Laser Physics Related content PAPER Combined influence of azimuthal and axial magnetic fields on resonant electron acceleration in plasma To cite this article: Arvinder Singh et al 2017 Laser Phys. 27 110001 - Multi-GeV electron acceleration by a periodic frequency chirped radially polarized laser pulse in vacuum Harjit Singh Ghotra and Niti Kant - Electron energy enhancement by frequency chirp of a radially polarized laser pulse during ionization of low-density gases Kunwar Pal Singh, Rashmi Arya, Anil K Malik et al. - Topical Review R Bingham, J T Mendonça and P K Shukla View the article online for updates and enhancements. This content was downloaded from IP address 129.8.242.67 on 27/10/2017 at 09:47 Laser Physics Astro Ltd Laser Phys. 27 (2017) 110001 (6pp) https://doi.org/10.1088/1555-6611/aa8759 Combined influence of azimuthal and axial magnetic fields on resonant electron acceleration in plasma Arvinder Singh1, Jyoti Rajput1,2 and Niti Kant2 1 Department of Physics, National Institute of Technology Jalandhar, Jalandhar, Punjab, India Department of Physics, Lovely Professional University, G.T. Road, Phagwara 144411, Punjab, India 2 E-mail: nitikant@yahoo.com Received 8 April 2017 Accepted for publication 20 August 2017 Published 26 October 2017 Abstract Resonant enhancement in electron acceleration due to a circularly polarized laser pulse in plasma, under the combined influence of external azimuthal and axial magnetic fields, is studied. We have investigated direct electron acceleration in plasma by employing a relativistic single particle simulation. The plasma is magnetized with an azimuthal magnetic field applied in the perpendicular plane and an axial magnetic field applied along the direction of laser beam propagation. Resonance takes place between electron and electric field of the laser pulse for the optimum value of the combined magnetic field, which supports electron acceleration to higher energies, up to the betatron resonance point. The optimum value of these magnetic fields is highly sensitive to laser initial intensity and laser initial spot size. The effects of laser intensity, initial spot size, and laser pulse duration are taken into consideration in optimizing the magnetic field for efficient electron acceleration. Higher electron energy gain, of the order of GeV, is observed by employing terawatt circularly polarized laser pulses in plasma under the influence of combined magnetic field of about 10 MG. Keywords: electron acceleration, circularly polarized laser, magnetic field, plasma (Some figures may appear in colour only in the online journal) 1. Introduction ultrashort femtosecond lasers can generate magnetic fields of the orders of several hundred MG—nearly 350 MG [11]. The azimuthal magnetic field is generated by both linearly polarized and circularly polarized lasers, whereas an axial magn etic field is produced only in circularly polarized lasers [12]. In case of circularly polarized lasers, electrons spiral about the propagation direction, and hence produce axial magnetic field via the azimuthal current [13]. The self-matching resonance acceleration (SMRA) regime has been proposed with the aid of analytical modelling and 3D-PIC simulations. With the combination of the self-generated magnetic fields (in both axial and azimuthal directions) and the laser pulse, relativistic electrons undergo two processes: trapping and resonance [14]. Researchers [15–17] have shown that suitable external magnetic fields also have impressive effects on electron acceleration, due to resonance. Direct electron acceleration in vacuum has been analysed in the presence of axial magnetic Recent advancement in short pulse laser technology has discovered new scopes of laser-plasma interaction, which have proved successful in explaining many exciting facts about plasma—e.g. laser confinement fusion, laser induced particle acceleration [1, 2, 5] etc. The first idea for accelerating particles in plasma with the help of various fields was given more than six decades ago [3]. However, Tajima and Dawson [4] proposed an idea that electrons can be accelerated to very high energies by short pulse lasers. Accordingly, an additional acceleration can be obtained from the interaction of these electron beams with external or self-generated magnetic fields. Such an electron gains additional energy due to spontaneous magnetic fields [6–9]. These resultant generated magnetic fields can be of the order of multiple gigagauss [10]. Recently, it has been experimentally observed that 1555-6611/17/110001+6$33.00 1 © 2017 Astro Ltd Printed in the UK A Singh et al Laser Phys. 27 (2017) 110001 where z0 is the initial position of the pulse peak, wave vec1/2 tor k = (ω/c) × (1 − ωp2 /ω 2 ) , ω is the angular frequency of laser pulse, r2 is given by (x2 + y2 ), r0 is the initial spot size of the laser beam, τ is laser pulse duration, group veloc1/2 ity vg = c(1 − ωp2 /ω 2 ) , c is the velocity of light in vacuum, ωp2 = 4πn0 e2 /m is relativistic plasma frequency, e and m = m0 γ are the charge and relativistic mass of the electron respectively (here m0 is the rest mass of the electron), and 1/2 γ = 1 + P2x + P2y + P2z is the Lorentz factor, where Px , Py and Pz are the x, y and z components of electron momentum. The electromagnetic fields related to the vector potential of the laser pulse are field using symmetry of radially polarized laser beams, which improves trapping and acceleration of electrons to the level of GeV [18]. Recently, the effect of axial magnetic fields on electron acceleration due to axicon focused Gaussian radially polarized laser has been investigated, and it was observed that electron energy can be enhanced effectively without the use of petawatt lasers [19]. The combined influence of frequency chirp and azimuthal magnetic field has been studied, employing linearly and circularly polarized lasers, and it is reported that the combined effect of chirping and azimuthal magnetic field not only enhances the electron energy gain, but retains this electron energy gain for longer distances [20, 23]. Results have shown that small energy spread of the order δε/ε 10−2 is obtained by laser induced electron acceleration in vacuum in magnetic field [21]. The purpose of the present study is to analyse the effect of external magnetic fields of the order of a few megagauss on direct electron acceleration in plasma, by optimizing the circularly polarized laser parameters to achieve resonance, so as to get higher electron energy gain and minimum values of radiation losses. Betatron oscillations are more prominent in magnetized plasma as compared to unmagnetized plasma. Energy transfer to an electron due to circularly polarized laser is much more efficient, as compared to linearly polarized laser in vacuum [22]. In circularly polarised laser, there is production of axial and azimuthal magnetic field; so, in this paper, we place emphasis on the electron acceleration process due to interaction between circularly polarized laser field and external axial and azimuthal magnetic fields in plasma. We study the dependence of the maximum electron energy on the laser intensity, spot size and external combined magnetic field. The results of a relativistic, single particle simulation for direct acceleration of electron are presented. It is observed that there exists an optimum value of these magnetic fields to establish resonance with the laser fields. Above this value, deceleration of the electron takes place. We have performed computer simulation for a single particle code to get the optimum values of magnetic fields. The value of magnetic fields employed in the paper is of the order of a few MG, which is experimentally available [26], hence favouring our model. The structure of the paper is as follows: The equations governing the motion of the electron for circularly polarized laser beam in plasma in the presence of azimuthal and axial magnetic fields are formulated in section 2. Numerical results for maximum electron energy gain γm with the optimum value of external magnetic fields parameter b0 , laser initial intensity parameter a0 , laser initial spot size parameter r0 and laser pulse duration parameter τ are discussed in section 3, and a conclusion is drawn in section 4. ∂A EL = − and BL = ∇ × A. (2) ∂t The external azimuthal magnetic field [23, 24] can be written as follows: B0 (yx̂ − xŷ). Bθ = (3a) r0 Further, the external axial magnetic field is given by Bz = ẑB0 . (3b) Here B0 is the magnitude of both azimuthal and axial magn etic field. So, the combined external magnetic field comp onent, which includes the azimuthal and axial magnetic fields, is given by the equation, B0 (yx̂ − xŷ) + ẑB0 . BT = (3) r0 When a charged particle moves through the electromagnetic fields, it experiences a force, which is known as the Lorentz force. The relativistic Lorentz force equations are: dPx ∂Ax ∂Ax B0 − e xvz − eB0 vy , =e + evz (4) dt ∂t ∂z r0 ∂Ay ∂Ay dPy B0 − e yvz − eB0 vx , =e + evz (5) dt ∂t ∂z r0 ∂Ay dPz ∂Ax B0 B0 − evy = −evx + e xvx + e yvy . (6) dt ∂z ∂z r0 r0 The equation for rate of change of the electron energy is given by ∂Ay d(γm0 c2 ) ∂Ax (7) . = evx + evy dt ∂t ∂t If the electron is at origin in the beginning with Px = Px0 and Py = Py0 , then the integration of the equations (4) and (5) yields 2. Relativistic analysis Consider the propagation of a circularly polarized laser pulse in plasma with a vector potential [27] A= B0 xz − eB0 y, r0 (8) Px = 2eAx + Px0 − 2eA0 exp(−z20 /v2g τ 2 ) − e 2 [t − (z − z0 )/vg ] r2 − 2 , A0 [x̂ cos(ωt − kz) + ŷ sin(ωt − kz) exp − τ2 r0 B0 Py = 2eAy + Py0 − e yz + eB0 x. (9) r0 (1) From equations (6) and (7) 2 A Singh et al Laser Phys. 27 (2017) 110001 3. Results and discussions 2 2 0 Pz − γm0 c − eB 2r0 (x + y ) = 0, (10) 2 2 0 Pz − γm0 c − eB 2r0 (x + y ) = C3 . d dt The combined laser fields and the external magnetic fields are employed to generate betatron oscillations in the elecIf the electron has momentum Pz = Pz0 in the beginning, trons. The intense laser field’s ponderomotive force facilitates 1/2 C3 = Pz0 − (P2x + P2y + P2z + m20 c2 ) . Therefore, from equa- electron motion in the forward direction in the accelerating tion (10), we may write platform, and the applied magnetic fields play an interesting role in enhancing its energy. So, here, we consider the comeB0 2 1/2 (x + y2 ). Pz = Pz0 + γm0 c − (P2x0 + P2y0 + P2z0 + m20 c2 ) + bined effect of these external magnetic fields on the electron 2r0 (11) energy. We observe the resonant enhancement in electron energy due to these magnetic fields. Throughout this paper, Using the dimensionless variables: we have chosen the normalized initial position of pulse peak a0 → eA0 /m0 c, ax → eAx /m0 c, ay → eAy /m0 c, b0 → eB0 /m0 ω, z0 = −100, normalized initial electron momentum Pz0 = 1 t → ωt, τ → ωτ , r0 → ωr0 /c, x → ωx/c, y → ωy/c, and normalized laser propagation constant k = 0.99. z → ωz/cz0 → ωz0 /c, Px0 → Px0 /m0 c, Py0 → Py0 /m0 c, Figure 1(a) shows the variation of electron maximum energy γm as a function of normalized magnetic field b0 . In Pz0 → Pz0 /m0 c, k → kc/ω, vx → vx /c, vy → vy /c, vz → vz /c. this figure, electron energy has been analyzed at different valIn terms of the above dimensionless variables, equations ues of normalized laser intensity a0 . Figure 1(b) shows the maximum electron energy gain calibration curve. The fig(7)–(11) can be written as follows: ure represents the variation of a0 and optimized magnetic field b0 dx 1 z2 0 parameter b0 as a function of the electron’s maximum energy = P + 2ax − 2a0 exp − 2 2 − x z − b0 y dt γ x0 r0 k τ gain γm . For each value of laser intensity there is a corre (12) sponding optimum value of magnetic field at which resonance is established which leads to maximum energy gain. It can be dy 1 b0 y z + b x = P + 2a − (13) y 0 seen from figure 1(a) that for laser intensity parameter a0 = 10 y0 dt γ r0 (corresponding to a value of I0 ~ 1.38 × 1020 W cm−2), ini tial spot size r0 = 80 (corresponding value is 12.7 µm) and dz 1 b0 2 2 2 2 1/2 2 (x = P + [γ − (1 + P + P + P ) ] + + y ) magnetic field parameter b0 = 0.09 (corresponding value of x0 y0 z0 dt γ z0 2r0 B0 ~ 9.6 MG), which is feasible, we can achieve a maximum (14) electron energy gain of 0.68 GeV. Similarly, for laser intensity dγ dx ∂ax dy ∂ay parameter a0 = 20 (corresponding value of I0 ~ 5.52 × 1020 = + (15) W cm−2) and the optimized magnetic field parameter b0 = 0.2 dt dt ∂t dt ∂t (corresponding value of B0 ~ 21 MG), the maximum energy 2 2 where, ax = a0 cos(t − k z ) exp[−(t − (z − z0 )/k ) /τ − gain is raised to 1.35 GeV. The practically achieved values 2 2 (r /r02 )] and ay = a0 sin(t − k z ) exp[−(t − (z − z0 )/k ) / of these magnetic fields are up to 100 MG [7]. Figure 1(b) 2 2 τ − (r /r02 )]. Equations (12)–(15) constitute a set of coupled shows that if we increase laser intensity parameter a0 , then the ordinary differential equations. The initial position of the elec- value of external magnetic field parameter b0 also increases, tron is assumed at origin with initial momenta (Px0 ,Py0 ,Pz0 ). to achieve higher values of maximum electron energy gain To examine the validity of our model, we have solved the coupled γm . Thus, for the higher values of laser intensity parameter differential equations (12)–(15) numerically by using computer a0 , optimum values of magnetic field parameter b0 to achieve simulation to find the conditions for maximum electron energy resonance condition, in order to get maximum energy gain, gain in the presence of external magnetic fields in plasma. appear to be higher. At resonance, the electron cyclotron freThe electron may lose a small amount of energy during quency and the Doppler-shifted frequency become comparadeceleration by means of radiations, which can be calculated ble, so that maximum energy is transferred to the electron. from Lenard’s result in terms of radiative power: Even after passing the laser pulse, electron energy gain is retained due to the effect of these magnetic fields [25]. Thus, 2 2 6 2 2 2e γ dv dv (16) . P(t) = − v × these magnetic fields help in retaining the resonance condition 3c dt dt for longer, and deflect the electron to keep it in accelerating In case of circular accelerators, such as synchrotrons and phase, so that effective increase in electron energy takes place. Figure 2(a) shows the variation of electron energy γ as a betatrons, momentum of the charged particle changes drastically in the direction of rotation of particle, but function of normalized time t for different values of laser iniin energy per revolution is very small as tial spot size r0 ; figure 2(b) shows the variation of normalized the change p initial laser initial intensity a0 and laser initial spot size r0 as d /dτ (= γ ω |p|) (1/c)dE/dτ . Here ω = cβ/ρ , nor a function of electron maximum energy gain γm . The magnimalized velocity β = v/c and ρ is the orbit radius. For high tude of axial and azimuthal magnetic fields is taken here as energies (β ≈ 1), the radiative loss/revolution can be esti- 9.6 MG. The other parameters are the same as taken in fig4 mated as δ E(MeV) = 8.85 × 10−2 [E(GeV)] /ρ(m). In a 10 ure 1. It is shown that the maximum electron energy increases GeV electron synchrotron, the loss/revolution is 8.85 MeV. with the laser intensity parameter a0 , because the longitudinal 3 A Singh et al Laser Phys. 27 (2017) 110001 (a) (a) (b) Figure 2. (a) Electron energy gain γ as a function of normalized time t for different values of normalized laser initial spot size r0 at b0 = 0.09. (b) Maximum energy gain γm calibration curve for normalized intensity a0 and normalized laser initial spot size r0 at b0 = 0.09. Other parameters are the same as in figure 1. (b) Figure 1. (a) Maximum energy gain γm as a function of normalized magnetic field parameter b0 for various values of normalized intensity a0 (b) Maximum energy gain γm calibration curve for a0 and b0 . Other parameters are Pz0 = 1 and k = 0.98, z0 = −100 and τ = 300 . Figure 3 shows the variation of electron energy gain γ as a function of normalized distance z for normalized laser pulse duration τ = 100 (corresponding laser pulse duration 53 fs), 300 (corresponding laser pulse duration 160 fs) and 500 (corresponding laser pulse duration 266 fs). The other parameters are optimized from figures 1 and 2. We observe that electron gains more energy due to 266 fs laser pulse duration as compared to 160 fs and 53 fs, for a set of optimized parameters. Energy exchange between electron and laser field in the applied magnetic fields is maximum for longer laser pulse durations. For the chosen optimized laser and magnetic field parameters, energy gain of 0.133 GeV is achieved for 53 fs laser pulse duration, 0.67 GeV for 160 fs and 1.01 GeV for 266 fs pulse duration. So, electron energy is sensitive to laser pulse duration τ in presence of magnetic fields. Figure 4 shows the variation of acceleration gradient dγ/dz and energy gain γ as a function of accelerating distance z . The acceleration gradient dγ/dz = −ev · E/vz is calculated, and is plotted as a function of accelerating distance z with a0 = 10 and b0 = 0.09. The maximum acceleration gradient of 60 GeV m−1 is observed with laser peak intensity a0 = 10. The maximum energy gain of 0.67 GeV is observed with peak momentum of electron depends on the square of laser intensity amplitude. So, energy gain increases with increasing amplitude of laser intensity. The maximum energy gain also depends on the laser initial spot size parameter r0 . It is clear from the figure 2(b) that maximum energy gain increases for smaller spot size. For higher values of laser intensity param eters, optimum values of laser spot size to achieve higher electron energy gain appear to be smaller. So, the electron will be accelerated more effectively for smaller spot size and higher intensity. At a0 = 10, under the influence of magnetic fields b0 = 0.09 and initial spot size parameter r0 = 80, we obtain energy gain of 0.68 GeV, which can be increased up to 1.3 9 at a0 = 20(corresponding value of I0 ~ 5.52 × 1020 W cm−2) and initial spot size parameter r0 = 40 (corre sponding value is 6.4 µm). So, the net electron energy gain during acceleration is sensitive to both the laser intensity and the laser spot size. 4 A Singh et al Laser Phys. 27 (2017) 110001 Figure 3. Electron energy gain γ as a function of z for a0 = 10, b0 = 0.09, and r0 = 80 for laser pulse duration τ = 100, 300 and 500. Other parameters are the same as in figure 1. Figure 6. Acceleration gradient as a function of distance z for tightly focused chirped laser pulse with intensity parameter a0 = 5, 10, and 15. Other parameters are the same as taken in figure 1. intensity 1.38 × 1020 W cm−2. With the higher energy laser pulse, a steeper acceleration gradient is observed. Figure 5 shows electron energy γ as a function of nor malized time t for different laser intensities under the influence of axial magnetic field and combined axial and azimuthal magnetic field. It has been observed that electron energy gain is higher in the presence of combined magnetic field, as compared to that in the presence of axial magnetic field only. In circularly polarised laser, there is production of self generated axial and azimuthal magnetic fields. Hence, the application of combined axial and magnetic fields acts as a booster in order to accelerate the electron up to GeV energy. For a0 = 10, in the presence of axial magnetic field of ~21 MG, energy gain of about 1.4 GeV is observed, whereas in the presence of combined magnetic field of ~9.6 MG, energy gain is about 2.4 GeV. Electron dynamics may be analyzed by the study of acceleration gradient. Figure 6 shows the variation of acceleration gradient as a function of distance z. Acceleration gradient may be calculated from the relation dγ/dz = −e (β · E) /βz , which can be easily derived from the Lorentz equation itself. Here, energy gradient is calculated for different values of nor malized laser intensity parameter a0 = 5, 10 and 20, for optimum values of magnetic field as taken from figure 1(a). Other parameters are also the same as in figure 1. We have observed acceleration gradients of about 32 GeV m−1 due to tightly focused chirped laser for a0 = 5. For a0 = 10 and 20, values of acceleration gradient rise up to 52 GeV m−1 and 83 GeV m−1 respectively. Thus, for higher values of the laser intensity parameter, a steeper acceleration gradient is observed. Figure 4. Acceleration gradient dγ/dz and energy gain γ as a function of accelerating distance z with a0 = 10 and b0 = 0.09. Other parameters are the same as in figure 1. 4. Conclusion Resonant enhancement in the acceleration of electrons by circularly polarized laser pulses in the presence of azimuthal and axial magnetic fields has been investigated. Effective exchange Figure 5. Electron energy gain γ as a function of t for a0 = 10 and 20 for b0 = 0.09 and bz = 0.2. Other parameters are the same as in figure 1. 5 A Singh et al Laser Phys. 27 (2017) 110001 [11] Gopal A et al 2013 MegaGauss magnetic field generation by ultra-short pulses at relativistic intensities Plasma Phys. Control. Fusion 55 035002 [12] Lei Z, Quan-Li D, Jun W S, Zheng-Ming S and Jie Z 2010 Quasistatic magnetic fields generated by a nonrelativistic intense laser pulse in uniform underdense plasma Chin. Phys. B 19 078701 [13] Kostyukov I V, Shvets G, Fisch N J and Rax J M 2002 Magn etic field generation and electron acceleration in relativistic laser channel Phys. 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Phys. 18 20 [20] Ghotra H S and Kant N 2015 Electron acceleration by a chirped laser pulse in vacuum under the influence of magn etic field Opt. Rev. 22 539 [21] Melikian R 2014 Acceleration of electrons by high intensity laser radiation in a magnetic field Laser Part. Beams 32 205 [22] Ghotra H S and Kant N 2016 Polarization effect of a Gaussian laser pulse on magnetic field influenced electron acceleration in vacuum Opt. Commun. 365 231 [23] Singh K P and Malik H K 2008 Resonant enhancement of electron energy by frequency chirp during laser acceleration in an azimuthal magnetic field in a plasma Laser Part. Beams 26 363 [24] Liu C S and Tripathi V K 2001 Relativistic laser guiding in an azimuthal magnetic field in a plasma Phys. Plasmas 8 285 [25] Gupta D N, Singh K P and Suk H 2012 Cyclotron resonance effects on electron acceleration by two lasers of different wavelengths Laser Part. Beams 30 275 [26] Singleton J, Mielke C H, Migliori A, Boebinger G S and Lacerda A H 2004 The national high magnetic field laboratory pulsed-field facility at Los Alamos national laboratory Physica B: Condensed Matter 346 614 [27] Singh K P 2004 Electron acceleration by circularly polarized laser in a plasma Phys. Plasmas 11 3992 [28] Sung J H, Lee S K, Yu T J, Jeong T M and Lee J 2010 0.1 Hz 1.0 PW Ti:sapphire laser Opt. Lett. 35 3021 of energy between the electron and laser electric field takes place in the resonance region, leading to enhanced electron energy gain. This resonance region is found more interesting than other regions of lower energy gain for various applications. In order to achieve higher maximum energy gain (~GeV), laser and plasma parameters, along with the combined magn etic fields, are optimized. It is also observed that maximum electron energy gain increases for smaller laser spot size for the optimum value of magnetic fields. 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