Excitation of accelerating plasma waves by counter-propagating laser beams

Generation of accelerating plasma waves using two counter-propagating laser beams is considered. Colliding-beam accelerator requires two laser pulses: the long pump and the short timing beam. We emphasize the similarities and differences between the conventional laser wakefield accelerator and the colliding-beam accelerator (CBA). The highly-nonlinear nature of the wake excitation is explained using both nonlinear optics and plasma physics concepts. Two regimes of CBA are considered: (i) the short-pulse regime, where the timing beam is shorter than the plasma period, and (ii) parametric excitation regime, where the timing beam is longer than the plasma period. Possible future experiments are also outlined. INTRODUCTION AND MOTIVATION Plasma is an attractive medium for ultra-high gradient particle acceleration because it can sustain a very high electric field, roughly limited by the cold wavebreaking field EWB = mcωp/e ≈ √ n[cm−3]V/cm, where ωp = √ 4πe2n/m is the plasma frequency and n is the electron density. To accelerate injected particles to velocities close to the speed of light c, this electric field has to be in a form of a fast longitudinal plasma wave with phase velocity vph ≈ c. The frequency of the fast plasma wave is ωp, and its wavenumber is kp ≈ ωp/c. Excitation of such plasma waves can be accomplished by lasers or fast particle beams [1–3]. Below we review the basics of the linear plasma wave excitation in very general terms, without restricting ourselves to the specifics. Let’s assume that plasma electrons are subject to the electric field of the fast plasma wave E, as well as other nonlinear forces FNL, for example, the ponderomotive force of one or more 1) This work was supported by the US DOE Division of High-Energy and Nuclear Physics laser pulses. The total current J = Jp + J2 which enters Ampere’s law ∇ × B = (1/c)∂t E + (4π/c)( Jp + J2) is intentionally split into two components. The first one, Jp = −en ve, where ve is the electron fluid velocity, is driven by the electric field E and satisfies ∂t Jp = e n E. The second component J2 is driven by the nonlinear ponderomotive force, or could also represent an external current provided by injected electron beam. Taking the time derivative of the Ampere’s law yields: ( ∂ ∂t2 + ω p0 ) E + c2∇×∇× E = −4π J2 ∂t , (1) where the ∇×∇× E term naturally vanishes in 1D. One can say that the science of making a plasma accelerator is about finding the most effective way of producing the appropriate J2z(z, t). Of course, not every functional form of J2z(z, t) is useful for relativistic particle acceleration. In the rest of this paper we concentrate on using two counter-propagating laser beams to excite J2z(z − ct). I COMPARISON OF SINGLE-BEAM AND COLLIDING BEAM ACCELERATORS The simplest laser-driven plasma accelerator is the plasma beatwave accelerator [1] (PBWA). It employs a pair of co-propagating laser beams with normalized vector-potentials a0,1 = e A0,1/mc 2 and frequencies ω0 and ω1 = ω0 − ωp. The nonlinear current J2z is driven by the ponderomotive force of the resulting electromagnetic beatwave according to ∂tJ2z = en∂z( a0 · a1). If the two laser-beams are detuned by the plasma frequency ωp, plasma wave is resonantly excited. From Eq. (1), to excite a plasma wave, one needs to deposit momentum into the plasma. The source of this momentum is, of course, the laser. However, since the typical laser frequencies ω0,1 ωp, it is impossible for a laser photon to impart its entire momentum to the plasma. What happens instead is that the frequency of a laser photon is down-shifted by the amount ωp, depositing the remainder momentum and energy into the plasma. In the case of the PBWA, the higherfrequency photons at ω0 are scattered into the lower-frequency photons at ω1 = ω0 − ωp. Schematically, this process is shown in the top Fig. (1). The phasors of the lasers lie on the ω = ω p + c k dispersion curve, and the vector difference of these phasors gives the phasor of the driven plasma wave. The total rate of the momentum transfer to plasma in PBWA is then proportional to the relative momentum transfer per photon η = ωp/ω0, times the rate of scattering which is proportional to the beam intensity. Since the relative amount of down-shifting η 1, high laser intensities are needed to ensure the high overall rate of the momentum transfer. Note that Fig. 1 (top) is also applicable to the laser wakefield accelerator (LWFA) which employs a single ultra-short (τL ≈ 2ω−1 p ) laser pulse. Broad bandwidth of the pulse implies that it contains a continuum of frequency plasma wake las er ω