Resonant interaction between laser and electrons ...

Under consideration for publication in J. Plasma Phys.

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Resonant interaction between laser and electrons undergoing betatron oscillations in the bubble regime A L E S S A N D R O C U R C I O1 , D A N I L O G I U L I E T T I2 G I U S E P P E D A T T O L I3 A N D M A S S I M O F E R R A R I O4 1

Physics Department of Roma University, ”La Sapienza” and Laboratori Nazionali di Frascati (INFN) 2 Physics Department of the University and INFN, Pisa, Italy 3 ENEA−Centro Ricerche Frascati,Roma,Italy 4 Laboratori Nazionali di Frascati (INFN) (Received 5 July 2015)

The betatron radiation in the bubble regime is studied in presence of resonant interaction between the accelerated electrons and the driver laser pulse tail. The calculations refer to experimental parameters available at the laser FLAME facility of the National Laboratories of Frascati (LNF) and represent the radiation spectra and spatial distributions which have to be expected in forthcoming experiments.

1. Introduction The interest for innovative X-ray sources, ultra-intense and ultra-short, is actually a hot topic. Nearly ten years ago the experimental evidence of the betatron radiation (Rousse, 2004) emitted by electrons accelerated in laser plasma channels opened new perspectives in that direction. After that, both experimental and theoretical efforts were made for the better-understanding of this phenomenon, which has been studied in more than one laser-plasma interaction regime (Cipiccia, 2011; Esarey,2002 ; Glinec, 2008). In this work we wanted to consider what were the characteristics of the betatron radiation emitted during electron laser-plasma acceleration at LNF laser facility. In particular we are interested in the case where, in the so-called bubble regime, the oscillating electric field of the laser radiation resonantly influences the transversal oscillation of the accelerated electrons. It can happen when the laser tail penetrates in a significant way the plasma bubble and the electrons experience an electric laser field which is greater (sometimes much greater) than the electrostatic focusing field provided by the positive charge inside the bubble. This condition induces beneficial effects in the emitted betatron radiation, increasing the intensity and moving the center of the emission spectrum toward higher frequencies. This is due to an increase of the betatron oscillation amplitude together with a blue-shift of the oscillation frequency, which in some cases can be approximately equal to a harmonic of the betatron frequency, determined by the plasma density and the accelerated electron energy. The calculations were carried out following an analytical method made agile by reasonable approximations that adapt to situations of interest for experiments in the process of programming at the LNF. These results will become certainly help in the design of these experiments and in particular of diagnostic dedicated to the detection of high energy electromagnetic radiation.

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Alessandro Curcio, Danilo Giulietti, Giuseppe Dattoli and Massimo Ferrario

2. Laser-driven betatron oscillations In the paper (Cipiccia, 2011) is considered the existence of three regimes of laser-driven betatron oscillations: non-resonant, weakly- resonant and strongly-resonant. In that work the three different regimes were explored by varying the plasma electron density. In fact, √ depending the plasma bubble radius R ∼ 2c a0 /ωp on the plasma electron density (Lu, 2007) through the plasma frequency ωp = (ne e2 /ε0 me )1/2 , the extension of the driver laser pulse tail inside the bubble can be set by controlling the dimension of the bubble withprespect to the laser pulse length. The normalized vector potential a0 ∼ 8.5 × 10−10 I0 [W/cm2 ]λ0 [µm]2 is related to the laser intensity and wavelength I0 and λ0 respectively. The velocity of light in vacuum is c , ne is the plasma electron density, e is the elementary charge, ε0 is the vacuum dielectric constant and me is the electron mass. The three regimes differentiate according to the entity of the interaction between the laser tail ad the accelerated electrons. In the non-resonant regime the electrons don’t interact with the laser pulse, therefore they undergo betatron oscillations due only to the restoring force field provided by the positive charges inside the bubble. In the weaklyresonant regime the electrons start to interact with the laser pulse and the form of their oscillations begin to deviate from the previous case. In the strongly-resonant regime the betatron oscillations are completely driven by the laser electromagnetic field. One of the most interesting result of (Cipiccia, 2011) is that all the three regimes can occur around the dephasing point, where the energy of the accelerated electrons can be considered approximately constant, before they start to decelerate. The electron motion around the dephasing point comes to be (Esarey, 2002): x(t) = rβ sin ωβ t z(t) = z0 + vz0 (t −

kβ2 rβ2 kβ2 rβ2 t− sin 2ωβ t) 4 8ωβ

(2.1)

where x(t) is the transverse coordinate and z the longitudinal one, along the laser axis. rβ is the betatron oscillation amplitude, z0 is the initial z coordinate, vz0 the initial longitudinal velocity, kβ and ωβ the betatron wave number and frequency respectively, where √ ωβ = ωp / 2γmax , γmax being the Lorentz factor of the accelerated electron at the point of maximum energy gain, which corresponds to the dephasing point. The relative motion between the accelerated electrons and the laser pulse is responsible for a red Doppler-shift of the laser frequency as witnessed by the electrons. We consider the interaction of an electron with an electromagnetic plane wave which phase is ξ = ω0 (t − z(t)/vφ ), where ω0 is the wave (laser) pulsation and vφ ∼ c(1 + ωp2 /2ω02 ) its phase velocity in through the plasma. By expressing z as z ∼ vz t the shifted frequency comes to be: ωs = ω0 (1 −

vz ) vφ

(2.2)

2 By using the fact that βz ∼ 1 − 1/2γmax and ωp > 1/2γmax holds. Therefore the laser red-shift frequency as witnessed by the accelerated electrons around the dephasing point comes to be ωs ∼ ωp2 /2ω0 . By p using the fact (Lu, 2007) that γmax ∼ a0 2ω02 /3ωp2 , we obtain ωp2 = ω0 ωβ 4a0 /3, i.e.:

Resonant interaction between laser and electrons in the bubble regime r ωs ∼

a0 ωβ 3

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(2.4)

The equation (2.4) shows that, when the laser tail provides the electrons a significant driving force, the transverse oscillations shift in frequency from ωβ to ωs , in fact, as shown in (Nemeth, 2008), the driving force imposes its frequency to the electron oscillatory motion. In the present work is showed, in agreement to what simulated and experimented in (Cipiccia, 2011), that the frequency of the driven oscillations is related both to the local laser intensity and to the betatron frequency, making possible the relativistic Dopplershift of the laser frequency to harmonics of the betatron frequency, with a following efficient exchange of energy between the electrons and the laser. A model of transverse trajectory in the interaction regime is described, which could be useful for fast calculations of the emitted radiation without passing before through numerical integration of the equations of motion. Furthermore the the expression (2.4) could be useful in order to set the experimental conditions for an efficient laser-driven betatron motion. The laser intensity experienced by the electrons varies with the time (it grows for γmax > ω02 /ωp2 ), then the resonance between the laser shifted frequency and the harmonics of the betatron frequency develops during time, with the possibility that ωs resonates with harmonics of ωβ higher than the first. The laser action on the accelerated electrons is duplex: when the electrons experience an intense driving electromagnetic field, their oscillation amplitude can be strongly enhanced; furthermore, being ωs > ωβ the betatron oscillation frequency increases. Both effects can play a significant role in determining an important shift of the radiation spectra towards higher energies. In fact the so-called critical energy 3 Ec ∼ 3γmax ~ωβ2 rβ /c, i.e. the value of the photon energy which separates the radiation spectrum in two parts corresponding to the same amount of radiated energy below and above it, grows as the square of the betatron frequency and is linear in the betatron oscillation amplitude. Let’s consider an electron at rest with respect to the laboratory frame: in order to be more important than the betatron motion, the quivering motion should occur in such a way that a0 >> rβ k0 . For typical values of the parameters k0 and rβ , a0 should be greater than many tens, which is not a currently available value for any laser system, therefore the laser electric field is actually never greater than the wakefield transverse field. Nevertheless, due to the relative motion between the laser and the electrons, the effective a0 as witnessed by the electrons is Doppler-shifted to larger values, making possible the enhancement of the driven oscillation amplitude. The Doppler-shifted a0 is calculated by substituting the laser pulsation ω0√with the Doppler shifted pulsation ωs , i.e. a00 ∼ eE/mcωs . Where E = E0 exp[−(2R)2 /( 2vg ∆τ )2 ] takes into account√of the tail of the electric field of a laser gaussian pulse E = E0 exp[−(z(t) − 2R − vg t)2 /( 2vg ∆τ )2 ] at the center of the bubble, where the electrons start to interact with the laser tail; vg is the laser group velocity inside the plasma. The new relativistic parameter a00 can be expressed √ 0 by utilizing the expression for ωs in this √ in terms of a0 = eE0 /mcω way:a00 ∼ 6a0 γmax ω0 /ωp exp[−(2R)2 /( 2vg ∆τ )2 ]. As soon the electrons overcome the bubble center, their distance from the bubble head (which at the center of the bubble is R) where the center of the laser is, become smaller and the gaussian part of a00 can grows very rapidly to the values of many tens. In fact the evolution of a00 can be evaluated in this way: p √ a00 (t) ∼ 6a0 γ(t)ω0 /ωp exp[−(z(t) − 2R − vg t)2 /( 2vg ∆τ )2 ] (2.5) The transverse excursion of an electron initially at rest in a laser pulse with a0 = 10 is

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Alessandro Curcio, Danilo Giulietti, Giuseppe Dattoli and Massimo Ferrario

about 1 microns, nevertheless the Eq. (2.5) justifies the fact that electron excursion can reach almost 10 microns when resonantly combining wakefield and laser field.

3. Transverse oscillations: comparison between numerical and analytic results The Eq. (2.1) are showed because in the present work the interaction is studied between the accelerated electrons in the bubble regime and the tail of the laser driving pulse. The interaction can occur near the dephasing point for suitable time duration of the laser pulse and electron plasma density (which determines the extension of the bubble). Near the dephasing point the acceleration is approximately zero, that’s why the electron motion, without the presence of the laser tail, can be described by Eq. (2.1). Due to the fact that we can consider γ(t) ∼ γmax , we can easily solve the equation of motion in the presence of the laser pulse tail, as we’re going to show. The accelerated motion without laser-electron interaction (Glinec, 2008), described by Eq. (3.1), reduces to Eq. (2.1) near the dephasing point where the accelerating fields are close to zero. In the review paper (Corde, 2013) the phenomenological model for the electron dynamics in the bubble regime presented in (Lu, 2007) is recalled, which consists in the following set of equations: x ˆ dγβx =− 2 dtˆ yˆ dγβy =− ˆ 2 dt dγβz (tˆ − tˆd ) =− 4γφ2 dtˆ (3.1) and the electron transverse motion is showed to be: x(t) = A Pν (τ ) + B Qν (τ )

(3.2)

where the variables with a hat are normalized by the choice m = c = e = ωp = 1. The parameters A and B are constants which have to be determined by initial conditions. Pν (τ ) and Qν (τ ) are the Legendre functions of the first and second kinds respectively and ν is determined by the equation ν(ν + 1) = 4γφ2 with γφ the Lorentz factor relative to the normalized laser phase q velocity βφ = vφ /c. The parameter τ is determined by the equation τ = (tˆ − tˆd ) tˆ2d + 8γφ2 γi , where the dephasing time is tˆd = 2kp Rω02 /3ωp2 and γi is the electron initial Lorentz factor, depending on A and B. The fields in the bubble are taken into account through the parameter τ , related to the longitudinal energy gain, in fact γz ∼ γmax (1 − τ 2 ). The result (3.2) reduces to the result (2.1) for t ∼ td (when the acceleration length is comparable to the dephasing length). In order to take into account for the presence of the laser tail near the dephasing point, one can solve the equation for a charged particle under the action of a electromagnetic wave packet, specializing to the case of a gaussian packet with an effective ponderomotive force at the frequency ωs , according to what done in the previous section. The relativistic quivering motion corresponding to the situation in which a particle interacts, starting at rest, with only the laser is xq = a0 c/ω0 cos ξ (Landau, 1971), from which we define the effective quivering force fq = ma0 cω0 cos ξ. For our case, in which the particle is approaching the laser with high velocity, we substitute ω0 with ωs and ξ = ωs t, according to what done in the previous section. Finally we set the following equation, which holds when the

Resonant interaction between laser and electrons in the bubble regime

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laser-electron interaction is much more important than the electron-wakefield interaction and it also takes into account of the temporal profile of the laser: ωp2 a0 e d2 x = − x+ dt2 2γmax

−

(z(t)−2R−vg t)2 2 ∆τ 2 2vg

γmax

cωs

cos(ωs t)

(3.3)

By considering the temporal profile of the laser slowly varying with respect to the transverse oscillations, the solution of (3.3) comes to be: r (z(t)−2R−vg t)2 − 6eE0 a0 2 ∆τ 2 2vg e ωβ t (3.4) x(t) = x0 cos ωβ t + cos a0 2 a0 | 3 − 1|mωp 3 This solution holds just near the dephasing point. The solution of motion which takes into account of the acceleration before the dephasing point is recovered by substituting the first term x0 cos ωβ t with the solution in terms of the Legendre functions. In fact, during the laser-electrons interaction they nearly coincide, that’s why this substitution is justified. At the end we obtain: r (z(t)−2R−vg t)2 − a0 6eE0 2 ∆τ 2 2vg ωβ t (3.5) x(t) ∼ A Pν (τ ) + B Qν (τ ) + e cos a0 2 a0 | 3 − 1|mωp 3 where E0 is the amplitude of the laser electric field, ∆τ is half of the laser pulse duration, vg = c(1 − 3ωp2 /2ω02 ) is the laser group velocity inside the plasma (Lu, 2007), z(t) is p obtained by integrating βz = 1 − 1/γd2 (1 − τ 2 )2 with γd = γi + tˆ2d /8γφ2 (Corde, 2013). p In this way when tˆ ∼ tˆd , i.e. near the dephasing point, βz reduces to the value 1 − 1/γd2 , then at first order in kβ rβ , we recover the result of 2.1 for z(t). For the result (3.4) we have supposed ∆τ >> 1/ω0 and vg ∆τ