling along a low- comparable to the crush pressure of the material used here, and index crystal direction undergo many highly correlated, small- would naturally compact to near a fully dense state due to self- angle scattering events; the effect of these interactions is to guide gravity. Therefore, compaction cratering is not expected to be or `channel' (refs 1±8) the particles through the lattice. Channel- common on large asteroids. ling effectively focuses positive particles into the interstitial High porosity may even be a ¯eeting characteristic of Mathilde- regions of the crystal: nuclear collisional processes such as sized asteroids. As shown by laboratory experiments13,14, and by Rutherford backscattering are suppressed, while the number of Mathilde, highly porous bodies can withstand multiple large interactions with valence electrons increases. The interaction of impacts without disruption. Each impact locally compresses the channelled positrons with electrons produces annihilation radia- asteroid, because its volume decreases by the crater volume, while all tion that can in principle9±12 serve as a quantitative, spatially mass is retained. Formation of the ®ve largest craters on Mathilde selective probe of electronic charge and spin densities within the
(DM . 20 km), increased its bulk density by ,20%. Hence, crystal: in the interstitial regions, two-photon annihilation is Mathilde's initial density may have been even lower than the present enhanced relative to single-photon annihilation, because the value, especially considering that additional large craters may exist latter process requires a nuclear recoil to conserve momentum. on the unobserved half of its surface. Over time, porous bodies may Here we report observations of single- and two-photon annihilation be compacted by impacts to the point of being fully dense. Ejection from channelled positrons, using a monoenergetic beam ¯ux of velocities would thenletters increase, to allowing nature escape of some debris and 105 particles per second. Comparison of these two annihilation
regolith build-up. In contrast, on a highly porous asteroid, onlyformation...... of ejecta blankets around large craters, much as we expect modes demonstrates the ability of channelled positrons to selec- small impacts (perhaps D , 1 km for Mathilde) produce ejectafor compact, rocky bodies. M tively sample valence electrons in a crystal. Useful practical deposits outside the crater rim (like shot 1648), whereas blankets Spatial sampling of crystal electrons would be absent around large craters. Only a small amount of ejecta by in-¯ight annihilation of implementation of the technique will require the development would escape Mathilde. Such asteroids would liberate signi®cantReceived 12 April; accepted 17 September 1999. 7 meteoritic material only from catastrophic impacts that shatter and fast positrons of more intense positron beams with ¯uxes approaching 10 particles disperse the whole body. 1. Veverka, J. et al. NEAR's ¯yby of 253 Mathilde: Images of a C asteroid. Science 278, 2109±2112 (1997). High porosity does not guarantee formation of compaction A. W. Hunt*², D. B. Cassidy*², F. A. Selim³, R. Haakenaasen§, per second. craters. For example, dry sand has a porosity of 35%, but sand2. Housen,T. E. Cowan K.²,R., R. H. Schmidt, Howell², K. R. G. M. Lynn &k Holsapple,& J. A. Golovchenko K. A.*¶# Crater ejecta scaling laws: Fundamental forms based These experiments were performed at the recently constructed 3- craters form primarily by excavation, with signi®cant ejecta blankets on dimensional analysis. J. Geophys. Res. 88, 2485±2499 (1983). at all sizes. Compaction in sand is minimal because it is already near * Department of Physics, Harvard University, Cambridge, Massachusetts 02138, MeV monoenergetic positron beamline at Lawrence Livermore a `fully dense' state (the most ef®cient packing of particles). In this3. Chapman,USA C. R., Merline, W. J. & Thomas, P. Cratering on Mathilde. Icarus (submitted). case, compaction cratering could only occur by crushing of the4. Veverka,² Lawrence Livermore J. et al. National NEAR Laboratory, Encounter Livermore, with asteroidCalifornia 94550, 253 Mathilde:USA Overview. Icarus (submitted). National Laboratory. An electrostatic accelerator was modi®ed to constituent sand grains, which requires stresses much higher than ³ Engineering Physics Department, Alexandria University, Alexandria 21544, 22 those experienced by most of the cratered material. Most granular5. Asphaug,Egypt E. et al. Disruption of kilometre-sized asteroids by energetic collisions. Nature 393, 437±440 accommodate a 109 mCi Na positron source, a tungsten moderator, silicate materials are at their fully dense state when their bulk density (1998).§ Norwegian Defense Research Establishment, 2027 Kjeller, Norway is in the range 2±3 g cm-3. Thus, compaction cratering in silicates k Department of Physics, Washington State University, Pullman, as well as appropriate focusing optics in the terminal. The positrons -3 6. Schmidt, R. M. & Holsapple, K. A. Theory and experiments on centrifuge cratering. J. Geophys. Res. can only occur if the bulk density is well below ,2 g cm . We note Washington 99164, USA that large craters on the martian moons Phobos and Deimos 85,¶ Division235±252 of Engineering (1980). and Applied Sciences, Harvard University, Cambridge, were accelerated to 2.65 MeVand then passed through two bending (densities ,1.9 g cm-3) do not show strong evidence of compaction Massachusetts 02138, USA BULK-PLASMON DISPERSION SPECTRUM OF Be. . . V05 7. Holsapple, K. A. & Schmidt, R. M. Point source solutions and coupling parameters in cratering effects12, probably because they are close to the fully dense state. # The Rowland Institute for Science, Cambridge, Massachusetts 02142, USA magnets, two solenoid focusing lenses and two quadrupole mechanics. J. Geophys. Res. 92, 6350±6376 (1987). et aI, the existence of a plasmon Furthermore,line was not re- evenCompton initiallyband highlyis observed porousas a asteroidsweak bump. about ten times ...... astigmatism correctors, all designed, constructed or modi®ed for ported, even for small values of thelargermomentum than Mathilde'sThe observed diameterline is woulddue to the havebulk plasmon lithostaticand stresses Energetic, positively charged particles travelling along a low- transfer. This, we assume, is due to their very no attempt has been made to detect higher-order 8. Holsapple, K. A. The scaling of impact processes in planetary sciences. Annu. Rev. Earth Planet. Sci. 21, large experimental width which iscomparableof the order of to theplasmons. crush pressureHowever in ofthe thecase materialof Q = 6. 5 usedtt here, and index crystal direction undergo many highly correlated, small- 22 eV and therefore would overlap wouldthe plasmon naturally compactmight be possible to nearto attribute a fullythe densebump at statethe far due to self- 333±373angle scattering (1993). events; the effect of these interactions is to guide band. In our experiments the experimentalgravity.width Therefore,left compactionto a second-order crateringplasmon. is not expected to be or `channel' (refs 1±8) the particles through the lattice. Channel- is of the order of 11 eV. 9. Housen, K. R., Schmidt, R. M. & Holsapple, K. A. Laboratory simulations of large scale fragmentation common on large asteroids. ling effectively focuses positive particles into the interstitial In conclusion, we observe that in Fig. 2 the peak ACKNOWLEDGMENT events. Icarus 94, 180±190 (1991). of the plasmon line is well separated Highfrom TDS porosity may even be a ¯eeting characteristic of Mathilde- regions of the crystal: nuclear collisional processes such as and Compton lines which overlap. sizedIn Fig. asteroids.3 the AsThe shownauthor bywould laboratorylike to thank experimentsProfessor F. 13,14, and by10. Kieffer,Rutherford S. W. backscattering From regolith are suppressed, to rock by while shock. theMoon number13, of301±320 (1975). spectra for P = 17', 20', and 25' allow the separate Alexopoulos for his encouragement and suggestions observation of the TDS and the plasmonMathilde,peak. The highlyin porouscarrying out bodiesthis work. can withstand multiple large11. Feves,interactions M., Simmons, with valence G. electrons & Siegfried, increases. R. W. The in The interaction Earth's of Crust. Its Nature and Physical Properties (ed. 75 magnet Beam impacts without disruption. Each impact locally compresses the channelled positrons with electrons produces annihilation radia- ° Heacock, J. G.) (Geophysical9±12 Monogr. 20, American Geophysical Union, Washington, DC, 1977). dump *Work supported in part by the Nationalasteroid,Hellenic becauseRea- its~ON. volumeSwanson, decreasesJ. Opt. Soc. byAm. the~54 crater1130 (1964). volume, while all tion that can in principle serve as a quantitative, spatially search Foundation. mass is retained. Formation~D. Pines, E/ementary of the ®veExcitations largestin Solids craters(Benja- on Mathilde12. Thomas,selective probe P. C. of Ejecta electronic emplacement charge and on spin the densities martian within satellites. the Icarus 131, 78±106 (1998). Slits ~G. Priftis, K. Alexopoulos, and A. Theodossiou, min New York 1964), NaI Phys. Letters 27A, 577 (1968). (DM . 20 km), increasedJ. Ziman, itsPrincip/es bulkof the densityTheory of bySolids,(Cam-20%. Hence, crystal: in the interstitial regions, two-photon annihilation is G. Priftis, Phys. Rev. B 2, 54 {1970).Mathilde's initial densitybridge U. mayP. , Cambridge, have beenEngland, even1964). lower than the present13. Love,enhanced S. G., relative HoÈrz, to F. single-photon & Brownlee, D.annihilation, E. Target porositybecause the effects in impact cratering and collisional Final focusing SA. Tanokura, N. Hitota, and T. Suzuki, J. Phys. ~3V. Krisham and R. H. Ritchie (private communica- Soc. Japan ~22 515 (1969). value, especially consideringtion) . that additional large craters may exist disruption.latter processIcarus requires105, a nuclear216±224 recoil (1993). to conserve momentum. solenoid 4D. Bohm and ~4D. DuBois, Ann. D. Pines, Phys. Rev.on~92 the609 unobserved(1953). halfF. of its surface.Phys. Over(N. Y. time,) ~7 174 porous{1959); bodies may Here we report observations of single- and two-photon annihilation 5D. Pines and D. Bohm, Phys. Rev. ~85 338 (1952). ~8 24 (1959). ~~B. 14. Stewart, S. T. & Ahrens, T. J. Porosity effects on impact processes in solar system materials. Vol. 30,, C. J. Powell, Proc. Phys. Soc. {London)be compacted~76 593 by impactsW. ¹inham to theet pointa/. , Phys. ofRev. being145, fully209 (1966). dense. Ejection from channelled positrons, using a monoenergetic beam ¯ux of (1960), velocities would then'6M. increase,Hasegawa and allowingM. Watabe, escapeJ. Phys. ofSoc. someJapan debris and CD-ROM105 particles (Lunar per second. Planetary Comparison Science of Conference these two annihilation, Houston, Texas, 1999). P. Nozjeres and D. Pines, Phys. Rev. 113, 1254 ~27 1393 (1969). HPGe {1959). formation of ejecta blankets'7%. Schlilke, aroundU. Berg, largeand 0, craters,Brummer, muchPhys. as we expect modes demonstrates the ability of channelled positrons to selec-4 "D. Bohm, The Many-Body Pxoblem (Dunod, Paris, Status Solidi 35, 227 (1969). for compact, rocky~8Y. bodies. M tively sample valence electrons in a crystal. Useful practical 1959), Ohmura and ¹ Matsudaira, J. Phys. Soc. Japan ! 9A. Glick and Channeled Annihilation Imaging letters to nature J. R. A. Ferrell, Ann. Phys. (¹Y.) 19 1355 (1964). implementation of the technique will require the development Il 359 (1960). Received 12 April; accepted 17 September 1999. Acknowledgementsof more intense positron beams with ¯uxes approaching 107 particles 1.regolith Veverka, J.build-up.et al. NEAR's ¯ybyIn contrast, of 253 Mathilde: on Images a highly of a C asteroid. porousScience asteroid,278, 2109±2112 only (1997)...... per second. Goniometer 2.small Housen, impacts K. R., Schmidt, (perhaps R. M. & Holsapple,D , 1 km K. A. Crater for Mathilde) ejecta scaling laws: produce Fundamental ejecta forms based These experiments were performed at the recently constructed 3- 66 cm 208 cm 79 cm PHYSICAL REVIEW B on dimensionalVOLUME analysis.3, NUMBERJ. Geophys.3 Res. 88, 2485±2499 (1983).1 FEBRUARY 1971 ThisSpatial work sampling was supported of crystal by NASA's electrons Planetary Geology and Geophysics Program. and target 3.deposits Chapman, outside C. R., Merline, the W. crater J. & Thomas, rim (like P. Cratering shot on 1648), Mathilde. whereasIcarus (submitted). blankets MeV monoenergetic positron beamline at Lawrence Livermore 4.would Veverka,Channeling be J. absentet al. NEARof around EncounterPositrons large with craters. asteroid 253 Only Mathilde: a small Overview. amountIcarus (submitted). of ejecta byNational in-¯ight Laboratory. annihilation An electrostatic of accelerator was modi®ed to 5.would Asphaug, escape E. et al. DisruptionMathilde. of kilometre-sized Such asteroids asteroids would by energetic liberate collisions. signi®cantNature 393, 437±440 22 Figure 1 Experimental set-up. Shown is a diagram of the ®nal leg of the 3-MeV J. U. Andersen and %. M. Augustyniak Correspondenceaccommodate a 109 and mCi requestsNa positron for source, materials a tungsten should moderator, be addressed to K.R.H. (1998). Be// TelephonemeteoriticLaboratories, materialMgrray onlyHi//, fromÃesy &ersey catastrophic07974 impacts that shatter and fastas well positrons as appropriate focusing optics in the terminal. The positrons 6.disperse Schmidt, the R. M. whole & Holsapple, body. K. A. Theory and experiments on centrifuge cratering. J. Geophys. Res.(e-mail: [email protected]). monoenergetic positron beamline at Lawrence Livermore National Laboratory. 85, 235±252 (1980). were accelerated to 2.65 MeVand then passed through two bending High porosity does not guarantee formation of compaction 7. Holsapple, K.E. A.Uggerhgj & Schmidt, R. M. Point source solutions and coupling parameters in cratering A. W.magnets, Hunt*², D. B. two Cassidy solenoid*², F. A. focusing Selim³, R. Haakenaasen lenses and§, two quadrupole &nstitute ofcraters.physics,mechanics.&nk ForersityJ. Geophys. example,of &arhus, Res. 92, dry80006350±6376 sand&arhus (1987). hasC, Denmark a porosity of 35%, but sand T. E. Cowan², R. H. Howell², K. G. Lynnk & J. A. Golovchenko*¶# (Received 7 July 1970) astigmatism correctors, all designed, constructed or modi®ed for 8.craters Holsapple, form K. A. primarily The scaling of by impact excavation, processes in with planetary signi®cant sciences. Annu. ejecta Rev. Earth blankets Planet. Sci. 21, © 1999 Macmillan Magazines Ltd Axia]. and planar channeling333±373in thin (1993).single-crystalline gold foils has been investigated by wide-angle scattering ofatmonoenergetic all sizes. Compactionpositrons. The inbeam sandwas isobtained minimalby accelerating because it is already near NATURE* Department of| Physics,VOL Harvard 402 | University,11 NOVEMBER Cambridge, Massachusetts 1999 | 02138,www.nature.com 157 © 1979 IEEE. Personal use of this9. material Housen, is K. permitted. R., Schmidt, However, R. M. & permission Holsapple, to K. reprint/republish A. Laboratory simulations this material of large scale fragmentation the positrons emitted froma `fullya Co dense'source in statea 1-MeV (theVan mostde Graaff. ef®cientThe results packingare in ofgood particles). In this USA for advertising or promotional purposesevents. or forIcarus creating94, 180±190new collective (1991). works for resale or redistribution to servers agreement with corresponding measurements for protons. For the planar case, classical ² Lawrence Livermore National Laboratory, Livermore, California 94550, USA or lists,calculations or to reuse anyare copyrightedcompared10.case, Kieffer,to componentcalculations compaction S. W. of From thisbased regolithworkon cratering inthe tootherdynamical rock works by could shock. musttheoryMoon be only ofobtaineddiffraction.13, occur301±320 from the by (1975).The IEEE. crushing of the LLNL 2.65 MeV positron ³ resultsIEEEare Transactionsvery similar11.constituentexcepton Feves, Nuclearfor M.,the Simmons, sandScience, "wiggles" G. grains, &Vol.due Siegfried,to NS-26, whichwave R.interference, W.No. inrequiresThe 3, Earth's Junewhich stresses Crust. 1979appear Its Nature muchin and higher Physical Properties than (ed.Engineering Physics Department, Alexandria University,75° magnet Alexandria 21544,Beam the quantum-mechanical calculation. These, however, are difficult to resolve experimentally. thoseHeacock, experienced J. G.) (Geophysical by most Monogr. of 20, the American cratered Geophysical material. Union, Most Washington, granular DC, 1977).Egypt22 5 dump Na source –Slits10 /s 12.silicateCHANNELING Thomas, materials P. C.RADIATION Ejecta are emplacement atFROM their POSITRONS on fully the martian dense satellites. state whenIcarus 131, their78±106 bulk (1998). density § Norwegian Defense Research Establishment, 2027 Kjeller,NaI Norway 13. Love, S. G., HoÈrz, F. & Brownlee,-3 D. E. Target porosity effects in impact cratering and collisional Final focusing INTRODUCTION is in the range 2±3angular g cm distribution. Thus, compactionof electrons and crateringpositrons in silicates k Department of Physics, Washington State University, Pullman, M. J. Alguard,* R. 1. Swent,* disruption.R. H. PantellIcarus 105,,* 216±224B. L. (1993).Berman,t S. D. Bloom,+ and S. Datztt3 solenoid can only occur if theemitted bulkfrom densityCu isimplanted well belowin copper,2 gsingle cm-crys-. We note Washington 99164, USA The aim of this experiment is 14.to shed Stewart,some S. T.light & Ahrens,tals, T. J. Porositya quantitative effects oncomparison impact processeswith intheory solar systemor with materials. Vol. 30,,¶ Division of Engineering and Applied Sciences, Harvard University, Cambridge, on the question Radiationof applicability fromof 56-MeVthatclassicalCD-ROM large positronschannel- (Lunar craters Planetarychanneledheavy-particle on Science the along Conference martian the, Houston,(llO), moons Texas, Phobos 1999). and Deimos -3 channeling was difficult because of Massachusetts 02138, USA HPGe ing theory to themu,directional and (100)effects(densities planesobserved and,for 1.9along g cmthethe radiation)
oOl$-9499/79/0600-3865$00.750 1979IEEE Positron Channeling Radiation / Undulator 5 Volume 57, number 1 PHYSICS LETTERS 17 May 1976
Volume 57,Volumenumber57,1 number 1 PHYSICS PHYSICSLETTERSLETTERS Volume 57, number 1 17 May 197617 May 1976 PHYSICS LETTERS 17 May 1976 ON THE THEORY OF ELECTROMAGNETIC RADIATION OF CHARGED PARTICLES IN A CRYSTAL
ON THE THEORY OF ELECTROMAGNETICM.A.RADIATIONKUMAKHOV OF ON THE THEORY OF ELECTROMAGNETICInstitute of NuclearRADIATIONPhysics, Moscow State University.OF MoscowON THE117234,THEORYUSSR OF ELECTROMAGNETIC RADIATION OF CHARGEDCHARGEDPARTICLESPARTICLESIN A CRYSTALIN A CRYSTAL CHARGED PARTICLES IN A CRYSTAL Received 9 February 1976
M.A.ItKUMAKHOVis predictedM.A. thatKUMAKHOVtherc can be a new type of radiation of relativistic channeled particles. A possibility of using M.A. KUMAKHOV Institute of Nuclear Physics,this radialionMoscowin physicsStateis indicated,University.in particular,Moscowfor creating117234,a nuclear-,USSRlaser. Institute of Nuclear Physics, Moscow State University. Moscow 117234, USSR Institute of Nuclear Physics, Moscow State University. Moscow 117234, USSR Consider a relativistic positron moving in a planar where I is taken from eq. (3). Equation (4) is derived Received 9 February 1976 channelReceivedof a9crystal.FebruaryTo first1976approximation, the poten- on the basis of common relations of electrodynamicsReceived 9 February 1976 tial in the channel may be taken as V 2 [1],where for the harmonic field. x is reckoned from the middle of the 0channel;x z being Since at I , 1(w) abruptly increases, the It is predicted that therc can be a new type of radiation of relativistic channeled particles. A possibility of using It is predicted that therc can be a new typedirectedof radiationalong theofchannel.relativisticThe equationchanneledof motionparticles.in AIt(effective)possibilityis predictedcrossofthatsectionusingthercforcannuclearbe aexcitationnew type(whichof radiation of relativistic channeled particles. A possibility of using this radialionthisinradialionphysics inis indicated,physics is inindicated,particular,suchinaparticular,fieldfor creatingis readilyforasolved:nuclear-,creating a laser.nuclear-, laser. for a single resonance is described by the Breight— this radialionWigner formula)in physicsis manyis indicated,orders of magnitudein particular,higher for creating a nuclear-, laser. x(t)x m sin(wt+~9~),______for the radiation. being considered than for the brerns- Consider a relativistic positron moving in a planar where I is taken from eq. (3). Equation (4) is derived Consider a relativistic positron moving in awhereplanarw2=(2 V where I is taken2i3~is thefromphase;eq. (3). strahiung.Equation (4) is derived — Consider a relativistic positron moving in a planar where I is taken from eq. (3). Equation (4) is derived channel ofchannela crystal.of aTocrystal.first approximation,To first approximation,thet the time,poten-mthe0poten-/m0on)sJlthev~/cbasison theof commonbasis of commonrelationsrelationsof electrodynamicsof electrodynamics tial in the channel may be taken2 [1],whereas V 2 [1],where0 the mass,forXmthetheharmonicforamplitudethe harmonicoffield.oscilla-channelfield.of a Becausecrystal.theToradiusfirstis small,approximation,we get a super-powerfulthe poten- on the basis of common relations of electrodynamics tial in thexchannelis reckonedmay frombe takenthe middleas V of thetions0channel;x (i.e. the particlez beingoscillates periodically).Since atThe in- radiation.I , 1(w)Forabruptlyexample, increases,a typical valuethe2off[1],wherein eq. (3) for the harmonic field. x is reckoned from the middle of the 0channel;x z being Since at tialI ,in1(w)the abruptlychannel mayincreases,be takentheas V stantaneous radius of curvature for a particle isx is reckonedis aboutfrom106thetimesmiddlegreater forof athesingle0channel;xparticle thanz beingthe Since at I , 1(w) abruptly increases, the directed alongdirectedthealongchannel.the Thechannel.equationThe equationof motionofinmotion(effective)in (effective)cross sectioncrossforsectionnuclearforexcitationnuclear excitation(which (which R = i)212 ~•V1 directed(2~/ alongradiationmodernthesynchrotronwechannel.are able DESYtoTheobtain(Eequationr~(0.1in the most—10)ofpowerfulXmotion10~eV). in (effective) cross section for nuclear excitation (which such a fieldsuchis readilya field issolved:readily solved: for a singleforresonancea single resonanceis describedis describedby the Breight—by the Breight— where ~ is a change of the transverseWigner4/2.velocity.Henceformula)suchforIt canthea isfieldmanyWetheisnoteregionreadilyordersthatof suchnosolved:ofmodernshortmagnitudewavelengthssynchrotronhigherascanthoseoperateof thein for a single resonance is described by the Breight— ______Wigner formula) is many. orders of magnitude higher x(t)x m sin(wt+~9~), be readily shown that (t3~)2= x~wfor the radiation being considered than for the brerns- x(t)x m sin(wt+~9~),______intensity of radiationforof atherelativisticradiation. particlebeingwe obtainconsideredchanneledthanparticlefor radiation.the brernsBesides,- this radiation is Wigner formula) is many orders of magnitude higher (u~~C) more monochromatic______than synchrotron radiation which . where w2=(2 V 2i3~is the phase; strahiung. x(t)x m sin(wt+~9~), for the radiation being considered than for the brerns- where w2=(2 V 2i3~is the phase;4 2 4 2 ~ strahiung. is also important. 0/m0)sJl — v~/c 1w xmy e /3c , (3) . . . . t the time, m ______where w2=(2ThisV radiation will also occur2ini3~theisaxialthe channelingphase; strahiung. t the time, m 0/m0)sJl — v~/c 0 the mass, Xm the amplitude=of oscilla-— Because the radius is small, we get a super-powerful 0 the mass, Xm the amplitude ofwhereoscilla-y l/\/I v~/c2Because.Here, the classicalthe radiuscalculationt theis time,small,aridmwealsoget0for/maelectrons.0super-powerful)sJl — Furthermore,v~/c the radiation of tions (i.e. the particle oscillates periodically).is justified asThethe numberin- of levelsradiation.of a relativisticForpar-example,this typea typicalwill takevalueplace (withoff differentin eq. (3)frequencies and tions (i.e. the particle oscillates periodically). The in- radiation. For example, a typical0 valuethe mass,off Xmin eq.the(3)amplitude of oscilla- Because the radius is small, we get a super-powerful stantaneous radius of curvature for atideparticleis muchisgreater than unity. isA relativisticabout 106quantumtimes greatera numberforof apeculiarities)single particlein the transitionthan thefrom the stantaneous radius of curvature for a particle theoryis (to be publishedis aboutseparately)106yieldstimesa resultgreatertions (i.e.fortheaxiala singleparticlechannelparticletooscillatesthe planarthanone.periodically).theElectrons dechannelThe in- radiation. For example, a typical value off in eq. (3) close to (3). Although the particle oscillates with a fre- strongly since under channeling they pass near nuclei R = i)212 ~•V1 (2~/ radiationmodern synchrotronwestantaneousare able DESYtoradiusobtain(Eofr~(0.1incurvaturethe most—10)forpowerfulX a10~particleeV). is is about 106 times greater for a single particle than the R = i)212 ~•V1 quency of(2~the/ order radiationmodern1014 _1015synchrotronsecwe1 are,the ablefrequencyDESYto obtain(Eandr~(0.1inundergothe most—10)inelasticpowerfulX scattering.10~eV).Still, an effect of ra- of radiation w significantly increases due to the Doppler diation will take place as the channeled electrons, being where ~ is a change of the transverse4/2.velocity.Hence forIt canthe Wethe noteregionthatRof=suchnoi)212modern~•Vshort1 wavelengthssynchrotronascanthoseoperateof thein (2~/ radiationmodern synchrotronwe are able DESYto obtain(E r~(0.1in the most—10) powerfulX 10~eV). where ~ is a change of the transverse4/2.velocity.HencetransverseforIt cantheeffect, reachingWethe noteregionthe valuethatofwmsuchno (1modern+shortv~/c)wywavelengthssynchrotron2. in Rosette asmotioncanthoseoperatego alongof thethein strings. At planar chan- intensitybe readilyofshownradiationthat of(t3~)2a relativistic= x~w particleIf the crystalwethicknessobtain is suchchanneledthat the numberparticleof radiation.neling of electronsBesides,this thiseffectradiationalso takes placeis as elec- be readily shown that (t3~)2= x~w where ~ is a change of the transverse4/2.velocity.Hence forIt canthe Wethe noteregionthatof suchno modernshort wavelengthssynchrotronascanthoseoperateof thein intensity (u~of radiation~C) of a relativistic particle oscillationswe obtainN~’1, thenchanneledthe spectralmoreparticledistributionmonochromaticradiation.will be Besides,tronsthanoscillatesynchrotronthisin radiationthe field ofradiationplane.is Forwhichprotons and r 2 -i be readily shownmesons thethateffect(t3~)2must =alsox~woccur but the power of (u~~C) dI 3 wi more0)monochromaticIw\i intensityp than synchrotronof radiation.. . radiationof a relativisticwhichparticle we obtain channeled particle radiation. Besides, this radiation is 4 2 4 2 ~ = I—• —i 1 ---2 — + is2 i——also iimportant.I ; w w radiation will be small since the mass of these particles 1w xmy e /3c , dw w W L(3) w \W .~J . . m . 4 2 4 2 ~ ______m m is alsomimportant.m (u~~C) more monochromatic than synchrotron radiation which 1w xmy e /3c , (3) . . This. radiation. will also occur in the axial channeling ______This radiation1 will also(4) occuris great.in theIn theaxialfrequencychannelingrange 0.1 Wm
Figure 6. A 2-TeV electron–positron collider based on laser- driven plasma acceleration might be less than 1 km long. Its electron arm could be a string of 100 acceleration modules, Laser TeV electrons material science 100 modular stages each with its own laser. A 30-J laser pulse drives a plasma wave in each module’s 1-m-long capillary channel of pre- formed plasma. Bunched electrons from the previous module Gas jet gain 10 GeV by riding the wave through the channel. The chain begins with a bunch of electrons trapped annihilation from a gas jet just inside the first module’s Capillary 1 TeV TeV positrons plasma channel. The collider’s Laser 1 TeV positron arm begins the same spectroscopy way, but the 10-GeV elec- Laser trons emerging from its first Motivation module bombard a metal medicine / channeling undulators / anti-matter radiation reaction / anti-Hydrogen-Positronium etc … target to create positrons, which are then focused and Plasma-based Collider – laser-driven Laser 100 modular stages 10 GeV injected into the arm’s string Figure 6. A 2-TeV electron–positron collider based on laser- of modules and accelerated driven plasma acceleration might be less than 1 km long. Its just like the electrons. electron arm could be a string of 100 acceleration modules, Gas jet Laser TeV electrons − 100 modular stages each with its own laser. A 30-J laser pulse drives a plasma e wave in each module’s 1-m-long capillary channel of pre- e+ formed plasma. Bunched electrons from the previous module Positron production target Gas jet gain 10 GeV by riding the wave through the channel. The chain begins with a bunch of electrons trapped from a gas jet just inside the first module’s Capillary 1 TeV TeV positrons plasmawould channel. be created The collider’s by trapping in the vicinity of a gas jet at the ramics are being developed for military and civil applica- positron arm begins the same Laser-Plasma Laser 1 TeV entrance of the first module’s plasma channel. After that ini- tions. Laser systems operating in so-called burst mode (a few tial trapping,way, but the the 10-GeV laser elec- and plasma parameters must be cho- seconds active, followed by minutes of cooling) have ap- trons emerging from its first Laser Collider effort sen somodule that there bombard is no a metal further trapping of plasma background proached 100-kW average power, but not yet the operating electronstarget in to createthe rest positrons, of the first module’s channel or in any parameters needed for LPAs. Diode-based lasers are being subsequentwhich are module. then focused and developed to reach greater than 50% wall-plug efficiency, Laser 100 modular stages 10 GeV Afterinjected the into laser the arm’spulse string propagates through the plasma chan- which would be essential for both light-source and collider of modules and accelerated nel of justa single like the module, electrons. it would have lost most of its energy. So applications. Gas jet The ever-increasing performance of laser systems has e− it will be necessary to inject a fresh 30-J drive pulse into each of e+ the 10-GeV accelerating modules. Transporting the laser pulse contributed much to the blossoming of laser-driven plasma Positron production target to the channel with conventional optics would require a 10-m acceleration over the past decade. So has the increasing power of computer simulation and, of course, the develop- distance between stages to avoid having excessive light inten- would be created by trapping in the vicinity of a gas jet at the ramics are being developedsity for damage military the and focusingcivil applica- optics. That 10-m spacing would ment of ingenious concepts for mastering the physics of entrance of the first module’s plasma channel. After that ini- tions. Laser systems operatinggreatly in so-called lengthen burst the mode overall (a few machine and thus reduce its aver- laser–plasma interactions. We believe that short-term appli- tial trapping, the laser and plasma parameters must be cho- seconds active, followedage by accelerating minutes of cooling) gradient—a have ap- key figure of merit. To avoid that, cations such as ultrafast hyperspectral radiation sources will proached 100-kW average power, but not yet the operating sen so that there is no further trapping of plasma background the LPA community is exploring novel concepts that would soon come to fruition. Reaching the high average-power lev- electrons in the rest of the first module’s channel or in any parameters needed for LPAs. Diode-based lasers are being subsequent module. developed to reach greaterallow than the 50% spacing wall-plug between efficiency, stages to be less than a meter. els required for particle-physics colliders is a daunting but which would be essential for both light-source and collider After the laser pulse propagates through the plasma chan- Several groups around the world, including ours, plan not insurmountable task that requires a revolution in laser nel of a single module, it would have lost most of its energy. So applications. to explore those and other issues using petawatt laser technology. it will be necessary to inject a fresh 30-J drive pulse into each of The ever-increasing performance of laser systems has systems with repetition rates as high as 10 Hz. Spurring that the 10-GeV accelerating modules. Transporting the laser pulse contributed much to the blossoming of laser-driven plasma We thank all past and present members of the LOASIS program at to the channel with conventional optics would require a 10-m acceleration over the pasteffort decade. is the So has commercial the increasing development—most notably in power of computer simulation and, of course, the develop- LBNL, especially Csaba Toth, Carl Schroeder, and Cameron Geddes, distance between stages to avoid having excessive light inten- France—of sophisticated petawatt-class systems. sity damage the focusing optics. That 10-m spacing would ment of ingenious concepts for mastering the physics of for their contributions to this article. To achieve the desired collider luminosity, a laser– greatly lengthen the overall machine and thus reduce its aver- laser–plasma interactions. We believe that short-term appli- age accelerating gradient—a key figure of merit. To avoid that, cations such as ultrafast hyperspectralplasma collider radiation would sources need will a repetition rate of about 15 kHz. References the LPA community is exploring novel concepts that would soon come to fruition. ReachingThat means the high anaverage-power average laser lev- power of half a megawatt per els required for particle-physics colliders is a daunting but 1. T. Tajima, J. M. Dawson, Phys. Rev. Lett. 43, 267 (1979). allow the spacing between stages to be less than a meter. module, which is still far beyond the performance of today’s 2. For a review, see E. Esarey et al., IEEE Trans. Plasma Sci. 24, 252 Several groups around the world, including ours, plan not insurmountable task that requires a revolution in laser to explore those and other issues using petawatt laser technology. lasers. Current high-peak-power lasers can operate with an (1996). systems with repetition rates as high as 10 Hz. Spurring that average power of 100 W at most, with a wall-plug efficiency 3. C. E. Clayton et al., Phys. Rev. Lett. 54, 2343 (1985). effort is the commercial development—most notably in We thank all past and present members of the LOASIS program at 4. C. G. R. Geddes et al., Nature 431, 538 (2004). LBNL, especially Csaba Toth,of Carl about Schroeder, 0.1%. and Cameron Geddes, France—of sophisticated petawatt-class systems. 5. S. P. D. Mangles et al., Nature 431, 535 (2004). for their contributions to this article.On a less grandiose scale than TeV colliders, LPAs offer To achieve the desired collider luminosity, a laser– attractive prospects for driving light sources. Their potential 6. J. Faure et al., Nature 431, 541 (2004). plasma collider would need a repetition rate of about 15 kHz. References 7. W. P. Leemans et al., Nat. Phys. 2, 696 (2006). That means an average laser power of half a megawatt per 1. T. Tajima, J. M. Dawson,advantages Phys. Rev. Lett. 43 over, 267 (1979). light sources based on conventional linacs 8. I. Blumenfeld et al., Nature 445, 741 (2007). module, which is still far beyond the performance of today’s 2. For a review, see E. Esareyinclude et al., IEEE compactness Trans. Plasma Sci. and24, 252cost, intrinsic synchronization be- 9. C. Joshi, Phys. Plasmas 14, 055501 (2007). lasers. Current high-peak-power lasers can operate with an (1996). tween the e– beams and drive-laser pulses, and the femtosec- 10. H.-P. Schlenvoigt et al., Nat. Phys. 4, 130 (2008). 3. C. E. Clayton et al., Phys. Rev. Lett. 54, 2343 (1985). average power of 100 W at most, with a wall-plug efficiency – of about 0.1%. 4. C. G. R. Geddes et al., Natureond 431duration, 538 (2004). of the e beam pulses. But the relatively low av- 11. D. Umstadter, J. K. Kim, E. Dodd, Phys. Rev. Lett. 76, 2073 (1996). On a less grandiose scale than TeV colliders, LPAs offer 5. S. P. D. Mangles et al., Natureerage431 laser, 535 (2004). power of today’s high-peak-power lasers places 12. E. Esarey et al., Phys. Rev. Lett. 79, 2683 (1997). 6. J. Faure et al., Nature 431, 541 (2004). attractive prospects for driving light sources. Their potential severe limitations on the average power of the electron beam 13. S. V. Bulanov et al., Phys. Rev. E 58, R5257 (1998). 7. W. P. Leemans et al., Nat. Phys. 2, 696 (2006). 14. A. Pukhov, J. Meyer-ter-Vehn, Appl. Phys. B 74, 355 (2002). advantages over light sources based on conventional linacs 8. I. Blumenfeld et al., Natureand445 therefore, 741 (2007). on the brightness of the radiation. include compactness and cost, intrinsic synchronization be- 9. C. Joshi, Phys. Plasmas 14, 055501 (2007). 15. J. Faure et al., Nature 444, 737 (2006). tween the e– beams and drive-laser pulses, and the femtosec- 10. H.-P. Schlenvoigt et al., Nat. Phys.From4, 130 various (2008). quarters, there’s considerable emphasis on 16. C. G. R. Geddes et al., Phys. Rev. Lett. 100, 215004 (2008). ond duration of the e– beam pulses. But the relatively low av- 11. D. Umstadter, J. K. Kim,creating E. Dodd, Phys. more Rev. Lett. capable76, 2073 (1996). pulsed lasers. High-average-power 17. A. Butler, D. J. Spence, S. M. Hooker, Phys. Rev. Lett. 89, 185003 12. E. Esarey et al., Phys. Rev. Lett. 79, 2683 (1997). erage laser power of today’s high-peak-power lasers places diode pump lasers and new amplifier materials based on ce- (2002). severe limitations on the average power of the electron beam 13. S. V. Bulanov et al., Phys. Rev. E 58, R5257 (1998). 14. A. Pukhov, J. Meyer-ter-Vehn, Appl. Phys. B 74, 355 (2002). and therefore on the brightness of the radiation. 15. J. Faure et al., Nature 444, 737 (2006). From various quarters, there’s considerable emphasis on 16. C. G. R. Geddes et al., Phys.www.physicstoday.org Rev. Lett. 100, 215004 (2008). March 2009 Physics Today 49 creating more capable pulsed lasers. High-average-power 17. A. Butler, D. J. Spence, S. M. Hooker, Phys. Rev. Lett. 89, 185003 diode pump lasers and new amplifier materials based on ce- (2002). Downloaded 28 Jun 2012 to 128.62.100.79. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms
www.physicstoday.org March 2009 Physics Today 49
Downloaded 28 Jun 2012 to 128.62.100.79. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms
Downloaded 28 Jun 2012 to 128.62.100.79. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms
Downloaded 28 Jun 2012 to 128.62.100.79. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms
Downloaded 28 Jun 2012 to 128.62.100.79. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms Downloaded 28 Jun 2012 to 128.62.100.79. Redistribution subject to AIP license or copyright; see http://www.physicstoday.org/about_us/terms LETTER doi:10.1038/nature16525
Multistage coupling of independent laser-plasma accelerators S. Steinke1, J. van Tilborg1, C. Benedetti1, C. G. R. Geddes1, C. B. Schroeder1, J. Daniels1,3, K. K. Swanson1,2, A. J. Gonsalves1, K. Nakamura1, N. H. Matlis1, B. H. Shaw1,2, E. Esarey1 & W. P. Leemans1,2
Laser-plasma accelerators (LPAs) are capable of accelerating charged of 1 TeV in a single stage, a plasma density of about 10 15 cm−3 is particles to very high energies in very compact structures1. In theory, required. This would result in an acceleration length of 1 km, a low therefore, they offer advantages over conventional, large-scale acceleration gradient, 10 kJ of required laser pulse energy, and an elec- ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7747particle accelerators. However, the energy gain in a single-stage LPA tron bunch that is not suitable for collider applications9. However, can be limited by laser diffraction, dephasing, electron-beam loading staging using multiple petawatt laser systems would allow for the use lectron–positron (e À /e þ ) plasmas are emitted, in the form different approach is foreseen in confining low-energy positrons 1 of ultra-relativistic winds or collimated jets, by some of the using radioactive sources with Penning traps11,15and. The laser-energy proposed depletion . The problem of laser diffraction can of much higher plasma densities, and hence the generation of higher 2 Emost energetic or powerful objects in the Universe, such as APEX experiment12 builds on this idea, accumulatingbe addressed a large by using laser-pulse guiding and preformed plasma accelerating gradients; this would result in a reduction in the total black holes 1,2, pulsars3 and quasars4. These plasmas are number of positrons in a multicell Penning trap, before injection associated with violent emission of gamma-rays in the form of into a stellarator plasma confinement device. Thewaveguides major challenge to maintain the required laser intensity over distances of LPA-based linear-accelerator length to a few hundred metres, as well short-lived (milliseconds up to a few minutes) bursts, which are of these schemes is the recombination of thesemany separate electronRayleigh lengths3; dephasing can be mitigated by longitudinal as more favourable laser parameters and electron-bunch charges9. To among the most luminous events ever observed in the Universe. and positron populations. Alternative schemes have been 4 These phenomena represent an unmatched astrophysical proposed in which electrons and positronstailoring are generated of the plasma density ; and beam loading can be controlled obtain such a compact set-up, coupling distances of the order of the laser- laboratory to test physics at its limit and, given their immense in situ16–21, thus avoiding the aforementioned recombination 5 distance from Earth (some more distant than several billion light issues. Despite the intrinsic interest of theseby results, proper the low shaping of the electron beam . To increase the beam energy depletion length, at the 1-metre scale, are assumed. Because the flu- years), they also provide a unique window on the very early stages percentage of positrons in the electron–positronfurther, beam (of it the is necessary to tackle the problem of the depletion of laser ence restrictions of conventional laser optics require them to be posi- of our Universe5–7. Arguably, one of the most intriguing order, if not o10%) and the low-density reported (collision-less questions is how these gamma-ray bursts are produced. It is skin depth much greater than the beam size, forbiddingenergy, plasma- by sequencing the accelerator into stages, each powered by tioned several metres away from the focal plane of the laser, plasma generally accepted that gamma-ray bursts should arise from like behaviour) prevent their application to thea laboratory separate study laser pulse6. Here, we present results from an experiment mirrors23 have been proposed instead as the final steering optics24. synchrotron emission of relativistic shocks generated within an of e À /e þ plasmas. All these previous experimental attempts have 8,9 electron–positron beam . This radiative mechanism requires a thus not been able to generate e À /e þ beams thatthat present demonstrates charge such staging. Two LPA stages were coupled over Such a compact staging set-up is also important to photon sources (for strong and long-lived ( 1; 000 1, with being the neutrality and a plasma-like behaviour, both fundamental pre- 25 t opÀ op a short distance (as is needed to preserve the average acceleration example when using γ-rays to inspect materials) , where it can be electron–positron plasma frequency) magnetic field; however, requisites for the laboratory study of this state of matter14. Weibel-mediated shocks generate magnetic fields that should We report here on the first experimentalgradient) evidence of the by a plasma mirror. Stable electron beams from a first used to decelerate electrons after photon production to mitigate shield- 1 9 decay on a fast timescale t opÀ due to phase-space mixing . generation of a high-density and neutral electron–positron plasma Also, diffusive Fermi acceleration,ð ’ aÞ proposed candidate for the in the laboratory. Its high density n 10LPA16cm 3wereimplies focused to a twenty-micrometre radius—by a discharge ing needs. e À =e þ À 7 8 acceleration of cosmic rays9, requires magnetic field strengths that that the collision-less skin depth in the plasma’ iscapillary-based smaller than the active plasma lens —into a second LPA, such that Here we demonstrate coupling of, and acceleration in, two sepa- are much higher than the average intergalactic magnetic field plasma transverse size effectively allowingÀÁ for collective effects to (CnT)10. These and other questions could be addressed by ad occur. These characteristics, together with the chargethe beams neutrality, interacted with the wakefield excited by a separate laser. rately powered LPA stages. Two synchronized laser pulses were applied hoc laboratory experiments; however, the extreme difficulty in small divergence y 10 20 mrad , and high average e À =e þ À Staged acceleration by the wakefield of the second stage is detected to drive two acceleration stages in series (Fig. 1). The first stage gen- generating e À /e þ populations that are dense enough to permit Lorentz factor (gAVE15 with a power-law spectral distribution, collective behaviour11,12 is still preventing laboratory studies and comparable to whatÀÁ observed in astrophysical jetsvia22) finallyan energy open gain of 100 megaelectronvolts for a subset of the erated electron beams from a gas-jet target with a central energy of the properties of this peculiar state of matter are only inferred up the possibility of studying the dynamics of e Àelectron/e þ plasmas beam. in a Changing the arrival time of the electron beam with from the indirect interpretation of its radiative signatures and controlled laboratory environment. 120 MeV (see Methods). To maximize the coupling efficiency to the from matching numerical models. The intrinsic symmetry respect to the second-stage laser pulse allowed us to reconstruct the second stage, these electron beams were refocused by a first discharge between negatively charged (e À ) and positively charged (e þ ) 8 particles within the plasma makes their dynamics significantly Results temporal wakefield structure and to determine the plasma density. capillary, acting as an active plasma lens , to the entrance of a second different from that of an electron-ion plasma or from a purely Experimental setup. The experiment (shownOur schematically results in indicate that the fundamental limitation to energy electronic beam. In the first case, the mass symmetry of the Fig. 1a) was carried out using the ASTRA-GEMINI laser system discharge capillary, serving as the second-stage target. The acceler- oppositely charged species induces different growth rates for a at the Rutherford Appleton Laboratory23, whichgain delivered presented a laser by laser depletion can be overcome by using staged ation fields in the second capillary were excited by the second laser 13 series of kinetic and fluid instabilities , and significantly affects beam with a central wavelength lL 0.8 mm, energy on target ¼ ± acceleration, suggesting a way of reaching the electron energies pulse, reflected by a tape-based plasma-mirror a few centimetres away. the possibility of generating acoustic or drift waves. In the second ELE14 J and a duration of tL 42 4 fs. An f/20 off-axis 6,9 case, the overall beam neutrality forbids the generation of parabola focussed this laser beam¼ (focal spotrequired with full-width for collider applications . Depending upon the relative timing of the two laser pulses, an energy current-driven magnetic fields that would hamper the onset of half-maximum (27±3 mm) containing B60% of theThe laser limitations energy, of conventional particle-accelerator technology10 are transverse instabilities. resulting in a peak intensity of C3 1019 W cm 2) onto the gain of about 100 MeV might be observed, with a charge-coupling  À 7 Different schemes have been proposed for the laboratory edge of a 20-mm-wide supersonic He gas jet dopedmotivating with 3.5% of the development of advanced particle-acceleration tech- efficiency of 3.5%. Continuous scanning of the relative timing of the generation of e À /e þ plasmas: in large-scale conventional+ N-2. A backing pressure of 45 bar was found to be optimum in accelerators, the possibility of recombining high-qualitye electron-etermsshower of maximum electron energy/ staging and charge ofniques, the accelerated such– tech as laser-plasma is ripe acceleration, ! which could see a broad laserLETTER pulsesRES allowedEARCH us to reconstruct the femtosecond-scale temporal and positron beams via magnetic chicanes14 is envisaged and a electron beam as resulting from ionization injectionrange24,25 ofin applications—ranging the from particle colliders that produce 12 9 field structure of the second-stage wake, providing an important wake energies beyondStageg I: 10 Plasmelectronvoltsa (TeV) to compact free-electron diagnostic. Numerical modelling confirms the effective trapping of the gasgas jjeett γlenlens Plasma-mirrorPlasma-mirror ×106 lasers and Thomson -ray sources.tapetape Within the past few years, tre- 8 electron beam in the second-stage wake structure, and provides evi- LANEX Singlemendous spectra progress in LPA development has been made. After the first Solid target 6 Plas. lead – Average dence for the femtosecond duration of the first-stage electron beams. e 4 demonstration of per-cent-level energy spread and small divergence 0.8 T / MeV LaserLaser 1 MMagneticagnetic – The electron beams generated in the first stage were transported to
e 2 Gas Pb Pb in millimetre-scale plasmas in 2004 (refs 11–13), electron beamssspspectrometerectrometer with 8 γ 0 the second-stage target using a pulsed active plasma lens . Radially LASER Pb 0.2 0.4 0.6 0.8 1 1.2 9 14 energies of 10 eV (GeV) were obtainedStageStage II with 40-terawatt laser pulses . E – (GeV) II: Pb Pb e 0.100 dischargedis capillary symmetrical focusing was achieved in a gas-filled, 15-mm-long cap- + Magnet gap ) 10 cm e Subsequently, electronpe beamsV with multi-GeV energies were reported a 230 Me µ Tape T illary with a diameter of 500 m, using an axial discharge current of 10 cm 70 cm 0.2 0.5 1 LanexLanexx scscrscreeneen
5 cm 5 cm centre LaserLaser 2 LANEX with0.055 petawatt-class( laser systems and plasmas of a (removable)(removable)few centimetres in 15–17 650 A, which produced an azimuthal focusing magnetic field. The Jet lengthJet (centre) . Controlling130 MeVMetheV injection of electrons into plasma waves Figure 1 | Experimental setup. (a) The laser wakefield-accelerated electrons (green spheres) impact onto a solid target, initiating a quantumSpot size (mm) 110 MeV Lanex screen high field strengths produced (∼0.5 tesla) re-focused electrons with Lens 90 MeV Capillary 18–21 electrodynamic cascade involving electrons, positrons (red spheres) and photons (blue sinusoids). The escaping electrons and positronsenables are0.00 separated the and accelerator to be precisely tuned . spectrally resolved using a magnetic spectrometer (details in the text) and a pair of LANEX screens. Plastic and lead shielding was inserted0.00 to reduce0.02 0.04 0.06 0.08 energies of 75–125 MeV within a distance of 25 mm through the z (m) the noise on the LANEX screens as induced by both the low-energy electrons and gamma-rays generated, at wide angles, during the laser–gasThe accelerating and gradient, Ez, of a single-stage LPA scales with the µ Figure 1 | The experimental set-up. In stage I, a pulse of laser light12/ is energy-dispersed (as part of a dipole spectrometer)plasma-mirror electron profiles. The tape to an energy-dependent spot size of 20–30 m electron–solid target interactions. (b) Typical measured spectra of the electron beam without the solid target. Dashed green lines depictplasma single-shot density, ne, as Enz ∝ e . The single-stage length, Lstage, is given electron spectra, whereas the solid brown line is an average over five consecutive shots. (c) Typical positronfocused signal, on asa gas recorded jet, producing by the LANEX an screen,electron for beam. This beam is then inset shows how the diameter of the waist (the ‘spot size’) of the electron − / (r.m.s.) at the second plasma stage (Fig. 1 inset). The divergence 0.5 cm of Pb. The image is to scale. The white dashed lines depict the projection of the magnet gap, whereas the grey dashed lines depict the position 32 transported to the entranceby of thestage laser-depletion II by a discharge capillary, length, which L deplete: LLbeamstaged evolves≈∝ epalonglete then beame path. Thus, (z), simulated the foracceptance different electron- of the lens was 5 mrad. of 0.2, 0.5 and 1 GeV positrons on the LANEX screen. is acting as an active plasma lens. In stage II, the beam enters a second beam energies produced by the first stage, according to ref. 8. Energies 1 discharge capillary. A secondLenergy laserETTER pulse gain further per accelerates stage scales the electrons; as ∝in the interval. With 75–125 the help MeV are of focuseddoi: particle-10.1038/nature16525 at the entranceThe of the second-stage stage II LPA target was formed by a separate discharge 2 ARTICLE NATURE COMMUNICATIONS | 6:6747 | DOI: 10.1038/ncomms7747 | www.nature.com/naturecommunications Wstage this laser is coupled to theL secondETTER discharge capillary via a plasma-mirror capillaryne to spot sizes of the orderdoi:10.1038/nature16525 of the input-laser spot size (18 µm). & capillary structure (see Methods). The discharge current created a Received 4 Apr 2014 | Accepted 24 Feb 2015 | Published2015 23Macmillan Apr 2015 PublishersDOI: 10.1038/ncomms7747 Limited. All rights reserved.OPENtape. Lanex screens are used to detect the energy integrated and 22 ARTICLE in-cell simulations, it has been shown that, in order to reach an energy pre-formed plasma that served as a waveguide, guiding the driving Generation of neutral and high-density Multistage coupling of independent laser-plasma Received 4 Apr 2014 | Accepted 24 Feb 2015 | Published 23 Apr 2015 DOI: 10.1038/ncomms7747 OPENlaser pulse over many RayleighMultistage lengths, minimizing coupling diffraction of independent and expectations for laser-plasma the experimental parameters, including laser intensity electron–positron pair plasmas in the laboratoryextending the accelerationaccelerators1Lawrence length. TheseBerkeley target National systems Laboratory, are well 1 char Cyclotron- and Road, plasma Berkeley, density. California 94720, USA. 2University of California–Berkeley, Berkeley, California 94720, USA. 3Eindhoven University of Generation of neutral and high-density 3,14 acterized , and a modelacceleratorsS.Technology, Steinke has previously1, J. van PO Tilborg Box been1, C.513, B enedettideveloped 5600MB1, C. G. Eindhoven,that R. Geddes permits1, C. TheB. Schroeder Netherlands.To 1,investigate J. Daniels1,3, K. theK. Swanson influence1,2, A. J. of Gonsalves the second-stage1, wakefield on the 1 2 2 3, 4 1 1 1 5,6 1 1 1,2 1 1,2 G. Sarri , K. Poder , J.M. Cole , W. Schumaker w, A. Di Piazza , B. Reville , T. Dzelzainis , D. Doria , L.A. Gizzi , K. Nakamura1 , N. H. Matlis1 , B. H. Shaw1 , E. Esarey & 1W. P. Leemans 1 1,3 1,2 1 electron–positron pair plasmas in the laboratorythe wakefield amplitudeS. Steinketo be ,determined J. van Tilborg , C. by Benedetti means, C. of G. R.the Geddes spectral, C. B. Schroeder electron, J. Daniels beam, K.in K. detail, Swanson we, subtractedA. J. Gonsalves ,the reference spectrum result- G. Grittani5,6, S. Kar1, C.H. Keitel4, K. Krushelnick3, S. Kuschel7, S.P.D. Mangles2, Z. Najmudinredshift2, N. Shukla of8 ,the transmittedK. Nakamura laser15,126, N.,27 H.. We Matlis used1, B. H. a Shawfeedback-controlled,1,2, E. Esarey1 & W. P. Leemans ing1,2 from an unperturbed beam (positive delay) from the spectrum L.O.G. Silva Sarri81,, D.K. Poder Symes2,9 J.M., A.G.R. Cole Thomas2, W. Schumaker3, M. Vargas3,w, A.3, DiJ. Vieira Piazza84&, B. M. Reville Zepf11,7, T. Dzelzainis1, D. Doria1, L.A. Gizzi5,6, Laser-plasma190 | NATURE accelerators (LPAs) | are VOL capable of 530 accelerating | 11charged F E ofB 1RUAR TeV in a singleY 2016stage, a plasma density of about 10 15 cm−3 is tape-based plasma mirror (see Methods) to combine the injected1 at each delay, to emphasize the effect of the− second laser pulse while G. Grittani5,6, S. Kar1, C.H. Keitel4, K. Krushelnick3, S. Kuschel7, S.P.D. Mangles2, Z. Najmudin2, N. Shukla8, Laser-plasmaparticles to very accelerators high energies (LPAs) in veryare capable compact of structures accelerating. In charged theory, ofrequired. 1 TeV in This a single would stage, ©result 2016 a plasmain an Macmillanacceleration density of lengthabout Publishers 10of 151 km, cm a3 lowis Limited. All rights reserved electron beam with the particlestherefore,laser driver to verythey high offer in energies the advantages second in very over compact stage. conventional, structures1 large-scale. In theory, required.accelerationmaintaining This gradient, would absolute result 10 kJ ofin requiredan accelerationcharge laser pulseinformation. length energy, of 1 and km, an a low elecThe - resulting electron dis- 8 9 3 3 8 1,7 9 L.O. Silva , D. Symes , A.G.R. Thomas , M. Vargas , J. Vieira & M. Zepf The laser pulses reflectedtherefore,particle off accelerators. they the offer plasma However, advantages mirror the overenergy wereconventional, gain inguided a single-stage large-scale in the LPA accelerationtrontributions bunch gradient, that areis not 10 plotted kJsuitable of required for in collider laserFig. pulse 2bapplications energy,in the and .form However, an elec -of a waterfall plot of elec- particlecan be limited accelerators. by laser However, diffraction, the dephasing,energy gain electron-beam in a single-stage loading LPA tronstaging bunch using that multiple is not suitable petawatt for laser collider systems applications would allow9. However, for the use 1 Electron–positron pair plasmas represent a unique state of matter, whereby there exists parabolic plasma channelcanand be laser-energycreated limited by laserin depletion the diffraction, discharge. The dephasing, problem capillary of electron-beam laser diffraction with loading an can stagingoftron much using spectra,higher multiple plasma petawattwhere densities, laser each and systems hence horizontal wouldthe generation allow forline of the higher usecorresponds to an energy 2 an intrinsic and complete symmetry between negatively chargedRecent (matter) and positivelyprogress in Laser-Plasmaandbe addressed laser-energy by usingdepletion Accelerator laser-pulse1. The problem guiding of and laser preformed diffraction tech plasma can ofaccelerating much higher gradients; plasma densities, this would and result hence in the a reductiongeneration in of thehigher total energy transmission of waveguides85%. Matched to maintain propagation the required laser intensity2of a transversely over distances of LPA-basedspectrum linear-accelerator that is averaged length to a few over hundred five metres, shots. as well Background-subtracted chargedElectron–positron (antimatter) pair particles. plasmas These represent plasmas a unique play a statefundamental of matter, role whereby in the dynamics there exists of be addressed by using laser-pulse guiding and preformed plasma accelerating gradients; this would result in a reduction in the total Gaussian laser pulse inwaveguides manya plasma Rayleigh to maintainwith lengths a3 the; dephasingtransverse required can laser be parabolicintensity mitigated over by distanceslongitudinal density of LPA-basedastwo-dimensional more favourable linear-accelerator laser parameters chargelength to and a mapsfew electron-bunch hundred for metres, the charges first as well9. Totwo peaks and valleys of ultra-massivean intrinsic astrophysical and complete objects symmetry and arebetween believed negatively to be associated charged (matter) with the and emission positively of manytailoring Rayleigh of the lengths plasma3 density; dephasing4; and can beam be mitigated loading can by longitudinalbe controlled asobtain more such favourable a compact laser set-up, parameters coupling and distances electron-bunch of the order charges of the9 laser-. To ultra-brightcharged (antimatter)gamma-ray bursts. particles. Despite These extensive plasmas theoretical play a fundamental modelling, role our in knowledge the dynamics of this of profile can be obtained—attailoringby proper oflow shaping the plasmalaser of the density powerelectron4; and beam and beam5. To intensity—ifloading increase can the be beam controlled energy the obtaindepletionthe such blue length,a compact curve at theset-up, 1-metrein coupling Fig. scale, 2adistances are, also assumed. of the averaged order Because of the the laser- over flu - five shots, are shown 5 stateultra-massive of matter is astrophysical still speculative, objects owing and to theare extremebelieved difficulty to be associated in recreating with neutral the emission matter– of input-laser spot size, wbyfurther,0 proper (which it shaping is necessary corresponds of the to electron tackle the beam problemto. Toa increaseradius of the depletionthe whereby beam energyof laser depletionencein restrictionsFig. length, 2d–g atof theconventional. The1-metre presence scale, laser are optics assumed. requireof the Because them second-stage to the be fluposi- - laser results in a further,energy,2 itby is sequencing necessary to the tackle accelerator the problem into stages,of the depletion each powered of laser by encetioned restrictions several metres of conventional away from laser the optics focal planerequire of them the laser, to be plasma posi- antimatterultra-bright plasmas gamma-ray in the bursts. laboratory. Despite Here extensive we show theoretical that, by modelling, using a compact our knowledge laser-driven of this the laser intensity is 1/eenergy,a separate compared by sequencinglaser pulse to6 .the theHere, accelerator on-axis we present into resultsvalue), stages, from each equals an powered experiment the by tionedmirrorsreduction several23 have metres been in proposedawaytotal from beam instead the focal chargeas planethe final of bythe steering laser,up plasmatooptics a 24factor . of three (Fig. 2a). state of matter is still speculative, owing to the extreme difficulty in recreating neutral matter– 6 2 23 24 setup, ion-free electron–positron plasmas with unique characteristics can be produced. Their matched spot size, rm. (Forathat separate demonstratesa parabolic laser pulse such .plasma Here, staging. we present Twoprofile, LPA results stages n (fromr )were = an ncoupled experiment0 + α overr , mirrorsSuchFor a compactappropriate have been staging proposed set-up timing isinstead also important ofas the the final to second-stage photonsteering sources optics (for. laser, however, charge chargeantimatter neutrality plasmas (same in amount the laboratory. of matter Here and weantimatter), show that, high-density by using a and compact small laser-drivendivergence thata short demonstrates distance (as such is needed staging. to Two preserve LPA stages the average were coupled acceleration over Suchexample a compact when staging using γset-up-rays isto also inspect important materials) to photon25, where sources it can (for be where n0 is the on-axis density,a short distance r is the(as is transverse needed to preserve spatial the average coordinate acceleration in examplewas detected when using γ beyond-rays to inspect the materials) energy25, cut-offwhere it can of be the input electron spec- finallysetup, open ion-free up the electron–positron possibility of studying plasmas electron–positron with unique characteristics plasmas in can controlled be produced. laboratory Their gradient) by a plasma mirror. Stable electron beams from a first used to decelerate electrons after photon production to mitigate shield- gradient)LPA were byfocused a plasma to a mirror.twenty-micrometre Stable electron radius—by beams from a discharge a first useding needs.to decelerate electrons after photon production to mitigate shield- charge neutrality (same amount of matter and antimatter), high-density and small divergence the plasma channel, and α is the parameter controlling the depth of trum, that is, >200 MeV. This charge accelerated beyond the cut-off experiments. LPAcapillary-based were focused7 active to a twenty-micrometre plasma lens8—into radius—bya second −LPA, a discharge such that ing Hereneeds. we demonstrate coupling of, and acceleration in, two sepa- 7 8 1/4 finally open up the possibility of studying electron–positron plasmas in controlled laboratory the channel; the matchedcapillary-basedthe beamsspot interacted size active is givenwith plasma the wakefieldbylens r—intom = excited a( απsecond byre )aLPA, separate such, with laser.that ratelyofHere the powered we demonstrateinput LPA stages.spectrum coupling Two synchronized of, (red and acceleration and laser pulsesyellow in, were two areas appliedsepa- in Fig. 2b, d, f), which −13 the beams interacted with the wakefield excited by a separate laser. rately powered LPA stages. Two synchronized laser pulses were applied experiments. re = 2.8 × 10 cm beingStaged the acceleration classical by electron the wakefield radius.) of the second In our stage experi is detected- toindicates drive two acceleration acceleration stages in seriesin the (Fig. second 1). The first stage. stage gen The- integrated charge of Stagedvia an accelerationenergy gain by of the 100 wakefield megaelectronvolts of the second for stage a subset is detected of the toerated drive electrontwo acceleration beams from stages a gas-jetin series target (Fig. with1). The a central first stage energy gen -of mental conditions, rm =viaelectron 45 an energyµ beam.m, and gain Changing ofthe 100 thelaser megaelectronvolts arrival spot time sizeof the for electronat afocus subset beam ofwas withthe erated1201.2 MeV electronpC (see in Methods). beams this fromregion To a maximize gas-jet represents target the couplingwith a central the efficiency chargeenergy to ofthe trapped in the acceler- = µ electronrespect to beam. the second-stage Changing the laser arrival pulse time allowed of the us electron to reconstruct beam with the 120second MeV stage, (see Methods).these electron To maximize beams were the refocused coupling by efficiency a first discharge to the w0 18 m, leading to mismatchedrespect to the second-stage propagation laser pulse and, allowed hence, us to reconstruct to varying the ating phase of the wake, corresponding to a trapping efficiency of temporal wakefield structure and to determine the plasma density. secondcapillary, stage, acting these as electron an active beams plasma were lens refocused8, to the entrance by a first ofdischarge a second temporal wakefield structure and to determine the plasma density. 8 peak intensities and wakefieldOur results strengths indicate that along the fundamental the capillary. limitation The to char energy- capillary,discharge3.5%. acting capillary,At asdelays an servingactive of plasma as λ thep /2 lenssecond-stage after, to the entrancethe target. times ofThe a second acceler of maximum - energy gain, Ourgain resultspresented indicate by laser that depletion the fundamental can be overcome limitation by using to energy staged discharge capillary, serving as the second-stage target. The acceler- acteristic oscillation lengthgain presentedof the laser by laser spot depletion size canis given be overcome by λ byOS using = π stagedzRM, ationroughly fields in 1 the pC second of additional capillary were chargeexcited by was the second detected laser around 110–150 MeV 2 acceleration, suggesting a way of reaching the electron energies ationpulse, fields reflected in the by second a tape-based capillary plasma-mirror were excited a fewby the centimetres second laser away. =/πλacceleration,λ= suggestingµ a way of6 ,reaching9 the electron energies pulse, reflected by a tape-based plasma-mirror a few centimetres away. where zrRM m , andrequired for 0.8 collider m isapplications the central6,9 . wavelength of the Depending(Fig. 2e,g upon). the This relative could timing ofcorrespond the two laser pulses, to an electrons energy that have deceler- requiredThe limitations for collider of conventional applications particle-accelerator. technology10 are Depending upon the relative timing of the two laser pulses, an energy laser. For our parameters,The λ OSlimitations = 25 ofmm. conventional Wake particle-accelerator excitation under technology these10 are gainated, of about or to 100 electrons MeV might bethat observed, have with been a charge-coupling deflected by the transverse wake motivating the development of advanced particle-acceleration tech- gain of about 100 MeV might be observed, with a charge-coupling motivating the development of advanced particle-acceleration tech- efficiency of 3.5%. Continuous scanning of the relative timing of the conditions was confirmedniques, by such measuring as laser-plasma optical acceleration, spectra which of could the see trans a broad- efficiencyfields ofinto 3.5%. the Continuous spectrometer scanning of the acceptance. relative timing ofThe the broad energy spread of niques, such as laser-plasma acceleration, which could see a broad laser pulses allowed us to reconstruct the femtosecond-scale temporal range of applications—ranging from particle colliders that produce laser pulses allowed us to reconstruct the femtosecond-scale temporal mitted laser pulse, showingrange of applications—rangingan increasing redshift from particle with colliders increasing that produce fieldthe structure first-stage of the second-stage electron wake,beam providing prevents an important unambiguous wake observation of the energies beyond 1012 electronvolts (TeV)9 to compact free-electron field structure of the second-stage wake, providing an important wake energies beyond 1012 electronvolts (TeV)9 to compact free-electron diagnostic. Numerical modelling confirms the effective trapping of the γ diagnostic. Numerical modelling confirms the effective trapping of the plasma density in the channel.laserslasers andand ThomsonThomson Quantitative γ-ray-ray sources.sources. analysis Within Within the the of past past the few few spectra years, years, tre tre-- electrondecelerating beam in the phase second-stage of the wake wake structure, under and provides these evi conditions.- mendous progress in LPA development has been made. After the first electron beam in the second-stage wake structure, and provides evi- 28,29 revealed a maximum relativemendous redshift progress in ofLPA 3% development with respect has been made.to the After central the first dencedenceNumerical fo for rthe the femtosecond femtosecond modelling duration duration of of the theperformed first-stage first-stage electron electron with beams. beams. the code INF&RNO demonstration of per-cent-level energy− spread and small divergence demonstration of per-cent-level18 energy 3spread and small divergence TheThe electron electron beams beams generated generated in in the the first first stage stage were were transported transported to to 1 2 wavelength of the laser atinin millimetre-scalemillimetre-scalea density of plasmas plasmas 2 × 10 in in 2004 2004 cm ( (refsrefs 11–13 .11–13 This),), electron electron corresponds beams beams with with allows detailed analysis of the interaction.8 Figure 3a shows reference- School of Mathematics and Physics, The Queen’s University of Belfast, Belfast BT7 1NN, UK. The John Adams Institute for Accelerator Science, Blackett 9 − 14 thethe second-stage second-stage target target using using a apulsed pulsed active active plasma plasma lens lens8. Radially. Radially Laboratory, Imperial College London, London SW7 2BZ, UK. 3 Center for Ultrafast Optical Science, University of Michigan, Ann Arbor, Michigan 48109-2099, energiesenergies ofof 10109 eV eV (GeV) (GeV) were were obtained obtained with with1 40-terawatt 40-terawatt laser laser pulses pulses14. . 1 2 to an average field amplitude of about 17 MV mm if wake excitation symmetricalsymmetricalsubtracted focusing focusing electron was was achieved achieved spectra in in a agas-filled, gas-filled, as a 15-mm-long 15-mm-longfunction cap cap of- - the delay between the USA.School4 Max-Planck-Institut of Mathematics fu and¨r Kernphysik, Physics, The Saupfercheckweg Queen’s University 1, 69117 of Belfast, Heidelberg, Belfast Germany. BT7 1NN,5 UK.IstitutoThe Nazionale John Adams di Ottica, Institute Consiglio for Accelerator Nazionale Science, delle Ricerche, Blackett Subsequently,Subsequently, electronelectron beams beams with with multi-GeV multi-GeV energies energies were were reported reported µ 3 26 illaryillary with with a a diameter diameter of of 500 500 µ m,m, using using an an axial axial discharge discharge current current of of 56124Laboratory, Pisa, Italy. Imperial6 INFN, College Sez. Pisa, London, Largo London B. Pontecorvo, SW7 2BZ, 3-56127 UK. Center Pisa, Italy.for Ultrafast7 Helmholtz Optical Institute Science, Jena, University Fro¨belstieg of Michigan, 3, 07743 Ann Jena, Arbor, Germany.occurs Michigan8 GoLP/Instituto 48109-2099, over the full lengthwithwith ofpetawatt-classpetawatt-class the capillary laserlaser systems systems. and and plasmas plasmas of of a a few few centimetres centimetres in in arrival of the electron bunch and the laser pulse. The simulations USA. 4 Max-Planck-Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany. 5 Istituto Nazionale di Ottica, Consiglio Nazionale delle Ricerche, 15–17 650650 A, A, which which produced produced an an azimuthal azimuthal focusing focusing magnetic magnetic field. field. The The de Plasmas e Fusa˜o Nuclear, Instituto¨ Superior Te´cnico, Universidade de Lisboa, Lisbon, Portugal. 9 Central Laser Facility, Rutherford Appleton Laboratory, lengthlength15–17.. ControllingControlling the the injection injection of of electrons electrons into into plasma plasma waves waves 56124 Pisa, Italy. 6 INFN, Sez. Pisa, Largo B. Pontecorvo, 3-56127 Pisa, Italy. 7 Helmholtz Institute Jena, Fro¨belstieg 3, 07743 Jena, Germany.To8 GoLP/Institutocontrol the phasing of the electron beam in the18–21 plasma wake of highhighshow field field strengthsthat strengths the produced produced observed ( ∼(∼0.50.5 tesla) tesla)energy re-focused re-focused modulations electrons electrons with with depend on the phasing Didcot, Oxfordshire OX11 0QX, UK. w Present address: SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA. enables the accelerator to be precisely tuned18–21 . de Plasmas e Fusa˜o Nuclear, Instituto Superior Te´cnico, Universidade de Lisboa, Lisbon, Portugal. 9 Central Laser Facility, Rutherford Appleton Laboratory, enables the accelerator to be precisely tuned . energies of 75–125 MeV within a distance of 25 mm through the Correspondence and requests for materials should be addressed to G.S. (email: [email protected]). The accelerating gradient, E , of a single-stage LPA scales with the energies of 75–125 MeV within a distance of 25 mm through the Didcot, Oxfordshire OX11 0QX, UK. Present address: SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, Californiathe second-stage 94025, USA. LPA, weThe varied accelerating the gradient, delay E zbetweenz, of a single-stage the LPA laser scales pulses with the plasma-mirrorof the electron tape to bunchan energy-dependent within the spot wake. size of 20–30 The µ mperiodicity of the modu- w ∝ 1212// plasma-mirror tape to an energy-dependent spot size of 20–30 µm Correspondence and requests for materials should be addressed to G.S. (email: [email protected]). plasmaplasma density,density, n nee,, as as En Enzz∝ ee . .The The single-stage single-stage length, length, L Lstagestage, ,is is given given that drive the first and second stages, with femtosecond precision,− / (r.m.s.)(r.m.s.)lation at at the isthe determinedsecond second plasma plasma stage stageby (the Fig.(Fig. 1plasma 1inset). inset). The The density divergence divergence and is consistent with the NATURE COMMUNICATIONS | 6:6747 | DOI: 10.1038/ncomms7747 | www.nature.com/naturecommunications 1 by the laser-depletion length, L : ≈∝−3232/ . Thus, the by the laser-depletion length, Ldepletedeplete: LLLLststagageded≈∝epepleletete nnee . Thus, the acceptanceacceptance of of the the lens lens was was 5 5mrad. mrad. NATURE COMMUNICATIONS | 6:6747 | DOI: 10.1038/ncomms7747& 2015 Macmillan | www.nature.com/naturecommunications Publishers Limited. All rights reserved. 1 with an optical-delay stageenergy in gain the per laser stage scalesbeam as Wline∝ of11 .the With injector the help of stage.particle- experimentalTheThe second-stage second-stage LPA LPA observation. target target was was formed formed However, by by a aseparate separate thedischarge discharge amount of post-accelerated energy gain per stage scales as Wststagagee∝ n . With the help of particle- & 2015 Macmillan Publishers Limited. All rights reserved. Electron spectra were recorded as a function of thene edelay between the capillarycapillarycharge structure structure decreases (see (see Methods). Methods). in the The laterThe discharge discharge accelerating current current created created phases a a of the wake as a result in-cell simulations, it has been shown2222 that, in order to reach an energy two laser pulses. In thein-cell case simulations, of a positive it has been delay,shown that, the in first-stageorder to reach an elec energy- pre-formedpre-formedof increasing plasma plasma that that wake served served curvature. as as a awaveguide, waveguide, The guiding guiding fact the the drivingthat driving the linearity of the wake 11LawrenceLawrence BerkeleyBerkeley National National Laboratory, Laboratory, 1 1 Cyclotron Cyclotron Road, Road, Berkeley, Berkeley, California California 94720, 94720, USA. USA. 2 University2University of of California–Berkeley, California–Berkeley, Berkeley, Berkeley, California California 94720, 94720, USA. USA. 3Eindhoven 3Eindhoven University University of of trons propagated withoutTechnology,Technology, the influencePO PO Box Box 513, 513, 5600MB 5600MB of Eindhoven, Eindhoven, the second The The Netherlands. Netherlands. laser pulse. After appears to be preserved in the experimental results could be attributed the second laser pulse arrived (negative delay), the electron spectra to a deviation from the parabolic plasma channel. We have found that, 190190 || NATURENATURE || VOLVOL 530530 || 1111 FFEEBBRUARRUARYY 20162016 were periodically modulated in energy (Fig. 2a). The© ©period 2016 2016 Macmillan Macmillan of Publishers Publishersthe Limited. Limited.for All example,All rights rights reserved reserved simulating a quartic plasma density profile yields a charge modulation was 80 ± 6 femtoseconds, consistent with a plasma wave- distribution similar to that obtained in the experiment (Extended Data 18 −3 length λp of 24 µm, at a density of (1.9 ± 0.3) × 10 cm . The constant Fig. 1). Simulations performed assuming matched guiding conditions, periodicity of the observed modulation as a function of delay behind and a more-energetic injector beam with reduced energy spread, indi- the driver pulse further indicates a quasilinear wake, consistent with cate that roughly 90% trapping can be achieved (Extended Data Fig. 2).
11 FE B RUARY 2016 | VOL 530 | NATURE | 191 © 2016 Macmillan Publishers Limited. All rights reserved 8 positron-electron showers
§ showers – > MeV electrons on converter Maxwellian spectrum target
§ although “some” authors have claimed so: shower ≠ beam pair-plasma ≠ beam
§ positrons NOT isolated orders-of-magnitude § positrons still divergent roll-off at § un-localized in momentum space high-energies 2 2 scheme-A)scheme-A) or a high-temperature or a high-temperature quasi-monoenergetic quasi-monoenergeticplasma acceleratedplasma accelerated electrons electrons with a with 0.6 GeV a 0.6 peak GeV en- peak en- distributiondistribution [25] (in scheme-B) [25] (in scheme-B) is coupled is into coupled the second into the secondergy interactingergy interacting with a 5-10 with mm a 5-10 thick mm Pb thick target. Pb target. From From stage. In thisstage. positron-acceleration In this positron-acceleration stage, a significant stage, a significantGEANT4GEANT4 / FLUKA / FLUKA simulations simulations of laser-plasma of laser-plasma accel- accel- number of particlesnumber of of particles the diverging of the shower diverging are shower trapped are in trappederated in multi-GeVerated multi-GeV electrons electrons [31], an [31 order-of-magnitude], an order-of-magnitude a large incidenta large spot-size incident laser-driven spot-size laser-driven plasma-wave plasma-wave [28]. higher [28]. positronhigher positron number number per electron per electron [35] is [ observed.35] is observed. An ultra-relativisticAn ultra-relativistic quasi-monoenergetic quasi-monoenergetice+-“beam”e+-“beam” is Increasing is Increasing the electron the electron energy produces energy produces higher positronhigher positron + + acceleratedaccelerated by the plasma by the fields. plasma This fields. ultra-short This ultra-shorte - numbere - pernumber incident per incident electron, electron, but the butshowers the showers retain their retain their 2 beam can bebeam further can beaccelerated further accelerated to higher energies to higher using energies usingspectral characteristicsspectral characteristics peaking peaking at a few at MeV a few [10 MeV] [10] more stagesmore [29, stages30]. [29, 30]. 2 scheme-A) or a high-temperatureIn scheme-B, quasi-monoenergeticIn scheme-B, the sheath-accelerated theplasma sheath-accelerated accelerated quasi- electrons quasi- with a 0.6 GeV peak en- Earlier worksEarlier have works modeled have the modeleddistribution trapping the [ trappingof25] plasma (in scheme-B) of plasma is coupled into the second+ ergy+ interacting with a 5-10 mm thick Pb target. From scheme-A) or a high-temperaturemonoenergetic quasi-monoenergeticmonoenergetic spectrum spectrumeplasmae e acceleratedshowere shower is modeled electrons is modeled with with with a 0.6 GeV peak en- electrons withelectrons high with temperatures high temperaturesstage. [32] in In a this [laser-driven32] positron-acceleration in a laser-driven stage, a significant GEANT4 / FLUKA simulations of laser-plasma accel- distribution [25] (in scheme-B) isa coupled drifting intoa Maxwelliandrifting the second Maxwellian withergy interacting with10 MeV10 with kinetic MeV a 5-10 kinetic energy mm energy thick Pb target. From plasma-wave,plasma-wave, unlike the unlike injection thenumber injectionof particle-showers of particles of particle-showers of the diverging shower are trapped in⇠ erated16 ⇠ 3 multi-GeV16 3 electrons [31], an order-of-magnitude stage. In this positron-accelerationand stage, 200and keV a significant 200 temperature keV temperatureGEANT4 and 10 and /cm FLUKA 10 density.cm simulationsdensity. This of This laser-plasma accel- with MeV-scalewith MeV-scale temperatures temperatures considereda large incident considered here. spot-size This here. laser-driven This plasma-wave [28]. higher positron number per electron [35] is observed. number of particles of the divergingmodels shower themodels are experimentally trapped the in experimentallyerated observed multi-GeV observed 1010 positrons electrons 1010 positrons [ [3125],] an [25 order-of-magnitude] work di↵erswork significantly di↵ers significantly from positronAn from ultra-relativistic acceleration positron acceleration us- quasi-monoenergetic us- e+-“beam” is Increasing the electron energy produces higher positron a large incident spot-size laser-driven(shot plasma-wave C)(shot where C) a [ where28 305]. J, ahigher+ 3050 =1 J, positron .0540 =1µm, number.054⌧pµm, per10ps,⌧p electron10ps, [35] is observed. ing beam-drivening beam-driven plasma-waves plasma-waves [33accelerated, 34] in [33 essential, 34 by] in the wave- essential plasma wave- fields. This+ ultra-short e - number per incident⇠ electron, but the showers retain their An ultra-relativistic quasi-monoenergetic8 50µm8e -“beam” FWHM50µm FWHM is spot-sizeIncreasing spot-size laser the was laser electron incident was energy incident on produces⇠ a on a higher positron particle interactionparticle interaction physics, whileaccelerated physics,beam crucially while can by be the eliminating crucially further plasma accelerated eliminating fields. This to ultra-short higher energiese+- usingnumber perspectral incident characteristics electron, but the peaking showers at retain a few their MeV [10] more stages [29, 30]. 1mm-thick1mm-thick 2mm diameter 2mm diameter Au target. Au target. the dependencethe dependence on GeV-scale on GeV-scale RFbeam accelerators. can be RF further accelerators. Although accelerated Although to higher energies using spectral characteristicsIn scheme-B, peaking the at a few sheath-accelerated MeV [10] quasi- Earlier works have modeled the trapping of plasma sheath-accelerationsheath-acceleration which usesmore which the stagesthermal uses the [29 expansion, thermal30]. expansion of of monoenergetic spectrum e+ e shower is modeled with electrons with high temperatures [32] in a laser-drivenIn scheme-B, the sheath-accelerated quasi- an electron-ionan electron-ion sheath [25] sheath at theEarlier [ sound25] at the speed,works sound have has speed, driven modeled has driven the trapping of plasma + monoenergetica drifting spectrum Maxwelliane e shower with is modeled10 MeV with kinetic energy a few 10s ofa MeV few 10s quasi-monoenergetic of MeV quasi-monoenergeticelectronsplasma-wave, with beams, high scaling unlike beams, temperatures it the scaling injection [32 it] in a of laser-driven particle-showers ⇠ 16 3 a driftingand Maxwellian 200 keV temperature with 10 MeV and kinetic 10 cm energy density. This plasma-wave,with MeV-scale unlike the temperatures injection of considered particle-showers here. This ⇠ 10 to GeV-energiesto GeV-energies has not been has shown. not been Lastly, shown. produc- Lastly, produc-⇤ corresponding⇤ corresponding author: and author:[email protected] [email protected] keV temperature the experimentally and 1016cm observed3 density. 10 Thispositrons [25] + + withwork MeV-scale di↵ers significantlytemperatures from considered positron here. acceleration This us- ing e e ingparticlee e showersparticle di showers↵ers from di↵ acceleratingers from accelerating a [1] aB. Richter,[1] B. Richter,SLAC-PUB-240,SLAC-PUB-240,models Nov. the(shot experimentally (1966) C) Nov. where; (1966) M. Tigner, a; observed M. 305 Tigner, J, 10 010=1positrons.054µm, [25]⌧p 10ps, + + working di↵ beam-driveners significantly+ plasma-waves+ from positron [33, acceleration34] in essential us- wave- ⇠ e -“beam”.e The-“beam”. energy The distribution energy distribution of e e of show-e e show-Nuovo Cimento,Nuovo Cimento,37,iss.3,pp.1228-31(1965)(shot37,iss.3,pp.1228-31(1965) C)8 where50µ am 305 FWHM;J.R.Rees, J, ;J.R.Rees, spot-size=1.054µm, laser⌧ was10ps, incident on a ingparticle beam-driven interaction plasma-waves physics, [33, while34] in essential crucially wave- eliminating 0 p ers peaks arounders peaks a few around MeV a [ few10, MeV24] with [10, an24] order-of- with an order-of-SLAC-PUB-1911,SLAC-PUB-1911, Mar.8 (1977) Mar.50;µ1mm-thickSLAC-R-418am (1977) FWHM; SLAC-R-418a 2mm spot-size (1993) diameter (1993) laser was Au target. incident⇠ on a particlethe dependence interaction physics, on GeV-scale while[2] crucially RFErikson, accelerators.[2] eliminating R.Erikson, (ed.), Although R.SLAC-R-714 (ed.), SLAC-R-714 (1984);J.R.J.Bennett, (1984);J.R.J.Bennett, magnitudemagnitude roll-o↵ in positron roll-o↵ in number positronsheath-acceleration every number 10 MeV. every which 10 MeV. uses the thermal expansion1mm-thick of 2mm diameter Au target. the dependence+ on+ GeV-scale RF accelerators.R. Billinge,R. Although M. Billinge, H. Blewett, M. H. Blewett, P. Brahmam, P. Brahmam, et. al., CERN- et. al., CERN- The characteristicsThe characteristics of laser-driven of laser-drivene e showere e areshower are sheath-accelerationan electron-ion which sheath uses [25] the at the thermal77-14 sound (1977) expansion speed,77-14; CERN-LEP-84-01 (1977) has of driven; CERN-LEP-84-01 (1984) (1984) dictated bydictated the parameters by the parameters ofan the electron-iona first-stage. few of 10s the of sheath first-stage. MeV These [25 quasi-monoenergetic] in- at These the sound[3] in-J.-E. speed, Augustin,[3] hasbeams,J.-E. driven Augustin, A. scaling M. Boyarski, A. it M. Boyarski, M. Breidenbach, M. Breidenbach, F. Bu- F. Bu- clude the peakclude electron the peak energy electron and energy charge and in charge scheme-A, in scheme-A, a fewto 10s GeV-energies of MeV quasi-monoenergetic has not beenlos, shown. beams, J. T.los, Dakin, scaling Lastly, J. T. et. it Dakin, produc- al., Phys. et. al., Rev.Phys.⇤ Lett.corresponding Rev.33 Lett.,1406(1974)33 author:,1406(1974)[email protected] + the kJ laserthe energy kJ laser and energy intensity andto GeV-energiesining intensity scheme-Be ine scheme-B in hasparticle addition not been showersin addition shown.[4] diM.↵ers Lastly,L.[4] from Perl,M. produc- G. accelerating L. S. Perl, Abrams, G. S.⇤ a Abrams, A.corresponding M.[1] Boyarski, A.B. M. Richter, author: Boyarski, M. Breiden-SLAC-PUB-240,[email protected] M. Breiden- Nov. (1966); M. Tigner, ++ + to the targetto composition the target composition anding dimensions.ee -“beam”. ande dimensions.particle The showers energy di distribution↵ers frombach, accelerating et. ofbach, al.,e Phys. et.e a al., Rev.show-Phys.[1] Lett.B. Rev.35 Richter,,1489(1975) Lett.Nuovo35SLAC-PUB-240,,1489(1975) Cimento, 37,iss.3,pp.1228-31(1965) Nov. (1966); M. Tigner, ;J.R.Rees, + [5] J. P.+ Lees et. al. (BABAR Collaboration) Phys. Rev. In scheme-A,In thescheme-A, shower the hase shower an-“beam”.ers anisotropic peakshas an The around anisotropic relativistic energy a few distribution relativistic MeV [10 of, e24[5]] withJ.e P.show- an Lees order-of- et. al. (BABARNuovo Cimento,SLAC-PUB-1911, Collaboration)37,iss.3,pp.1228-31(1965)Phys. Mar. Rev. (1977); SLAC-R-418a;J.R.Rees, (1993) MaxwellianMaxwellian momentum momentum distributionersmagnitude peaks distribution function around roll-o in function ap few↵-spacein MeV positron in p [10-space, 24 numberLett.] with109 an everyLett.,101802,(2012) order-of- 10109 MeV.,101802,(2012)SLAC-PUB-1911,[2] Erikson, Mar. R. (ed.), (1977)SLAC-R-714; SLAC-R-418a (1984) (1993);J.R.J.Bennett, W 33 GeV [6] The ALEPH[6]+ The Collaboration, ALEPH Collaboration,[2] Erikson, et. al., Physics R.R. et. (ed.), Billinge, al., Physics ReportsSLAC-R-714 M. Reports427 H. Blewett,, (1984)427,;J.R.J.Bennett, P. Brahmam, et. al., CERN- 60-- " ' [ " " 1 " " 1 " " 1 I'- magnitudeThe characteristics roll-o↵ in positron of laser-driven number everye 10 MeV.e shower are where p =(wherep ,pp)=( andp p,pis) parallel and p is to parallel the laser to propa- the laser propa- R. Billinge,77-14 M. H. (1977) Blewett,; CERN-LEP-84-01 P. Brahmam, et. (1984) al., CERN- ? I iss.+ 5-6, pp.iss. 257-4545-6, pp. (2006)257-454; arXiv:hep-ex/0412015 (2006); arXiv:hep-ex/0412015;K. ;K. k ?k k Thedictatedk characteristics by the parameters of laser-driven ofe the first-stage.e shower are These in- gation axisgation [26, 27x axis],x [26, 27], 77-14[3] (1977)J.-E.; CERN-LEP-84-01 Augustin, A. M. (1984) Boyarski, M. Breidenbach, F. Bu- . I Abe et al.,AbePhys. et al., Rev.Phys. Lett. Rev.70,2515(1993) Lett. 70,2515(1993) I dictatedclude by the the peak parameters electron of energy the first-stage. and charge These in scheme-A, in- [3] J.-E. Augustin, A. M. Boyarski, M. Breidenbach, F. Bu- I: los, J. T. Dakin, et. al., Phys. Rev. Lett. 33,1406(1974) I: [7] R. Wideroe,[7] R. Wideroe,Archiv frArchiv Elektrotechnik fr Elektrotechnik21,iss.4,p.38721,iss.4,p.387 I J I I X clude the peak electron energy and charge in scheme-A, I the kJ laser energy and intensity in scheme-B in addition los, J.[4] T.M. Dakin, L. Perl, et. al., G.Phys. S. Abrams, Rev. Lett. A.33 M.,1406(1974) Boyarski, M. Breiden- X (1928); E. O. Lawrence, M. S. Livingston, Phys. Rev. 1: (1928); E. O. Lawrence, M. S. Livingston, Phys. Rev. x x I 2 2 2 the2 kJ laser2 energy and2 2 intensity2 in scheme-B in addition [4] M. L. Perl,bach, G. S. et. Abrams, al., Phys. A. Rev. M. Boyarski, Lett. 35 M.,1489(1975) Breiden- x I to the target composition and dimensions. f(p)=C (p +x p )exp 1+p + Ap (1) 40,p.19(1932); f f(I p)=C I (p + p )exp 1+p + Ap (1) 40,p.19(1932); I {’ ’ i f ? to? the target? composition and dimensions. bach,[5] et. al.,J. P.Phys. Lees Rev. et. Lett. al. (BABAR35,1489(1975) Collaboration) Phys. Rev. I- k ? k In scheme-A,? k the? showerk has[8] anW. anisotropic D.[8] Kilpatrick,W. D. relativistic Kilpatrick,Review ofReview Scientific of Scientific Instruments Instruments28, 28, h q i [5] J. P. LeesLett. et. al.109 (BABAR,101802,(2012) Collaboration) Phys. Rev. InMaxwellian scheme-A,h q the momentum shower has distribution ani anisotropicp.824 (1957)function relativistic in p-space 2 1 2 1 - 1 1 p.824 (1957) Lett. 109,101802,(2012) where =wheremec T =,mAec= TMaxwellian T, A,= longitudinalT T momentum , longitudinalp distributionand p and function in p-space [6] The ALEPH Collaboration, et. al., Physics Reports 427, Fig. 2. Positron flux? in tungsten per incident electron?? vs z for incident energyst ?ofk ?where pk =(? p ,pk) and p kis[9] parallelH. Bethe,[9] toH. the W. Bethe, Heitler, laser W. propa-Proc. Heitler,[6] The ofProc. the9 ALEPH Royal of the Collaboration, Society Royal Society A 146 et., A al.,146Physics, Reports 427, 33 GeV. The different curves are for successively bigger cutoffs in maximum positron ? k k iss. 5-6, pp. 257-454 (2006); arXiv:hep-ex/0412015;K. transverseenergy of 5, 10,20, 50, and transversep100 MeV. Themomenta minimum energyp cutoff is aremomenta 21 MeV. The- normalized wherestagez arep =(– normalizedpositronp to,pme)c and, trans- to-pproductionmisec parallel, trans-pp. to thestage 83-112 laserpp. (1934) propa- 83-112; H. (1934) J. Bhabha,; H. J. Bhabha, W. Heitler, W. Heitler,Proc. ofProc. the of the bins are one radiation length. Note? the shower maximum is around? seven radiation gation axis? [26k , 27], k iss. 5-6, pp. 257-454 (2006); arXiv:hep-ex/0412015;K. lengths for this energy. The calculation covers the first eleven radiation lengths. Abe et al., Phys. Rev. Lett. 70,2515(1993) verse T andverseW 33 longitudinal GeV T and longitudinalT temperaturesgationT axistemperatures [26 are, 27 in], eV are and in eV andRoyal SocietyRoyal A Societypeak159,iss.898,p.432(1937) ~ A 2.5159 AbeMeV,iss.898,p.432(1937) et al., Phys.; Rev. L. Landau, Lett.; L.70 Landau,,2515(1993) ? ? k kanisotropic relativistic Maxwellian [7] R. Wideroe, Archiv fr Elektrotechnik 21,iss.4,p.387 C 2.0is the normalizationC isSLAC the-PUB normalization-4484 constant [ constant26, 27]. [ As26, defining27]. As definingG. Rumer,G.Proc. Rumer, ofProc. the[7] Royal ofR. the Wideroe, Society Royal(1928) Society AArchiv;166 E.,iss.925, fr AO. Elektrotechnik166 Lawrence,,iss.925, M.21 S.,iss.4,p.387 Livingston, Phys. Rev. 2 2 p.213 (1938)2 2temperature(1928); E. O. Lawrence, M. S. Livingston, Phys. Rev. an averagean drift average velocity drift diverges velocity from divergesf(p)= a kineticC2 from(p 2 a theory+ kineticp )exp theory 2 1+pp.2132+ Ap (1938) (1) 40,p.19(1932); f(p)=C (p + p?)expk [10] 1+?S.p Ecklund,+ Ap? SLAC-PUB-4484(1)k ~ 200keV40,p.19(1932) (1987);; R. Krause- approach, aapproach, drifting relativistic a drifting relativistic bi-Maxwellian bi-Maxwellian? isk not ap- is not? ap- ? [10] S. Ecklund, SLAC-PUB-4484[8] W. D. (1987) Kilpatrick,; R.Review Krause- of Scientific Instruments 28, h q k i [8] W. D. Kilpatrick, Review of Scientific Instruments 28, 1.0 h q Rehberg,Rehberg, S.i Sachert, S. Sachert, G. Brauer, G. Brauer, A.p.824 Rogov, (1957) A. Rogov, K. Noack, K. Noack, plied here [27plied]. The here peak [27]. particle The peak number particle is at number an energy21 is1 at an en- 1 1 1 Cwhere = 2m c T , A = 2T T , longitudinal p and+ where = mec Te , = mec TApplied , A = SurfaceAppliedT T shower= Science Surface e 252density Sciencep.824,pp.3106-3110(2006)[9] (1957)252H.,pp.3106-3110(2006) Bethe, W. Heitler,;F. Proc.;F. of the Royal Society A 146, ? ? ?k k ? k k of 2.3 MeVergy corresponding of 2.3 MeV to correspondingdf (1ptransverse)/dp =0with topdf (momentap?)/dpT ==0with are normalizedk to ?m1e-c10, trans-× 10[9]16 H.cm Bethe,-3 W. Heitler, Proc. of the Royal Society A 146, , longitudinalk ? p andk k transverseBulos,p momenta H.Bulos, DeStaebler, H. are DeStaebler, S. Ecklund, S. Ecklund, R.pp. Helm, 83-112 R. et. Helm, al.,(1934)IEEE et.; al., H. J.IEEE Bhabha, W. Heitler, Proc. of the 5.0 MeV anda transverse a transverse temperature, temperature,verse? T T T=0and=0.2 longitudinal MeV.2k MeV. and A T= 25.temperatures? are in eV and pp. 83-112 (1934); H. J. Bhabha, W. Heitler, Proc. of the 40 60 80 100 k TransactionsTransactions on Nuclear on Nuclear Science Science32Royal,iss.5,pp.1832-3432 Society,iss.5,pp.1832-34 A 159,iss.898,p.432(1937); L. Landau, E (MeVI normalized? ? to? mec, transverse Tk and longitudinal T Royal Society A 159,iss.898,p.432(1937); L. Landau, The openingThe angle opening of the angle particle ofC theis shower particle the normalization is shower 6 .The is 6 constant.The? (1985) [26, 27(1985)]. Ask defining G. Rumer, Proc. of the Royal Society A 166,iss.925, + Fig. 3. Yield per l-MeV energy+ (E) b in versus E at z = 6 radiation lengths. temperatures are in eV and C is the normalization con- G. Rumer, Proc. of the Royal Society A 166,iss.925, e e showers5 are modeled withan electron average densities drift velocity be- diverges[11] G. Stange, from aInternal kinetic report,theory DESY Sp.213 l-73/4 (1938) (1973);A.V. e e showers arestant modeled [26, 27 with]. As electron defining densities an average be- drift[11] velocityG. Stange, di- Internalp.213 report, (1938) DESY S l-73/4 (1973);A.V. 15 3 15 17theory3 3 17 3 PIC [10] S. Ecklund, SLAC-PUB-4484 (1987); R. Krause- tween 10 cmtween and 10 cm 10 cmand 10withapproach,cmf + /fwith a drifting-ratiof + /f be- relativistic-ratio be-Kulikov, bi-Maxwellian S.D.Kulikov, Ecklund, is S.D. not Ecklund,[10] E.M. ap- S. Reuter, Ecklund, E.M. Reuter,Proc.SLAC-PUB-4484 ofProc. IEEE of Par- IEEE Par- (1987); R. Krause- verges frome a kinetice e theorye approach, a drifting rela- Rehberg, S. Sachert, G. Brauer, A. Rogov, K. Noack, plied here [27]. The peak particleticle number Acceleratorticle is at Accelerator an Conference, energy Conference,Rehberg, pp.2005-2007 S. pp.2005-2007 Sachert, (1991) G. (1991) Brauer, A. Rogov, K. Noack, tween 0.1 totween 0.4. 0.1 to 0.4. tivistic bi-Maxwellian is not applied here [27]. The peak Applied Surface Science 252,pp.3106-3110(2006);F. [12] R. H.[12] Helm,R. H. J. Helm, E. Clendenin, J. E.Applied Clendenin, S. Surface D. Ecklund, S. Science D. Ecklund, A.252 V.,pp.3106-3110(2006) A. V. ;F. The aboveThe laser-driven above laser-driven particle-showerof particle-shower 2.3 MeV parameters corresponding parameters are to aredf (p)/dp =0withT = Bulos, H. DeStaebler, S. Ecklund, R. Helm, et. al., IEEE particle number is at an energy of 2.3 MeVKulikov, correspondingkKulikov, R. Pitthan, R. Pitthan,IEEEk Bulos, ParticleIEEE H. Particle AcceleratorDeStaebler, Accelerator S. Confer- Ecklund, Confer- R. Helm, et. al., IEEE modeled after experimental data5 [.240]. MeV Positron and a number transverse temperature, T =0.2 MeV. Transactions on Nuclear Science 32,iss.5,pp.1832-34 modeled after experimentalto df (p)/dp data=0with [24]. PositronT =5 number.0 MeVence, and pp.500-502 a transverseence,? pp.500-502 (1991) H. (1991)Transactions Braun,H. A. Braun, Kulikov, on Nuclear A. Kulikov, R. Science Pit- R.32 Pit-,iss.5,pp.1832-34 9 9 15 The3 opening15 k 3 anglek of the particle shower is 6 .The (1985) of 10 overof 1 10 MeVover with 1 MeV 10 temperature,cm with 10densitycmT were=0density.2 mea- MeV. were The mea- opening angle of the (1985) + ? than, M.than, Woodley, M. Woodley,IEEE ParticleIEEE Particle Accelerator Accelerator Confer- Confer- particlee showere showers is 6 .The are modelede+ e showers with electron are modeled densities[11] be- G. Stange,[11] G.Internal Stange, report,Internal DESY report, S l-73/4 DESY (1973) S l-73/4;A.V. (1973);A.V. sured withsured an anisotropic with an anisotropic relativistic relativistic Maxwellian15 3 Maxwellian dis- 17 dis-3ence, pp.1845-47ence, pp.1845-47 (1991) (1991) tween 10 cm 9 and 10 cm15 with3 f + /f17 -ratio3 be- Kulikov, S.D. Ecklund, E.M. Reuter, Proc. of IEEE Par- tribution.tribution. These showers These are showerswith produced electron are produced bydensities 10 laser- between by 109 10laser-[13]cmI. Curie[13]and andeI. 10 Curie F.cme Joliot, and F. Comptes Joliot,Kulikov, Comptes Rendus S.D. Ecklund,Rendus Hebdomadaires Hebdomadaires E.M. Reuter, Proc. of IEEE Par- tween 0.1 to 0.4. ticle Acceleratorticle Accelerator Conference, Conference, pp.2005-2007 pp.2005-2007 (1991) (1991) with fe+ /fe -ratio between 0.1 to 0.4. The above laser-driven particle-shower parameters[12] are R. H.[12] Helm,R. H. J. Helm, E. Clendenin, J. E. Clendenin, S. D. Ecklund, S. D. A. Ecklund, V. A. V. The above laser-driven particle-shower parameters are Kulikov, R. Pitthan, IEEE Particle Accelerator Confer- modeled after experimental data [24]. Positron number Kulikov, R. Pitthan, IEEE Particle Accelerator Confer- modeled after experimental data [24]. Positron number ence, pp.500-502ence, pp.500-502 (1991) H. Braun,(1991) A.H. Kulikov, Braun, A. R. Kulikov, Pit- R. Pit- of9 109 over 1 MeV with15 10153 cm 3 density were mea- of 10 over 1 MeV with 10 cm density were mea- than, M.than, Woodley, M. Woodley,IEEE ParticleIEEE Accelerator Particle Accelerator Confer- Confer- suredsured with with an anisotropic an anisotropic relativistic relativistic Maxwellian Maxwellian dis- dis- ence, pp.1845-47ence, pp.1845-47 (1991) (1991) 9 tribution.tribution. These These showers showers are produced are produced by 109 bylaser- 10 laser-[13] I. Curie[13] andI. Curie F. Joliot, and Comptes F. Joliot, Rendus Comptes Hebdomadaires Rendus Hebdomadaires
Dr. Aakash A. Sahai Dept. of Physics Dr.& AakashJohn Adams A. Sahai Institute for Accelerator Science Dept. of Physics & John Adams Institute for Accelerator Science
1234 Dr. Aakash A. SahaiEsarey, Schroeder, and Leemans: PhysicsDetailed of derivation laser-driven of these fluid equations plasma-based can be found in - electron … Dept. of Physics A. A. Sahai PhD thesis (Appendix) 12341234 & John AdamsEsarey,Esarey, Institute for Accelerator Schroeder, Schroeder, Science and Leemans: Leemans: Physics PhysicsDetailedOn of ofcertain Dr. laser-drivenderivation laser-drivenAakash non A. -Sahailinear of these and relativisticfluid plasma-basedequations plasma-based effects can in be plasma found- basedin - electron particle electron accelerators.…… A. A. Sahai Dept. ofPhD Physics thesis (Appendix) 1234 ProQuestDr.& AakashJohn Dissertations Adams A. Sahai Institute for Accelerator Publishing, Science 2015. 3719664. Esarey, Schroeder, and Leemans: Physics On of certain laser-drivenDept. ofnon Physics-linear and relativistic plasma-based effects in plasma-based electron particle accelerators.… Dr. Aakash A. Sahai Dr. Aakash A. 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Shvets plasma and2 wave2 Li,22ch 1999 on͒. axis For adecreases wide ͑ p asch ␦1n͒ para-/␦n 0 =1 potential in the ␥2 ␥␥ + 1+2 −2 /k r , where =z−ct is the͑ distance͒ Dept.͑ of behind͒ Physics theand la-The thepeak shape plasma of then /potentialn0 = fluiddependsЌ+Ќ +͑ upon͑1+ quantities1+ the amplitude͒ ͒ ,͒ of the potential are which is ͑21͒ thethe plasma plasma2 4 wave wavebolic,bolic,bolic, on onnnn==n axisn axis͑01+͑͑1+1+rr decreasesr/ decreasesr//rther͒,͒,͒ plasma2 plasma, plasma2 plasma2 2p4 4ch wave channel channel as as channel on␦n axis the the the decreases amplitude//␦␦ amplitudenn 00 =1 as=1 of of␦n͑ of͒/electronand␦andn͑ compression0 the͒ the=1 plasmaphase plasma fluid fluid2 2Ќ quantities2 quantities22 are2 are 2 0 0 2 2ch−chch2−2/k/kr r ,, where where==zz͑−͑−ctct͒͒isis the␦ the͑͑ distance distance͒͒␦ & John behind EquationAdams Institute for plasma the the for Accelerator-wave la- la-shown density Science to from depend 1D fluid uponnn /theory.n/ ␥then Ќvalue= =␥+ of a0.͑+1+2 In͑ ͑the1+1+ figure͒ below͒͒ , the, variation2 of the numerically ͑21͑͒21͒ thethethe plasma plasma2 plasma2 4 wave wave wave on on on2 axis axis2p axisp4chch decreases decreases decreases as as as␦n͑␦potentialn͒/͑␦͒n͒//͑ 0␦inn͒n the=1͑͑00 ͒͒=1=1and the plasma2 fluid2020 quantitiesЌ 2 2222 are −2 /k r , wherebolic, n==nz͑1+−ctr /ris͒, theser plasma pulse distance͑ channelAndreev behindet the al., amplitude 1997͒. thepotential of la- in the obtainedn / potentialn =profile␥␥ plotted + along+͑1+ the͑ 1+longitudinal͒ coordinate͒ , co-moving with laser ͑21͒ 0 −2 /k r , where =z− ct is the distanceThe behind peak shape the la- of the potentialn␥/n ␥dependsЌ=+ 1+2͑2 upon1+1+ the͒ amplitude, of the potential which is 21 the plasma2 2p4 4ch wave− on2 / axisk r , wheredecreasesch =pz−ch ct asis the␦n distance͑͒/␦n͑ behind0͒=1 the la-The peak shapeis presentedof then /potential␥ n(dotЌ00 located+= ͑dependsЌ1+02 atЌ theЌ +peak͑ upon͑͒ of1+ the͑ the field amplitude͒ in the͒2͒ electron,22 of comp the ressionpotential phase) which. is ͑ ͒ ͑21͒ 2 2 thethethe plasma2 plasma2 plasma2 2 24p 4 wavech wave wave on onser onser axis pulse axis pulse axis decreases͑ decreasesAndreev͑ decreasesAndreevetet as as al. al. as,␦, 1997electronn 1997͑␦n͒͒͒͒/.͑/. compression␦␦n͒n/͑electron͑0␦0͒n͒=1=1͑ compression0 ͒phase=1 phase 2 2 2 2 2 22 potential in the PIC Simulations 2͑1+͒ −2−2/k/kr r ,, where where−−2−22=//=kk/zkzr−r−,,ctct where, where whereisis the the==z=z− distancez distance−ct−ctctisisis the the the distanceet distance distancebehind al. behind behind the the the la- la- the theshown la-The peak la- la- shownto shape depend toof dependthen /potentialuponn n upon/n=n// ␥the nndepends/ ␥then value =+value=+=͑ ␥␥upon1+ ͑of of1+ a0. +thea0.2 In+ ͑2 ͑amplitude ͒theIn1+1+͑͒ ͑figurethe1+1+, figure belowof͒͒ the, thepotentialbelow͒,͒ variation, thewhich, of variation theis numerically of the numerically͑21͒ ͑21͑͒21͒ ͑21͒21 2p pchch the2 plasma22 2 22p4 4pch wavech onser axis pulse decreases͑Andreev as ,␦ 1997n͑͒͒/electron.␦n͑ compression0͒=1 phase 0 20Ќ0Ќ00 ЌЌ 222 2 2 2 ͑ ͒ 2 4 ser pulse ͑Andreev et al., 1997͒. obtained potential profile plotted along ␥the longitudinal− 1+ coordinate2 2 co-moving with laser ser pulse ͑Andreev−−2−22/k//kkretr , al., where, where where, 1997==z=z͒−z.−ct−ct ctisisis the the the distance distance distance behind behind the the the la- la-obtainedshown la- to depend potential uponnn/n/ then profile value=n=/␥n of a0.plotted +=2 In͑ ͑the1+1+ figure 2along͑21+ 2below͒͒21+Ќ the, the ͑,longitudinal͒ variation2,͒ of2 the numerically coordinate co-moving with͑21 laser͒21 21 −2 /k r , where 2 =2pzpp4ch−ch ct is the distance behind the la- is presented (dotn0 0/locatedn Ќ0 at= the peak of2 the2͑ 21+field in the2͒ electron comp, ression phase). ͑ ͒ ͑ ͒ 21 et al. u =␥␥͑2−͑1+1+͒ ͒, 22 ͑ ͒ serserser pulse pulse pulse͑AndreevAndreev͑Andreevetet al. al., 1997,, 1997 1997͒. ͒.. obtained potential profile plotted 0 along the longitudinalz Ќ −coordinate͑ ͑1+͒ co͒-moving with laser ͑ ͒ p ch −2 /k r , where͑ =z− ct is the distance͒ behind thePIC la- isSimulations presented (dotn/n located= at2 the͑21+1+ peak͒ of2 Ќthe, field in 2the2 electron compression phase). 21 serser pulse pulse͑Andreev͑Andreevetpetch al. al.,, 1997 1997͒͒.. is presented (dot locatedlow0 ata0 the peak– self of the͑u- fieldmodulated=2 in␥͑ the1+2͒ −electron2 1+ ͒ comp wakefieldression, phase). is USABLE ͑ ͒ 22 ser pulse Andreev etC. al. Nonlinear, 1997 plasma. waves PIC Simulations electron compression phase uzz = Ќ ͑͑ ͒ , ͑ ͑͒22͒ serser pulse pulse͑Andreev͑Andreevetet al. al., 1997, 1997͒͒. ͒. PIC Simulations 2 2͑1+2͒ of the plasma-wave 2͑1+͒ C.C. Nonlinear Nonlinear plasma plasma waves waves u␥z = 2͑21+͑1+͒ ͒ , ͑22͒ ser pulse ͑Andreevser pulseet al.͑Andreev , 1997et͒. al. , 1997͒. 2 2 Ќ2− ͑1+2 ͒ 2 2 2 1+2 2 C. Nonlinear plasma waves ␥ − ͑1+͒ ͑ ͒ 2u2Ќ =␥␥Ќ␥−−͑−1+2͑21+2͑1+͒ ͒ ,2 ͒ ͑22͒ z 2 ЌЌ 2 In the linear regime, EӶE0, the plasma wave is a␥ 2 2 ␥ + ͑1+͒2 2 uz =u2␥ЌЌ−=−͑␥1+͑1+−2͒21+1+2͒Ќ , , 2 2 ͑22͒ 22 Ӷ uz = ͑͑2 ͒ , ͑ ͑͒22͒ C. Nonlinear plasmaIn wavesIn the the linear linear regime, regime, EEӶEE00,, the theelectron plasma plasma compression wave wave phase is is a au z =␥Ќ␥␥ =−␥␥−Ќ͑Ќ+1++͑͑1+͑1+͒͒͒ ͒. , ͑23͒ ͑22͒ u = ␥ z Ќ 2 , 2 22 C. Nonlinear plasma wavessimple sinusoidal oscillation withelectron compression frequencyof the plasma phase - wave ppotentialanduzz in= the a Ќ −2͑͑1+1+2 Ќ ͒ , 2 ͑ ͑͒22͒ C.C. Nonlinear Nonlinear plasma plasmaIn waves waves the linear regime, EӶE0, the plasma wave is a uz = ␥␥2==͑2␥1+͑Ќ1++2͑͑1+1+͒ ͒ ,͒ . . ͑23͑͒23͑22͒ ͒ simple sinusoidal oscillation withof the frequency plasma-wave electronand compression a phase C. Nonlinear plasmasimple waves sinusoidal oscillation with frequency pp anduz =uThe au rest=2 of͑2= the1+␥͑1+ properties␥ −͒2 ͒of ͑͑2 the,1+1+ 1+ plasma-wave ͒also change, , with the amplitude of the ͑22͒ ͑22͒22 C. Nonlinear plasmaC.C. Nonlinear Nonlinear waves plasma plasma waves waveswave phase velocity v p ͑theelectron phase compression velocity phase is determined zz Ќ2=͑1+2͑͑1+͒ ͒ ͒ . ͑23͒ ͑ ͒ simple sinusoidal oscillation with frequency p and apotential2 and1+ a0. In the figure below all these properties are presented for the referee to C. Nonlinear plasma waveswave phase velocity v ͑the phaseof the velocityplasma-wave is determined ͑ 2 ͒ 2͑1+͒2 wave phase velocity vpp͑the phase velocity is determinedufurther=2 understandThe the above2 ground͑21+͑-work1+ expressions of the analytical͒ ͒ model, for developed the by coldthe author. fluid motion u and ␥ ͑22͒ C. Nonlinear plasmaIn waves the linear regime,by the driverEӶE, e.g.,, the = plasma cos wavez/v −t is. a When Ez ␥ + 1+2 C. Nonlinear plasma waves wave phase velocity͒ 0 the phase0 velocity͓ p͑ p is determined͔͒ 2 Ќ ͑ ͒ In the linear regime, EӶE0, thevp plasma͑Excellent agreement is waveseen between numerical is solution a and PIC simulation for the same ␥ The+The͑1+2 above above͒ expressions expressions2 2 for for the the cold cold fluid fluid motion motionu andu and␥ ␥ byby the the driver driverEӶE,͒, e.g., e.g., set of= laser= and0 plasmacoscos parameters͓p (slight͑zz// mismatchvvp− ist due͔͒ to.. the numerical When When solution beingEE 2 2Ќ 2 InIn the the linear linear regime, regime, EӶ͒E00,, the the plasma plasma0 ͓ p wave wave͑ p is is͔͒ a a ␥ =also␥␥Ќ +2 describe+͑͑1+1+͑1+2 Page ͒ the5 of 33͒ ͒ single. particle motion of an electron ͑p˜ ͑23͒ C. Nonlinear plasmaIn waves the linear regime,տ E0E, E theӶӶE plasmaE , the wave plasmain a becomes1D approximation wave whereas highly PIC simulations is nonlinear.a are in 2D). Wake-␥ ␥ +The1+2 aboveЌ expressions2 for the cold fluid motion u and ␥ Insimple the linear sinusoidal regime,by oscillation the driver00͒,, the e.g., with plasma frequency=0 cos wave͓p͑z/ isvpppotential− aandt͔͒. in Whenthe a E= 2␥ЌЌ also+͑ ͑1+ describe͒ ͒ . the single particle motion of an electron͑23͒ p˜ simpleIn the sinusoidal linear regime,տ oscillationտEE , theE plasmaӶ withE , wave the frequency becomes plasma highly waveand nonlinear. is a a Wake- ␥␥=also␥ +2 describe2͑͑1+1+2 Page the5͒ of 33 . single2 particle motion of an electron͑ ͑23͑p˜ ͒ In the linear regime, 0,0E theӶ plasmaE , the0 wave plasma becomes wave highlyppotential is nonlinear. ain the Wake-␥ =andЌ ␥˜ , initially at. rest in the potentials a and . ͑23͒ In the linearsimplesimple regime, sinusoidal sinusoidalEӶE oscillationfield oscillation, generation the0 plasma with with in the frequency frequency nonlinear wave 1Delectronp is regimeandand compression a a a can phase be ␥== ex-␥Ќ +also2͑͑1+1+␥ describe ͒ . . the single ͒ particle motion of an͑ ͒ electron͑23͑͒23͑͒ ͑͒p˜ տE , the plasma wave becomes highlyelectron compressionPagep nonlinear. 6 of 33 phase Wake- + ͑1+ ͒ simplesimple sinusoidal sinusoidal oscillation oscillation0 0 with with frequency frequency p andand a a The rest of the properties␥ =and of2 2␥ the˜Ќ,͑ 1+ initiallyplasma-wave͒ atalso. rest changein with the the potentials amplitude of thea and . 23 wave phase velocityfieldfield generationv generationp ͑the phase in in the the velocity nonlinear nonlinear isp 1D 1D determined regime regimeThe rest can can of the be properties be␥ = ex- ex- of 2 the͑ 1+ plasmaand-2wave␥˜2͒,͑ initially 1+also. change͒ withat rest2 the͒ 2 amplitude͒ in the of the potentials ͑a͒͑͒ and23͑ ͒͑͒.͑ ͒ simplewave phase sinusoidal velocity oscillationaminedv p ͑the by phase assuming with velocity frequency that the is determined drive beamand is nonevolving, apotential and a0. In the2 figure͑1+Analytical below͒ all these properties solutions are presented to Eq. for͑ 20the͒ refereein terms to of͑ elliptic͒ in- In the linearsimple regime, sinusoidal EӶӶ oscillationE , the with plasma frequency wave and isp a apotential and a0. In the figure belowand␥ all ␥these˜ ,+ initially properties1+ are at presented rest͒ forin the the referee potentials to a͑͒ and ͑͒. In the linear regime,wave phaseE velocityEfield0,v generation the ͑the plasma phase in the velocity nonlinear wave isp 1D determined is regime a can be ex- ␥ = ␥2͑1++͑ ͑1+͒ ͒ ͒ . ͑23͒ wavewave phase phase velocity velocityaminedv0v ͑pthep͑the by phase assuming phase velocity velocity that the is determined drive is determined beam is nonevolving,further understandThe 2the͑ above ground1+Analytical2ЌAnalytical-Ќwork͒ expressions of the analytical solutions solutions model for todeveloped to Eq. the Eq. ͑by20 cold͑the20͒ author.in͒ in terms fluid terms of motion of elliptic elliptic in-u and in- ␥ simple sinusoidalwave phase oscillation velocityaminedvp withp͑the by phase frequency assuming velocity that the is determined driveand beamfu isr ather nonevolving, understandThe the above ground-work expressionstegrals of the analytical can model be for d foundeveloped the2 by for coldthe author. square fluid laser motion pulseu profilesand ␥ Bu- by the driver͒, e.g.,i.e., the drive=0 beamcos ͓z isp͑ az function/vt p −ptpotential͔͒. of When only in theE the E coordinate ͑ In the linear regime,waveby the phase driver E velocityӶ͒,E e.g.,amined,v the͑Excellent=the by assuming0 agreement plasmaphasecos͓ is seenp velocity͑between that/v numericalp wave the− ͔͒solution drive is. determinedand When beamPIC is simulation a is for nonevolving, the same ␥ = ␥Analytical solutions . to Eq. ͑20͒ in terms of elliptic in- ␥ 23 wave phase velocity v0pi.e.,͑Excellentthep the agreement phase drive is seen beam velocitybetween numerical is a solution functionis determinedand PIC simulation of for only the same the coordinateThe aboveTheThe␥ above above expressions=tegralstegralsЌ + expressions2 expressions can can͑͑1+1+ be be for found found the͒ for for cold for for. thesquare square the fluid cold cold laser motion laser fluid pulse fluid pulse motionu profilesand motion profiles␥u͑Bu-and͑uBu-and ␥ ͑ ͑͒23͒ simple sinusoidalbybyby the oscillation the driverdriver driver͒,͒,͒ e.g.,, e.g.,i.e., e.g.,with the=set driveof= laser frequency=cos and0 plasma beamcoscos͓ parameters͓͓͑z is/pv͑ (slight͑ azz function−// mismatchvvt͔͒pelectron−. ist due When͔͒and to. of. compressionthe numerical When When onlyE a solution the phase beingEE coordinateThe above expressionslanov et al., for 1989 the; Berezhiani cold fluid and motion Murusidzeu and 1990␥ ; simple sinusoidalby the oscillation driver͒, e.g., with=setz of= −laserv andp frequency0t0 plasma,cos where0 parameters͓pp͑ (slightzv/p mismatchvഛpc− ist dueis͔͒p to.p the numerical Whenand phase solution beingE a velocity also of describealso the describetegralsPage the5 of 33 can singlePage the5 of be 33 found single particle for particle motionsquare laser motion of an pulse electron of profiles an electronp˜Bu- ͑p˜ տտEE , the plasmai.e., wavein the a 1D inapproximation drivea becomes1D approximation whereas beam whereasPIC si ishighlymulations PIC a simulations function are in 2D nonlinear.) .a rep in 2D) of. onlyThe Wake-the rest coordinate of theThe properties above of 2 the expressions 1+ plasma-wave also for change the cold with the fluid amplitude motionu of the␥͑u͑ and ␥ wave phase velocityby the0,0 thev driver plasmathe͒, phase wave e.g.,=z becomes−v= velocityt, wherecos highly͓zv ͑ഛ iszc nonlinear./vt determinedis− thet͔͒. phase When Wake-E velocityE The of above the␥ = expressionslanovlanov2͑etet͑1+ al. al., for, 1989 1989͒ the͒; ;Berezhiani. coldBerezhiani fluid and motion and Murusidze Murusidzeand 1990 1990; ; ͑23͒ by the driverp ͑ ͒, e.g.,=z−=vpt0p,cos where0 ͓ p͑vp/ppഛvpc− is͔͒p . the When phase velocityalso of describealso the describeSprangle the singlePage the 5 ofet 33 single particleal., 1990a particle motion, 1990b motion. of As an the electron ofplasma an electron wavep˜ am-͑p˜ ˜ simple sinusoidalտտտ oscillationEEE,,0 the the, the plasma plasma plasma wave with wave waveplasma becomes becomes frequencybecomes wave. highly For highly highly laser nonlinear. drivers, nonlinear. nonlinear.andv Wake-p Ӎv a Wake-g Wake-potential, wherealso andvg a0.is describealso theIn the describe lanovfigurePage the5 of 33below et single theal. all, these 1989 single particle properties; Berezhiani particle motion are presented͒ motion and of an for Murusidze electronthe of referee an͑ electronto 1990͑ p˜ ; ͑p տE0,0 the plasma wave=z becomes−vpt, where highlyvp ഛc nonlinear.is thep phase Wake- velocity and of the␥˜and, initially␥˜Sprangle, initially at restet al. at,in rest 1990a the͒ ,in potentials1990b the͒. potentials As thea plasmaanda͑ wave͒and. am-͑͒. wavewave phase phase velocity velocityfieldfield generationv generation ͑thethe phase phase inplasma in theplasma the nonlinear velocity velocity wave. nonlinearwave. For For 1D laser laser is 1D regime determined determined drivers, drivers, regime canvvp Ӎ can bevvg,, ex- bewhere ex-alsovvg isis describealso the the describeSprangle the2 single͑1+ theet single͒ al. particle,͒ 1990a particle, motion1990b͒ motion. of As an͑ the͒ electron plasmaof an͑ electron wave͑͒p˜ am-͑p˜ տտEE0,0 the,v thep plasmap͑ plasma wave wavelaser becomes becomes pulse group highly highly velocity.Page nonlinear. 6 of 33Page nonlinear. 6 of The 33 quasistatic Wake-p Ӎ Wake-g further approximation understandg The˜ the␥˜ above groundplitude-work expressionset becomes of al.the analytical nonlinear, model for developed the the plasma by coldthe waveauthor. fluid steepens motion u and ␥ fieldfield generation generation inplasma in the the nonlinear wave. nonlinear For 1D laser 1D regime drivers, regime can can be ex-, be where ex-andandis␥and the␥˜and,, initially initially␥˜,Sprangle, initially initially at at rest rest at͒ atin͒ rest,in rest1990a the the͒ in͒ potentials,in potentials1990b the the potentials͒. potentials Asa͑ thea͒͑and plasma͒ anda͑a͒͑͑ waveand͒͒.͑and͒. am-͑ ͒.͑͒. by the driverfield͒field, e.g., generation generation= in0 in thelasercoslaser the nonlinear pulse͓ pulse nonlinearp group͑ groupz/v 1D velocity.p velocity.− 1D regimet͔͒ regime. The The When can quasistatic quasistaticvp Ӎ can bevg ex- be E approximation ex-vg AnalyticalAnalyticalplitudeplitude solutions becomes becomes solutions to Eq. nonlinear, nonlinear, to͑20 Eq.͒ in the͑ the terms20 plasma͒ plasmain of terms wave elliptic wave of steepens steepens in- elliptic in- wave phase velocityfieldfieldaminedamined generationv generationp ͑ byExcellentthe by assuming assuming agreement phase in in the theis͑ Sprangle thatseen nonlinear velocity thatbetween nonlinear the numerical theet drive al. 1D drive, solution 1990a beamis 1D regime determinedand beam, regime 1990bPIC is simulation nonevolving, can is͒ can nonevolving, can be for bethe ex- besame applied ex-and suchTheThe␥˜and, that initially above␥ above˜ ,plitude initiallyand at its expressions rest becomes expressions period at͒ in rest thelengthens. nonlinear,͒ in potentials the for for potentials the thea plasma the͑͒ coldand colda wave͑ fluid͒͑and steepens͒ fluid. motion͑ motion͒. u andu and␥ ␥ by the driver͒, e.g., set of laser= and plasmalasercos parameters͓ pulse (slight͑ groupz/ mismatchv velocity.− ist due͔͒ to. the The numerical When quasistatic solution beingE approximation AnalyticalAnalyticaland solutions its period solutions to lengthens. Eq. to͑20 Eq.͒ in͑ terms20͒ in of terms elliptic2 of in- elliptic in- by the driver͒,aminedaminedamined e.g.,amined by by by assumingby assuming= assuming assuming00cos͑Sprangle that͑ thatSprangle͓ that that the thepp͑ theetz the driveet drive al./ al.v drive drive,,pp 1990a 1990a beam beam−t beam͔͒ beam,,1990b.1990b is nonevolving, When is͒͒ cancan nonevolving, be be applied appliedE such suchtegralsAnalyticalalso that thatAnalytical can describeand be solutionsIn found its the periodPage solutions region the5 for of 33 to lengthens. square behind Eq. single to͑20 the laserEq.͒ in drive particle͑ terms20 pulse beam,͒ in profiles of termsa motion elliptic=0, an ofBu- analysis in-elliptic of an in- electron ͑p˜ տE , the plasmai.e.,i.e., wave the the drivein a drive becomes1D approximation beam beamthe iswhereas a is plasma highly function PIC a functionsimulations fluid nonlinear. ofar quantitiese in only of2D). only the are thecoordinate also Wake- coordinate assumed toTheAnalytical be func-tegrals aboveand solutions can its expressions period be found to lengthens. Eq. for20 square forin terms the laser of cold ellipticpulse2 2 fluid profiles͑ in- motion͑Bu- u and ␥ by the0 driver͒,aminedamined e.g.,i.e., the by by drive assuming= assuming beamcos͑Sprangle that isthe͓ that a plasma the functionp͑ thezet drive/ al. fluidv drive,p 1990a of− beam quantitiest only͔͒ beam, .1990b is the When nonevolving, areis͒ coordinatecan nonevolving, also be assumed appliedE to suchtegralstegrals be func- thattegralsAnalytical can can be becanIn foundIn found the bethe solutions region foundregion for for square behind square behind for to͑ square the Eq.laser͒ the laser drive͑ drive20 pulse laser pulse͒ beam,in beam, profiles terms pulsea profiles=0,a =0, profilesan of͑Bu- an analysis ellipticBu- analysisBu- in- ˜ տE , the plasmai.e.,i.e.,i.e., wave the the the drive drive drivebecomes beam0 beam beamthe istions a is plasma is highly function a a only function function fluid of thenonlinear. of quantities comoving only of of only only the are variable the coordinate also Wake- coordinate coordinate assumed. The 1D tolanov bealso limitalso func-tegrals ap- describeet describe al.ofIn can, Eq. the 1989 be region͑Page16 the; found ͒ the5 Berezhianiofindicates 33 behind single single for thatthe square particle driveand the particle electrostatic beam, Murusidze laser motiona pulse2 =0, motion potential an 1990 profiles analysis͑ of; os-͑ of an͑Bu- an electron electron͑p ͑p˜ տE ,0 the plasma wave==z−zv−pvt, becomest, where wherethevp ഛv plasma highlycഛisc fluidis the the nonlinear. quantities phase phase velocity are velocity also Wake- assumed of the of to thetegrals be func-lanov can beofet foundEq. al.͑16,͒ for 1989indicates square; Berezhiani that laser the pulse electrostatic and profiles Murusidze potential͑Bu- os- 1990; field0 generationi.e.,i.e., in thez the the drivet drivep nonlinear beam beam istionstionsഛ a ispc function aonly only function 1D of of the regimethe of comoving only comoving of only the can variable thevariable coordinate coordinate be.. ex- The The 1Dlanovlanovalso limit limitandtegralslanov ap- ap-et describeet␥˜ al. al., canet,ofcillates initially, 1989 Eq. al. 1989 be,͑; in16 thefound 1989;Berezhiani the͒ Berezhianiindicates atrange; single forBerezhiani rest square that and͒ഛ particle andin theഛ Murusidze laser the electrostatic and Murusidzeand pulse potentials Murusidze motion the 1990 axial potentialprofiles 1990; electric of; 1990a os-͑ anBu-͑; ͒ electronand ͑͒p.˜ տE , the plasma= wave=z−=−vzpv−tp,v becomes, wheret, where wherevvppഛpliesv highlycഛisis toc thebroadis thePage nonlinear. the 6 phase phaseof drivers, 33 phase velocitykprЌ velocityӷ Wake-1, where of of the the ofrЌ theis theSprangle charac-lanovetofet al. Eq., al. 1990a͑16,͒ 1989indicates, 1990b; Berezhiani that.min As the the electrostatic plasmamax and wave Murusidze potential am- os- 1990; ͑ 0 plasma=z−vp wave.tp, where Fortions laservpliesppഛ onlyc todrivers, broadis of the the drivers,v comovingӍ phasev ,k wherer variable velocityӷ1, wherev is. The theofr is 1D the thelanov limitand charac- ap-Sprangleet␥˜ , al. initiallycillates,cillates 1989et in; al. in theBerezhiani the, at 1990a range range rest, ͒min1990b͒ഛin andഛഛ͒ the.ഛ Murusidze Asmax and potentials theand the plasma the axial 1990axial electric wave; electrica͑͒ am-and ͑͒. fieldfield generation generation= inz inplasma− thev thept, where nonlinear wave. nonlinearv Forppliesഛc laser tois 1D 1D broad the drivers, regime regime phase drivers,p vp velocitygӍkpp canr canvЌЌgӷ, where1, be be where ofg ex- the ex-vgrЌisis the theSprangle charac-andlanov␥˜et,et al. initiallyfield, al. 1990a oscillates, 1989, 1990b at; inBerezhiani the rest. As rangemin͒ thein−E plasma the andmaxഛEഛ potentials wave MurusidzeE . am- The values 1990a͑; ͒ and ͑͒. plasma=z−vp wave.t, where For laservpteristicഛc drivers,is radial thev dimensionӍ phasev , where velocity of thev is drive ofthe beam. the Sprangle TheSprangleAnalytical 1D etcillates al.et, 1990a al. in, the 1990a solutions, range1990b, ͒1990bmin͒. Asഛ͒ to.ഛ the Asmax Eq.max plasma theand plasma20 the wavemax axialin wave electric am- terms am- of elliptic in- amined by assumingplasmaplasmaplasma wave. that wave. wave. For the For For laserplies drive laserteristic laser to drivers, broad drivers, radialdrivers, beam drivers,v dimensionppӍvv isvpggӍk, nonevolving,pv wherevrЌg,,ӷ of where1, the wherevgg driveisvvg therЌisis beam.is the the theplitude The charac-Sprangle 1D␥˜ becomesfieldfieldet oscillates oscillates al. nonlinear,, 1990a in in the the, range the1990b range plasma−E−͒maxE. AsഛഛE wave theഛE͑EഛmaxE plasma steepens͒. The. The values wave values am- field generation inlaser the pulse nonlinear group velocity.teristic 1D radial The regime quasistatic dimensionp Ӎ cang approximation of the be drive ex-g beam.Sprangleand Theplitude 1D,et initially al., becomes 1990aand , at1990b nonlinear,, rest denoted͒. As͒ in by the the the plasmamax, are plasma potentials͑Esarey wavemax wave and am- Pilloff, steepensa͑ ͒ and ͑ ͒. plasmalaser wave. pulse For group laser velocity.quasistatic drivers, The fluidvp Ӎ quasistatic momentumvg, where approximationv andg is continuity the plitude equationsSprangleAnalytical becomesfieldminet oscillates al. nonlinear,, solutions 1990amax in the, the1990b range plasma to−͒.E Eq. Asmaxm ഛ wave theE͑20ഛ plasmaE͒ steepensmaxin. The terms wave values am- of elliptic in- aminedamined by by assuming assuminglaserplasmalaser pulse pulse that that wave. group group the the For velocity. velocity. driveteristic drive laserquasistatic The radialdrivers, The beam beam quasistatic fluid quasistatic dimensionv ismomentump Ӎ nonevolving,vg approximation, of where the and drive continuityvg is beam. the equations Theplitudeandplitude 1D itsplitudeAnalytical period becomesmin becomes becomes lengthens.andand nonlinear,max solutions nonlinear,,, nonlinear,denoted denoted the by by plasma the tom the, Eq.are plasma, are plasma͑ waveEsarey͑͑Esarey20 wave steepens͒ and wavein and Pilloff, steepens terms Pilloff, steepens of elliptic in- i.e., the drive beamlaser͑Spranglelaser pulse pulse is aet group group functional., 1990a velocity.quasistatic velocity.give, 1990b of The The fluid only͒ can quasistatic quasistatic momentum be the applied coordinate approximation and such continuity that plitude equationstegralsand becomes its1995 canmin period͒ nonlinear, be lengthens.max foundPage 8 of 33 the for plasma squarem wave steepens laser pulse profiles ͑Bu- laser͑Sprangle pulse groupet al. velocity.,quasistatic 1990a The, 1990b fluid quasistatic͒ momentumcan be approximation applied and continuity such that equationsandandAnalyticalplitude its its period period1995min becomes lengthens.͒ lengthens.and solutionsmax nonlinear,, denoted by to the Eq.m, are plasma2 ͑͑Esarey20͒ wavein and terms Pilloff, steepens of elliptic in- amined by assuming͑Spranglelaser͑SprangleSprangle pulsethatetet group theal.et al., al., 1990a 1990a drive, velocity.give 1990agive,,1990b1990b, beam1990b The͒͒ cancan quasistatic iscan be be nonevolving, applied applied be applied approximation such such such that that that tegralsInandand the its region its can1995 period period͒ behind be lengthens. lengthens. found the drive for beam, squarea =0, laser an2 analysis pulse profiles ͑Bu- i.e.,i.e., the the drive drive beam beam͑theSprangle͑ plasma is is a aet function fluid function al., quantities 1990agive , of of1990b are only only also͒͒ can the assumed be coordinate coordinate applied to be such func- thatandtegrals its periodIn1995 the can lengthens.͒ region be2 behind found the for drive2 square2 beam,22 1/2a laser=0, an pulse analysis profiles ͑Bu- ͑Spranglethethe plasma plasmaet al. fluid, fluid 1990a quantities quantities, 1990bu − are͒acan arealso=0, be also assumed applied assumed to such be to func- that be func- InInlanovand the the͑13 region͒ its region periodet behind behind al.= lengthens.,Eˆ the 1989 the/2 drive ± drive ;͓͑1+ beam,Berezhiani beam,Eˆ a/2͒=0,a −=0,1 an͔ 2 an analysis, 2 andanalysis Murusidze͑24͒ 1990; =z−vpt, wherethe͑Sprangle plasmavp ഛ fluidetc al.is, quantities 1990a the, 1990b phaseЌ areЌ also͒ can velocity assumed be applied to be of such func- the that tegralsof Eq.In16 the canindicates region bem ˆ 2 behind found2max that the thep for electrostatic driveˆ 2 squaremax2 beam,22 2 1/2 potentiala1/2 laser=0, an pulse os- analysis profiles ͑Bu- i.e., the drive beamthetionsthe plasma is plasma only a function of fluid fluid the comovingquantities quantitiesu ofuЌ−−a onlya variableЌ are=0, are=0, also alsothe. assumed assumed The coordinate 1D limit to to be be ap- func- func- In the͑1313͒In region͑ the͒ region behindm ==EEˆmax behind the/2/2 ± drive ±p͓͑1+ the1+ beam,E drivemaxEˆ /2a͒/2− beam,=0,1−͔1 an, ,a analysis=0, an͑24 analysis͒24 =z−v t, wherethetions plasmav ഛ onlyc fluid ofis quantities the the comovingЌ phase areЌ also variable velocity assumed. The to of be 1D func- the limit ap-oflanovlanov Eq.͑of͑16͒ Eq.͒ etindicateset͑16 al.͒ al.m indicates,ˆ,2 that 1989max 1989 the; thatp electrostatic;͓͑BerezhianiBerezhianiˆ the2 max electrostatic2͒ potential1/2͔a2 and and potential os- Murusidze Murusidze͑ ͒ os- 1990 1990; ; =z−vptp, wheretionsthetionsv plasmap onlyp onlyഛc of of fluid theis the comoving the comovingquantitiesuЌ phase− a variableЌ variable are=0, also velocity.. assumed The The 1D of limit limit to be the ap- ap- func- ofcillatesSprangle Eq.͑of13In͒ Eq.͑ in16 the the͒ ͑indicates16 region range͒metindicates=ˆE al.max behind that,/2 1990aഛ ±  that thep the͓͑ഛ1+ electrostatic the drive,Emax1990b electrostaticand/2 beam,͒ the− 1͔͒. axial potential As, =0, electricpotential the an os- analysis͑ plasma24͒ os- wave am- plasma wave. Fortionspliestions laser toonly only broad of of drivers, the drivers, the comoving comovingk␥vp−rpЌӍӷu variable1, variablev−g where,=1, wherer.Ќ. The Theisv theg 1D charac-is limit limit the ap- ap-oflanov Eq.of14͑16 Eq.͒etindicateswhere͑16 al.͒ ˆ,indicatesEmax that 1989min=Emax the;/ thatE electrostatic0Berezhianiഛandmax the theഛ electrostaticϮ give potentialmax andand potential os-min Murusidze, re- os- 1990; =z−vpt, wheretionsvpliesp onlyഛ toc of broadis the the comoving drivers, phasek r variablekӷpr z ӷ velocity1,. where Ther 1Dr of limitis the the ap- charac-cillatescillatesSprangleof͑cillates in Eq. in͒ the thewherewhere16 range in rangeet theindicatesE al.Eˆmax rangemin,==E 1990aഛEmaxഛ/ thatഛE/minE0ഛand,maxand the1990b theand the electrostaticandϮ theϮmaxgive͒ thegive. axialand As axialmax electric theand the potentialandelectric axialmin plasma, re- electric, re- os- wave am- plasma wave. Forpliestionsplies to laser onlyto broad broad of drivers, drivers, the drivers, comovingk␥v␥pr−ЌЌӍӷppu1, variablev1,Ќz − where where, =1, wherer.Ќ Theisv the theЌ 1Dis charac- charac- limit the ap- fieldSprangle͑14 oscillates͒ ͑spectively.͒ inetˆ the al.max range Formin, 1990amaxE −Eഛ/0E,maxտഛഛ1990b1,E Eq.ഛE͑16͒͒..indicates The Asmax values the thatmin plasma the wave am- plasma wave. 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The 1D The 1DfieldSpranglecillates oscillates inspectively.electricet the in al. the range field, range For 1990a departsEmax−maxEഛ/,Emax0 from1990bഛഛ1,E Eq.max aഛ simplemaxE͒͑and.16max As͒ .indicates the The sinusoidal themax axial values that plasma electric form the wave am- plasma wave. Forteristicpliesquasistatic laser to radial broad drivers, fluid dimension drivers, momentumvpkӍp ofrЌv theӷg and,1, where drive where continuity beam.rЌvgisis equations the The the charac- 1D plitudeminfieldand oscillatesspectively.max becomes, denoted in For theE by nonlinear,max rangemin/Em0,տ− are1,E Eq.͑Esareymaxഛ͑16 theE͒ഛindicates andE plasma Pilloff,. The that values the wave steepens laserlaser pulse pulse group groupteristicteristic velocity.teristic velocity. radial radial radial dimension The The dimension dimension quasistatic quasistaticn of͑p the− of ofz͒ the thedrive=  approximationpn drive drive0. beam. beam. The The 1DThe 1D 1Dfieldplitude oscillatesfield͑15͒ oscillateselectricelectric in becomes the field field range in departsthe departs− nonlinear,Erangemax from fromഛ−EmaxE aഛmax a simpleE simplemaxഛ the.E The sinusoidalഛ sinusoidalmaxE values plasmamax. The form form values wave steepens quasistaticquasistatic fluid fluid momentum momentumnn͑p−− andz͒== and continuitynpn0.. continuity equations equationsminand͑and15and15͒min itsmaxelectricand͑Akhiezer period,, denoted denotedmax field, and denoted lengthens. departs by by Polovin,m, from are by, are 1956͑Esarey am͑;Esarey, simpleBulanov are and͑Esarey sinusoidal andet Pilloff, al. Pilloff,, 1989 and form; Be- Pilloff, ͑Sprangle et al.quasistaticteristic,give 1990a radial, fluid1990b dimension momentum͒ can͑ p of be andz͒ the appliedp continuity drive0 beam. such equations The that 1D plitude1995minfield͑͒ ͒ oscillates͑max becomesAkhiezerPage 8 of 33 in and the Polovin, nonlinear, rangem 1956−Emax; Bulanovഛ theEഛetE plasmaet al.max, al. 1989. The; Be- values wave steepens laser pulse groupquasistatic velocity.quasistatic fluid The fluid momentum quasistatic momentumn͑p −  andz͒ =  and continuityp approximationn0. continuity2 equations2 2 equations minand͑and15min͒ itsandmaxand period͑rezhianiAkhiezer, denotedmax, and, denoted lengthens. and denoted by Murusidze, Polovin,m by, are by 1956m 1990͑Esarey,; are,;Bulanov areSprangle͑Esarey and͑Esarey Pilloff,et and, al. 1989, and 1990a Pilloff,; Be- Pilloff,, =k2 ͑ thatequationsn/n −1͒ can1995 beand͒ writ-1995min its͒͑Akhiezer periodmaxPage and 8 of lengthens. 33 Polovin, 1956; mBulanov et al., 1989; Be-2ץ/ such ץSprangle et al.,quasistaticgive 1990a, 1990b fluid momentumThecan Poisson be applied equation and continuity ͑Sprangle͑ et al.,give 1990agive , 1990b͒͒ can be applied22 such22 2 p that0 1995͒ rezhiani and Murusidze, 1990; Sprangle et al., 1990a, ==kkp equations͑nn//nn0−1͒ can be be writ- writ-1995minInand therezhiani1990b regionmax͒., In and denoted particular, Murusidze, behind by the electric the 1990m, are; driveSprangle field͑Esarey exhibits beam,et and al. the, 1990a char- Pilloff,a =0,, an analysisץץ//ץץquasistatic fluid momentumTheThe Poisson Poisson equation equation and continuity the plasma fluidgivegivegive quantities areten also as ͑Berezhiani assumed and2 Murusidze to2 be2p͑ func- 19920 ; Esarey,͒ 1995and Ting,͒ 1995 itset ͒͒ˆrezhiani period1990b2 . In and particular, lengthens. Murusidze,ˆ 2 the electric2 1990; 1/2Sprangle field exhibitset al. the, 1990a char-2 ,2 =kp͑n that 1992/n0͑−13; 1Esarey,͒͒ can be Ting, writ-Inet the= E 1990b region/2͒ ±͒. In͓͑ particular, behind1+E /2 the͒ the electric− 1͔ drive, field beam, exhibits͑24 thea͒ char-=0, an analysisץ Murusidze/ such ץSprangle et al., 1990au ,−1990ba =0, ͒Thecanten Poisson as be͑Berezhiani equation applied and͑ thethe plasma plasma fluid fluidgive quantities quantitiesЌ Ќ are aretenal. also also as, 1993͑Berezhiani assumed; assumedTeychenné andet Murusidze to toal., 1993 be be͒ func- func- 1992; Esarey, Ting,1995Inmet ͒ˆ theˆ21990b2maxacteristic region͒ˆ.2 Inp particular, “sawtooth” behindˆ 2ˆ max2 the profile2 electricˆ22 the1/2 associated1/2 field2 drive exhibits1/2 with beam, wave the char- steep-2a =0, an analysis tions only of theu comovinguЌ−u−Ќaa−Ќ=0,a=0,Ќ =0, variabletenal. as, 1993͑Berezhiani; Teychenné. The andet Murusidze 1D al., 1993 limit 1992͑͑13;13 ap-Esarey,͒͒ ͑13͒ Ting,ofmet= Eq.=EEmaxacteristicacteristic͑/2=16/2 ±E ±͒ p “sawtooth”indicates͓͑ “sawtooth”/2͓͑1+1+ ± EmaxE͓͑1+/2 profile profile͒/2E that͒− 1− associated͔1 associated/2͔,͒ the,− 1 with͔ electrostatic with, wave͑ wave24͑ steep-͒24 steep-͒ ͑24 potential͒ os- the plasma fluid quantitiesЌ Ќ areal. also, 1993; assumedTeychenné et to al., 1993 be͒͒ func- Inm theˆ 2 maxmening regionˆ 2 2max andp the behind densityˆ p2 max oscillationsˆ22 themax2 1/2 drive2 2 become1/2 beam,1/2 highly peakeda =0, an analysis uЌ −uЌaЌ−=0,aЌ =0,al., 1993; Teychenné et al., 19932 ͑13−1/2͒ ͑13͒ ofm Eq.= Emacteristic͑ening=16/2E͒ ±ˆ andindicatesp/2 “sawtooth”͓͑ the1+ ±  densitypE͓͑1+/2 profile oscillations thatE͒ ˆ− 1 associated/2͔ the͒ , become− 1 electrostatic͔ with, highly wave͑24 peaked steep-͒ ͑24 potential͒ os- -2.. The The 1D 1D limit limit␥͒ ap- ap-͑13͒whereof Eq.Eˆ maxmening=͑͑as=16EˆE illustrated2max͒max and/indicatesE/2 the ±and density inpmax Fig. the͓͑1+8Ϯ oscillationsˆ maxinE2 thatgivemax Sec./22III.A the͒ become−͒and11/2.͔ electrostaticFurthermore,, highly, re- peaked the ͑24 potential͒ osץtionstions only only of of the the comoving comovinguЌ − aЌ =0, variable variable u a −2 2 2 22Ќ −−1/21/2 cillatesˆ maxening inEmax and the the0 range density oscillationsE ഛ becomemaxഛ highlyminand peaked the axial electric - = ␥ rЌ is1 − the␥Ќ charac-͑14−͑͒131 ͒,whereof Eq.͑16Eˆ͒ m=͑16=asEˆ illustratedmaxindicates/E/2 ±and inp͓͑ the Fig.1+Ϯ8 thatingiveminmax Sec./2 theIII.A͒ − and͒1. electrostatic͔ Furthermore,,max, re- the ͑24 potential͒ os ץplies to broad drivers,␥Ќ−−pЌukz=0,−rЌ=1,ӷ1,k where2 .2 The2 2p p 1D limit2␥2Ќ 2− ap-1/2 where Emax͑͑=periodasEEmax͒ illustrated/ ofE0 theand in nonlinearE Fig. the 8Ϯingive plasma Sec.ϮmaxIII.A wave͒and. Furthermore, increasesmin , re- as the the 22ץtions only of the comovingp variable−2−p = ␥p p ͫ1 − ␥␥͑1+͒ ͬ͑14−͒1 , spectively.cillates͑16whereˆ͒ max͑ inas For illustratedmax themaxE= range/0maxE inտ/ Fig.1,0and Eq.8minin the Sec.͑ഛ16͒III.Aindicatesഛmaxgive͒. Furthermore,maxmax thatandminand the the themin, axial re- electricץ ץpuz − =1, kp − ␥ ␥ −  uk− r =1,ӷ k 2= ␥ rͭ 1 − 2p Ќ 2 ͑14−͒1 ͮ, cillates16 periodˆ in the ofmax the range0 nonlinear plasmaഛ waveഛ increases and as the the axial electric -2 2prЌp ͫis the the␥2 ͑1+ charac- charac-͒2 ͬ ͑14where͒ where͑ Emax͒ =EperiodEˆ max=/E of0 theand/E nonlinear theandϮ thegivemin plasmaϮmaxgive waveand increasesmaxminand, re- as, the reץ pliesplies to to broad broad drivers, drivers,␥ p−kz ppruzЌ−ӷ1,=1,1, where− wherep2 p Ќ k = ␥ ͭЌ 1 − p − 1 ͮ, spectively.field16 oscillates ForamplitudemaxE /maxE increases. inտ1,0 the Eq. The range͑16 nonlinear͒ indicatesϮ−E plasmamax thatഛ wavelength theEഛminE . The values - p p ͫ ␥ ͑1+͒ ͬ͑14͒ ͑ where͒ E max=E 0 /E and the givemax and max, reץ u − =1, p  − ␥ teristic radial dimension␥ p− z u − of =1, the drive2 ͭ beam.2p The2 1D͑ͮ14͒spectively.period FormaxE of the/maxE nonlinearտ1,0 Eq. plasma͑16͒ indicates wave increasesmax that as the themin p z spectively.ˆ Formax E0 /E տ1,ഛ Eq.ഛ16 indicates that the 2 2 ͫ ␥2 ͑1+͒ ͬ 2 −1/2 ͑14͒cillateselectric field inamplitude the departs range increases. frommax The0 amin simple nonlinear sinusoidal͑ plasma͒ max wavelengthand form the axial electricץ ,puz − =1 − ␥ plies to broad drivers, k r ӷ1, where rͭ is thep charac- ͮ where Eamplitude=E increases./տE and The the nonlinearϮ give plasmaand wavelength , re- n  −pЌ =  nwhere. ␥2 =1+Ќu2 =1+a , ␥ =͑1−2 ͒ 15, andspectively.electricfield=spectively.v /c oscillatesfield. Forinmax departs theE Formax limit/maxEE in␥0 fromp ӷ the1,/01E is Eq. aտ͑ rangeBulanov simple1,͑16 Eq.͒ indicateset− sinusoidal͑ al.16E,͒max 1989indicatesmax;ഛ thatBerezhianiE form theഛ thatEmin andmax the. The values teristic radial dimension␥ −͑ p puz z−͒ of =1,p the0 drive2 Ќ 2 beam.Ќ 2 p The2 p 1D−͑1/2 ͑͒14͒ electricfieldp spectively.p field oscillatesamplitudein the departs limit For increases.␥Emax inӷ from1 is the/E0Bulanov The aտ range simple1, nonlinearet Eq. al.,− sinusoidal 1989͑16E plasma͒;maxBerezhianiindicatesഛ wavelength formEഛ and thatEmax the. The values teristic radial dimensionn  −  of=  n thewhere. drive␥Ќ =1+uЌ beam.=1+a2 , ␥p = The͑1−p͒− 1D151/2, and ͑pAkhiezer=vpelectric/c. and andin the field Polovin, limit departs␥p,maxӷ denoted 19561 is͑ ;0Bulanov fromBulanov a byet simple al.et, 1989 al.,, 1989; aresinusoidalBerezhiani ; Be-͑Esarey and form and Pilloff, spectively.vminpand/c field. Murusidze, departs FormaxE fromp 1990/E; aSprangle͑տ simple1, Eq.et sinusoidal͑ al.16,͒m 1990aindicatesഛ, form1990bഛ that͒ Np theץ=/pץquasistatic fluidn momentum͑ p−  z͒=  np where0.The and axial␥2Ќ =1+ continuity electricu2Ќ =1+ fielda2 of, ␥ thep equations=͑ wake1−2p−͒ is͑1/215E,͒= and−Eelectric teristic radial dimension͑ np ͑ z−͒ of ͒p= thewhere0 n drive.␥ =1+u beam.=1+a , ␥ The=͑1− ͒ 1D͑ ,z͒ and͑150͑Akhiezerfield͒ =velectric/c. oscillates andin the field Polovin, limit departs␥ inpmaxӷ 1956 the1 is ͑;0 fromBulanovBulanov range aetet simple− al. al.etE, al. 1989max, 1989; sinusoidalBerezhiani;EBe- E and formmax. The values ,minpandelectricandMurusidze, andMurusidze, field Polovin,max, departs 1990 denoted 1990 1956; ;Sprangle1/2Sprangle; fromBulanov by aet simple al.,etm 1990a,, al. 1990a are,, 1989 sinusoidal1990b, ͑1990bEsarey; Be-͒/2 Np form and Pilloffץ/͑pAkhiezerץn͑ − p ͒ = z n . Thep 0 2 axialЌ electric2 Ќ2 field of thep wakep is͑15Ez͒=−E0 , and͑Akhiezerand and=͑2/ Murusidze, and͒ ͑1+Page Polovin,, denoted8 of 199033͒ E; 1956͑!Sprangle͒,; byBulanov whereet, al.Eet are,͑! 1990a al.͒=,͐/2͑ 1989Esarey,͒d͑Np;1 Be- and Pilloffץrezhiani/ץquasistatic fluid momentumnp ͑ z−  ͒p= 0The andthen axial. plasma continuity electric fluid field quantities of the equations are wake is Ez =−͑15E0͒ 1995electricandmin andMurusidze, field Polovin,max departsp 1990 1956max; 1/2Sprangle; fromBulanov2 aet simple al.et, al. 1990am, 1989 sinusoidal, 2 1990b; Be-0͒ Np formץ/ Akhiezer͒͑ץ2 =k2 field͑n/n of0 − the1 equations͒ wakecan be is E writ-=−E͑15ץ continuity/ electric 2ץ.quasistatic fluidThe momentumn Poisson͑ p−  equation͒z= The andnp 0 axial give p z pthe0 2 plasma2 fluid2 p quantities are z 0 rezhiani͑Akhiezer and͒ =͑ Murusidze,2/2 and͒2p͑1+ Polovin,1/2max 1990͒ 1/2E; 19562͑!Sprangle͒,; Bulanov whereet al.Eet,2͑ 1990a! al.͒=,͐ 19890 , d/2͑;1 Be- ,2 fluid=kp2 quantities͑n/n2 0 −1 equations͒ arecan be writ- ͑15͒ rezhiani1990bmin͑Akhiezer.and In and particular,=−͑!2/ Murusidze,sinmax and͒p͑,1+ Polovin, thedenotedismax the electric1/2 1990͒ completeE2 1956;͑!Sprangle field͒, by; ellipticBulanov where exhibitsm, integralet are al.E theet2,͑! 1990a al.of͑ char-͒Esarey=, the͐/20 1989, sec-d͑1; Be- and Pilloffץ/ continuity ץquasistatic fluidThe momentumn Poisson͑ −  equation͒ =  andthen . plasma , 2 fluid=k2p quantities͑n/n0 −1͒ are2can be−1/2 writ-et 1995rezhiani͒͒ =͑2/2 and͒2 ͑1/21+͒ Murusidze, ͒ E ͑!͒ 1990, where; SprangleE ͑!͒et=͐ al.d,͑ 1990a1ץ/plasma2 ץThe Poissonp equationz pthe0 -2 = 1992k2 ͑n;/Esarey,n0 −1͒ can Ting, be writ-rezhiani1990b1995͑Akhiezer. In and͒ particular,−! Murusidze,sin2 and2p͒ 1/2 Polovin, theismax the electric 1990 complete 19562; Sprangle field;2 ellipticBulanov exhibitset integral al. theet,2−2 1990a al. of char-, the 19890 , sec-; Beץ/ 2ץgive tenThe as Poisson͑Berezhiani equation and Murusidze , - =k22p͑n 1992/ 2np0 −; 1Esarey,͒ ␥can2Ќ be Ting,−1/2 writ-et 1990brezhiani͒͒. In particular,−ond!2 andsin kind2 Murusidze,1/2͒ with theis argumentthe electric complete 1990! field=1; ellipticSprangle− exhibits1+ integral theetor ofal. char-, the 1990a secץ Murusidze/2 ץgive Theten Poisson as Berezhiani equation and , = 1992k ͑n;/Esarey,n␥02Ќ−1͒ can− Ting,1/2 beet writ- acteristicrezhiani “sawtooth”−! sin and Murusidze,is profile the complete associated 19902 elliptic; with͑Sprangle integralmax wave−2͒ of steep-et the al. sec-, 1990aץ␥/nץtenThe as Berezhiani Poisson͑ equation and Murusidzen -2p ͒=p k21p͑−n/ 2n␥20 −1͒2−can1/2 − bep writ-, 1990b1995͒͑.1990b17͒ In͒ particular,ond͒. In kind particular,2͒ with the argument electric the! electric field=12 − exhibits͑1+2 fieldmax͒ exhibitsthe−or2 char- the charץ/=0 1993, /ץ.give Theal., 1993 Poisson͑ ; Teychenné equationet al tenten as Berezhiani as ͑Berezhiani and Murusidze andn2/n0 Murusidze= ␥2 2p 1992ͩ1p−; 1992␥Esarey,Ќ ; Esarey, Ting,ͪ − et Ting,p , acteristicet rezhiani͑17͒ “sawtooth”ond and kindˆ Murusidze, withprofile argument associated 19902! =1; withSprangle−ˆ͑1+ wavemax−2͒2 steep-etor al.,1/2 1990a, ͑ p ͫ 2␥͑1+ ͒ ͬ 1990b . In particular, the electric field exhibits the char- - =k1 −n/np Ќ−1 2can− be writ-, acteristic17 “sawtooth”͒ profile associated2 with wave steepץ␥/n ,= 1993ץ/.Theal., 1993 Poisson; Teychenné equationetn al uЌ − aЌ =0,al.,ten 1993 as; Teychenné͑Berezhianiet and al.,0 Murusidze 19932p p͒ ͫͩp͑ 1992␥2 ͑01+; Esarey,͒͒2 ͪ ͑ Ting,13pͬ͒ et ening͑1990b and͒ mond the͒.= In kind densityE2 particular, with/2 oscillations argument ±  theEp͓͑! electricE1+ become=1−E2͑1+ field highlymax/2͒ exhibits͒Eor peaked−E1Ӷ͔ the, char- ͑24͒ ten as ͑Berezhiani andn/n0 Murusidze= ␥ p͒ ͩ1 − 1992␥ p ; Esarey,ͪ −  Ting,p , etacteristic͑17acteristic͒ “sawtooth”ˆ “sawtooth”max profile1+3 associated͑ profilemax/ 02͒ associatedˆ/16, withmax wave2 withmax steep-/ wave1/20 1 steep- p ͫ 2 ͑1+ ͒ ͬ 2 2 u − a =0,al., 1993al., 1993; Teychenné2 ; Teychennéet al.,et 1993 al.,2͒ 1993ͩ ͒ −␥1/2p 2 ͪ 13 ening1990bacteristic and the͒.= In densityE “sawtooth” particular,ˆ =/2 oscillations1+3 ±  profile͑ theEmax electric1+/E become0 associated͒ /16,E2 ˆ field highly/2 exhibits withEmax peaked−2/E1 wave0 Ӷ the1 ,1/2 steep- char- 24 ͫ ͑1+ ͒ ͑ ͬ͒ ening and the densityNp p oscillationsE͓͑ E become highly͒ E peaked͔E Ӷ ͑ ͒ ; Teychenné andet Murusidze al.,␥ 1993 1992p ; Esarey,͑ Ting,13͒ et ͑as illustratedacteristicm = in “sawtooth”Emax Fig. /28 1+3in ± Sec.p͑ profilemax͓͑III.A/1+02͒ associated͒/16,.maxE Furthermore,/2 with͒ max−/ wave the10 ͔ 1 steep-, ͑24͒ ץu Ќ− a Ќ=0,tenal. as, 1993͑Berezhiani Ќ Ќ al. −2 2 2 et al. 2Ќ ͒ −1/2 2 1/2 ening andening them and densityNp2 themax= densityp oscillations1+3ͭ͑2/͑E͒͑ oscillationsmaxEpmax/E become/0E͒ 0/16,+ E20/Emax become highlymax͒, E2Emax peakedmax highly/E/E0 0Ӷ1/2ӷ11, peakedͮ , 1993; Teychenné ,2 1993͒ −1/2 ˆ = ˆ - = ␥  1 − ␥2Ќ − 1 , ␥2Ќ ͑161/2͒ ͑as illustratedacteristic in “sawtooth” Fig.Np 8ͭp in͑2/ Sec.͒͑ profileEmaxIII.A/E0 associated+͒.E Furthermore,0/Emax͒, withEmax/E wave the0 ӷ 1,ͮ steep ץk 2 u − a =0, −2 2 13 E  E ; Teychenné22 p p etu al.=2,␥␥2 1993Ќ 1+2͒2−1/2 −−1/21 − ␥2Ќ͑ ͒1/2 , ͑periodas illustratedening18 ofm and= the in theNp nonlinearmax= Fig. density/2p 8ͭ ±͑in2/ plasma oscillationsSec.͒͑pE͓͑max1+III.A/E wave0 +͒E.max become0 Furthermore,/ increasesEmax/2͒͒, E highly−max as1/E͔ the0 the peakedӷ,1,ͮ ͑24͒ 2ץЌ Ќ al., 1993p −2 2 z 2 ͑ ͒ p ening͑ ͒ andˆ the density oscillations become highly peaked ␥p p ͫ1 − ␥␥͑␥21+p ␥͒ ͬ − 1 , 2 ␥2Ќ ͑1621/2͒ ͑as illustrated͑as illustrated in Fig. in8 ͑in2/ Fig. Sec.͒͑E8maxIII.Ain/E Sec.0 +͒.E Furthermore,0III.A/Emax͒͒, .E Furthermore,max/E the0 ӷ 1,͑25͒ the= ץץ ץkp 2 u 2= 1+2 −−1/21 − , 18 ͭ ͮ k =2 ␥ ͭ 1 − z p Ќ͑ Ќ2 ͒ p − 1 ͮ, 16 periodwhere͑ of͒ theE nonlinear=E plasma/E waveand increases the Ϯ asgive the and , re- p ͩ ␥ ͑ ͪ ͒ −2 −2 2 2p ͫ 2␥͑1+ ͒ ͬ p p 2 ening andmax the densitymax oscillations0 become highly peakedmax min ͫ uz ␥=2 ␥͑2p1+͑1+␥2͒2͒ͬp −−1/21 − 2 p Ќ , period͑͑as18͒ of illustrated theˆ nonlinear in Fig. plasma8 in Sec. waveIII.A increases͒. Furthermore, as the͑25͒ the ץ22ץ ␥ −  u − =1,k k = ␥ =ͭ␥ 1− 1 p− Ќ ͫ − 1ͩ ͮ, −␥ 1͑1+, ͒16214ͪ ͬ 16amplitude increases. The nonlinear plasma wavelength ͑25͒ p2 p 2ppͫ puz␥= ␥͑1+͑1+␥͒͒ ͬp − 1 − p ͑͑ ͒ ͒,͑ ͒ ͑18͑as͒ illustratedEˆ E in Fig.E8 in Sec. III.AϮ͒. Furthermore, the −ץ p z p -p 2 ͩ ␥ ͪ where = / and the give and , re 2 2ץ 2 p 22 ͫ 2−1/2 2 ͑1+ ͒ ͬ period of the nonlinear plasma wave increases as the 2 ͭ Ќ ͮ 2 period of the nonlinear plasma wave increases as the −2 2 p max max 0 max͑25͒ min - = ␥2pͫ p ͫ␥12 −͑1+␥␥͑͒1+ͬͫ2͒−ͬ1/2ͩ −␥ 1͑1+, ͒14ͪ ͬ͑16͒amplitudeamplitudewhere͑as illustrated increases. increases.EwheremaxE= inmax TheE Fig.Theismax the nonlinear8 nonlinear/ peakEin0 Sec. electricand plasmaIII.A plasma field the͒ wavelength. of Furthermore,Ϯ wavelength thegive plasma wavemax theand min, reץ2 ץpץu − =1,k  − ␥ 2 ͭ␥  p 2 2 ͮ p ͑ ͒ p z k = ͭ 1 − p Ќ − 1 ͮ, 16 spectively.periodˆ of the For nonlinearE plasma/E տ wave1, Eq. increases16 indicates as the that the −2 2 ͑ ͒in the limitwhere␥ ӷ1E is Bulanovis the peaket electric al., 1989 field; Berezhiani of the plasma͑ and͒ wave -1+ up2Ќ =1+p ͫa2 , ␥␥p2=͑1+͑1−2͒p2−͒ͬ1/2 , and p2=͑v14p /c͒1/2. amplitudewhereperiod increases.Ewhere ofp the=EmaxE͑ The nonlinearis the nonlinear/maxE peakand plasma electric0 plasma the field waveϮ wavelength of thegive increases plasma wave asand the , re=2ץpuz − =1,wherep ␥2Ќ − ␥ k = ␥ ͭ 1 − p − 1 ͮ, ␥2 1/216 amplitudeandmax increases.=2maxmax/k =2 Thec0/ . nonlinear For a square plasma laser pulse wavelength profile,max min =1+u2p =1+p ͫa2 , ␥␥ 2=͑1+1−2͒ −ͬ1/2, and  =Ќv /c.͑ ͒inspectively. the limit ␥p ӷ1 ForE isp ͑BulanovEp et/E al.p ,տ 19891,; Berezhiani Eq. 16 andindicates that theץ␥wherep 2 ␥  Ќ2 Ќͭ p2p ͑ p2͒ ͮ␥p2 p 1/2 periodwhere of themax nonlinearisk the peakc plasma electric field wave of the increases͑ plasma͒ wave as the − u − where=1, ␥ =1+u =1+a ,␥␥= ␥= 1+1− 1 − , and1 −  Ќ͑=14v /͒c. . in theamplitude19 limitand␥p ӷ increases.p1=2 is ͑/Bulanovp =2max The/ etp. nonlinear For0 al., a 1989տ square; Berezhiani plasma laser pulse wavelength profile, and pand21/2 Murusidze,spectively.͑ ͒ and 1990p;=2 ForSprangle/kpE=2maxc/et/pE. al. For0, 1990aa square1,, Eq.1990b laser͑ pulse16͒ ͒Np profile,indicates that theץ/2␥2pץ electric2 2Ќ field2 ͫ2 of␥ thep2 2͑p͑1+͑ wake2͒͒p−͒ isͬ1/2E2p=−1/2−Eץp z The axial2Ќ ͭ ␥ = 2␥p 1+ 1 − z 1 − ͮ0 Ќ . in theelectricamplitude19 limit ␥ withӷ field1 is increases. optimalBulanov departs length Theet al. for, nonlinear 1989 from plasma; Berezhiani wave a plasma simple excitation and wavelength sinusoidal͑L form where ␥ =1+u =1+a , ␥ =p͑1−͒ ,p andͩ ␥ = /cͪ. ͑ ͒ p ␥͑ ӷ ͑ ͫ͒ 2␥͑1+v ͒ ͬ in theand limit =2/k1=2 is͑cBulanov/ . For aet square al., 1989 laser; pulseBerezhiani profile, and whereЌ ␥ =1+Ќ u =1+ap , ␥ = 1− , andp p2 = /Murusidze,c. 1990; Sprangle et al., 1990a, 1990b͒ andͪp vp. spectively.amplitude͑19͒ with increases.p For optimal1/2p Ep length The/etE forp2 nonlinear al.տ plasma1,2 Eq. wave− plasma1/2 excitation͑16/2 wavelength͒Npindicates͑L that theץ/p Ќ 2ץ␥The axial electric2Ќ field2Ќ ␥ of= the␥22p͑1+ wakep͒ ͑p1 is−E2zpp=−͒ͩ11/2−−E0 ͫ ͑1+ ͒2 ͬ Murusidze, 1990; Spranglemax 0 , 1990a, 1990b͒ ͑and15ͪ=v͒. /c. =electric͑2/͑in19͒͒ the ͑1+ limitwithӍ field ␥ optimal/2p͒ӷ͒, departs1EE is͑!͑ length/BulanovE͒,=͑2a where for/2 from͒͑et1+ plasma al.2a /2,− 1989E a͒1/2 wave͑! simplefor;͒=Berezhiani͐ a excitation linearlyd͑Np1 sinusoidal po- and͑L formץ/andp ץn͑ −  ͒The=thewhere axial plasman . electric2␥ fluid=1+ fieldu2 quantities=1+ of␥ = thea␥2p,͑1+ are wake␥ =͒͑11 is−−E2pz͒=ͩ−11/2−−E,␥0 p z p 0 p ͫ 2 ͑1+ ͒p ͬp p maxNp max2 0 2 /2 Ќ Ќ p p 2 Murusidze,electricin thewith 1990 limit field optimal; ␥1/2Sprangleӷ1 departs length is Bulanovet for al., plasma from 1990aet al., wave, 19891990b a excitation simple; Berezhiani0 L sinusoidal and form -vpand/c=. ͑2/Akhiezer͒Murusidze, ͑1+ӍNp/2͒ ͒ and,p 19901/2EEmax͑!;/͒ Polovin,E,Sprangle0͑=͑a where/22 ͒͑1+aet 1956/22E al.͒ ͑,!−;1/2͒for 1990a=Bulanov͐ a linearly͒d, /2͑1990bNp1 po-͑ et͒ al.Np , 1989; Beͬץ=/andͪp͒ץץand͑1+−/E =ץ␥ThewheretheThe axial plasma axial␥ electric2Ќ =1+ fluid electricu field2 quantitiesЌ =1+ field of thea2 of, are wake␥ thep =2͑ͫ wake1 is−E2zp=−͒ͩ is1/2−EE,0 n͑p − z͒ = pn . In the limit ␥2 ӷ1, Eq. ͑16͒z simplifiesp 0͑15 to͒ ͑Bulanov͑2inet al. the2p, 1/2 limitӍlarizedmaxNp␥/2ӷ laser.͒, E1E2max is!͑/BulanovE0 =͑2a /2͒͑et1+ al.2a /2,2− 1989E͒1/2 !for; Berezhiani0 a linearlyd po- and v͒pand/c. =electric−͑!2/Murusidze,sin͒ p͒͑1+Ӎ fieldismax the/2p 19901/2͒͒, E complete departs;2/͑SprangleE͒=,͑a elliptic/2 where͒͑ from1+eta integral al./2͒, a 1990a2͑for simple of͒= a, the͐ linearly/201990b sec-͑1 sinusoidal po-͒/2Np formץ=/͑p15ץ n͑ −  ͒ =thewhereThe plasman 0. axial␥Ќ =1+ fluid electricu quantitiesЌ =1+ fielda , of are␥ thep =␥͑p wakeӷ1−p͒ is E,= and−E -͑Bulanovand=͑͑2/Akhiezer2etMurusidze,͒ al.2,p͑1/21+larizedmaxNp and͒ laser. 1990Emax2͑ Polovin,!;͒1/2,Sprangle0 where 1956etE al.;2͑,!Bulanov͒ 1990a=͐ d, 1990b͑1 et al.͒ ,Np 1989; Beץ/ toץp z theThep plasma0 axial fluid electric quantities fieldIn the of are limit2 the2 p wake−1/21, Eq. is ͑E16z͒=simplifies−E0 n͑ −  ͒ =  then . plasma2 fluidIn2 quantities1989 the; limit2␥Berezhiani␥2p areӷ1, Eq. and͑ Murusidze,16͒z simplifies0͑15 1990 to͒;͑BulanovSprangle−!͑Akhiezersin2etet= al. al.2͑,͒2/, 1/2islarized͒ theThep͑1+ and complete laser. lengtheningmax1/2͒ Polovin,2E elliptic2 of͑! the͒, integral plasma 1956 where− wave2; ofBulanov the periodE0 2 sec-͑!͒= can͐/20 beetd͑ al.1 , 1989; Be- ,͑BulanovSprangleand −ond!2etrezhianietMurusidze,sin al. kind al.2 ,,1/2͒ withlarizedisThe argumentthe and 1990 laser. lengthening complete; Murusidze,Sprangle1/2! =1 of! elliptic the− 1+ plasmaet integral al. 1990, wave 1990aor; ofperiodSprangle, the!1990b can sec- be͒/2Npet al., 1990a;ץ 1990/ toץp z Thep 0 axial electric2 In field1989 the of; limitBerezhiani2 theЌ ␥p wakeӷ−1,1/2 Eq. and is͑ Murusidze,16Ez͒ =simplifies−E0 -1989 quantities=;kBerezhiani␥2Ќ͑n/ aren−1/2− and1͒ Murusidze,can be 1990 writ-; Sprangle−!Akhiezersinet= al.͑2/, 2 ͒is2pThe͑ the1+1/2 and complete lengtheningmax Polovin,͒ 2E2 elliptic͑ of͒, the͑ 1956 integral plasmawheremax−2͒; waveBulanov ofE the2 period͑ ͒ sec-=͐ can0 etd be al.͑1 , 1989; Beץ/ fluid ␥ץ The Poisson equationthen plasman the plasma/ 0 = 22p fluidp quantities119892−1990a; 2Berezhiani2␥2p, 1990b are2−͒1/20 and− Murusidze,p , 1990͑; 17Sprangle͒ ond͑rezhianiet kind al.=−,͑!2/͒ withsin͒importantThe and argumentp͒͑1+ lengtheningis Murusidze,max inthe1/2 plasma-based!͒ complete=1E2 of2−͑! the͑1+͒, plasma accelerators. ellipticmax 1990 where͒ wave−or2; integralSprangle periodForE example,2͑! of can͒= the͐/2 be0 inet sec-d͑ al.1 , 1990a, the plasman/n0 =2␥2 fluidp ͩ quantities12−1990a␥k2 ,Ќ1990bn aren2 ͒ ͪ −−1/2p , ͑17͒ ondrezhiani= kind͑2/2 with͒important2 ͑1/21+ andargument in Murusidze, plasma-based͒ E2! ͑=1!͒−,͑1+ accelerators. wheremax 1990−2͒ or For;ESprangle example,͑!͒=͐ ind͑1et al., 1990a, canͬ be writ- p max 2 2 −1͒ ͒ / 1+͑͑␥2 = ץͫ/ p ץ The Poisson equation ␥  1990ap, 1990b͒2 0  −! sin2 important2 1/2is the in plasma-based complete elliptic accelerators. integral For example, of the0 sec- in -ͩ1 1990a−=␥k2 ,p1990bЌ͑n/␥2n2 ͪ −1−1/2͒ canp , be writ-͑17͒ ond1990b kind! withimportant͒.͒ Inargument particular, in plasma-based! =12 −͑1+ the2 accelerators.max electric͒ or For− fieldexample,2 exhibits in the charץ/2ppץ= The Poisson equationn/n0 ͫ 2͑1+ ͒͒Ќ0 ͬ − sin ͒ isE theE complete ellipticE integralE Ӷ of the sec- ten as ͑Berezhianin/n and0 = ␥2  Murusidzep 2ͩ12− ␥ pp 19922 ͪ ; −−Esarey,1/2p , Ting,͑17͒et rezhianiond2 kind2 and1/21+3 with͑ Murusidze, argumentmax/ 02͒ /16,2! =1 1990−͑1+max; max/Sprangle−20͒ or1 et al., 1990a, p ͫ 2 ͑1+ ␥͒ ͬ p 2 1990b! ͒. In particular,2 the electric field exhibits the char- -2pͩp =␥Rev.k1 p− Mod.͑n2/␥ Phys.,nЌͪ0 Vol.−1 81,͒ No.can 3,− July–September bep , writ- 2009 ͑17͒ 1990b−ond sin= kind.1+3͒ In withis͑ particular,E argumentthemax/E complete0͒ /16,! the=12 elliptic−͑ electric1+Emax integral/E0͒ Ӷ −or field12 of the exhibits sec- the charץ␥/=n0/ץThe Poisson equationn ten as Berezhiani and Murusidzeͫ ͑1+ 19922͒ ; 2Esarey,−1/2 ͬ Ting, et Np ͒ p E E Emax E Ӷ Ќ 1+3 / /16, / 1 ten as Berezhiani͑ andn/n0 Murusidze= ␥2 p ͩRev.1p− Mod.␥ 1992 Phys., Vol.; 81,Esarey,ͪ No. 3,− July–Septemberp , Ting, 2009 et͑17͒ acteristic ond= kind with “sawtooth”͑ argumentmax 02͒ 2! profile=1−͑1+ associatedmaxmax−2͒0 or with wave steep- ͑ p ͫ 2␥͑1+ ͒2 1/2 ͬ al., 1993; Teychennén/n et= ␥ al.2 , 1993Rev.1 − Mod. Phys.,p Ќ Vol. 81,2 No. 3,− July–September , 2009 ͑17͒ Np p 1+3ͭ͑2/͑E͒͑maxEmax/E/0E͒ 0/16,+ E0/E2max͒, EEmaxmax/E/E0 0Ӷӷ11,ͮ 0 p p ͩ ͒␥2 ␥2 ͪ p 1990bondNp =͒ kind. Inp with particular, argument ! the=1 electric−͑1+max͒ fieldor exhibits the char- ͫ ͑1+ ͒2 1/2 ͬ acteristic2/ “sawtooth” E /E E+ E profile/EE , E associated/E ӷE1, E withӶ wave steep- ten as ͑Berezhiani andn/n = Murusidze2␥  Rev.1 − Mod. 1992 Phys.,p Vol.; 81,Esarey,Ќ No. 3,− July–September Ting,, 2009et 17 ͭ͑ ͒͑ 1+3max ͑0 max0/ 02max͒ /16,͒ max 0 maxͮ/ 0 1 0 p ͩ ␥ ͪ p ͑ ͒ = al. et al.p ͫ 2 ͑1+ ␥͒2 1/2 ͬ acteristicNp p 2/ “sawtooth” E /E + E profile/E , E associated/E ӷ 1, with wave steep- , 1993; Teychennéu = ␥ 1+, 1993 ͒− p1 − 2 Ќ , 18 ͭ͑ ͒͑1+3maxE 0 /E 0 /16,2max͒ maxE 0 /E ͮӶ 1 al., 1993; Teychenné zet al.2p͑ , 1993ͩ͒ p ␥ 2 ␥2 ͪ 21/2 ͑ ͒ ening and= the density͑ max 0͒ oscillationsmax become0 highly peaked ͫ ͒ ͑1+ ͒ Ќ ͬ Np2/ p E /EE+ E /EE , E /E ӷE1,͑25E͒ Ӷ u = ␥2 1+  − p1 − , 18 ͭ͑ ͒͑ 1+3max ͑0 max0/ 02max͒ /16,͒ max 0 maxͮ/ 0 1 z p͑ ͒ p ͩ ␥ ͪ ͑ ͒ acteristic “sawtooth” profile associated with wave steep- ͫ 2␥͑1+ ͒ ͬ = al., 1993; Teychenné2 uz =et␥2p al.͑1+, 1993͒ p2͒− 1 − −p1/2Ќ 22 , 1/2 ͑18͒ ening andNp thep 1+3ͭ͑ density2/͑E͒͑maxEmax/E/ oscillations0E͒ 0/16,+ E0/Emax͒, becomeEEmaxmax͑25/E/͒E0 0Ӷӷ1 highly1,ͮ peaked ͫ ͩ ␥2 ͑1+͒2 ͪ ͬ ening and= the density oscillations become25 highly peaked uz = ␥p͑1+2͒ p − ͩ1 − ␥ p ␥2Ќͪ , 1/2 ͑18͒ as illustratedNp p ͑2/ in͒͑Emax Fig./E0 8+ Ein0/E Sec.max͒, III.AEmax/E͑0.ӷ Furthermore,͒ 1, the ͮ ͒ ͭ ͫ ␥ 2 ͑1+ ͒ ͬ ͑ ץ −2 2 2 u = ␥ 1+2Ќ − −1p1/2− ␥2Ќ 1/2 , 18eningwhere E andNp =is thep ͭ peak͑ density2/ electric͒͑Emax/ oscillationsE field0 + E of0/ theEmax plasma͒, becomeEmax͑ wave25/͒E0 ӷ highly1,ͮ peaked 2 ͑ ͒ ͩ ␥ ͪ ͑ ͒ max z ͫ 2 p −1/2͑1+ ͒ ͬ 2 p p 2 ␥ = ␥  u 1= ␥2− 1+␥Ќ  − 1 − −22Ќ1 1/2,21/2 , 16 18where͑as illustratedEmax is theͭ peak͑2/ in electric͒͑E Fig.max/E field80 +inE of0 Sec./ theEmax plasma͒III.A, Emax wave͒./E Furthermore,0 ӷ 1,͑ͮ25͒ the ץ k −2 2 p z p͑ ͒ p ͩ ␥ ͪ ͑ ͑͒ ͒ as illustrated in Fig. 8 in Sec. III.A . Furthermore, the p p ␥ ͫ 2␥͑1+ ͒ ͬ ͑ ͒ u = ␥ 1+22Ќ − 2−11/2− p Ќ 2 , 18 whereperiodEmax is of the the peak nonlinear electric field of plasma the plasma wave wave increases͑25͒ as the 2 ץ −k2 2 = ␥2  z 1 2−2p͑ ͒ ͫ p ͩ ␥2␥−2͑Ќ1+1 ,1/2͒2 ͪ ͬ 16 ͑ ͒and Ep =2/kp =2c/p. For a square laser pulse profile, 25 p up ͫ= ␥ ͑1+␥␥͑͒1+ − ͒1ͬ− ␥p2 1/2 , ͑ ͒͑18͒whereand͑as illustrated=2max/isk the=2 peakc/ in. electric For Fig. a square field8 in of laser Sec. the pulse plasmaIII.A profile, wave͒. Furthermore,͑ ͒ theץ ץp 2 ␥z ␥ p 2 p 2ͩ ␥ Ќ ͪ p p p k = ␥ ͭ 1= −2 ͑1+ p͒Ќ1ͫ− 1 − 2 −͑1+1 ͮ͒, . ͬ 16͑19͒ period of the nonlinear plasma wave increases as the −2 2 p p p 2 ␥p 2 ͑ ͒ k c ͑25͒ 2 p ␥ = 2␥ ͑1+␥2͒1+1ͫ−  2ͩ1 − ␥ 2͑Ќ1+2͒1/2ͪ . ͬ ͑19͒ andwithperiodwherep optimal=2 / E ofpmax=2 length theis the/ fornonlinearp. peak For plasma a electric square wave plasma laser field excitation ofpulse the wave profile, plasma͑L increases wave as theץ p ͫ ͑ p͒ ͬ p ͩ ␥ ͪ ␥  ͫ 2␥p͑1+ ͒ ͬ andamplitude =2/k =2c increases./ . For a square The laser nonlinear pulse profile, plasma wavelength k = ͭ 1 − p − 1 ͮ2, 16 p p␥ͫ= ␥2p͑1+␥͑͒ 1+1 − p͒1ͬ− p Ќ 2 . 1/2 ͑ ͑19͒ ͒ withwherep optimalEpmax lengthis the forp peak plasma electric wave field excitation of the plasma͑L waveץ p 2 2 2ͩ ␥2 ͪ 2 2 −1/2 ͭ p ͫ ͑1+ ͒ͮ2 ͬ period of the nonlinear plasma wave increases as the ␥ = ␥p͑1+͒ 1 − p ͩ1 − ␥ p ␥2 ͪ . 1/2 ͑19͒ withӍamplitude whereand optimal/2͒,EE=2max length/ increases.E/isk= the͑=22a for/2 peakc͒͑/1+ plasma2a. The electric For/2−͒1/2 a wave nonlinear squarefor field a excitation of linearly laser the pulse plasma plasma po-͑L profile, wave wavelength ͫ 2 ͑1+ ͒ Ќ ͬ Np pmax 0 p p 1+͒ ͬ 2p −1/2 2 with optimal length for plasma wave excitation L͑ 22␥ ͫ 2 2 ץ ͭ p ͮ2 1/2 amplitudewhere/2 , EEmax/Eis= increases. thea /22 peak1+a electric/22 The−1/2 fieldfor nonlinear a of linearly the plasma po-͑ plasma wave wavelength 2 ͩ ␥ ␥ ͪ Ӎ Npand͒ =2/k ͑=2c͒͑/ . For͒ a square laser pulse profile, ͫ ͑1+ ͒ ͬ pmax 0 p␥ ӷ p ␥ = ␥2 1+ 1 −  1 − Ќ . 19 in the limit 1 is ͑Bulanov et al., 1989; Berezhiani and where ␥2Ќ =1+uIn2Ќ the=1+ limita2p͑␥,2 ӷ␥1,p͒= Eq.͑1͑−16p͒2psimplifiesp−͒1/22 ␥,2 and to ͑2Bulanov1/2p =vetp / al.c,.͑ Ӎ͒larizedNpand/2 laser.͒,Epmax=2/E/0k=pp͑=22a /2c͒͑/1+p2a. For/2−͒1/2 a squarefor a linearly laser pulse po- profile, 2 2 ␥ = 2␥p͑1+p ͒ 1 − p 2ͩ1 − ␥ Ќ ͪ . ͑19͒ amplitudein the/2with, E limit optimal/E increases.␥= aӷ length/21 is1+aBulanov for The/2 plasma nonlinearforet a wave linearly al., 1989 excitation po-plasma; Berezhiani͑L wavelength and In the limit2 ␥2 ӷ1, Eq.ͫ 16 simplifies−1/22␥͑1+ to ͒Bulanovͬ et al., ӍlarizedNpand laser.͒ max=2/0k ͑=2c͒͑/ ͑. For͒ a square laser pulse profile, where ␥ =1+u =1+a , ␥ = 1−͑ ͒ , and͑2  = /c. p Ќ InЌ the␥ limit= ␥2p͑␥1+pӷ1,p͒ Eq.1͑− 16p ͩ1psimplifies͒− ␥2 p Ќ to Bulanovͪ p . vetp al., ͑19͒larizedinwith the laser.p optimal limit p␥ lengthӷ1 is forp2 ͑Bulanov plasma2 wave−1/2et al. excitation, 1989;͑BerezhianiL and where ␥ =1+u 1989=1+; Berezhiania ,2 ␥ = andͫ1−͑ Murusidze,͒ ,͑1+ 1990 and;͒͑2Sprangleͬ=vet/ al.c., The lengtheningp of the plasma wave period can be pand͑19͒ Murusidze,withӍ optimal/2͒, E 1990 length/E =;͑2a Sprangle for/2͒͑1+ plasma2a /2−͒1/2et wavefor al. a excitation, linearly 1990a po-, ͑L1990b͒ Npץ/.pץThe axial2Ќ electric2Ќ ␥ field= ␥2p͑1+ ofp thep͒ 1͑ wake− p 2ͩ1p−−͒ is1/2␥Ep =−Eͪ In the limit ␥ ӷ1, Eq.ͫ ͑16͒ simplifies2 ͑1+ to ͑͒Bulanovͬ et al., larized laser.Np max 0 1989; Berezhiani and Murusidze,pz 1990; 2Sprangle0 et al., The lengthening␥ ofӷ the plasma wave period can be where ␥ =1+u =1+a , p␥ =2 ͑1−ͩ ͒ ␥ , and ͪ =v /c. inMurusidze,with theӍNp limit optimal/2͒, E p 1990 length/E 1=;͑ isaSprangle for͑/22Bulanov͒͑1+ plasmaa /22 ͒ et waveet−1/2for al. al., a, excitation 1989 linearly 1990a; Berezhiani po-, ͑L1990b and Ќ Ќ1989; Berezhianip andͫ Murusidze,p ͑1+ 1990;͒ Spranglepͬ pet al., The lengtheningmax of0 the plasma wave period can be ͒ andet al.important, Murusidze,ӍlarizedNp in/2 laser.͒ plasma-based, E 1990/E =;͑2a accelerators.1/2Sprangle/2͒͑1+2a /2 For−͒1/2et example,for al. a, linearly 1990a in po-, 1990b/2 Np ץto /Bulanov ץThe axial electric1990aIn field the, 1990b limit of͒ the␥ ӷ wake1, Eq. is16Esimplifiespz =−E0 etand al., importantThe=Ӎ͑2/ lengthening in/2͒͒ plasma-based, E͑1+max of/E the=0 ͑a accelerators. plasma͒/2͒͑E1+ wave͑a!/2͒ For,͒ period example,for where can a linearly in be E po-͑!͒=͐ ͒d͑Np1ץ/͑ץThethe axial plasma electric fluid19891990a field quantities; Berezhiani, 1990b of͒ the2 andp are wake Murusidze, is͑ E͒ z 1990=−;ESprangle0 andet al.,importantMurusidze,larizedNp in laser. plasma-basedp max 19900max; 1/2Sprangle accelerators.2 Foret example, al., 1990a in , 2 1990b/20͒ Npץto͑/Bulanov ץ The axial electric1990a fieldIn, the1990b of limit͒ the␥2 p ӷ wake1, Eq. is͑16E͒ simplifies=−E the plasma fluid1990aIn quantities1989 the, 1990b; limitBerezhiani͒ ␥2p ӷ are1, Eq. and͑ Murusidze,16͒ simplifiesz 19900 to;͑BulanovSprangle etet al. al.important,,=͑2/larized2The in͒2 plasma-basedp laser. lengthening͑1+1/2max͒ accelerators. of1/2E the2͑! plasma͒, For wherewave example, period in E can2͑! be͒=͐0 d/2͑1 the plasma fluidIn quantities1989 the; limitBerezhiani␥p ӷ are1, Eq. and͑ Murusidze,16͒ simplifies 1990 to;͑BulanovSprangle etet al. al.,, =−larized͑!2/Thesin͒ laser. lengtheningp͑1+ismax the of1/2͒ the completeE plasma2͑!͒, wave elliptic where period integral canE be2͑! of͒= the͐/20 sec-d͑1 the plasma fluid quantities1989Rev.1990a Mod.; Berezhiani, Phys.,1990b Vol. are͒2 81, and No. 3, Murusidze,July–September−1/2 2009 1990; Sprangle et al.,=͑2/2importantThe͒2 lengthening͑1/21+͒ in plasma-based͒ ofE the͑! accelerators. plasma͒, where wave For period example,E can͑! in be͒=͐ d͑1 1989Rev.1990a Mod.; Berezhiani, Phys.,1990b Vol.␥͒ 81, and No. 3, Murusidze, July–September 2009 1990; Sprangle et al., −!importantThesin2 lengthening2p͒ in1/2 plasma-basedismax the of complete the2 accelerators. plasma2 elliptic wave For period example, integral can2− in be2 of the0 sec- 2 Rev.1990a Mod., Phys.,1990b Vol.͒2 81,Ќ No. 3, July–September−1/2 2009 −ond!2importantsin kind2 1/2͒ with in plasma-basedis argument the complete accelerators.! =1 elliptic−͑1+ For example, integral͒ inor of the sec- n/n = ␥ Rev.1990a Mod.1, Phys.,−1990b Vol.␥͒2 81,Ќ No. 3, July–September−1/2 −  2009, 17 −!importantsin ͒ in plasma-basedis the complete accelerators.2 elliptic For example, integralmax−2 in of the sec- 0 2p p 2␥2Ќ 2−1/2 p ͑ ͒ ond kind with argument ! =12 −͑1+max͒ −or2 n/n0 = ␥2 p Rev.ͩ1 − Mod.␥ Phys., Vol. 81, No.ͪ 3, July–September− p , 2009 ͑17͒ ond kind with argument 2! =1−͑1+max−2͒ or p ͫ 2␥͑1+ ͒ ͬ n/n = ␥  1 − p Ќ 2 −  , 17 2 0 2p p ͫͩRev. Mod.␥2 Phys.,͑1+ Vol. 81,͒2 No.ͪ 3, July–Septemberpͬ 2009 ͑ ͒ ond kind with argumentE !E=1−͑1+max͒ Eor E Ӷ n/n0 = ␥ p Rev.ͩ1 − Mod.␥ Phys.,p Vol. 81, No.ͪ 3, July–September− p , 2009 ͑17͒ 1+3͑ max/ 02͒ /16, max/ 0 1 p ͫ 2 ͑1+ ͒ ͬ ͩRev. Mod.␥ Phys.,p Vol. 81,2 No.ͪ 3, July–September 2009 = 1+3͑Emax/E0͒ /16,2 Emax/E0 Ӷ 1 ͫ ͑1+ ͒ ͬ Np p E E E E Ӷ p 1+3͑ max/ 02͒ /16, max/ 0 1 2 1/2 Np = p 1+3ͭ͑2/͑E͒͑maxEmax/E/0E͒ 0/16,+ E0/Emax͒, EEmaxmax/E/E0 0Ӷӷ11,ͮ ␥ Np = p 2 2Ќ 1/2 ͭ͑2/͒͑Emax/E0 + E0/Emax͒, Emax/E0 ӷ 1,ͮ u = ␥ 1+  − 1 − ␥2Ќ 1/2 , 18 Np = p ͭ͑2/͒͑Emax/E0 + E0/Emax͒, Emax/E0 ӷ 1,ͮ z 2 ͑ ͒ p ͑ ͒ u = ␥2 p 1+  − 1 − 2 ␥2Ќ 21/2 , 18 ͭ͑2/͒͑Emax/E0 + E0/Emax͒, Emax/E0 ӷ 1,͑ͮ25͒ z p͑ ͒ p ͩ ␥ ͪ ͑ ͒ ͫ 2␥͑1+ ͒ ͬ u = ␥ 1+  − 1 − p Ќ 2 , 18 ͑25͒ z 2p͑ ͒ ͫ p ͩ ␥2 ͑1+͒2 ͪ ͬ ͑ ͒ 25 uz = ␥p͑1+͒ p − ͩ1 − ␥ p ͪ , ͑18͒ ͑ ͒ ͫ 2 ͑1+ ͒ ͬ p 2 ͑25͒ ͫ ͩ ␥ ͑1+͒ ͪ ͬ where Emax is the peak electric field of the plasma wave p 2 1/2 wherewhereEEmax isis the the peak peak electric electric field field of of the the plasma plasma wave wave 2 ␥2Ќ 1/2 whereand Ep =2maxis/k thep =2 peakc/ electricp. For a field square of laser the plasma pulse profile, wave ␥ = ␥ 1+ 1 −  1 − ␥2Ќ 1/2 . 19 and p =2max/kp =2c/p. For a square laser pulse profile, 2p͑ ͒ p 2 ␥2 21/2 ͑ ͒ and p =2/kp =2c/p. For a square laser pulse profile, ␥ = 2␥p͑1+͒ 1 − p ͩ1 − ␥ Ќ ͪ . ͑19͒ with optimal length for plasma wave excitation ͑L ͫ 2␥͑1+ ͒ ͬ and =2/k =2c/ . For a square laser pulse profile, 2 ␥ = ␥2p͑1+͒ 1 − p ͩ1 − ␥2 p Ќ ͪ . ͑19͒ withp optimalp length forp2 plasma2 wave−1/2 excitation ͑L ͫ ͑1+ ͒2 ͬ ␥ = ␥p͑1+͒ 1 − p ͩ1 − ␥ p ͪ . ͑19͒ withӍ optimal/2͒, E length/E =͑2a for/2͒͑1+ plasma2a /2−͒1/2 wavefor a excitation linearly po-͑L ͫ 2 ͑1+ ͒ ͬ Np max 0 p 2 2 ͩ ␥ ͪ withӍNp optimal/2͒, E length/E =͑a for/22 ͒͑1+ plasmaa /22 ͒ wave−1/2for a excitation linearly po-͑L ͫ ͑1+ ͒ ͬ max 0 p In the limit ␥2p ӷ1, Eq. ͑16͒ simplifies to ͑Bulanov et al., ӍlarizedNp/2 laser.͒, Emax/E0 =͑2a /2͒͑1+2a /2−͒1/2 for a linearly po- In the limit ␥2 ӷ1, Eq. ͑16͒ simplifies to ͑Bulanov et al., ӍlarizedNp/2 laser.͒, Emax/E0 =͑a /2͒͑1+a /2͒ for a linearly po- In1989 the; limitBerezhiani␥2 pӷ1, Eq. and͑ Murusidze,16͒ simplifies 1990 to;͑BulanovSprangle etet al. al.,, larizedThe laser. lengthening of the plasma wave period can be In1989 the; limitBerezhiani␥ pӷ1, Eq. and͑ Murusidze,16͒ simplifies 1990 to;͑BulanovSprangle etet al. al.,, larizedThe laser. lengthening of the plasma wave period can be 19891990a; Berezhiani, 1990bp and Murusidze, 1990; Sprangle et al., importantThe lengthening in plasma-based of the accelerators. plasma wave For period example, can in be 19891990a; Berezhiani, 1990b ͒ and Murusidze, 1990; Sprangle et al., importantThe lengthening in plasma-based of the accelerators. plasma wave For period example, can in be 1990a, 1990b͒ ͒ important in plasma-based accelerators. For example, in 1990a, 1990b͒ important in plasma-based accelerators. For example, in Rev. Mod. Phys., Vol. 81, No. 3, July–September 2009 Rev. Mod. Phys., Vol. 81, No. 3, July–September 2009 Rev. Mod. Phys., Vol. 81, No. 3, July–September 2009 Rev. Mod. Phys., Vol. 81, No. 3, July–September 2009 Strong Optical Shock excitation in the mismatched regime of bubble plasma-wave based Strong Optical Shock excitation in the mismatched regime of bubble plasma-wave based LWFA LWFA A. A. Sahai1 1 A. A. Sahai1 Department of Physics and John Adams Institute for Accelerator Science, Blackett Laboratory, Imperial College London, SW7 2AZ, United Kingdom⇤ 1Department of Physics and John Adams Institute for Accelerator Science, Blackett Laboratory, Imperial College London, SW7 2AZ, United Kingdom⇤ A strongly mismatched regime of laser-plasma acceleration using a self-guided laser-driven bubble- shaped nonlinear plasma electron wave is introduced and modeled. In this regime the radial envelope A strongly mismatched regime of laser-plasma acceleration using a self-guided laser-drivenof a laser-pulse bubble- incident at the plasma entrance is mismatched to the nonlinear electron response shaped nonlinear plasma electron wave is introduced and modeled. In this regime theexcited radial envelope by it, in contrast to the established understanding. A nonlinear laser envelope equation is of a laser-pulse incident at the plasma entrance is mismatched to the nonlinear electronderived response to show that as the strength of the mismatch is increased, the envelope oscillations steepen and excited by it, in contrast to the established understanding. A nonlinear laser envelopebecome equation increasingly is asymmetric, exhibiting shorter and tighter radial squeeze phases. The sharply derived to show that as the strength of the mismatch is increased, the envelope oscillationsincreasing steepen intensity and in a shortened squeeze phase results in the slicing of the longitudinal laser become increasingly asymmetric, exhibiting shorter and tighter radial squeeze phases.envelope, The sharply driving a strong optical-shock. The optical shock results in an elongating bubble shape with increasing intensity in a shortened squeeze phase results in the slicing of the longitudinalsignificantly laser higher peak plasma fields and a novel self-injection mechanism which produces beams envelope, driving a strong optical-shock. The optical shock results in an elongating bubbleof high shape transverse with qualities. The behavior of peak beam energies from Particle-In-Cell simulations and significantly higher peak plasma fields and a novel self-injection mechanism which producesself-guided beams multi-GeV experimental data are in good agreement with the predictions of an adjusted-a0 of high transverse qualities. The behavior of peak beam energies from Particle-In-Cellmodel simulations and significantly and exceed the matched regime predictions. self-guided multi-GeV experimental data are in good agreement with the predictions of an adjusted-a0 model and significantly exceed the matched regime predictions. 18 2 19 3 I. INTRODUCTION was 2.5 10 Wcm (a0=1.1) at n0 = 2 10 cm , the matched⇥ spot-size is 3µm whereas11 the⇥ launched spot- 18 + 2 19 3 ' I. INTRODUCTION was 2.5 10 Wcm (a0=1.1) at n0 = 2 10 cm , the size was 12µm (FWHM 20µm). The predicted en- PIC-based⇥ – eThe laseracceleration strength parameter⇥ in eq.1 ,results is its value in vac- ' ' matched spot-sizeuum is as3 itµm is launchedwhereas the or incident launched at spot- the plasma entrance. ergy from eq.1 is 40 MeV but experiments obtained a size was 12µm (FWHM' 20µm). The predicted en- The laser strength parameter in eq.1, is its value in vac- In plasma, the value of a0 is known to significantly vary spectral peak at 70 MeV. Similarly in [4], the intensity ' ' 18 2 18 3 uum as it is launched or incident at the plasma entrance. ergy from eq.1 isover 40 MeVthe acceleration but experiments length obtained due to several a non-linear was 3.2 10 Wcm (a0=1.3) at n0 = 6 10 cm , the spectral peak at 70 MeV. Similarly in [4], the intensity ⇥ ⇥ In plasma, the value of a0 is known to significantly vary laser-plasma interactions e↵ects such as the localized matched spot-size is 5µm whereas the launched spot- 18 2 18 3 ' over the acceleration length due to several non-linear was 3.2 10 Wcm (a0=1.3) at n0 = 6 10 cm , the size of 12.5µm (FWHM 21µm). The expected beam ⇥ variation of the wavelength⇥ profile, group velocity pro- ' laser-plasma interactions e↵ects such as the localized matched spot-sizefile is and5µ pumpm whereas depletion the launched of the laser spot- pulse. Thus, this energy is 155 MeV but the spectral peak was at 175 MeV. size of 12.5µm (FWHM' 21µm). The expected beam variation of the wavelength profile, group velocity pro- equation' best models a scenario where a0 is relatively The analysis in our work gains additional signifi- file and pump depletion of the laser pulse. Thus, this energy is 155 MeVconstant but the over spectral the acceleration peak was at length,175 MeV. as is argued to be cance due to its relevance to many other self-guiding equation best models a scenario where a0 is relatively The analysisthe in our case work under gains a matched additional spot-size. signifi- results. Some examples are, Austin-2 GeV data [16]: 17 3 constant over the acceleration length, as is argued to be cance due to its relevance to many other self-guiding W = 275µm, a = 0.6, n = 5 10 cm matched-w 0 0 0 ⇥ 0 ' the case under a matched spot-size. results. Some examples are, Austin-2 GeV data [16]: 12µm; Nebraska-0.3 GeV data [17]: W0 = 17µm, a0 = 2.2, Energy Spectrum17 3 dN / [d / ] (arb. units) 18 3 W0 = 275µm, a0 = 0.6, n0 = 5 10 cm matched-wE E0 n0 = 2.5 10 cm matched-w0 10µm; Gwangju-3 ⇥ ' ⇥ ' 17 3 12µm; Nebraska-0.3 GeV data [17]: W0 = 17µm, a0 = 2.2, GeV data [18]: W0 = 25.5µm, a0 = 5.0, n0 = 5 10 cm Energy Spectrum dN / [d / ] (arb. units) 18 3 ⇥ E E n0 = 2.5 10 cm matched-w0 10µm; Gwangju-3 matched-w0 20µm and Strathclyde-125MeV data [19]: ⇥ ' E = 5.176% 3 ' 19 3 GeV data [18]: W = 25.5µm, a = 5.0, n = 5 10 cm W0 = 20µm, a0 = 1.5, n0 = 1 10 cm matched- 0 0 0 E ⇥ (1) ⇥ matched-w0 20µm and Strathclyde-125MeV data [19]: w0 5µm. E = 5.6% ' 19 E3= 6.3% ' W0 = 20µm, a0 = 1.5, n0 = 1 10 cm matched- A non-uniform laser focal-spot (a large 2-number) E (1) ⇥ E M w0 5µm. a↵ects the transverse characteristics of the plasma wave E = 6.3% ' E = 4.1% A non-uniform laser focal-spot (a large 2-number) [20], leading to non-optimal acceleration and focusing E E a↵ects the transverse characteristics of the plasmaM wave field profiles apart from faster laser energy loss due to E = 4.1% It is important to note that in the GeV-scale energy gain the tendency for faster di↵raction of the higher modes. [20], leading toexperiments non-optimal that acceleration our analysis and is focusing based on [14, 15], dur- E A reduction in experimental artifacts for a larger than field profiles aparting from fixed faster incident laser spot-size energy density loss due scans to the maximum It is important to note that in the GeV-scale energy gain the tendency for faster di↵raction of the higher modes. matched spot-size focal spots would lead to higher qual- experiments that our analysis is based on [14, 15], dur- beam energy gains are observed within a density range ity of the plasma wave. The quality of the plasma wave A reduction in experimentalfor which the artifacts interaction for is a largerin the mismatchedthan regime. ing fixed incident spot-size density scans the maximum matched spot-size focal spots would lead to higher qual- has been indirectly inferred using the laser focal profile beam energy gains are observed within a density range We show below that at these higher intensities, the mis- at the plasma entrance [16] and exit [15]. The presence ity of the plasmamatch wave. exaggerates The quality the of the di↵ plasmaerence between wave eq.1 and the for which the interaction is in the mismatched regime. has been indirectly inferred using the laser focal profile of multiple hot-spots in the incident focal spot may be We show below that at these higher intensities, the mis- experimental data. inferred by observing the laser exit mode, as seen in at the plasma entranceIt is also [16] important and exit [ to15]. note The the presence pertinence of this work match exaggerates the di↵erence between eq.1 and the of multiple hot-spots in the incident focal spot may be Fig.3(c)-(h) in [15]. experimental data. on mismatched regime to the first results on self-guided inferred by observinglaser-driven the laser plasma-based exit mode, quasi-mono-energetic as seen in elec- We show using a nonlinear envelope equation of a It is also important to note the pertinence of this work18 -3 n = 10 cmFig.3(c)-(h), E in~ [15 tron5]. -10J, acceleration. w In= [340um,], the incident a laser~ intensity1.5 self-guided laser driving a bubble plasma wave that the on mismatched regime to the first results0 on self-guided L 0 0 oscillations of the spot-size become increasingly asym- We show using a nonlinear envelope equation of a laser-driven plasma-based quasi-mono-energetic elec- metric in response to an increasing degree of mismatch, self-guided laser driving a bubble plasma wave that the tron acceleration. In [3], the incident laser intensity with shorter (and tighter) “squeeze” phases and longer oscillations of the spot-size become increasingly asym- ⇤ corresponding author: [email protected] “relaxation” phases of the laser spot-size. This behavior metric in response to an increasing degree of mismatch, with shorter (and tighter) “squeeze” phases and longer ⇤ corresponding author: [email protected] “relaxation” phases of the laser spot-size. This behavior
Dr. Aakash A. Sahai 735 Blackett Laboratory Department of Physics & John Adams Institute for Accelerator Science are transversely lost as they interact with corresponding defocusing regions of the plasma wave.
Electrons are lost in the electron compression region of the plasma wave, where positrons are focused. Positrons are lost in the ion cavity region of the plasma wave, where electrons are focused.
This is reason why the electrons and positrons get separated out.
The p1p2 and p1x2 phase-spaces of this process of the separation of the shower electrons and shower positrons are now presented using time evolution of the transverse momentum and transverse displacement presented as movies in the Supplementary Materials.
However, following this valid criticism from the referee a comment is now added in the manuscript to explicitly state this point - The wave focusing fields longitudinally segregate the e+-beam from electrons (see Supplementary Materials).
For the convenience of the referee a snapshot of the p1p2 phase space is presented in the figure below. This shows the transverse dispersion of the accelerated particles (once again the full movie of both the p1p2 phase space and the p1p2 phase space are in the 12 + Supplementarye Material)–LPA - PIC-based beam phase-space
shower-e+ shower-e-
plasma-e-
Page 22 of 33
13 Tunability - PIC-based parameter scan
critical to understand shower-wave interactions
tuning of
e+-beam spectral characteristics
with laser and plasma properties 14 PIC-based - positron beam characteristics
(a) (b) ultra-short positron bunch
~ 107-8 - e+ / bunch
(c) (d) long. dim ~ 5 – 7.5 μm
tran. dim ~ 5 – 7.5 μm
open. angle ~ 5 – 10 mrad 15 Primary Challenges
Experimental challenges
§ characterize ultrashort positron-electron shower produced by BNL-ATF beam
§ control the interaction of shower and wave (coupling the laser)
Physics challenges
§ Extending trapped charge – from shower to the beam
§ Cooling the positron beam etc. – shower particles are divergent
Technological challenges
§ Channeling undulators – couple the beam into sample
§ Annihilation spectroscopy etc. - can the beam help in material science 16 Proposed Milestones
Yr. 1 – demonstration of positron-electron jet production in metal target, its characterization over the sub-ps electron beam parameter-space (spot-size, charge, current) and its interaction with laser-ionized plasma
Yr. 2 – demonstration of coupling high-power CO2 laser pulse within the plasma-cell simultaneously with positron-electron jets
Yr. 3 – demonstration of tuning of the characteristics of the positron beam by scanning over electron beam, CO2 laser and plasma properties. 17 Conclusions
§ BNL-ATF facility is uniquely poised for the first demonstration of laser- driven tunable positron beam
§ Applications of ultrashort positron beams can benefit atomic-scale material characterization.
§ Collaboration between CUD, Tech-X, UMich, UCI, Fermilab, LLNL, LANL has been setup. Each collaborator brings a unique set of capabilities. Electron Beam Requirements Parameter Units Typical Values Comments Requested Values Beam Energy MeV 50-65 Full range is ~15-75 MeV with highest beam quality at nominal values 60-80
Bunch Charge nC 0.1-2.0 Bunch length & emittance vary with charge 0.1-2.0 Compression fs Down to 100 fs A magnetic bunch compressor available to compress bunch down to 100-250, 500 (up to 1 kA peak ~100 fs. Beam quality is variable depending on charge and amount of current) compression required.
NOTE: Further compression options are being developed to provide bunch lengths down to the ~10 fs level Transverse size at IP (s) µm 30 – 100 It is possible to achieve transverse sizes below 10 um with special 5-100 (dependent on IP permanent magnet optics. position) Normalized Emittance µm 1 (at 0.3 nC) Variable with bunch charge 1-3 Rep. Rate (Hz) Hz 1.5 3 Hz also available if needed 1.5 Trains mode --- Single bunch Multi-bunch mode available. Trains of 24 or 48 ns spaced bunches. Single bunch CO2 Laser Requirements Configuration Parameter Units Typical Values Comments Requested Values
CO2 Regenerative Amplifier Beam Wavelength µm 9.2 Wavelength determined by mixed isotope gain media Peak Power GW ~3 Pulse Mode --- Single Pulse Length ps 2 Pulse Energy mJ 6 M2 --- ~1.5 Repetition Rate Hz 1.5 3 Hz also available if needed Polarization --- Linear Circular polarization available at slightly reduced power
CO2 CPA Beam Wavelength µm 9.2 Wavelength determined by mixed isotope gain media 9.2 Note that delivery of full power Peak Power TW 5-10 ~5 TW operation is planned for FY21 (requires further 0.1-10 pulses to the Experimental Hall is in-vacuum transport upgrade). A 3-year development presently limited to Beamline #1 effort to achieve >10 TW and deliver to users is in only. progress. Pulse Mode --- Single Pulse Length ps < 2 2-0.5 Pulse Energy J ~5 Maximum pulse energies of >10 J will become available 0.1-2.5J (y1, y2), in FY20 5J (y3) M2 --- ~2 2 Repetition Rate Hz 0.05 0.05 Polarization Linear Adjustable linear polarization along with circular linear polarization will become available in FY20 Other Experimental Laser Requirements Stage I Stage II Ti:Sapphire Laser System Units Values Values Comments Requested Values Stage I parameters should be achieved by mid-2020, while Stage II Central Wavelength nm 800 800 parameters are planned for late-2020. FWHM Bandwidth nm 20 13 Transport of compressed pulses will initially include a very limited Compressed FWHM Pulse fs <50 <75 number of experimental interaction points. Please consult with the Width ATF Team if you need this capability. Chirped FWHM Pulse Width ps ³50 ³50 Chirped Energy mJ 10 200 Compressed Energy mJ 7 100 Energy to Experiments mJ >4.9 >80 Power to Experiments GW >98 >1067
Nd:YAG Laser System Units Typical Values Comments Requested Values Wavelength nm 1064 Single pulse Energy mJ 5 Pulse Width ps 14 Wavelength nm 532 Frequency doubled Energy mJ 0.5 Pulse Width ps 10 Special Equipment Requirements and Hazards
• Electron Beam • Plasma capillary discharge system – laser • Transverse deflecting cavity • Permanent magnet quadrupole • Stark-line shift measurement setup (plasma density vs. gas pressure) • Mask for beam splitting (beam-driven active plasma beam dump)
• CO2 Laser • Mirror with hole delivery (5 J, sync. with e-beam) into the capillary / gas-jet • Tape reflector delivery (5 J, sync. with e-beam) into the capillary / gas-jet
• Superconducting Magnetic field • Can a superconducting magnet be setup on the beamline ? Experimental Time Request CY2020 Time Request Capability Setup Hours Running Hours Electron Beam Only Laser* Only (in FEL Room) Laser* + Electron Beam 24 120 can run in time-shared mode with plasma beam dump experiment
Time Estimate for Full 3-year Experiment (including CY2020) Capability Setup Hours Running Hours Electron Beam Only Laser* Only (in FEL Room) Laser* + Electron Beam 72 360
* Laser = Near-IR or LWIR (CO2) Laser Diagnostics
Laser and Particle Beams Figure 8: positron spectrometer & MeV bremsstrahlung X rays from intense laser gamma-ray diagnostics interaction with solid foils cambridge.org/lpb Laser and Particle Beams 505 S. Palaniyappan1, D. C. Gautier1, B. J. Tobias1, J. C. Fernandez1, J. Mendez1, T. Burris-Mog1, C. K. Huang1, A. Favalli1, J. F. Hunter1, M. E. Espy1, D. W. Schmidt1, Research Article R. O. Nelson1, A. Sefkow2, T. Shimada1 and R. P. Johnson1 Cite this article: Palaniyappan S et al (2018). MeV bremsstrahlung X rays from intense laser 1Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA and 2University of Rochester, New York interaction with solid foils. Laser and Particle 14627, USA Beams 36,502–506. https://doi.org/10.1017/ S0263034618000551 Abstract Received: 16 November 2018 Laser-based compact MeV X-ray sources are useful for a variety of applications such as radi- Revised: 5 December 2018 ography and active interrogation of nuclear materials. MeV X rays are typically generated by Accepted: 6 December 2018 impinging the intense laser onto ∼mm-thick high-Z foil. Here, we have characterized such a Key words: MeV X-ray source from 120 TW (80 J, 650 fs) laser interaction with a 1 mm-thick tantalum Intense laser–plasma; MeV X-ray radiography foil. Our measurements show X-ray temperature of 2.5 MeV, flux of 3 × 1012 photons/sr/shot, beam divergence of 0.1 sr, conversion efficiency of 1%, that is, 1 J of MeV X rays out of Author for correspondence: ∼ ∼ ∼ S. Palaniyappan, Los Alamos National 80 J incident laser, and source size of 80 m. Our measurement also shows that MeV X-ray −5 Laboratory, Los Alamos, NM-87545, USA, yield and temperature is largely insensitive to nanosecond laser contrasts up to 10 . Also, Fig. 5. MeV X-ray radiograph. (a) 10 cm × 10 cm × 6 mm tungsten object called “R2DTO” with radial slots used for measuring the MeV X-ray source size, (b) radio- E-mail: [email protected] preliminary measurements of similar MeV X-ray source using agraph double-foil of R2DTO scheme, taken using where MeV X rays from the double-foil scheme with 1:1 magnification, (c) radiograph of the same object using MeV X rays from the single-foil the laser-driven hot electrons from a thin foil undergoing relativisticscheme transparency with 6.2 × magnification. impinging onto a second high-Z converter foil separated by 50–400 m, show MeV X-ray yield more than an order of magnitude lower compared with the single-foil results. 1 mm-thick tantalum foil. For this setup, we added an extra We believe that better understanding of the hot electron transport 4 mm-thick tungsten filter to block X rays below 240 keV (<1% from the thin foil to the converter foil could help optimize the transmission below 240 keV). Hence, the 67 and 210 keV X-ray double-foil scheme. Also, reducing the gap between the two peaks are missing in Figure 3a–c. One-dimensional HELIOS rad- foils down to 20 µm or lower as indicated in Sefkow et al. Introduction hydro (MacFarlane et al., 2006) simulation showed that these pre- (2011) could help mitigate the hot electron transport issues. pulses produce a pre-plasma density gradient of only a few The integrated X-ray spectrum from the double-foil scheme yields Compact MeV X-ray sources are useful for several applications such as radiography and active 10 interrogation of nuclear materials (Courtois et al., 2011). Intensemicrons lasers at the can front generate surface of the target due to lower sound 3.6 × 10 photons/sr per shot. However, the yield varied by more speed of the heavier ions. Also here we use s-polarized laser than one order of magnitude from shot-to-shot. multi-MeV hot electrons when interacting with a ∼mm-thick high-Z foils such as tungsten or tantalum. In these targets, the laser couples its energy to thewhere hot electrons the laser mainlyE-field on the has no component along the pre-plasma Figure 5a shows the R2DTO object, a 10 cm × 10 cm × 6 mm target surface via several physical processes (Malka and Miquel,gradient1996; that Wilks could and Kruer,reduce the vacuum heating in the pre-plasma tungsten object with radial slots, used for quantifying the MeV 1997; Santala et al., 2000). Subsequently, the hot electrons generate(Brunel, MeV1987 bremsstrahlung). X X-ray source size from the X-ray radiograph of the object using rays as they traverse through the rest of the high-Z foil. Several experimentsFigure have 4a shows characterized the electron spectrum (dashed red line) from the Bayesian-Inference-Engine (BIE) analysis. Figure 5b shows such intense laser-driven bremsstrahlung X-ray sources (Perry theet al 110., 1999 nm; Edwards aluminumet al., foil measured using the magnetic spec- R2DTO radiograph using the X rays from the double-foil source 2002; Clarke et al., 2006; Hayashi et al., 2006; Galy et al., trometer.2007; Courtois Theet electron al., 2011; spectrum peaks around 12 MeV. The with 1:1 magnification. Figure 5c shows the same radiograph Courtois et al., 2013; La Fontaine, 2014; Liang et al., 2016; Chen etinstrument al., 2017; Yang hadet al a., 2017 lower). (higher) energy cut-off of 0.2 (28.2) using the X rays from the single-foil source with 6.2 × magnifica- Here, we have characterized such a MeV bremsstrahlung X-rayMeV. source It from is possible 120 TW (80that J, the low-energy electrons below 3 MeV tion. The images show that the edges of the radial slots are blurred 650 fs) Trident laser at the Los Alamos National Laboratory interactingdo not escape with a the 1 mm-thick target (Cobble et al., 2016). The measured elec- when using the double-foil X-ray source. The BIE analysis of these tron spectrum matches12 well with a Maxwell–Jüttner distribution f images shows that the MeV X-ray source size was 35–195 and 270– tantalum foil. Our measurements show X-ray temperature of 2.5 MeV, flux2 of 3 × 10 pho- 2 tons/sr/shot, beam divergence of ∼0.1 sr, conversion efficiency of(E∼)=(1%,E thatβ/θ is,K2∼(1/1 Jθ of))exp(-E/ MeV θ) with θ = kT/mc = 12, where β = v/c, 600 µm in the single-foil and double-foil schemes, respectively. The X rays out of 80 J incident laser, and source size of 80 µm. Ourand measurementkT is the electron also shows temperature. Figure 4b shows the typical details of the BIE analysis is discussed elsewhere (Tobias et al., that MeV X-ray yield and temperature is largely insensitive to nanosecondX-ray spectrum laser contrast retrieved up from the measured X-ray transmission 2017). The larger source size in the double-foil scheme could to 10−5. data from the double-foil targets with ∼200 µm spacing between come form the hot electron transport issues that could increase In contrast, numerical simulations have shown that a double-foilthem via scheme, Expectation-Maximization where laser- ation algorithm (Lange and the size of the electron beam reaching the converter foil. driven hot electrons from a thin foil that undergoes relativisticCarson, transparency1984; Zhang (Kaw andet al., 2007). Varying the distance between foils from 50 to 400 µm did not seem to affect the X-ray spectra Dawson, 1970; Palaniyappan et al., 2012, 2015) impinging onto a separate high-Z converter Conclusions and yield in any consistent manner. The X-ray spectrum foil, could generate more efficient Kα X rays than the single-foil scheme (Sefkow et al., 2011). The same reasoning can also be extended to MeV X-raypeaks generation. at 210 Our keV preliminary and 2.9 MeV with a temperature of 2.1 MeV. In conclusion, we have characterized MeV bremsstrahlung X-ray measurements of MeV X-ray source using the double-foil scheme,Additionally, where the the laser-driven X-ray spectrum shows a tantalum kα peak at source from 120 TW (80 J, 650 fs) Trident laser interaction with a hot electrons from a thin foil (110 nm aluminum foil) undergoing67 keV relativistic (Fig. 4c transparency). However, the 67 and 210 keV X-ray peaks have 1 mm-thick tantalum foil. Our measurements show X-ray tem- 12 (Kaw and Dawson, 1970; Palaniyappan et al., 2012, 2015; Cobblenegligibleet al., 2016 X-ray) impinging energy content compared with the broader perature of 2.5 MeV, flux of 3 × 10 photons/sr/shot, beam diver- onto a second high-Z converter foil separated by 50–400 µm, show2.9 MeV MeV peak. X-ray yieldIt is more most likely that the 210 keV X-ray peak is gence of ∼0.1 sr, conversion efficiency of ∼1%, that is, ∼1 J of than an order of magnitude lower compared with the single-foilan apparent results discussed peak above. due to significant X-ray filtering from the MeV X rays out of 80 J incident laser, and source size of We believe that a better understanding of the hot electron transport0.5 mm-thick in vacuum tantalum from the foil. 80 µm. Our measurement also shows that MeV X-ray yield and © Cambridge University Press 2019 thin foil to the converter foil could help optimize the double-foilFigure scheme. 4d Also,shows reducing the simulation results when an electron beam temperature is largely insensitive to nanosecond laser contrast 5 the gap between the two foils down to 20 µm or lower as indicatedwith in the Sefkow measuredet al.(2011 electron) spectrum in Figure 4a traverses up to 10− . Also, preliminary measurements of similar MeV could also help mitigate the hot electron transport issues. Althoughthrough understanding a 1 mm-thick and opti- tantalum converter foil using the code X-ray source using a double-foil scheme, where the laser-driven mizing the double-foil scheme could potentially yield a betterMonte MeV bremsstrahlung Carlo N-Particle X-ray (MCNP) (Forster and Godfrey, 1985). hot electrons from a thin foil undergoing relativistic transparency source, it is beyond the scope of the present work. The simulation results show a two temperature (1 and 4 MeV) impinging onto a second high-Z converter foil separated by 50– X-ray spectrum that has no obvious spectral peaks. The fact 400 µm, show MeV X-ray yield more than an order of magnitude
that the X-ray spectrum from the MCNPX simulation (Fig. 4d) lower compared with the single-foil results. Despite the interest, is much hotter than the measured X-ray distribution (Fig. 4b) further optimization and complete understanding of the double- seems to provide evidence that there may be issues in the trans- foil scheme using comprehensive Particle-In-Cell (PIC) simula- port of the hot electrons from the thin foil to the converter foil. tions is beyond the scope of the work presented here.