The Backward Raman Amplifier (BRA) is proposed as a possible scheme for improving laser driven plasma wakefields. One- and two-dimensional particle-in-cell code simulations and a 3-Wave coupling model are presented and compared to demonstrate how the BRA can be applied to the laser wakefield accelerator (LWFA) in the non-relativistic regime to counteract limitations such as pump depletion and diffraction. This article provides a discussion on optimal parameters for the combination of BRA and LWFA and a prescription for a BRA pump frequency chirp to ensure coupling beyond the particle dephasing limit. Simulation results demonstrate a reduction or alleviation of the effects of diffraction and an increase in wake amplitude and sustainability and provide direct insights into new methods of controlling plasma wakes in LWFA and other applications.

The ability to create large amplitude plasma waves traveling near the speed of light using a laser pulse has led to several scientific breakthroughs such as laser wakefield acceleration (LWFA),1 betatron x-ray sources,2–4 and terahertz generation.5 In general, the amplitude and application of these plasma waves can be limited by laser diffraction, depletion, and particle dephasing in the LWFA. In the strongly nonlinear, blow-out regime of the LWFA, dephasing of the relativistic electrons from the accelerating fields, the small charge of the accelerated bunch, and the inability to accelerate positive charge for uses such as electron-positron colliders are limitations of this blow-out regime. Several different approaches have been proposed to overcome these drawbacks such as combining the blow-out and the direct laser acceleration scheme.6 Staged LWFA accelerators7 involve a pre-accelerated electron beam from the plasma “bubble” and a long wake field wave in the second acceleration stage. The alternative scenarios of LWFAs usually involve several laser beams and often employ linear wakes over a long acceleration length.8 Inevitably, they result in an increase in the particle beam emittance and come at a cost of increased size and complexity of the accelerator.

In this article, we propose another approach, using a Backward Raman Amplifier (BRA) to maintain the driving laser pulse and hence to enhance and control the wakefield generation. Backward Raman amplification and compression9–11 has been proposed as a laser amplification scheme; it has, however, seen mixed success in experimental demonstrations.12–14 The experimental efficiency of the BRA scheme is reported to be less than 10%, which is below various theoretical predictions.15 Clearly, the theoretical understanding of the BRA and of related physical processes remains incomplete. The role of wakefield generation that is emphasized in our paper is one of these physical processes that have not been discussed before in the context of the BRA. This in spite of the fact that wake generation is an inevitable feature of short pulse propagation in a plasma, which as we will demonstrate, causes a frequency shift of the seed pulse and thus affects the BRA coupling. While the overall goal for BRA until now has been to maximize laser pulse amplification, we consider its use as a control mechanism during plasma wake generation. Specifically, we examine the first application of BRA to amplify and sustain a short seed laser pulse while simultaneously enhancing wakefield generation in a plasma. This article will review short pulse propagation in the context of LWFA (Sec. II A), followed by a theoretical model for the BRA (Sec. II B), and present the results of combining the two (Sec. III A). We will describe the wake's effects on the BRA (Sec. III B) and use a chirped pump in the BRA scheme in order to enhance the wakefield generation (Sec. III C). Finally, our results are summarized in the conclusion (Sec. IV).

The two processes, wake generation and BRA, have been extensively studied on their own. Before examining their nonlinear coupling and interactions, we will first review the relevant properties of the linear wake and the backward Raman amplifier. The summary of known results presented below will be useful for further developments of the BRA for the wake field enhancement scenario.

The primary simulation tool in our studies is the relativistic particle-in-cell (PIC) code SCPIC16 which has been written on the basis of the code Mandor17 and has already been used in many high intensity laser-plasma applications for electron acceleration and LWFA.18–21 As in similar studies,22–24 we employ a moving window with the speed of light c. Simulations presented in this article will be labeled with a case number, corresponding to the seed parameters listed in Table I. In all BRA cases, the plane wave pump intensity is 1 × 1014 W/cm2, and the wave length is 1.064 μm. In addition to PIC simulations, we will consider a reduced description of the BRA based on a three wave coupling model. The wave coupling equations provide a useful description of the BRA which can be apply to long time and large distances of laser pump and seed interactions in cases where the multidimensional effects, related, for example, to laser pulse diffraction, are not dominant.

TABLE I.

List of all simulations cases.

Case123456
Density ne/nc 0.0035 0.015 0.05 0.0035 0.0035 0.0035 
Te (eV) 100 100 50 100 100 100 
f-number 30 30 40 15 13 10 
ZR (μm) 3204 3462 7193 801 602 356 
tpulse (fs) 30 30 100 30 30 30 
λp/2c (fs) 30 14.5 30 30 30 
γ0/ωpe 0.016 0.0126 0.009 0.016 0.016 0.016 
a0 0.0091 0.0091 0.011 0.0091 0.0091 0.0091 
a1 0.205 0.205 0.011 0.205 0.205 0.205 
Case123456
Density ne/nc 0.0035 0.015 0.05 0.0035 0.0035 0.0035 
Te (eV) 100 100 50 100 100 100 
f-number 30 30 40 15 13 10 
ZR (μm) 3204 3462 7193 801 602 356 
tpulse (fs) 30 30 100 30 30 30 
λp/2c (fs) 30 14.5 30 30 30 
γ0/ωpe 0.016 0.0126 0.009 0.016 0.016 0.016 
a0 0.0091 0.0091 0.011 0.0091 0.0091 0.0091 
a1 0.205 0.205 0.011 0.205 0.205 0.205 

The focus of our work is on short nonrelativistic pulses with a pulse duration, tpulse [full-half-width-maximum (FWHM)], comparable in spatial extent to the plasma wavelength ctpulseλp (with λp = 2π/kp and kp = ωpe/c, where ωpe is the plasma frequency and c is the speed of light). The primary effect of short pulses on the background plasma is wake generation in the form of a longitudinal plasma wave at wave length λp and phase velocity vp ≈ c. We denote in the following such pulses as “seed” pulses with the field amplitude a1. In the linear approximation, the plasma electron density perturbation associated with the wake, δnw, is given by1 

δnwn0(ζ,y,z)=c/vg14kpζdζsin[kp(ζζ)]ζ,y,z2|a1|2,
(1)

where ζ = xvg1t denotes the coordinate in the frame where the seed pulse moves with the group velocity vg1 in the x-direction, n0 denotes the background plasma electron density, the seed electric field amplitude is given by a1=eE1mecω1=8.55×1010(I1λ12[Wcm2μm2])1/2 in terms of the intensity of the short seed laser pulse I1 and its wavelength λ1 (λ1 = 1.13 μm, when ne = 0.0035nc, where nc is the critical electron density for this laser frequency), and kp=ωpe/vg1. The operator ζ,y,z2=ζ22+y22+z22 is taken along the axis. To illustrate the pertinence of this expression, Eq. (1), we compare it in Fig. 1(a) with results from SCPIC simulations for a seed pulse having initially a Gaussian envelope with full-half-width-maximum (FWHM) of tpulse = 30 fs in time (and ls = ctpulse ≈ 9 μm in length) and the peak seed pulse of amplitude a1 = 0.205 in a plasma with density ne/nc = 0.0035 (cf. Ref. 25). This simulation corresponds to case “1” (see Table I for all simulation cases discussed).

FIG. 1.

In the short pulse regime (case “1”) (see Table I): (a) 1D cut in x at y = 0 of fields and density from 2D PIC and theory, (b) longitudinal wakefield from PIC, and (c) transverse wakefield from PIC. In the self-modulated regime (case “2”): (d) 1D cut of fields and density from PIC, (e) longitudinal wakefield from PIC, and (f) transverse component of the wakefield from PIC. In both cases, the initial seed amplitude is a1 = 0.205, and the initial seed duration is 30 fs. Plasma densities are ne/nc = 0.0035 (case “1”) (a)–(c) and ne/nc = 0.015 (case “2”) (d)–(f), taken at time t = 1.16 ps in the short pulse regime and t = 13.19 ps in the self-modulated regime. Both PIC simulations were run with 30 cells/μm resolution and 9 particles/cell, and the simulation domains were 150 × 300 μm for case “1” and 200 × 200 μm for case “2.”

FIG. 1.

In the short pulse regime (case “1”) (see Table I): (a) 1D cut in x at y = 0 of fields and density from 2D PIC and theory, (b) longitudinal wakefield from PIC, and (c) transverse wakefield from PIC. In the self-modulated regime (case “2”): (d) 1D cut of fields and density from PIC, (e) longitudinal wakefield from PIC, and (f) transverse component of the wakefield from PIC. In both cases, the initial seed amplitude is a1 = 0.205, and the initial seed duration is 30 fs. Plasma densities are ne/nc = 0.0035 (case “1”) (a)–(c) and ne/nc = 0.015 (case “2”) (d)–(f), taken at time t = 1.16 ps in the short pulse regime and t = 13.19 ps in the self-modulated regime. Both PIC simulations were run with 30 cells/μm resolution and 9 particles/cell, and the simulation domains were 150 × 300 μm for case “1” and 200 × 200 μm for case “2.”

Close modal
FIG. 2.

Wake amplitude vs propagation distance reconstructed from subsequent moving window frames of 2D PIC simulation case “4.” The laser is focused to the start of the plasma at x = 100 μm and propagates from left to right, leaving a wake in the plasma. The theoretical spot size of the laser form is overlaid in black lines with a corresponding Rayleigh length ZR ≈ 800 μm. After propagating 3ZR, the wake amplitude is reduced to approximately 1/4 its initial amplitude.

FIG. 2.

Wake amplitude vs propagation distance reconstructed from subsequent moving window frames of 2D PIC simulation case “4.” The laser is focused to the start of the plasma at x = 100 μm and propagates from left to right, leaving a wake in the plasma. The theoretical spot size of the laser form is overlaid in black lines with a corresponding Rayleigh length ZR ≈ 800 μm. After propagating 3ZR, the wake amplitude is reduced to approximately 1/4 its initial amplitude.

Close modal

The generation of the wake is not a resonant process, but Eq. (1) predicts the maximum response for ls = λp/2. Alternatively, longer pulses ls > λp give rise to “self-modulated” (SM) solutions where the wake Langmuir wave is generated at the front of the laser pulse and couples resonantly to the seed via forward Raman instability.

The simulation shown in Figs. 1(d), 1(e), and 1(f), corresponding to case “2” (see Table I), illustrates the regime of a self-modulated seed pulse with a large-amplitude wake. Except for the higher plasma density, ne/nc = 0.015, resulting in a shorter plasma wave length λp, case “2” has the same conditions as case “1.”

To obtain sufficiently high intensities for wakefield generation, laser pulses must first undergo optical focusing. As a result of focusing, the spot size radius of the laser in a vacuum evolves as can be described by r(x)=r01+x2/ZR2, where r(x) is the radius of the laser spot size that depends on the propagation distance x and on the Rayleigh length ZR determined by ZR = πf2λ, where f is the mirror's f-number and λ the laser wave length. The Rayleigh length indicates the distance of laser propagation where the laser spot size is doubled in area, reducing its intensity by a factor of 2/2/1 in 3D/2D/1D. Since the amplitude of the wake depends on a12 in Eq. (1), optimum wake generation requires the driving pulse to be focused close to its diffraction limit r0 over as long a distance as possible. This can be achieved through the use of large f-number optics; however, physical constraints on the focusing distance limit the maximum f-number available in laboratory, and therefore, pulse diffraction can be the significant limiting factor in non-relativistic wakefield acceleration, as seen in Fig. 2.

Although electron acceleration in the wake fields will not be discussed at length, we consider a dephasing time and the length as a characteristic physical constraint for the BRA coupling. Depending on the plasma density and laser wavelength, particle dephasing can occur when particles are accelerated to velocities higher than the phase velocity of the wake. These particles eventually overshoot the accelerating phase and are decelerated. Particle dephasing places a length constraint in the region of acceleration. For a highly relativistic electron, in a wake of radial extent much greater than λp, the linear dephasing length can be calculated using the relative velocity difference between the particle and the wake, the particle beam moves at velocity approximately c, while the wake's phase velocity vph=c1ne/nc. This results in the dephasing constraint (cvph)td=λp/2. Defining the dephasing length as Ld = ctd results in λp/2=Ld[1(1λ2/λp2)1/2] being Ldλ2/2λp2 for ne/nc ≪ 1, so that the dephasing length can be approximated as1Ldλp3/λ2. For a given laser wavelength and plasma density, the dephasing length is a constant barrier on particle acceleration that is underutilized due to the diffraction of low f-number lasers. To ensure that the wakefield is maintained over the extent of the dephasing length, we propose the use of a backward Raman amplifier which allows for corrections to be applied to the wake generating laser during its propagation in the plasma.

Backward Raman Amplification (BRA) involves the use of a long pump pulse (ω0, k0) counterpropagating with respect to the short seed pulse (ω1 = ω0ωpe and k1 = kpk0). When the two lasers overlap, they beat at the plasma frequency, exciting a Langmuir wave (ωL = ωpe and kL ≈ 2k0kp) that resonantly couples to the seed. Energy flows from the pump to the seed, resulting in amplification and compression of the seed.

BRA is typically modeled in 1D with a set of three coupled equations for the slowly varying field envelopes of the waves, the pump wave a0, the seed a1, and the plasma (Langmuir) wave aL.10 We will compare in the following the results from PIC simulations of the BRA with solutions of this 3-wave coupling model defined by the following set of equations:10 

[τ(|vg0|+vg1)ζ]a0=γ0aLâ1,
(2)
τâ1=γ0a0aL*,
(3)
[τvg1ζ+ν]aL=γ0a0â1*,
(4)

where a0,1eE0/(me0,1) are the normalized amplitudes of the electromagnetic fields E0,1 of the pump (0) and the seed (1) wave, with â1=(ω0/ω1)1/2a1 and with (ω0,1,k0,1) as the corresponding frequencies and wave vectors; aL=(ωL/ω0)1/2eEL/(mecωp) denotes the normalized amplitude of the Langmuir wave electric field, EL, with (ωL,kL), and the coupling constant reads γ0=(kLc/4)ωp/(ωLω1)1/2. The equations are written in the stationary frame of the seed pulse, using the variables ζ = xvg1t and τ = t, where vg0 and vg1 are the pump and seed wave group velocities, and the Langmuir wave group velocity has been neglected. The damping coefficient ν has been added to the Langmuir wave equation (4). Its functional form and magnitude can simply be the linear Landau damping νL=(πωpe3/2k2)fM/v|v=ωL/k evaluated at the Maxwellian distribution function, fM. On the long time scale, ν may be modified due to nonlinear evolution of Langmuir waves that may include electron trapping and wave coupling. Such long time effects do not usually arise during BRA coupling for short seed pulse durations as in the example of Fig. 3.

FIG. 3.

Comparison of field amplitudes between PIC simulation (color and broadened lines as a result of enveloping the fields) and 3-wave coupling model (solid black lines): line (2) for the seed (with amplitudes × 1/2), line (1) for the laser pump, and line (3) for the Langmuir wave, at interaction time t = 3.57 ps. Amplitudes are normalized to the incoming pump amplitude. The PIC simulation was run with a simulation domain of 400 μm, 135 cells/μm resolution, and 1024 particles/cell.

FIG. 3.

Comparison of field amplitudes between PIC simulation (color and broadened lines as a result of enveloping the fields) and 3-wave coupling model (solid black lines): line (2) for the seed (with amplitudes × 1/2), line (1) for the laser pump, and line (3) for the Langmuir wave, at interaction time t = 3.57 ps. Amplitudes are normalized to the incoming pump amplitude. The PIC simulation was run with a simulation domain of 400 μm, 135 cells/μm resolution, and 1024 particles/cell.

Close modal

Figure 3 illustrates the comparison between PIC simulation and the 3-wave coupling model for parameters corresponding to case “3,” leading to the π-pulse solution10 of Stimulated Raman Scattering (SRS). The seed pulse amplitude at time t = 3.57ps has already experienced amplification with respect to its initial amplitude a1 = 0.011. Amplification is due to the pump laser—propagating from the right to the left—initially at a0 = 0.011 through coupling with the Langmuir wave. The seed pulse duration tpulse is 100 fs at FWHM, and the plasma density and electron temperature are ne/nc = 0.05 and Te = 50 eV, respectively. The example of case “3,” and previous studies based on PIC simulation with similar parameters,22,24,27 has been performed in order to validate 3-wave coupling models. Among these studies, Ref. 27 dealt with long time self-similar evolution of 3-wave solutions10 albeit under idealized conditions of cold background plasma.

The good agreement between the 3-wave coupling model (2), (3), and (4) and PIC simulation shown in Fig. 3 characterizes the initial short time evolution of BRA. As we will discuss below, plasma wake generation and subsequent seed pulse frequency shift and self-modulation of the relatively long electromagnetic pulse are among nonlinear effects that limit the application of a straightforward 3-wave coupling model. Such limitations motivate a need for PIC simulations when the reduced wave coupling models of BRA are derived.

The different modes in k-space are illustrated in Fig. 4 that shows the electric field spectrum with electrostatic components in orange and electromagnetic components in blue: the laser pump mode at k0, the wake at kp = ωpe/c, the seed at k1 = k0kp, and the Langmuir waves from Raman coupling kL ≈ 2k0kp. The one dimensional Fourier transform is taken in space and is presented as “FFT(Ey)” and “FFT(Ex)” for the Fourier transform of Ey and Ex, respectively, into kx.

FIG. 4.

Electric field power spectrum vs wavenumber, showing electrostatic components in orange and electromagnetic components in blue. (1) Wake at the plasma wavenumber kp = ωpe/c, (2) seed at k1 downshifted by kp from the pump, (3) pump defined to be k0, and (4) Langmuir waves from Raman coupling kL ≈ 2k0kp. Simulation parameters correspond to case “2.”

FIG. 4.

Electric field power spectrum vs wavenumber, showing electrostatic components in orange and electromagnetic components in blue. (1) Wake at the plasma wavenumber kp = ωpe/c, (2) seed at k1 downshifted by kp from the pump, (3) pump defined to be k0, and (4) Langmuir waves from Raman coupling kL ≈ 2k0kp. Simulation parameters correspond to case “2.”

Close modal

In this section, we will demonstrate the control and enhancement of the laser produced wakefield by combining it with the backward Raman amplifier scheme. For this purpose, we present the results of simulations where we bring together both effects, BRA and wakefield, in order to optimize and sustain the laser generated wakefield. We first present the results of 2D PIC simulations for the plasma conditions corresponding to case “1” from Table I except for the plane wave limit of the seed and the pump pulses. The discussion of these results will introduce important physical processes related to BRA and wake coupling. Next, we will consider BRA as the mechanism that overcomes diffraction of the seed pulse and extends wake generation to at least the timescales on the order of the particle dephasing length (Sec. III A). In Sec. III B, we will present results on timescales much greater than the dephasing length and consider the effects of wakefield generation on the resonance coupling of the BRA. We will also revise a theoretical model for both laser wakefield generation and backward Raman amplification and consider the chirp of the pump (cf. Sec. III C) as the mechanism mitigating frequency and wavelength dephasing due to the wake generation.

The results below correspond to a 30 fs FWHM driving a seed laser pulse of intensity 5 × 1016 W/cm2 at best focus and a flat top plane wave pump of intensity 1 × 1014 W/cm2, in a plasma of density ne = 0.0035nc. The parameters are chosen such that the seed efficiently excites plasma waves of the wake (ctseed = λp/2) in the non-relativistic regime (a1 = 0.205), and the pump intensity is sufficiently low such that it does not produce backscattered SRS from the particle noise before the interaction with the seed pulse. The spatial SRS gain coefficient of a pump wave, G=Lxγ02|a0|2/|νLvg1|, requires the length Lx ≈ 70 mm to reach G > 1 values for the parameters of this example. This estimate corresponds to the background electron temperature of 100 eV and the SRS Langmuir wave of the BRA at kL ≈ 0.46kD. Recent work26 has proposed the use of a “flying focus” on the BRA pump to combat parasitic precursors, and such a setup could possibly support larger pump intensities and hence larger amplification of the seed.

While the low electron density of this example is an optimal choice for the wake generation by the 30 fs laser pulse, the electron temperature of 100 eV is consistent with the experimental conditions of gas jet plasmas.28 The resulting large ratio kL/kD ≈ 0.46 of the BRA Langmuir wave characterizes the kinetic regime of the SRS29,30 and of the Langmuir wave nonlinear evolution.31 This leads to strong linear Landau damping, νL ≈ 0.14ωpe, and in the nonlinear regime to electron trapping and to trapped particle modulational instability.29–31 However, the nonlinear physics of Langmuir wave evolution—clearly identifiable in simulation results below—becomes important only on time scales longer than the short seed pulse duration, and therefore, it has negligible effects on the BRA and the wakefield generation. Consistently, simulations run at different temperatures (50 eV, kL ≈ 0.32kD and 200 eV, and kL ≈ 0.65kD) show only small variations in the resulting seed and wake amplitudes [see Figs. 9(a) and 9(b)].

Figure 5 shows the results of a 2D plane wave seed and pump simulation with parameters of case “1,” the transverse dimension of 10 μm, and the transverse periodic boundary conditions. The three panels in Fig. 5 illustrate the main physical processes involved in the simultaneous BRA and the wakefield generation. Figure 5(a) displays the longitudinal electric field of the Langmuir waves due to SRS at kL ≈ 0.46kD and the wake at kp = ωpe/c ≈ 0.014kD. Superimposed is the 1D cut at y = 5 μm of the transverse electric field due to the seed and the pump lasers. At the time of Fig. 5, t = 8 ps, the wake is well developed and extends hundreds of microns behind the seed. We have estimated the bounce frequency of trapped electrons in the SRS driven Langmuir wave. At the maximum amplitude of the Langmuir wave electric field such that eEL/mcωL ≈ 0.02 (see Fig. 6), the bounce frequency ωb=(kLeEL/m)1/29.3×10131/s, and therefore, the characteristic trapping time τb = 2π/ωb = 67.5 fs is longer than the time of the seed pulse duration and the BRA coupling. On the other hand, the freely propagating (to the left) SRS Langmuir waves experience effects of the trapped particles which result in the transverse modulations of their wavefronts on the longer time scale corresponding to the trapped particle modulational instability.29–31 This is seen on the left of Fig. 5(a), far behind the leading seed pulse, and in the two dimensional Fourier transform of the longitudinal fields in Fig. 5(b). The two dimensional Fourier transforms are taken in space and are presented as “fft2(Ex)” and “fft(Ey)” for the Fourier transform of Ex and Ey, into kx and ky, respectively (see the figure label). The spectra are calculated over the whole length and width of the simulation window, and in addition to the wake (kx = kp = 0.059k0 and ky ≈ 0) and the SRS Langmuir wave (kx = kL = 1.94k0 and ky ≈ 0), one can see broad continua of transverse components, resulting from the trapped particle modulational instability29–31 and the second harmonic of the kL-wave. The detailed analysis of the spectra about kx ≈ 2k0 has revealed additional components at kL ± nkp and ωL ± p [cf. inset in Fig. 5(b)] which result from the coupling between the strong wakefield wave (kp, ωp) and the SRS driven Langmuir wave. Such coupling takes time, and we have found no evidence that it actually affects the BRA interaction due to short seed pulse duration. These extra spectral components in the Langmuir wave fields can modify damping and contribute to frequency shifts, but again, only on the longer time scale than the Raman amplification process.

FIG. 5.

From 2D plane wave PIC simulation with conditions of case “1”: (a) snapshot of the longitudinal electric field in the xy-plane at t = 8.0 ps (x: propagation direction; the red/blue color corresponds to positive/negative field values, respectively); the superposed black line shows the transverse field of the seed pulse and pump from a cut at y = Ly/2; (b) spectrum of the electrostatic field in the kx-ky plane with a subfigure showing a line-out at ky = 0, both at the same time instant as in panel (a); and (c) spectrum of the electromagnetic field (summed over ky) as a function of time. The PIC simulation was run with a simulation domain of 10 × 200 μm, 40 cells/μm resolution and 16 particles/cell.

FIG. 5.

From 2D plane wave PIC simulation with conditions of case “1”: (a) snapshot of the longitudinal electric field in the xy-plane at t = 8.0 ps (x: propagation direction; the red/blue color corresponds to positive/negative field values, respectively); the superposed black line shows the transverse field of the seed pulse and pump from a cut at y = Ly/2; (b) spectrum of the electrostatic field in the kx-ky plane with a subfigure showing a line-out at ky = 0, both at the same time instant as in panel (a); and (c) spectrum of the electromagnetic field (summed over ky) as a function of time. The PIC simulation was run with a simulation domain of 10 × 200 μm, 40 cells/μm resolution and 16 particles/cell.

Close modal
FIG. 6.

From 2D plane wave PIC simulation with conditions of case “1”: Comparison of field amplitude between PIC simulation (color and broadened lines as a result of enveloping the fields) and 3-wave coupling model (solid black line): line (1) for the pump + seed and line (2) for the Langmuir wave (with amplitudes × 100), at interaction time t = 2.61 ps. Amplitudes are normalized to the initial seed amplitude. The PIC simulation was run with a simulation domain of 10 × 200 μm, 40 cells/μm resolution, and 16 particles/cell.

FIG. 6.

From 2D plane wave PIC simulation with conditions of case “1”: Comparison of field amplitude between PIC simulation (color and broadened lines as a result of enveloping the fields) and 3-wave coupling model (solid black line): line (1) for the pump + seed and line (2) for the Langmuir wave (with amplitudes × 100), at interaction time t = 2.61 ps. Amplitudes are normalized to the initial seed amplitude. The PIC simulation was run with a simulation domain of 10 × 200 μm, 40 cells/μm resolution, and 16 particles/cell.

Close modal

The focus of this work is on coupling between two nonlinear processes, wakefield generation and BRA. We will demonstrate that wakefield generation can be extended over large plasma distances due to the BRA enhancing the amplitude of the seed pulse. On the other hand, the presence of several spectral components in the electrostatic fields can lead to deleterious effects such as producing different components at kL ± nkp and ωL ± p or adding frequency shifts and instabilities due to trapped particles. These effects influence the SRS Langmuir wave on the long time scale. We have also found that the time dependent density modifications due to the wake produce k-vector and frequency shifts of the electromagnetic seed pulse. This effect is illustrated in Fig. 5(c) and will become central to our discussions in Secs. III A–C [cf. Fig. 13(b)]. Despite our focus on short seed pulses (that extend half of a plasma wavelength) to maximize wakefield generation, the wake Langmuir waves are always created by high intensity femtosecond laser pulses, and therefore, the BRA schemes should verify whether the wake will introduce frequency and wavelength shifts and account for these decoupling effects in the reduced models based on the three wave approximation.

The characteristic feature of the electrostatic field spectra is large separation between the wake at kp and the SRS Langmuir wave at kL (cf. spectral components 1 and 4 in Fig. 4, respectively). On the other hand, the electromagnetic field components of the seed pulse (marked as 2 in Fig. 4) and of the pump (number 3) can overlap for lower densities than in case “2” (see table) of Fig. 4. This has been the condition of the current case “1” at ne/nc = 0.0035. Therefore, in the comparison between the three wave coupling model, (2), (3), and (4), and PIC simulations in Fig. 6, we have plotted the seed and pump wave amplitudes together. Note the remarkable agreement for the Langmuir wave amplitudes near the vicinity of the seed and discrepancy on the longer distance scale due to nonlinear effects including particle trapping and coupling to the wake. The three wave coupling model in Fig. 6 was solved with the Langmuir wave damping coefficient, ν = νL = 0.14ωp. In general, we have found a weak dependence at the early time, i.e., until the first maximum in the Langmuir wave amplitude, on the damping coefficient ν (cf. also Ref. 30).

For optimal BRA coupling, the seed pulse duration should be long enough such that its spectral bandwidth is localized to the BRA resonance region; however, optimal wake generation requires the seed pulse duration to be of the order of half a plasma wavelength, ctpulseλp/2, and the SRS growth rate increases with plasma density (shorter plasma wavelength). A chirped pump has previously been used to extend coupling in the BRA to all spectral components of a short seed pulse.32 The effect of the broad seed pulse spectrum was examined in PIC simulations,33 showing Brillouin coupling34 in the BRA amplification and compression scheme. Since our seed pulses are short and of relatively low intensity, Brillouin coupling is expected to be unimportant. Our plasma density and seed pulse length parameters were chosen to be in a demonstrated region of optimal amplification,23 to be experimentally relevant, and to balance the effects of density on BRA coupling.

Figure 7 shows the layout of the f/15 simulations (case “4”), where the laser propagation is in the x-direction and polarization is in the y-direction. In Fig. 7, we compare the fields resulting from a simulation without BRA, Ex in Fig. 7(a) and Ey in Fig. 7(b), to the simulation with BRA [(Figs. 7(d) and 7(e), respectively], for case “4.” Figures 7(c) and 7(f) show the difference in each field component between both cases, namely, ΔEx(y) = Ex(y) (with BRA)–Ex(y) (without BRA). From the quantity ΔEy in Fig. 7(c), it can be seen that the seed pulse is enhanced as it passes through the plane wave pump (oscillation amplitude ∼ 9 × 10−3 outside the wake region), leaving a trail of depletion in the plane wave pump. Figure 7(f) shows enhancement to the wake due to the amplification of the seed.

FIG. 7.

Simulation results with f/15 at t = 0.776 ps. Laser polarization is in the y-direction and propagation is in the x-direction. (a) Ey with BRA, (b) Ey without BRA, (c) difference in Ey between the subcases with and without BRA, (d) Ex with BRA, (e) Ex without BRA, and (f) difference in Ex between the subcases with and without BRA. Subplots (c) and (f) are obtained by ΔEx(y)=Ex(y)(withoutBRA)Ex(y)(withBRA). Both PIC simulations of case “4” were run with a simulation domain of 150 μm × 300 μm, 30 cells/μm resolution, and 9 particles/cell.

FIG. 7.

Simulation results with f/15 at t = 0.776 ps. Laser polarization is in the y-direction and propagation is in the x-direction. (a) Ey with BRA, (b) Ey without BRA, (c) difference in Ey between the subcases with and without BRA, (d) Ex with BRA, (e) Ex without BRA, and (f) difference in Ex between the subcases with and without BRA. Subplots (c) and (f) are obtained by ΔEx(y)=Ex(y)(withoutBRA)Ex(y)(withBRA). Both PIC simulations of case “4” were run with a simulation domain of 150 μm × 300 μm, 30 cells/μm resolution, and 9 particles/cell.

Close modal

Figure 8 shows the time evolution of the seed's energy density with and without BRA. Without BRA, the f/15 seed quickly diffracts, resulting in much lower wake amplitude. With BRA, the f/15 seed still diffracts [Fig. 8(b)], and however, the energy density is nearly maintained by the BRA, allowing the seed to continue to produce a large wake.

FIG. 8.

Evolution of the energy density of an f/15 seed in a plasma vs propagation time and transverse space. (a) With no amplifier. (b) With a Raman amplifier. The laser is at maximum focus at t = 0.2 ps. Simulation parameters correspond to those of case “4.”

FIG. 8.

Evolution of the energy density of an f/15 seed in a plasma vs propagation time and transverse space. (a) With no amplifier. (b) With a Raman amplifier. The laser is at maximum focus at t = 0.2 ps. Simulation parameters correspond to those of case “4.”

Close modal

In the f/30 case, Figs. 9(a) and 9(d), the amplifier operates faster than the pulse diffraction. This leads to significant enhancement of the pulse and wake until decoupling from the wake induced frequency shift. In the f/15 case, Figs. 9(b) and 9(e), the amplifier provides additional energy, but at early times, the pulse diffracts quicker than it can be replenished. In the f/15 case, the competition between diffraction and amplification reaches a balance, and the wake is maintained at nearly the same or higher amplitude than the unamplified pulse as shown in Fig. 9(e). In the higher density plasma of case “2,” the seed experiences modulation by its wake [Fig. 1(d)] which—despite diffraction—results in a focusing that increases the seed amplitude with or without the BRA. When a BRA is applied to case “2,” there is improved BRA coupling due to the higher plasma density, and this results in a large enhancement to both the seed and wake amplitude as shown in Figs. 9(c) and 9(f).

FIG. 9.

Comparison of the seed and wake amplitudes with (red) and without (blue) BRA. (a) Seed amplitude comparison f/30, (b) seed amplitude comparison f/15 with theory from Eq. (5) shown by the dashed black line, (c) seed amplitude comparison SM f/30, (d) wake amplitude comparison f/30, (e) wake amplitude comparison f/15, and (f) wake amplitude comparison SM f/30. Simulation parameters correspond to those of case “1” for (a) and (d), case “4” for (b) and (e), and case “2” for (c) and (f).

FIG. 9.

Comparison of the seed and wake amplitudes with (red) and without (blue) BRA. (a) Seed amplitude comparison f/30, (b) seed amplitude comparison f/15 with theory from Eq. (5) shown by the dashed black line, (c) seed amplitude comparison SM f/30, (d) wake amplitude comparison f/30, (e) wake amplitude comparison f/15, and (f) wake amplitude comparison SM f/30. Simulation parameters correspond to those of case “1” for (a) and (d), case “4” for (b) and (e), and case “2” for (c) and (f).

Close modal

A simple model that is consistent with PIC simulations is proposed to describe the competition between diffraction effects and the BRA. Assuming that the seed pulse is well approximated by the Gaussian solution to the wave equation during the BRA interaction, we can model the intensity evolution of the seed, taking into account both the effects of diffraction and the BRA. Typical results of the BRA for the wake generation are illustrated in Fig. 9. Apart from case “3,” all our BRA simulations are in the superradiant regime9,35,36 (ωpe2ω0a0a1) where the laser pump is strongly depleted as it transfers energy to the seed. This can be clearly observed in Fig. 7(c), displaying the depleted pump behind the right propagating seed pulse. With the parameter 0 ≤ α ≤ 1 being the efficiency of the energy flow from the pump to the seed, the incremental energy (dE) transfers from the pump to the seed during time dt is dE1=dE0=2αI0A1(t)dt. Where A1(t) is the time dependent spot size area of the seed laser, the factor of 2 arises due to the counterpropagation of the pump and seed. Integrating these results in order to obtain the seed intensity evolution gives

I1(t)=[2αcls0tI0(t)[1+(ctZR)2]d12dt+I1(0)][1+(ctZR)2]1d2,
(5)

where d indicates the spatial dimension of the solution, i.e., d = 2 or = 3 for 2D or 3D respectively, ls = ctpulse, and ZR is the Rayleigh length of the seed pulse. The solution to Eq. (5) with α = 0.45 efficiency and constant pump intensity I0 = 1 × 1014 W/cm2 shows good agreement with PIC simulation case “4” in Fig. 9(b). We can estimate the energy transfer from the pump to the seed in order to balance intensity loss due to diffraction. To determine the pump intensity constraint, one can set I1t=0 to get the solution

I0(t)=I1(0)d12lsαZR[1+(ctZR)2]1ctZR.
(6)

One can verify this result by substituting Eq. (6) into Eq. (5) to confirm I1(t) = I1(0). Since diffraction results in a dynamic loss in seed intensity, the pump requirements I0(t) (6) are not constant in time but can be evaluated near the seed's Rayleigh length (t  ZR/c, where losses are the strongest) to find a constant pump intensity that ensures that diffraction is always overcome. As seen in PIC simulation case “4” [cf. Fig. 9(b)], Eq. (6) (with α = 1) predicts a BRA pump of intensity 3 × 1014 W/cm2, while our BRA pump intensity is 1 × 1014 W/cm2, and hence, it is impossible for our pump to overcome diffraction near the Rayleigh length (simulation time, t ≈ 4 ps), but our pump is more than sufficient to maintain the pulse at later times.

Figure 10 summarizes the results of 2D PIC simulations of BRA applied to LWFA. Efficient particle acceleration can be limited by diffraction of the pulse but is ultimately dictated by the particle dephasing length. For a range of experimentally relevant f-numbers, Fig. 10 demonstrates that BRA applied to LWFA can help maintain the wake amplitude of diffracting lasers and allow for better or full use of the particle dephasing limit. Simulation results in our standard parameters (cases “1,” “4,” “5,” and “6”) show that when a BRA pump of intensity I0 = 1 × 1014 W/cm2 is applied to seed lasers with the best-focus intensity I1(0) = 5 × 1016 W/cm2, the effects of diffraction are overcome for f-numbers of 15 or larger, but enhancement cannot continue indefinitely. In Sec. III B, we will discuss a new saturation mechanism of the BRA in context to wake generation.

FIG. 10.

Limiting lengths for particle energy gain (in 2D) as a function of the seed pulse aperture (f-number) for a 30fs driving pulse in a plasma of density ne = 0.0035nc. LWFA without BRA is restricted to the green region; when a BRA is applied, diffraction is overcome allowing access to the yellow region. Markers indicate results from PIC simulations (run until the dephasing time t ≈ 17 ps). The diffraction limit is defined to be the length at which diffraction causes the wake amplitude to be ≤75% of its maximum unamplified value. The 3D diffraction limit without BRA is shown by the dashed line; the dephasing length remains constant in 3D.

FIG. 10.

Limiting lengths for particle energy gain (in 2D) as a function of the seed pulse aperture (f-number) for a 30fs driving pulse in a plasma of density ne = 0.0035nc. LWFA without BRA is restricted to the green region; when a BRA is applied, diffraction is overcome allowing access to the yellow region. Markers indicate results from PIC simulations (run until the dephasing time t ≈ 17 ps). The diffraction limit is defined to be the length at which diffraction causes the wake amplitude to be ≤75% of its maximum unamplified value. The 3D diffraction limit without BRA is shown by the dashed line; the dephasing length remains constant in 3D.

Close modal

In this section, we examine the BRA applied to LWFA at times exceeding particle dephasing. For our standard parameters (cases “1” and “4”), particle dephasing happens at t ≈ 17 ps (t ≈ 1.5 ps for SM case “2”); however, BRA and LWFA applications that are not concerned about dephasing (ex: betatron x-ray sources) may still take advantage of the enhancement via BRA.

As shown in Figs. 9(a) and 9(d), at late times, the BRA eventually decouples from the seed, resulting in a loss of seed and wake amplitude. This decoupling can be attributed to the frequency and wavelength shifts in the seed pictured in Fig. 11.

FIG. 11.

Longitudinal wavenumbers of electromagnetic fields vs time from 2D PIC simulations: (a) case “1” and (b) case “2,” showing a decoupling between the pump and the seed attributed to the red-shifting of the seed. Line 1 is the pump at k0, and line 2 shows the mean wavenumber of the redshifting seed.

FIG. 11.

Longitudinal wavenumbers of electromagnetic fields vs time from 2D PIC simulations: (a) case “1” and (b) case “2,” showing a decoupling between the pump and the seed attributed to the red-shifting of the seed. Line 1 is the pump at k0, and line 2 shows the mean wavenumber of the redshifting seed.

Close modal

It has been shown37 that a short laser pulse that propagates in plasma with a density profile due to the wake's Langmuir wave, δnw/n0, will develop frequency shifts that can be described by

δk1=ωp2τ2ω1n0δnwζ,δω1=ωp22ω1δnwn0+δk1vg1,
(7)

with δk1 = xΦ = ζΦ and δω1 = –tΦ, where Φ(ζ,τ)=ωp2δnwτ/(2ω1n0). The characteristic linear dependence of δk1 in time, τ = t, is clearly seen in SCPIC simulations and well reproduced by Eq. (7). The resulting red shift in the wavelength and frequency follows from the geometry of the wake Langmuir wave, i.e., the seed pulse propagates together with the front of the wake experiencing constant and predominantly positive density gradient [cf. Eq. (7)]. The theoretical curve in Fig. 12(b) has been calculated using Eqs. (1) and (7), by taking the average of the wave-vector shift (7) over the seed pulse length

δk1=ls10lsdζδk1(ζ)=k0ωpt6π2nenc|a1(0)|2,
(8)

where we assumed kpls = π and vg1c.

FIG. 12.

In the short pulse regime (case “1”): (a) fields and density from 1D PIC and theory at t = 4.9 ps and theoretical curves overlaid in black and (b) spectrum of the transverse field from 1D PIC with theory from Eq. (8) shown in black. In the self-modulated regime (case “2”): (c) fields and density from 1D PIC at t = 17.2 ps and (d) spectrum of the transverse field from 1D PIC. In both cases, the initial seed amplitude is a1 = 0.205, and the initial seed duration is 30 fs. Plasma densities are ne/nc = 0.0035 and ne/nc = 0.015. Both PIC simulations were run with a simulation domain of 600 μm, 60cells/μm resolution, and 128 particles/cell.

FIG. 12.

In the short pulse regime (case “1”): (a) fields and density from 1D PIC and theory at t = 4.9 ps and theoretical curves overlaid in black and (b) spectrum of the transverse field from 1D PIC with theory from Eq. (8) shown in black. In the self-modulated regime (case “2”): (c) fields and density from 1D PIC at t = 17.2 ps and (d) spectrum of the transverse field from 1D PIC. In both cases, the initial seed amplitude is a1 = 0.205, and the initial seed duration is 30 fs. Plasma densities are ne/nc = 0.0035 and ne/nc = 0.015. Both PIC simulations were run with a simulation domain of 600 μm, 60cells/μm resolution, and 128 particles/cell.

Close modal

Note that the dramatic frequency cascading of the self-modulated pulse in Fig. 12(d) is entirely due to the coupling between the wake and subsequent stages of forward Raman scattering. This pulse never reaches intensities that would lead to relativistic self-focusing that is usually associated with the self-modulated regime.38 

To model the frequency shifts in the seed, the system of the 3-wave coupling equations,10 Eqs. (2)–(4), has to be modified by introducing in Eq. (3) the frequency shift term δω. Equation (3) in the system Eqs. (2)–(4) is hence replaced by

[τ+iδω]â1=γ0a0aL*,
(9)

where δω=ωpe22ω1δnwn0. This modification is necessary to account for the effects of the plasma wake on the BRA. For very short seed pulses as proposed in Refs. 9 and 39, the wake can have dramatic effects, leading to the loss of resonance between BRA modes.

Stationary solutions to Eqs. (2), (9), and (4) closely reproduce PIC simulations and are displayed in Fig. 13(b) to illustrate the need for including wake generation into theories involving short laser pulses such as BRA.

FIG. 13.

For the case “1” with a constant pump: (a) fields â1 and aL from PIC at t = 40 ps and (b) spectrum of the transverse electric field (containing both pump and seed components) from PIC and from theory: line 1 for the constant pump (line at k = k0), line 2 for the solution to Eq. (10) obtained from the amplitude of the seed in PIC, and line 3 for the solution to Eq. (10) obtained from a numerical solution to the 3-wave coupling model. For the ideally chirped pump: (c) fields â1 and aL from PIC at t = 75 ps and (d) spectrum of the transverse electric field from PIC: line (1) for pump the field spectrum prescribed by the matching condition and solid line (2) for the seed spectrum obtained from Eq. (11). Both PIC simulations were run with a simulation domain of 600 μm, 60 cells/μm resolution, and 128 particles/cell.

FIG. 13.

For the case “1” with a constant pump: (a) fields â1 and aL from PIC at t = 40 ps and (b) spectrum of the transverse electric field (containing both pump and seed components) from PIC and from theory: line 1 for the constant pump (line at k = k0), line 2 for the solution to Eq. (10) obtained from the amplitude of the seed in PIC, and line 3 for the solution to Eq. (10) obtained from a numerical solution to the 3-wave coupling model. For the ideally chirped pump: (c) fields â1 and aL from PIC at t = 75 ps and (d) spectrum of the transverse electric field from PIC: line (1) for pump the field spectrum prescribed by the matching condition and solid line (2) for the seed spectrum obtained from Eq. (11). Both PIC simulations were run with a simulation domain of 600 μm, 60 cells/μm resolution, and 128 particles/cell.

Close modal

We will demonstrate how BRA can be applied to enhance the plasma wake when a chirped pump is employed. In the absence of a frequency mismatch, i.e., δω = 0 in Eq. (3), the waves satisfy the usual matching conditions, ω0=ω1+ωL,k0=kL+k1 (k0 = –k1 + kL for the backscatter, with ka=|ka|,a=0,1,L). When we describe the application of BRA to wake enhancement, we will consider short pulses on the order of the plasma wavelength λp. Any application of the wave coupling model [Eqs. (2)–(4)] in the description of BRA must include a nonlinearity of the wake where δnw|a1|2 in the phase shift of Eq. (7).

Figure 13(a) shows the results of SCPIC simulations at t = 40 ps for the seed pulse amplitude a1 and the pump amplitude a0 and for the normalized amplitude of the electrostatic field, aL, associated with the wake and Langmuir waves participating in the BRA which are characterized by a short wavelength and are concentrated in the front part of the seed pulse. For this simulation with ne/nc = 0.0035, the initial seed pulse duration is τL = 30 fs measured as the FWHM of a Gaussian envelope and the left-propagating laser pump amplitude a0 = 0.0091 (case “1”). An important feature of our BRA application to enhance the plasma wake is the wavelength separation between two main Langmuir wave components, i.e., the wake and BRA excited perturbations. Our attempt to describe interactions, as seen in Figs. 13(a) and 13(b), i.e., BRA and the wake generation, by means of the 3-wave coupling model, Eqs. (2), (3), and (4) can be well approximated by a function for the nonlinear shift in Eq. (3) of the following form:

δω=g1ne/ncωp|â1|2,
(10)

where the numerical constant g1 = 1/(12π) was calculated for a sin model of the seed envelope. Figure 13(b) shows the results of this model compared with SCPIC results using numerical solutions to the 3-wave coupling model (line 3) and integrating Eq. (10) with the seed amplitude obtained from PIC (line 2). SCPIC simulations show decoupling between the pump a0 and the seed â1 due to the nonlinear evolution of the seed. This more complicated and nonuniform (within the pulse) frequency shift is absent in the 3-wave coupling model which shows further growth of the seed before saturation, leading to a larger frequency shifting rate. Figure 14 shows the wake amplitude versus seed propagation time from 1D simulations for different conditions. Figure 14(a) (line 2) illustrates the enhancement of the wake amplitude due to BRA coupling until the decoupling at t = 40 ps due to nonlinear evolution of the seed.

FIG. 14.

Wake amplitude versus seed propagation time from 1D simulations for (a) short Gaussian pulses with parameters of case “1” and (b) self-modulated pulses with parameters of case “2.” In (a), curves correspond to no pump (1), constant pump (2), δk1 = 0, linearly chirped pump (3), δk1 from Eq. (11) with b = 0, and ideally chirped pump (4), δk1 from Eq. (11) with b = 0.012 ps−1. In (b), curves correspond to no pump (1), constant pump (2), and linearly chirped pump (3).

FIG. 14.

Wake amplitude versus seed propagation time from 1D simulations for (a) short Gaussian pulses with parameters of case “1” and (b) self-modulated pulses with parameters of case “2.” In (a), curves correspond to no pump (1), constant pump (2), δk1 = 0, linearly chirped pump (3), δk1 from Eq. (11) with b = 0, and ideally chirped pump (4), δk1 from Eq. (11) with b = 0.012 ps−1. In (b), curves correspond to no pump (1), constant pump (2), and linearly chirped pump (3).

Close modal

To overcome this decoupling due to the shift in the seed, a frequency-chirped pump can be used. To first order, the frequency of the seed follows Eq. (7). In this approximation, a linearly chirped pump32 can prolong coupling and increase wake production as shown in Fig. 14(a) (line 3), illustrating the enhancement of the wake and delayed decoupling in this case. However, as the seed is amplified, it produces a larger amplitude wake, resulting in a non-linear change in frequency, and eventually, a linearly chirped pump will not suffice to maintain coupling.

We can estimate this nonlinear frequency change by using Eq. (7). This expression can be easily modified to include the effects of Raman amplification through a linear amplitude growth rate9,12 by substituting a1(0) with a1(t) = a1(0) (1 + bt), with b as a parameter. Substituting this into Eq. (8) and integrating lead to the following approximation of the seed's wavenumber shift:

δk1(t)=kp3k1(0)vg16π(a12(0)t+a1(0)bt2+13b2t3).
(11)

The pump frequency satisfies the usual matching conditions for coupling when ω0(t)=ω0(0)+cδk1(t), corresponding to a chirped pump with ω0(0) as the unshifted frequency. This result can also be obtained by applying similar steps to the theory given in Ref. 40. Equation (11) underestimates the rate of frequency reduction as it does not account for time evolution of the critical density defined by the changing seed pulse; however, by adjusting the parameter b, it was used to model the spectral evolution of the seed [Fig. 13(d) line 2] and to model a pump that would ensure coupling [Fig. 13(d), line 1]. Figures 13(c), 13(d), and 14(a) (line 4) demonstrate such a case where the seed and pump maintain coupling throughout the non-linear shifting process with the value of b = 0.012 ps−1. One can observe the strong wake enhancement in Fig. 14(a) (line 4) as the seed's frequency approaches the plasma frequency, and the seed is slowed down dramatically. Figure 13(c) shows a snapshot of the pulse and its wake at t = 75 ps, giving a quantitative value for the electric fields, and in this simulation, Ew ∼ 4.5 × 1010 V/m. Strong electric fields such as these may be capable of reflecting and accelerating ions.41 

To compare our result with realistic experimental conditions, we will consider the instantiation phase of the pump required BRA for coupling. We note that in the presence of a shift in the seed and frequency chirped pump, the Langmuir frequency ωL may deviate from resonance. However, since our parameters correspond to the superradiant regime9,35,36 (ωpe2ω0a0a1) where there is an increased bandwidth in the Langmuir waves, the interaction is still effective. To satisfy the wavenumber matching conditions k0=kL+k1, one may use the average shift of the seed wavenumber (8) [cf. Fig. 12(b)] to determine the instantaneous phase of the pump to sustain the original resonance condition: Φ0(x,t)=ω0(t+x/c)+α(t+x/c)2/2 with α=ω0ωpe12π2nenc|a1(0)2|, where α is the linear chirp coefficient and has a value of –2.3 × 1023 s−2 for our standard parameters (“1,” “4,” “5,” and “6”). Linear chirp coefficients up to 4.47 × 1023 s−2 have been obtained in experiments32 and could be applied to maintain coupling.

When the effects of diffraction are completely overcome and the seed pulse amplitude grows in time, it is necessary to use the non-linear “ideal chirp” prescription (11) to maintain coupling. For early times (t ≤ 40 ps), the “ideal chirp” prescription is well approximated by the linear shift [cf. Fig. 14(a), line 3], but for long durations, the non-linear shift is unlikely to be achieved by current CPA lasers. Density tapering42 could also potentially be used to extend the coupling duration between the pump and the seed. However, the increased plasma densities that are required also result in a greater red-shifting rate of the seed [Eq. (8)], such that the seed's frequency evolution becomes non-linear and requires a very large, non-linear density gradient that is unlikely to be achieved in experiments.

We have studied the application of the BRA to the enhancement and control of the linear wake generation over a long plasma region. This has been achieved by energy transfer to a wake driving pulse via Raman coupling with a laser pump of low intensity and of long pulse duration. Using PIC simulations, we have examined a wide range of background plasma conditions and considered results of the previous BRA optimization studies.23 Wake generation favors low background electron density, such as ne = 0.0035nc, where it is possible for a 30 fs laser seed pulse to have a FWHM comparable with half of a plasma wavelength λp/2. This optimal pulse length for wake generation at higher plasma densities would require shorter laser pulses and therefore will undermine the efficiency of the Raman amplification process. Also, wakes at the lower densities are more effective for particle acceleration,1 as, e.g., they correspond to a longer dephasing length. The physics of particle acceleration using wakes that were studied in this paper will be considered next in a separate publication. We have examined a range of electron temperatures and focused on the experimentally relevant28 choice of Te = 100 eV. By varying electron temperature, we could alter the properties of the resonant Langmuir wave driven by the SRS. At Te = 100 eV and ne = 0.0035nc, the k-vector of the plasma wave, kL ≈ 0.46kD, and BRA operates in the kinetic regime of SRS.29–31 However, at the short seed pulse durations, kinetic effects related to trapped particle dynamics are only relevant on much longer time scales than BRA coupling. In addition, the strong linear Landau damping limits the growth of SRS of the pump wave, thus allowing for the long scale plasma to be considered. With the above choice of plasma parameters, we have proceeded to examine the role of BRA in countering the limitations imposed by diffraction of the seed pulse and extending the wake generation to plasmas that are longer than the dephasing length. For a range of experimentally relevant f-numbers, Fig. 10 summarizes the results of 2D PIC simulations and demonstrates that BRA applied to wake generation can help maintain the wake amplitude of diffracting laser pulses.

A great deal of an initial theoretical insight and motivations for experimental measurements of BRA have been based on the solutions to the simple three wave coupling model.9–11 Such reduced models are useful in the understanding of BRA in large scale plasma experiments, provided diffraction of the laser pulses is not a factor and the effects of wake generation are included. We have demonstrated good agreement between PIC simulations and the reduced theoretical description at the early times of the BRA coupling. We have extended the validity of the wave coupling models by accounting for the impact of the wake on the SRS coupling. This has been accomplished by including frequency and wavelength shifts in the electromagnetic seed equation which are caused by the time dependent, wake related, plasma inhomogeneities. The loss of the resonance coupling in the BRA that follows has been mitigated by introducing a frequency chirp in the laser pump. Theory for the pump frequency chirp, in agreement with long time PIC simulations, showed the possibility of trapping the amplified laser seed pulse in a plasma. This results in a large increase in the wake amplitude and the generation of a shock like structure in the electrostatic field that will be further studied for the enhancement of particle acceleration.

W.R. would like to thank Dr. P. Alves for useful discussions. J.L. and S.W. would like to thank Arthur Pak for useful discussions. J.L. is grateful for support from LLNL through the summer scholar program. J.L. and W.R. would like to acknowledge partial support from NSERC. S.H. is grateful to the CPHT computer support team.

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