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Radiative Trapping in Intense Laser Beams

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Radiative Trapping in Intense Laser Beams Radiative trapping in intense laser beams J G Kirk Max-Planck-Institut f¨urKernphysik, Postfach 10 39 80, 69029 Heidelberg, Germany E-mail: [email protected] Abstract. The dynamics of electrons in counter-propagating, circularly polarized laser beams are shown to exhibit attractors whose ability to trap particles depends on the ratio of the beam intensities and a single parameter describing radiation reaction. Analytical expressions are found for the underlying limit cycles and the parameter range in which they are stable. In high-intensity optical pulses, where radiation reaction strongly modifies the trajectories, the production of collimated gamma-rays and the initiation of non-linear cascades of electron-positron pairs can be optimized by a suitable choice of the intensity ratio. PACS numbers: 12.20.-m, 52.27.Ep, 52.38.Ph Journal reference: Plasma Physics and Controlled Fusion 8, 085005 (2016) 1. Introduction The design of experiments on 10 PW laser facilities that will probe strong-field QED [1, 2, 3] requires an understanding of the behaviour of electrons and positrons in intense photon beams [4]. However, in all but the simplest configurations, such as a single plane wave [5], the dynamics are highly complex, displaying regions of both ordered and chaotic motion [6, 7, 8]. Radiative reaction plays a crucial role at these intensities, and has been implemented in numerical studies by several groups, using both a classical and a fully quantum approach [9, 10, 11, 12, 13, 14, 15]. Amongst other effects, it leads to a contraction of phase-space, which can cause particle trajectories to accumulate on attractors [8, 16, 17, 18, 19]. This \radiative trapping" is an important effect, because it may allow the electrons and the gamma-rays they radiate to be controlled. arXiv:1605.00822v2 [physics.plasm-ph] 8 Jul 2016 Attractors are closely related to limit cycles, which are periodic trajectories in phase space. When these are stable to infinitesimal perturbations, they correspond to a simple attractor, which confines trajectories that pass sufficiently close to the cycle. When they are unstable, an attractor with a more complicated structure (a strange attractor) may nevertheless be present in the immediate neighbourhood, and again lead to a concentration of particles in a restricted region of phase-space. The goal of this paper is to identify electron limit cycles analytically and to study their stability properties using both analytical and numerical tools. To achieve this, the treatment is restricted to classical electrodynamics. The chosen field configuration consists of two circularly Radiative trapping in intense laser beams 2 polarized, monochromatic, vacuum waves with arbitrary frequencies, amplitudes and propagation directions, but with opposite helicities. In classical electrodynamics, only two dimensionless parameters are needed to specify the relativistic dynamics in this set- up. These are the ratio λ of the wave amplitudes, and the classical radiation reaction parameter rc, defined in equation (15). To date, work in this field has concentrated on numerical integration of trajectories in a standing wave, which corresponds to λ = 1. In this case, circularly polarized waves exhibit an attractor only where the electric field vanishes, and strong-field QED effects, such as the production of hard gamma-rays and electron positron pairs, are negligible. In this paper, attractors are found on which the electric field amplitude and the particle energy are sufficient to permit strong-field QED effects in a limited range of λ. Expressions giving this range, the limit-cycle orbits and their linear stability properties as functions of λ and rc are presented in section 3. Following this, numerical integration of the equations of motion is used in section 4 to present stroboscopic sections of phase- space illustrating the character of the particle orbits close to these limit cycles. The remainder of the paper is organized as follows: section 2 introduces the notation, presents the field configuration and formulates the equations of motion, section 5 relates the results to previous work and briefly discusses some practical implications, and section 6 summarizes the conclusions. 2. Field configuration and equations of motion ~ Consider two monochromatic plane waves in vacuum, with wave vectors k1;2=c and ~ ~ ~ frequencies !1;2 = k1;2 . Assuming k1 6= k2, it is always possible to find a frame of reference in which these waves have the same frequency and oppositely directed wave vectors. In a system consisting of a single photon from wave 1, and a single photon from wave 2, this is the frame in which the total momentum is zero (the ZMF). In it, 1=2 p ~ ~ the waves have the frequency ! = !1!2 − k1 · k2 = 2. Our strategy is to look for limit cycles that lie entirely in the transverse plane, when viewed in the ZMF. On such orbits, the force exerted by the magnetic field must vanish. One obvious location is a plane in which the magnetic field itself vanishes, but this can occur only in very special configurations, such counter-propagating waves of equal amplitude and either aligned linear polarization or circular polarization of opposite helicity. More generally, one can search for orbits on which the particle velocity is always parallel to the magnetic field. Such orbits occur when the electric and magnetic field vectors corotate in the transverse plane such that their phase difference and amplitudes remain constant. In the remainder of this section, we show that this configuration is present in counter-propagating waves of arbitrary relative amplitude, provided they are of opposite helicity, and formulate the corresponding equations of motion. ~ The electric field is related to the dimensionless vector potential ~a by E1;2 = (mc=e) @~a1;2=@t, where m and e are the mass and charge of the positron. For circularly Radiative trapping in intense laser beams 3 polarized waves, one can choose a Lorentz gauge in which j~aj = constant and the electrostatic potential vanishes. The ratio of the magnitudes of the dimensionless vector potentials can be defined as λ = a2=a1, where, without loss of generality, 0 ≤ λ ≤ 1. Note that a1;2 and λ are Lorentz invariant, as is also the helicity of each wave. Thus, in terms of dimensionless independent variables (t ! t=!, x ! xc=k), the transverse components of the vector potentials in the ZMF can be written as a0 a = e±{(t−x+φ1) 1 2 λa0 a = e∓{(t+x+φ2) : (1) 2 2 (Here, and in the following, the complex notation a ≡ ~a · ~y^ + {~z^ is used for the transverse components of a vector.) In (1) the upper (lower) signs correspond to left- (right-)handed polarizations, the φ1;2 are arbitrary phases, and the x-axis is chosen to lie in the direction of propagation of the primary wave (wave 1). The constant a0, defined in equation (1) is the magnitude of the vector potential in the standing wave (λ = 1), and will be used below to normalize the particle four-momentum. At fixed x, the amplitude of the superposed fields is constant in time, provided waves of opposite helicity are selected. Selecting convenient phases, and choosing left-handed polarization for wave 1, the vector potential is a = a1 + a2 {t −{x {x = {a0e e + λe =2 : (2) Thus, in addition to the amplitudes, also the relative phase of the E~ and B~ vectors at a given location is constant in time: mc! a E~ = 0 1 + λ2 + 2λ cos (2x)1=2 e 2 mc! a B~ = 0 1 + λ2 − 2λ cos (2x)1=2 e 2 mc! 2 E~ · B~ = a2λ sin x cos x : (3) e 0 We now set m = c = 1 and write the (dimensionless) four-momentum, as (γ; ux; u?), 2 ∗ 1=2 where u? = uy + {uz and γ = (1 + ux + u?u? ) , and further introduce the canonical momentum in the transverse plane P? and its radial and angular components Pr and Θ according to {Θ P? ≡ sPre (4) = u? + sa ; (5) where s = q=e, with q the particle charge. In the absence of radiation reaction, the dynamics are Hamiltonian, with the three canonical momenta ux, Pr and Θ. The Radiative trapping in intense laser beams 4 Hamiltonian function H(ux;Pr; Θ; x) = γ a2 = 1 + u2 + P 2 + 0 1 + λ2 + 2λ cos 2x x r 4 1=2 + a0Pr [sin (t − Θ − x) + λ sin (t − Θ + x)] (6) does not contain the coordinates conjugate to Pr and Θ, indicating that these are constants of the motion. Ignoring the corresponding coordinates, the two remaining first-order equations of motion are dx = u /γ (7) dt x du 1 a P λa P x = λa2 sin x cos x + 0 r cos (t − Θ − x) − 0 r cos (t − Θ + x) : (8) dt γ 0 2 2 Interestingly, the choice Pr = 0 renders the Hamiltonian independent of time, so that the third constant of motion is γ and the system is integrable. To take account of radiation reaction, we adopt the approximation used in [20], in which the spatial components of the four-force, are anti-parallel to those of the four- momentum. This approximation arises from the leading term in an expansion in 1/γ of the radiation reaction force as formulated by [21] and is adequate to describe the orbits in large amplitude, circularly polarized waves. The radiation reaction terms are then d~u mc2 2 = − η2~u/γ; (9) dt RR h!¯ where 2e2! = ; 3mc3 and η is the parameter that determines the importance of QED effects, defined in terms of the electromagnetic field tensor, F µν, as µν η = jF uνj =Ecrit ; (10) 2 3 with Ecrit = m c =eh¯ the critical or Schwinger field.
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