On the Design of Experiments for the Study of Extreme Field Limits in The

Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42

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On the design of experiments for the study of extreme ﬁeld limits in the interaction of laser with ultrarelativistic electron beam

S.V. Bulanov a,n, T.Zh. Esirkepov a, Y. Hayashi a, M. Kando a, H. Kiriyama a, J.K. Koga a, K. Kondo a, H. Kotaki a, A.S. Pirozhkov a, S.S. Bulanov b, A.G. Zhidkov c, P. Chen d, D. Neely e, Y. Kato f, N.B. Narozhny g, G. Korn h,i a Kansai Photon Science Institute, JAEA, Kizugawa, Kyoto 619-0215, Japan b University of California, Berkeley, CA 94720, USA c Osaka University, Osaka 565-0871, Japan d Leung Center for Cosmology and Particle Astrophysics of the National Taiwan University, Taipei 10617, Taiwan e Central Laser Facility, STFC, Rutherford Appleton Laboratory, Didcot OX11 0QX, United Kingdom f The Graduate School for the Creation of New Photonics Industries, Hamamatsu, Shizuoka 431-1202, Japan g Moscow Engineering Physics Institute (State University), Moscow 115409, Russia h Max-Planck-Institut fur¨ Quantenoptik, Garching 85748, Germany i ELI Beamline Facility, Institute of Physics, Czech Academy of Sciences, Prague 18221, Czech Republic article info abstract

Article history: We propose the experiments on the collision of laser light and high intensity electromagnetic pulses Received 13 January 2011 generated by relativistic flying mirrors, with electron bunches produced by a conventional accelerator Received in revised form and with laser wake field accelerated electrons for studying extreme field limits in the nonlinear 31 August 2011 interaction of electromagnetic waves. The regimes of dominant radiation reaction, which completely Accepted 14 September 2011 changes the electromagnetic wave–matter interaction, will be revealed in the laser plasma experi- Available online 22 September 2011 ments. This will result in a new powerful source of ultra short high brightness gamma-ray pulses. Keywords: A possibility of the demonstration of the electron–positron pair creation in vacuum in a multi-photon Radiation damping processes can be realized. This will allow modeling under terrestrial laboratory conditions neutron star Nonlinear Thomson and Compton magnetospheres, cosmological gamma ray bursts and the Leptonic Era of the Universe. scattering & 2011 Elsevier B.V. All rights reserved. Quantum electrodynamics Multiphoton creation of electron–positron pairs Electron–positron avalanches

1. Introduction systems with its relation to the turbulence problem and in the achievements in nonlinear wave theory, which led to the formation According to a widely accepted point of view, classical and of a novel area in mathematical physics, thus demonstrating that quantum electrodynamics represent well understood and fully com- there cannot be a fully complete area of science, as known by plete areas of science whereas one of the forefronts of fundamental prominent physists [1]. In the beginning of the 21st century there physics is in string theory, the predictions of which will hopefully appeared a demand to understand the cooperative behavior of bring experimental physics to novel, higher than ever, levels. In the relativistic quantum systems and to probe the quantum vacuum. It second half of the 20th century and in the beginning of the 21st was realized that the vacuum probing becomes possible by using century humankind witnessed enormous progress in elementary high power lasers [2]. With increase in the laser intensity, we shall particle physics with the formulation and experimental proof of the encounter novel physical processes such as the radiation reaction Standard Model and in cosmology with the observational evidence of dominated regimes and then the regime of the quantum electro- dark matter and dark energy in the Universe. At the same time dynamics (QED) processes. Near the intensity, corresponding to the classical physics continued its vigorous development, which resulted QED critical electric ﬁeld, light can generate electron–positron pairs in the understanding of the nature of chaos in simple mechanical from vacuum and the vacuum begins to act nonlinearly [3]. There are several ways to achieve the higher intensity required for revealing these processes. n Corresponding author. One of the methods is based on the simultaneous laser frequency E-mail address: [email protected] (S.V. Bulanov). upshifting and pulse compression. These two phenomena were

0168-9002/$ - see front matter & 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nima.2011.09.029 32 S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 demonstrated within the Relativistic Flying Mirror (RFM) concept, configurations the electron bunch collides with a petawatt power which uses the laser pulse compression, frequency up-shift, and laser pulse. In particular, ideally the spot size of electron beam focusing by counter-propagating breaking plasma waves—relativistic should be matched to the focused laser spot, the length matched to parabolic mirrors [4]. In the proof of the principle experiments for this the Rayleigh length, and relative timing jitter minimized. Electron concept a narrow band XUV generation was demonstrated [5] with bunches generated by the LWFA can be optimally synchronized the high photon number [6]. with the counter-propagating petawatt laser pulse. In the third Another way was demonstrated in large scale experiments [7], configuration, Fig. 1c, LFWA-generated bunches collide with an where a 50 GeV bunch of electrons from SLAC interacted with a extremely intense electromagnetic pulse produced with the Rela- counter-propagating laser pulse of the intensity of approximately tivistic Flying Mirror (RFM) technique. The RFM reflecting counter- 5 1017 W/cm2. Gamma-rays with the energy of 30 GeV propagating laser pulse upshifts its frequency and shortens its produced in the multi-photon Compton scattering then subse- duration due to the double Doppler effect. It can focus the reflected quently interacted with the laser light creating the electron– radiation to the spot much smaller than the laser wavelength due to positron pairs. The conclusion on appearance of the gamma-rays higher reflected radiation frequency [4]. As we discuss later, in with the energy of 30 GeV has been obtained by analysing the these three configurations electron bunches colliding with counter- spectra of scattered electrons and positrons. The direct evidence propagating laser pulses produce gamma-rays via nonlinear Thom- of the gamma-ray emission through the spectral measurements, son or inverse multiphoton Compton scattering. The third config- so far, is of particular importance. The direct measurement similar uration is designed for the most intense interaction, i.e., for paving a to Ref. [8] would allow the selection of the non-linear processes way towards extreme field limits in the nonlinear interaction of from the consequent linear photon scattering and, therefore, the electromagnetic waves. quantitative verification of theoretical approaches. The paper is organized as follows. In the second section we In the present paper we propose table-top experiments on the present the key dimensionless parameters characterizing the collision of electromagnetic waves with electron bunch in three extreme field limits. Then the electromagnetic field parameters configurations, Fig. 1. In the first configuration, Fig. 1a, fast electrons required for probing the nonlinear vacuum are briefly discussed. are produced by a compact microtron affording hundred MeV The fourth section contains a consideration of the electron– bunches with the duration of picosecond. In the second configura- positron gamma-ray plasma generation via the multi-photon tion, Fig. 1b, electron bunches are generated in plasma, where a Breit–Wheeler process. Section 5 is devoted to the formula- high-intensity laser pulse excites wake waves accelerating elec- tion of possible approaches towards nonlinear vacuum probing trons [9]. This laser wake field accelerator (LWFA) uses the so called and multi-photon electron–positron pair generation with self-injected electrons, which enter into the accelerating phase of present-day lasers and charged particle accelerator systems. The the wake wave due to wave-breaking. LWFA can generate GeV final section is devoted to conclusions and discussions of the electron bunches with a duration of ten femtoseconds. In both results obtained.

γ-rays gas e- − electron bunch laser pulse#1 e γ -rays electron bunch + laser electron accelerator e e+ laser pulse pulse#2 wake wave

electrons gas e− gas driver γ-rays pulse#2 driver pulse#1 FM pulse laser e+ wake wave electrons wake wave for LFWA for FM

Fig. 1. Schematic setup of the proposed experiment on the electron–positron pair generation via the Breit–Wheeler process. (a) Relativistic electron bunch collides with the tightly focused laser pulse. (b) Same as (a), but laser pulse collides with the LWFA accelerated electron bunch; in contrast to (a) the electron bunch size can be comparable with the laser focus spot size and the Rayleigh length. (c) LWFA electron bunch collides with the ultra-high intensity EM wave generated with the relativistic ﬂying mirror. (d) Electron bunch interaction with focused laser pulse. S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 33

3 2. Extreme field limits radiation, omax ¼ 0:29o0ge , is proportional to the cube of the electron energy [17]; here ge ¼a. When we consider the Physical systems obey scaling laws, which can also be pre- electron moving in the plane of null magnetic field formed sented as similarity rules. In the theory of similarity and modeling by two colliding EM waves, o0 is the wave frequency. the key role is played by dimensionless parameters, which 3. Quantum electrodynamics (QED) effects become important, when characterize the phenomena under consideration [10]. The the energy of the photon generatedbyComptonscatteringisof _ 2 dimensionless parameters that characterize the high intensity the order of the electron energy, i.e. om ¼ gemec .Anelectron 2 Electromagnetic (EM) wave interaction with matter can also be with energy gemec rotating with frequency o in a circularly _ _ 3 found in Ref. [11]. The key in the extreme field limit parameters polarized wave emits photons with energy om oge .The 2 are as follows: quantum effects come into play when ge 4gQ ¼ðmec = _oÞ1=2 600ðl=1mmÞ1=2, i.e. in terms of the EM field normalized 1. Normalized dimensionless EM wave amplitude amplitude [15] eE| 2 2 2 a ¼ ð1Þ 2e mec 2 mec 2 1 m c2 aQ ¼ ¼ a ¼ a e ð6Þ e 3_2o 3 _o 3 rad where | ¼ c=o ¼ l=2p, corresponds to the intensity I ¼ 1:37 18 ð Þ2 2 ¼ 2 10 1mm=l a W=cm for linear polarization. At a 1, the where a ¼ re=|C ¼ e =_c 1=137 is the fine structure constant 11 laser electric field E acting on the electric charge e produces and |C ¼ _=mec ¼ 3:86 10 cm is the reduced Compton wave- 2 | 23 a work equal to mec over the distance . The quiver electron length.HerewetakeintoaccountthatinthelimitI410 energy becomes relativistic. For plane EM wave this parameter ð1mm=lÞ2 W=cm2 due to strong radiation damping effects the 2 1=4 is related to the Lorentz invariant, which being expressed via electron energy scales as mec ða=eradÞ . the 4-potential of the electromagnetic field, Am, with m¼0, 1, 2, 4. The limit of the critical QED field, also called the Schwinger 3, is equal to field qffiffiffiffiffiffiffiffiffiffiffi m 2 3 2 e AmA me c mec 16 V ¼ ð Þ ES ¼ ¼ ¼ 1:32 10 ð7Þ a 2 2 e_ e|C cm mec 2 29 2 corresponding to the intensity of cES =4p 4:6 10 W=cm , is characterized by the normalized EM field amplitude It is also the gage invariant (on the gage invariance in classical 2 and quantum physics see, e.g. [12]). Below we shall use mec 5 l aS ¼ 4:1 10 ð8Þ notations a0 and am for initial and maximum values of the _o 1 mm normalized electric field, respectively, which should not be confused with the zeroth and spacial components of the The electric field E acting on the electric charge e produces a 4-vector a ¼eA /m c2. S m m e work equal to m c2 over the distance equal to the Compton 2. For the EM emission by an electron the essential parameters e wavelength | [3,18–20]. are as follows. The first parameter is e , which is proportional C rad We note here that the electric field corresponding to the to the ratio between the classical electron radius, 2 2 13 condition a¼aQ, where aQ is given by Eq. (6), is equal to re ¼ e =mec ¼ 2:8 10 cm, and the EM wavelength 2 2e mec 2 2re EQ ¼ ¼ aES ð9Þ erad ¼ ð3Þ 3_2o 3 3|0 and does not depend on the EM wave wavelength with the EM field intensity I ¼ cE2 =4 ¼ 3:6 1024 W=cm2. Other two parameters are above introduced normalized ampli- Q Q p When the electric field strength approaches the critical QED tude, a, and the relativistic electron gamma-factor, ge. Accord- ing to the P-theorem of similarity and modeling theory [10], limit the electron–positron pair creation from vacuum x y z becomes possible. However, as is well known, the plane EM the characteristic dimensionless parameter has a form erada ge with the indices x,y,z, which finding requires additional con- wave does not create the electron–positron pairs, because this sideration. For example, for a circularly polarized EM wave in process is determined by the Poincare invariants [3] mn 2 2 mnrs frame of reference, where the electron is on average on rest, FmnF E B Fmne Frs EUB F ¼ ¼ and G ¼ ¼ ð10Þ the power emitted by the electron in the ultrarelativistic limit 4 2 4 2 is proportional to the forth power of its energy [13] which vanish for plane EM wave. Here Fmn¼@mAn@nAm is the 2 2 2 4 PC,g ¼ erado0mec ge ðge 1Þpge ð4Þ electromagnetic field tensor expressed via the electromagnetic mnrs field 4-vector potential Am, and e is the fully antisymmetric unit tensor where m, n, r, s are integers from 0 to 4. For the In the non-radiative approximation, the electron can aquire electron–positron pair creation to occure it requires non plane the energy from the EM field with the rate Eo m c2a , i.e. 0 e 0 EM wave configuration. g ¼a. The condition of the balance between the acquired and e 5. In QED the charged particle interaction with EM fields is lost energy yields a3 e1 , i.e. the indices x,y,z are x¼1, y¼3, rad determined by the relativistically and gage invariant para- z¼0, and the key dimensionless parameter is a3e . The rad meter radiation effects become dominant at qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = mn 2 1 3 ðF pnÞ a4arad ¼ erad ð5Þ we ¼ ð11Þ mecES i.e. in the limit I42 1023 W/cm2 for l¼1 mm [14,15]. If l¼1.5 108 cm, which corresponds to a typical wave-

length for the XFEL generated EM radiation (see [16]), The parameter we characterizes the probability of the gamma- the radiation effects become significant at a4aradE20, i.e. photon emission by the electron with Lorentz factor ge in the field 28 2 for I43 10 W/cm . The characteristic frequency of this of the EM wave [21–24]. It is of the order of the ratio E/ES in the 34 S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 electron rest frame of reference. Another parameter 3. Probing nonlinear vacuum qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mn 2 3.1. Electron–positron pair creation from vacuum _ ðF knÞ wg ¼ ð12Þ mecES Understanding the mechanisms of vacuum breakdown and polarization is challenging for nonlinear quantum field theories is similar to we with the photon 4-momentum, _kn, instead of the and for astrophysics [28]. Reaching the Schwinger field limit electron 4-momentum, pn. It characterizes the probability of the electron–positron pair creation due to the collision between under Earth-like conditions has been attracting a great attention for a number of years. Demonstration of the processes associated the high energy photon and EM field. The parameter we defined by Eq. (11) can be expressed via the electric and magnetic fields and with the effects of nonlinear QED will be one of the main electron momentum as challenges for extreme high power laser physics [2]. Vacuum nonlinearity is characterized by the normalized Poin- qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 care invariants we ¼ ðmecgeEþp BÞ ðpUEÞ ð13Þ ESmec F E2B2 G EUB ¼ ¼ ¼ ¼ ð Þ f 2 2 and g 2 2 15 For the parameter wg defined by Eq. (12) we have ES 2ES ES 2ES rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Electron–positron pair creation from vacuum by the EM field _ 2 og 2 wg ¼ Eþkg B ðkgUEÞ ð14Þ [18–20] is a tunneling process, which requires the parameter gK¼1/a ESmec c 2 to be much less than unity, i.e. ac1, and K0 ¼ 2mec =_o 2aS to be much larger than unity [29], with the number of pairs created per These processes acquire optimal rate at we,wg41. For the laser pulse collision with counterpropagating photon or electron this unit volume in unit time given by condition is satisfied at a4ð2=3Þag1e1 . 2 2 e,g rad e ES pb p In order to get into regimes determined by the above- We þ e ¼ ebcoth exp ð16Þ 4p2_2c e e mentioned parameters the required laser intensity should be of the order of or above I41023 W/cm2. This will bring us to Here e and b are the normalized invariant electric and magnetic fields experimentally unexplored domain. At such intensities the laser in the frame of reference, where they are parallel: interaction with matter becomes strongly dissipative, due to rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffi rqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi efficient EM energy transformation into high energy gamma rays e ¼ f2 þg2 þf and b ¼ f2 þg2f ð17Þ [15,25]. These gamma-photons in the laser field may produce electron–positron pairs via the Breit–Wheeler process [26]. Then when e-0, the pair creation rate We þ e tends to zero. For b-0we the pairs accelerated by the laser generate high energy gamma have quanta and so on [22–24], and thus the conditions for the 2 2 e ES 2 p avalanche type discharge are produced at the intensity þ ¼ ð Þ We e 2 e exp 18 24 2 3_ e E10 W/cm . The occurrence of such ‘‘showers’’ was foreseen 4p c by Heisenberg and Euler [19]. If the EM field varies sufficiently slowly in space and time, Relativistic mirrors [4–6] may lead to an EM wave intensifica- i.e. the characteristic scale of this variation is much larger than tion resulting in an increase of pulse power up to the level when the characteristic scale of pair production process, which is given the electric field of the wave reaches the Schwinger limit when by the Compton length, then the field can be considered as electron–positron pairs are created from the vacuum and the constant in each point of 4-volume and the number of pairs is vacuum refractive index becomes nonlinearly dependent on the equal to EM field strength [19,27]. In quantum field theory particle Z 4 creation from the vacuum under the action of a strong field has Ne þ e ¼ We þ e d x ð19Þ attracted a great amount of attention, because it provides a typical example of non-perturbative processes. In the future, where the integration is performed over the 4-volume. nonlinear QED vacuum properties can be probed with strong Since the pairs are produced near the maximum of the electric and powerful EM pulses. field e ¼ em, we express EM field dependence on time and space

Fig. 2. (a) The vector field shows r- and z-components of the poloidal electric field in the r, z plane for the TM mode. The color density shows the toroidal magnetic field 2 distribution, Bf(r,z). (b) The first Poincare invariant F normalized by the dimensionless laser amplitude am: FTMðr,z ¼ 0,t ¼ 0Þ. (c) FTMðr,z,t ¼ 0Þ. S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 35 coordinates locally as period in the multiphoton regime

2K x2 y2 z2 t2 p eNam N þ ¼ ð26Þ e em 1 ð20Þ e e 7 9=2 2dx2 2dy2 2dz2 2dt2 2 K 4

As we see it is always small, provided am 51, and K b1. and substitute it into Eq. (18) for the rate We þ e ; here dx, dy, dz, and dt are the characteristic sizes and duration of high-intensity part of EM field. Integrating it over the 4-volume according to 3.2. Electron–positron gamma-ray plasma generation via the multi- Eq. (19) we obtain for the number of pairs [30,31] photon Breit–Wheeler process cdtdxdydz 4 p Reaching the threshold of an avalanche type discharge Ne þ e ¼ e exp ð21Þ 4|4 m 64p C em with electron–positron gamma-ray plasma (EPGP) generation via the multi-photon Breit–Wheeler process [26] discussed in Pair creation requires the first invariant F be positive. This Refs. [21–24] requires high enough values of the parameters w condition can be fulfilled in the vicinity of the antinodes of e and w defined above by Eqs. (11) and (12). colliding EM waves, or/and in the configuration formed by several g In the limit of small parameter w 51 the rate of the pair focused EM pulses, [31,32]. Near the focus region this configura- g creation is exponentially small [21], tion can be modeled by the axially symmetric three-dimensional ! (3D) electromagnetic field comprised of time-dependent electric 3 e2m2c3 w 3=2 8 ð Þ¼ e g ð Þ W99 wg 3 exp 27 and magnetic fields. As is known, in the 3D case the analog of a 32 _ og 2p 3wg linearly polarized EM wave is the TM mode, with poloidal electric and toroidal magnetic fields. The EM configurations approximat- In the limit wg b1 the pair creation rate is given by ing the EM fields found in Ref. [32] near the field maximum are 7 2 2 3 3 2=3 27G ð2=3Þ e me c wg described by the solution to Maxwell’s equations in vacuum and W99ðw Þ¼ ð28Þ g 56p5 _3 2 expressed in terms of Bessel functions and associated Legendre og polynomials. The EM fields for the TM mode and the first Poincare Here _og is the energy of the photon with which the Breit– invariant are shown in Fig. 2. The second invariant, G ¼ðEUBÞ,is Wheeler process creates an electron–positron pair. equal to zero, i.e. b ¼ 0 for this EM configuration. Since for the large ge, the photon is emitted by the electron For the TM mode, in the vicinity of the electric field maximum, (positron) in a narrow angle almost parallel to the electron the z-component of the electric field oscillates in the vertical momentum with the energy of the order of the electron energy, direction, the radial component of the electric field is relatively the parameters we and wg are approximately equal to each other, small: although this is not necessarily so in the limit when the electron E emits photons according to classical electrodynamics, i.e. when m 2 2 1=2 1=2 E ezEm cosðo0tÞþer k rzcosðo0tÞð22Þ g ðm c =_o Þ ¼ a . In order to find the threshold for the 8 0 e o e 0 S avalanche development we need to estimate the QED parameter and the -component of the magnetic field vanishes on the axis j we, defined by Eqs. (11) and (13). The condition for avalanche being linearly proportional to the radius development corresponding to this parameter is that it should E become of the order of unity within one tenth of the EM field B e m k r sinðo tÞð23Þ f 8 0 0 period (e.g. see Refs. [22–24]). Due to the trajectory bending by the magnetic field the electron transverse momentum changes. It here we use the cylindrical coordinates r, j, z. Using expression is easy to find the trajectory in the (r,z)-plane assuming o0t{1 for the probability of electron–positron pair creation given by Eq. and amo0tc1 of the electron moving in the EM field given by (21) and expanding the first Poincare invariant, FTMðr,z,tÞ, in the Eqs. (22) and (23). It is described by the relationships vicinity of its maximum, we find that the pairs are created in a ð Þ¼ ð Þ small 4-volume near the electric field maximum with the follow- pz t mecamo0t 29 ing characteristic size along the r-, z-, and t-directions: a k r o t o t ð Þ¼ m 0 0 0 0 ð Þ 1=2 1=2 1=2 pr t mec 3=2 I1 3=2 30 | 5am | 10am 1 5am 2 2 dr 0 ,dz 0 ,dt o0 ð24Þ paS paS paS a r o t a r o t o t o t ð Þ¼ m 0 0 þ m 0 0 0 þ 0 ð Þ r t 3=2 I1 3=2 I0 3=2 I2 3=2 31 here am ¼eEm/meo0c is the normilized field amplitude. As shown 2 2 16 2 2 in [24] the first pairs can be observed for an one-micron where I (x) is the modified Bessel function [33] and r is the initial wavelength laser intensity of the order of In 1027 W=cm2, which n 0 electron coordinate, which is of the order of dr as given by corresponds to a /a ¼0.05, i.e. for a characteristic size approxi- m S Eq. (24). It is also assumed that p (0)¼p (0)¼0. As we see, the mately equal to drE0.04l . z r 0 trajectory instability is relatively slow with the growth rate equal We note here that in the case when the parameter g ¼1/a K to o /23/2. characterizing the adiabaticity of the process is large, g ¼ 1=ac1, 0 K According to Eq. (30) the radial component of electron i.e. a{1, which corresponds to the high frequency EM radiation, 2 momentum is proportional to the square of time and K ¼ 2mec =_o is large as well, the pair creation occurs in the 2 multiphoton regime with the number of pairs created per unit pr ðam=8Þk0drðo0tÞ ð32Þ volume in unit time given by [29] Assuming o t to be equal to 0.1p, we obtain from Eq. (12) that w 0 e 2K becomes of the order of unity, i.e. the avalanche can start, at 2c 4gK W þ ð25Þ n e e 4 5=2 a /a E0.08, which corresponds to the laser intensity I 2:5 p3| K eN 0 S C 1027 W=cm2. At that limit the Schwinger mechanism provides 7 Here eN¼2.718281828... is the Napier or the Euler number. approximately 10 pairs per one-period of the laser pulse focused to 3 Assuming the field distribution is given by formula (20) with a l0 region [31]. characteristic size of the order of | and integrating Eq. (25) over In the case of two colliding circularly polarized EM waves, the the 4-volume, we obtain for the number of pairs generated per resulting electric field rotates with frequency o0 being constant 36 S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42

1=3 in magnitude. The radiation friction effects incorporated in the For example, if a0 erad 450 the circularly polarized laser relativistically covariant form of equation of motion of a radiating pulse with the wavelength of 0.8 mm and intensity of I 2:2 electron [13] result in 1023 W=cm2 generates a burst of gamma photons with the energy of about 20 MeV with the duration determined either by the laser dum e 2e2 m c ¼ Fmnu þ gm ð33Þ pulse duration or by the decay time of the laser pulse in a plasma. e ds c n 3c In the regime when the radiation friction effects are dominant, Here the radiation friction force in the Lorentz–Abraham–Dirac b 1=3 i.e. when a0 erad , the angle f between the electron momentum form is determined by gm ¼ d2um=ds2umðdun=dsÞðdu =dsÞ, 3 1=4 n and the electric field is small being equal to ðerada0Þ , i.e. the um¼(g,p/mec) is the four-velocity, and s is the proper time, ds¼dt/g. electron moves almost opposite to the electric field direction. The For the electron rotating in the circularly polarized colliding electron momentum is given by expression (37), [15,24]. This EM waves the emitted power becomes equal to the maximal yields an estimation energy gain rate at the field amplitude a4e 1=3 (corresponding to ! rad 1=2 the laser intensity I 4:5 1023W=cm2) as was noted above (see a p a w 0 ? ¼ 0 ð39Þ Eq. (5)). In this limit the radiation friction effects should be taken e 2 aSmec aS erad into account. 2 3 In order to do this, we represent the electric field and the This becomes greater than unity for a0 4aS erad 1:6 10 , electron momentum in the complex form: which corresponds to the laser intensity equal to 5:5 1024W=cm2 Without radiation friction taken into account the E ¼ Ey þiEz ¼ E0expðio0tÞ and p ¼ py þipz ¼ pexpðio0tijÞ intensity requirements are much softer. This is the condition of ð34Þ the electron–positron avalanche onset. We see that the radiation friction effects do not prevent the EPGP cascade occurring in the where j is the phase equal to the angle between the electric field case of circularly polarized colliding laser pulses. Extending our vector and the electron momentum (see Fig. 3). In the stationary discussion on the XFEL generated photon beams, for which regime the radiation friction force is balanced by the force acting e E8 105, we find that the condition w 41requiresa 40.3. on the electron from the electric field. The electron rotates with rad e 0 As we see the condition corresponds to the electric field given by Eq. constant energy. The equations for the electron energy have the (9) with for which the intensity is equal to5.5 1024 W/cm2.The form charcteristic field for the avalanche onset is EwEaES,i.e.the 1 a2 ¼ðg21Þð1þe2 g6Þ and tanj ¼ ð35Þ quantum radiation effects become dominant at the field strength a 0 e rad e e g3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirad e factor 1/137 smaller than the QED critical field ES. 2 In order to further clarify the role of radiation friction in the with a0 ¼eE0/meo0c and ge ¼ 1þðp=mecÞ . In the limit of weak 1=3 charged particle interaction with a relativistically strong EM field radiation damping, a 5e , the absolute value of the electron 0 rad we consider the electron motion within the framework of momentum is proportional to the electric field magnitude approximation corresponding to the Landau–Lifshitz form of the 4 p? ¼ meca0 with pJ ¼ mecerada0 ð36Þ radiation friction force [13,33] (we notice that the radiation friction in the QED limit is discussed in Ref. [35]) b i.e. p? pJ, while in the regime of dominant radiation damping effects, i.e. at a be1=3, it is given by e @Fmn e 0 rad gm ¼ u u ðFmlF unðF ulÞðFnku ÞumÞ ð40Þ 2 l n l 2 nl nl k 1=4 1=2 mec @x mec pJ ¼ mecða0=eradÞ and p? ¼ mecða0eradÞ ð37Þ Retaining leading in the limit g -N term we can write it as i.e. p 5p . Here p and p are the momentum components e ? J ? J "# perpendicular and parallel to the electric field. For the momen- e 2 p p B 2 pUE 2 tum dependence given by this expression the power radiated by f ¼ erado0 geEþ ð41Þ meo0c g mec mec 2 e the electron is mec o0a0, i.e. the energy obtained from the driving electromagnetic wave is completely re-radiated in the form of For the case of two colliding circularly polarized EM waves in 1=3 high energy gamma rays. At a0 erad we have for the gamma the zero magnetic field plane, where the rotating electric field is photon energy given by E ¼ Ey þiEz ¼ðmeco0=eÞaexpðio0tÞ, this expression takes the form m c3 _o 0:45 _ e ð38Þ "# g e2 2 p 2 p? f ¼ erado0 a 1þ ð42Þ ge mec

Although the Landau–Lifshitz approximation for the radiation friction force implies the relative weekness of radiation; asymp- E e- totically at t-N for stationary solution it gives the same scaling c 1=3 as given by expression (37), when a erad .InFig. 4 we show the results of numerical integration of the electron motion equations. ϕ hk Here (a) electron trajectory; (b) electron momentum vs time; p (c) time dependences of perpendicular and parallel to the electric ﬁeld components of the electron momentum are shown. The

upper row corresponds to the dissipationless case with erad¼0 and to the normalized electric field equal to a¼5 103. For lower 8 3 row we have erad ¼ 1:46 10 , a ¼ 5 10 . As we see, the radiation friction qualitatively changes the electron dynamic. The momentum becomes approximately an order of magnitude less than in the case without dissipation. The perpendicular Fig. 3. Electron moving in the rotating electric field of colliding circularly 1/2 polarized EM waves emits EM radiation. The angle j between the electron momentum component, p? ¼mec(aerad) , becomes approxi- momentum and electric field is determined by Eq. (35). mately an order of magnitude less than the parallel component. S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 37

Fig. 4. Results of numerical integration of the equations of motion for an electron in a circularly polarized electric ﬁeld: (a) electron trajectory; (b) electron momentum vs. time; and (c) time dependences of the electron momentum perpendicular and parallel to the electric ﬁeld components. Upper row—no radiation friction; erad ¼0, 3 8 3 a0 ¼5 10 . Lower row—including the radiation friction; erad ¼ 1:46 10 , a0 ¼ 5 10 .

4. Possibility of nonlinear vacuum probing with present-day results in the reflected light intensification laser systems 2 2 I=I0 13gph,W ðS=l0Þ ð43Þ 4.1. Approaching the Schwinger field limit here S is the transverse size of the laser pulse efficiently reflected at the RFM. The reflected pulse power is the same as in the The concept of the flying mirror has been formulated in Ref. [4] incident pulse P ¼ P . If the singularity can be approximated by as a way for approaching the critical QED electric field, the 0 the Dirac delta-function, nðxÞdðxÞ, the reflection coefficient is Schwinger field limit. This concept is based on the fact that an proportional to g3 with reflected light intensification [3,30] EM wave reflected off a moving mirror due to the double Doppler ph,W effect undergoes frequency multiplication with the multiplication 3 2 I=I0 32gph,W ðS=l0Þ ð44Þ factor (1þbM)/(1bM) in the limit bM-1 proportional to the 2 1=2 square of the Lorentz factor of the mirror, gM ¼ð1bMÞ . This The reflected pulse power increases as P ¼ P0gph,W . makes this effect an attractive basis for a source of powerful high- We note here that utilization of the converging spherical frequency radiation. Here bM¼vM/c is calculated for the mirror Langmuir wave [39] may be suitable for a flying mirror with velocity vM. There are several other schemes for developing higher than (43) and (44) reflected light intensification: 2 5 compact, intense, brilliant, tunable X-ray sources by using the I=I0 10 gph,W . relativistic mirrors formed in nonlinear interactions in laser This mechanism allows generating extremely short, femto, plasmas, whose realization will open new ways in nonlinear attosecond duration pulses of coherent EM radiation with extre- electrodynamics of continuous media in the relativistic regime mely high intensity. (see articles, Refs. [4,36]). A demonstration of the Flying Mirror concept has been Here we consider the relativistic flying mirror [4–6] based on the accomplished in the experiments of Refs. [5]. Two beams of utilizationofthewakeplasmawaves(Fig. 1d). It uses the fact that the terawatt laser radiation interacted with a gas jet. The first laser dense shells formed in the electron density in a nonlinear plasma pulse excited the nonlinear wake wave in a plasma with the wake, generated by a laser pulse, reflect a portion of a counter- parameters required for the wave breaking. The achievement of propagating EM pulse. In the wake wave, electron density modula- this regime was verified by observing the quasi-mono-energetic tions take the form of a paraballoid moving with the phase velocity electron generation and the stimulated Raman scattering. The close to the speed of light in vacuum [37]. At the wave breaking the second counter-crossing laser pulse has been partially reflected electron density in the nonlinear wake wave tends towards infinity. from the relativistic mirrors formed by the wake plasma wave. The formation of peaked electron density maxima breaks the EM pulses with wavelengths from 7 nm to 15 nm and an esti- geometric optics approximation and provides conditions for the mated duration of a few femtoseconds were detected. reflection of a substantially high number of photons of the counter The experiments with the 0.5 J, 9 TW laser in the counter- propagating EM pulse. As a result of the EM wave reflection from propagating configuration [6] demonstrated dramatic enhance- such a ‘‘relativistic flying mirror’’, the reflected pulse is compressed in ment of the reflected photon number in the extreme ultraviolet the longitudinal direction. The paraboloidal form of the mirrors leads wavelength range. The photon number (and reflected pulse toareflectedwavefocusingintoaspotwiththesizedeterminedby energy), is close to the theoretical estimate for the parameters the shortened wavelength of the reflected radiation. of the experiment. The key parameter in the problem of the Relativistic Flying If the RFM is excited by the pulse of the same amplitude as the 2 Mirror is the wake wave gamma factor, gph,W, which plays a role incident on it one, then the condition a0 ¼ 2gph,W follows from the of the reflecting mirror gamma factor, gM. The number of photons requirement for the wake wave to be in the breaking regime. back reflected at the density singularity of the form nðxÞx2=3 is Using the expression for the reflected pulse intensity (43), we 4 proportional to 0:1 gph,W (for details see Ref. [38]), which obtain that the electric field in the reflected pulse scales as 38 S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42

20 Epgph,WE0. In order to calculate the threshold for the electron– factor E10 smaller. For the XFEL photon beam, it is smaller by a positron pair creation via the Schwinger mechanism, we take into factor E1012. 2 account that the upshifted frequency is o ¼ 4gph,W o0 and In the QED nonlinear vacuum two counter-propagating elec- reflected pulse is compressed and focused in the region with tromagnetic waves mutually focus each other [27]. The critical 2 2 2 longitudinal and transverse size equal to l0=4gph,W and l0/2gph,W, power Pc ¼ cE S0=4p, where S0 is the laser beam waist, for the respectively. Using these relationships and expression (21) we mutual focusing is equal to find the number of pairs created by two colliding pulses per one 45 P ¼ cE2l2 ð49Þ wavelength and one period cr 14a S 0 ! 4 24 1 S 2paS l0 For l0 ¼1 mm it yields Pc 2:5 10 W, which is beyond the Ne þ e ¼ exp ð45Þ 8 4 3 reach of existing and planned lasers. We notice here that as 2 p l0 a0 S shorter the EM wavelength, l0, the lower is the critical power. For 8 From (45) we find that the RFM concept utilization can provide example, if we take the XFEL wavelength, l0 ¼1.5 10 cm, the the first pairs to be detected at a0¼100, i.e. at the source laser critical power corresponds to 50 pW limit. intensity of the order of 1022 W=cm2. For the laser emitted radiation, if we take into account that the Experiments utilizing the EM pulse intensified with the Rela- radiation reflected by the RFM has a shortened wavelength, 2 tivistic Flying Mirror technique may allow studying regimes of lr ¼ l0=4gph,W and that its power is increased by a factor gph,W, higher-than-Schwinger fields, when E4ES. This may be possible we may find that for gph,WE30, i.e. for a plasma density 3 17 -3 1/2 because the light reflected by the parabaloidal FRM is focused into 10 cm because gph,W¼(ncr/n) , nonlinear vacuum properties a spot moving with a relativistic velocity and is well collimated can be seen for the laser light incident on the FRM with a power of within an angle 1/gph,W, [4]. The wave localization within the about 50 PW. This makes the Flying Relativistic Mirror concept narrow angle means that the wave properties are close to the attractive for the purpose of studying nonlinear quantum electro- plane wave properties to the extent of the smallness of dynamics effects. Moreover the collision of two bubbles with or the parameter 1/gph,W. In this case the first Poincare invariant of without accelerated electron bunches inside them will provide a 2 2 the EM field, F, has a value of the order of E =2gph,W . The second good table-top laboratory for studying a number of nonlinear QED Poincare invariant, G (10), vanishes. Therefore the electric field effects. amplitude in the reflected EM wave can exceed the Schwinger limit by factor gph,W. 4.2. Multi-photon creation of electron–positron gamma-ray plasma We note that a tightly focused EM wave cannot reach an in ultrarelativistic electron beam collision with the EM pulse amplitude above ES, due to the electron–positron pair creation [24], which in turn leads to the scattering of the EM wave. While Here we discuss the requirements for experimental realization creating and then accelerating the electron–positron pairs the of abundant electron–positron pair creation in the PW class laser laser pulse generates an electric current and EM field. The electric interaction with relativistic electron beam of moderate energy field induced inside the Lepton-Gamma Plasma (LGP) cloud with (see also [40]). a size of the order of the laser wavelength, l0 can be estimated to In the experiments [6] on the 527 nm terawatt laser interac- be Epol ¼2pe(n þ þn)l0. Here n þ En are the electron and tion with 46.6 GeV electrons from the SLAC accelerator beam positron density, respectively. When the polarization electric field positrons were observed. The positrons were generated in a two- becomes equal to the laser electric field, Epol¼Elas, the coherent step process in which laser photons were scattered by the scattering of the laser pulse away from the focus region occurs. electrons and then the generated high-energy photons collided This yields for the electron and positron density n þ En ¼ with several laser photons to produce an electron–positron pair. 3 Elas/4pel0. The particle number per l0 volume is about a0l0/re. This corresponds to the multiphoton Breit–Wheeler process This is a factor a0 smaller than that required for the laser energy þ No0 þog-e e , ð50Þ depletion from the creation of electron–positron pairs, which implies that the laser will be scattered before being depleted. where N is the number of laser photons colliding with the Due to the nonlinear dependence of the vacuum susceptibilities gamma-photon to produce the pair. on the electromagnetic-field amplitude the vacuum index of refrac- The parameter wg, which determines the probability of the tion becomes intensity dependent according to the expression electron–positron pair creation by photons, is given by Eqs. (12) and (14). For a plane EM wave it takes the form n ¼ 1þn2I, ð46Þ E _ ¼ ð k cÞð51Þ with wg 2 og g,x ES mec 2 2 2 In the counter-propagating configuration we have w 2ða=a Þ a 7 E 37 7 cm g S n2I ¼ ð11 3Þ 1:6 10 ð11 3ÞI ð47Þ _ 2 45p ES W ð og=mec Þ. 2 E _o _2o o _2o o here the EM wave intensity is I¼cE /4p. As we see, the nonlinear g ¼ 0 g ¼ 0 g ð Þ wg 2 2 2a 2 4 2a 2 4 52 vacuum susceptibility can approach unity for the laser intensity of Es mec me c me c 36 2 the order of 10 W=cm , i.e. higher than the Schwinger limit. The The r.h.s. term of this expression, in which the bars indicate well known Kerr constant of vacuum defined as KQED¼(n1)/ frequencies in another frame of reference, explicitly shows the (l 9E92) can be found to be [2,27] 0 relativistic invariance of wg. It becomes of the order of unity for the energy of the photon, which creates an electron–positron pair, 7a2 |3 ¼ C ð Þ equal to KQED 2 48 90p mec l0 m c2 _ ¼ 2 e ð Þ og mec _ 53 In order to compare it with the Kerr constant for water, which 2 o0a is equal to 4:7 107 cm2=erg ð5 1014 m2=VÞ, we estimate Using this expression we can find the number of laser photons

KQED for the EM wave wavelength l0 ¼1 mm and ﬁnd it be of the Nl required to create the electron–positron pair. In the reference order of 8:5 1028 cm2=erg ð1034 m2=VÞ, i.e. approximately a frame where the electron–positron pair is at the rest we have the S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 39

2 relationship Nlo~ 0 ¼ o~ g, which gives Nl ¼ o~ g=o~ 0 og=4o0g~ . laser ponderomotive force provided its energy is large enough Here the tilde is used for quantities in the pair rest frame and g~ g 4ct a=2w ð59Þ corresponds to the Lorentz transformation to this frame. For the e las ? _ 2 _ 2 photon energy we have Nl o~ 0 ¼ mec . Since o~ g ¼ mec and where w? is the focus width. As we see, for a one-micron, 1 PW, 2 4 _2 2 _ 2 o~ 0o~ g ¼ o0og ¼ me c wg= a we find g~ ¼ wgmec =2 o0a, which i.e. a¼10 , 30 fs duration laser pulse focused to a one-wavelength yields spot this condition requires ge 4500, i.e. the electron energy a above 250 MeV. Further we assume that this condition is Nl ¼ ð54Þ respected. wg When a counter-propagating electron collides with the laser The probability per unit time of the pair creation by photons is pulse its longitudinal momentum decreases. According to Eq. (56) given by Eqs. (27) and (28). In the region of wgE1, it is of the it is equal to order of W 0:1e2m2c3=_3o ¼ 0:1aðm c2=_Þð_o =m c2Þa ¼ e g e 0 e 2 _ a mec 0:1ao0a. The mean free path of the photon of the energy og p ¼ p þmec ð60Þ x 0 ½ð 2 2 þ 2Þ1=2 before pair creation is equal to lm.f.p¼c/W, 2 me c p0 p0 l l In the boosted frame of reference where the electron is at the l ¼ 0 220 0 ð55Þ m:f :p 0:2paa a rest the laser frequency is related to the laser frequency in the 2 laboratory frame, o ,as[43] (see also Ref. [44] where nonlinear It is about 2 mm for a¼10 . 0 Thomson scattering is discussed) Now we estimate the parameter w value, which characterizes e sffiffiffiffiffiffiffiffiffiffiffiffiffi the nonlinear Compton scattering. We consider the electron o0 1b0 interacting with the plane EM wave. As a result of the interaction o0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð61Þ 2 1þa 1þb0 with the laser pulse, the electron, being inside the pulse, acquires qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi longitudinal and transverse momentum components. Using the 2 2 2 - where b0 ¼ p0= me c þp0. The bar ‘‘ ’’ denotes the frequency solution presented in Ref. [13], we find that the component of the value in the frame of reference where the electron on average is at electron momentum along the direction of the laser pulse rest. The characteristic frequency of the radiation emitted by the propagation can be found from the equation 3 electron in this frame of reference is equal to om ¼ 0:3 o0a , [17]. _ ¼ _ 3 ðm2c2 þm2c2a2 þp2Þ1=2p ¼ðm2c2 þp2Þ1=2p ð56Þ The photon energy,pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiom 0:3 o0a , satisfies condition e e x x e 0 0 _ 2 2 _ om ¼ mec ge for a ¼ mec =0:3 o0, which corresponds to the We use the conservation of generalized momentum, which parameters when the QED description should be used. For lower yields for the component of the electron momentum parallel to laser amplitudes the photon generation can be described by the the laser electric field: py¼amec, provided the radiation friction nonlinear Thomson scattering process. effects are negligibly small. The x-component of the electron Dividing the radiation power given by Eq. (4) by the typical 3 momentum before collision with the laser pulse is negative and energy of the emitted photon, _om ¼ 0:3 _o0a , and multiplying 9 9 equal to p0 . For a plane EM wave propagating along the x-axis by 2p=o0 we find the number of photons emitted by the electron 1 0 with E¼E(xct)ey and B¼E(xct)ez, where E¼c A (xct) and during one wave period A(xct) is the y-component of the 4-vector potential and prime 3p denotes differentiation with respect to the variable xct, the N aa ð62Þ g 4 invariant we given by Eq. (13) takes the form The photon energy in the laboratory frame of reference is E p ¼ x ð57Þ we ge a3 1b ES mec _ ¼ _ 0 _ 2 ð Þ om 0:3 o0 2 1:2 o0ag0 63 1þa 1þb0 Substituting expression (56) to Eq. (57) we obtain We note that in the interaction of a GeV electron with a PW "#"# laser pulse the radiation friction effects must be incorporated. E ðm2c2 þp2Þ1=2p _o ðm2c2 þp2Þ1=2p w ¼ e 0 0 ¼ a 0 e 0 0 ð58Þ Here we shall consider it as a perturbation, i.e. in the Landau– e E m c m c2 m c S e e e Lifshitz form [13], which limits of applicability have been For 1 mm petawatt laser pulse focused to a few micron focus discussed in Refs. [41,42]. 2 In order to estimate o we should find the electron beam spot (a¼10 ) the parameter we becomes equal to unity for g 3 energy taking into account the radiation losses. Retaining the g0 ¼2.5 10 , i.e. for the electron energy of about of 1.3 GeV. The gamma-ray photons depending on the parameters of the main order terms in Eq. (40), we obtain the equation for the laser-electron interaction can be generated either via nonlinear x-component of the electron momentum Thomson scattering in the classical electrodynamics regime or via 2 dpx 2 px multiphoton inverse Compton scattering, when the quantum ¼erado0a ð2tÞ ð64Þ dt mec mechanical description should be used. Its solution is given by

4.3. Gamma-ray photon generation via the nonlinear Thomson p ð0Þmec p ðtÞ¼ x R ð65Þ x t 2 0 0 scattering mecþerado0pxð0Þ 0 a ð2t Þdt

At first we analyze nonlinear Thomson scattering. We consider If we assume a dependence of a(t) of the form 2 2 a head-on collision of an ultrarelativistic electron with a laser aðtÞ¼a0expðt =2t Þ, Eq. (65) can be rewritten as pulse. The laser ponderomotive pressure pushes the electron p ð0Þm c p ðtÞ¼ x pffiffiffiffiffiffiffiffiffiffiffiffie pffiffiffi ð66Þ sideways with respect to the pulse propagation and also changes x 2 mecþerado0pxð0Þta p=32½erf ð 2t=tÞ1 the longitudinal component of the electron momentum. In the 0 electron rest frame (see below) the laser pulse duration is here erf(x) is the error function equal to erf ðxÞ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi R 2 x 2 tlas tlas 1þa =2ge. The electron is not scattered aside by the ð p=2Þ 0 expðt Þdt [33]. Eq. (66) shows that for large enough 40 S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42

2500 here u¼wg/we, z¼(u/we)2/3, and Ai(z) is the Airy function. At small a e,γ 1/3 1=3 u the intensity is proportional to u , i.e. I og , and at large u it e decreases exponentially. The maximum of the intensity distribu- 2 5 E γ tion is at wg we for we 1 and at wg we if we b1, i.e. at 2 _ a 0.5 og mec a= .

100 1000 Using expression (61) for o0 we obtain that the parameter _2 2 4 _ 2 2 wg ¼ a o0og=me c equals wg ¼ 2g0a o0=mec .Ata¼10 it 3 becomes of the order of unity for g0E2.5 10 . We see that the QED consideration gives the same value E1.25 GeV for the 0 0 required relativistic electron energy as in the case of the classical approximation corresponding to nonlinear Thomson scattering. -5 05 t 4.5. Electron–positron generation in two laser beams interacting with plasmas Fig. 5. Time dependence of the electron and photon parameters for a 1.25 GeV electron beam interacting with a 15 fs PW laser pulse (a0¼100): the electron energy—black; parameter we—blue; laser pulse profile—red, parameter wg—green. In order to construct a compact source of the electron– (For interpretation of the references to color in this figure legend, the reader is positron-gamma-ray plasma it is desirable to produce gamma referred to the web version of this article.) rays in collisions of the laser accelerated electrons with the EM field of the laser pulse (see Fig. 1, where three versions of the 2 pxð0Þta0 the electron momentum tends to the limit of experimental setup are presented). As is known, the required GeV pffiffiffiffiffiffiffiffiffi range electron energy has been achieved in experiments involving - 2 pxðtÞ p=8ðmec=erado0ta Þð67Þ t-1 0 a 40 TW ultrashort laser pulse interaction with underdense in accordance with the theory formulated in Refs. [13,34] and plasma [47], when the quasi-mono-energetic electron bunches numerical solution of the equations of the electron motion in the have been generated in the laser wake field acceleration process. laser field with the radiation friction effects taken into account In the case of usage of the laser wake-field accelerated 2 electrons, the bunch is optimally synchronised with the laser [43]. For a0 ¼10 and o0t¼6 the electron momentum is about pulse. In addition, the electron density is substantially higher than px(N)/mec¼500. In Fig. 5 we present the time dependence of the electron and in the beam of electrons produced by conventional accelerators. photon parameters for a GeV electron beam interaction with the For example, a typical microtron accelerator generates a 150 MeV electron beam with the charge of 100 pC and duration of about Gaussian PW (a0 ¼100, t¼15 fs) laser pulse, when the ponderomotive force and radiation friction effects are incorporated into 20 ps. Being focused to a 50 mm focal spot it provides the electron 12 3 the electron equation of motion. The electron energy before and density nb 10 cm . As a result approximately 300 electrons after the interaction with the laser pulse equals 1.25 GeV and can interact with the laser pulse. According to Eqs. (54) and (55) the maximal number of electron–positron pairs which can be 0.25 GeV, respectively. The parameter we(t)E2(a(t)/aS)(px(t)/mec) reaches the maximum value of 0.3 inside the laser pulse. produced is equal to 300 per shot. However, since for a PW range _ 2 laser pulse the parameters we and wg are small compared to unity, The parameter wgðtÞ2ðaðtÞ=aSÞð ogðtÞ=mec Þ with og 0:3o0 2 2 2 2 3=2 the rate of electron–positron pair generation is exponentially low. ½ð1bxÞ=ð1þbxÞ½ðpy þpz Þ=me c reaches the maximum value of about 0.3. This requires a number of the laser shots to be accumulated in order to detect the electron–positron pairs. If we use a LWFA generated electron bunch with a transverse 4.4. Gamma-ray generation via the multi-photon Compton size of several microns, with 10 fs duration, e.g. see [45], and with scattering an electric charge of 1 nC, which corresponds to the optimal parameters [46], approximately 109 electrons interact with the For multi petawatt lasers with larger laser amplitudes of the laser pulse. For the parameters w and w of the order of unity we focused light, a43m c2=2 _o g 300, the gamma rays are e g e 0 0 may expect in the optimal case the generation of approximately generated in the multiphoton Compton scattering of ultrarelati- 109 pairs per shot. We note here that in the experiments [47] the vistic electrons electric charge of a 1 GeV electron beam is equal to 30 pC, i.e. the þ 7 No0 þog-e e ð68Þ expected number of pairs could be about 3 10 per shot. Utilization of the EM wave intensified by the relativistic flying The kinematic consideration of the multiphoton reaction leads mirror to collide with the LWFA accelerated electrons, as is shown to the relationship between the laser frequency, o0, and the in Fig. 1c, can further increase the efficiency of the electron– frequency of scattered photons, og. In the frame of reference, where the electron is at rest, we have (e.g. see [3,21]) positron pair generation. In this case the parameters we and wg increase by a factor equal to gph,W and can exceed unity, thus No o ¼ 0 ð69Þ providing conditions for the avalanche type prolific generation of g _ 2 2 1þðN o0=mec þa =4Þð1cosyÞ the electron–positron pairs. where y is the angle between the laser pulse and the photon We note that for the experiments discussed in our paper it is propagation directions, and a is the normalized laser field. At not critical to have high quality electron beams. However, if the y-p the scattered frequency tends to its maximal value discussed gamma-ray sources will find applications in the future, 2 the beam quality will become important depending on specific mec =_. In the laboratory frame of reference, the scattering cross section is maximal in the narrow angle in the backward applications. direction with respect to the laser propagation. Table 1 presents maximum values of the parameters we and wg In the rest frame of the electron the intensity of radiation is for different regimes of the electron bunch interaction with 30 fs given by [21] PW laser pulses. Z At present, the optical parametric chirped pulse amplification dI e2m2c3 u 1 2 u2 ¼ e ð Þ þ þ 0ð Þ ð Þ technique [48] produces E8 fs, 16 TW laser pulses and several fs 2 3 Ai y dy 1 Ai z 70 du 4p2_ ð1þuÞ z z 2ð1þuÞ petawatt lasers are under development [49]. If we take a 8 fs, S.V. Bulanov et al. / Nuclear Instruments and Methods in Physics Research A 660 (2011) 31–42 41

Table 1 [11] S.V. Bulanov, T.Zh. Esirkepov, D. Habs, F. Pegoraro, T. Tajima, European

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