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.

## I. INTRODUCTION

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 accelerators^{7} 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 compression^{9–11} 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).

## II. INTRODUCTION OF METHODS AND BASIC PROCESSES

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 SCPIC^{16} which has been written on the basis of the code Mandor^{17} 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 × 10^{14} W/cm^{2}, and the wave length is 1.064 *μ*m. 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.

Case . | 1 . | 2 . | 3 . | 4 . | 5 . | 6 . |
---|---|---|---|---|---|---|

Density n/_{e}n _{c} | 0.0035 | 0.015 | 0.05 | 0.0035 | 0.0035 | 0.0035 |

T (eV) _{e} | 100 | 100 | 50 | 100 | 100 | 100 |

f-number | 30 | 30 | 40 | 15 | 13 | 10 |

Z (_{R}μm) | 3204 | 3462 | 7193 | 801 | 602 | 356 |

t (fs) _{pulse} | 30 | 30 | 100 | 30 | 30 | 30 |

λ/2_{p}c (fs) | 30 | 14.5 | 8 | 30 | 30 | 30 |

γ_{0}/ω _{pe} | 0.016 | 0.0126 | 0.009 | 0.016 | 0.016 | 0.016 |

a_{0} | 0.0091 | 0.0091 | 0.011 | 0.0091 | 0.0091 | 0.0091 |

a_{1} | 0.205 | 0.205 | 0.011 | 0.205 | 0.205 | 0.205 |

Case . | 1 . | 2 . | 3 . | 4 . | 5 . | 6 . |
---|---|---|---|---|---|---|

Density n/_{e}n _{c} | 0.0035 | 0.015 | 0.05 | 0.0035 | 0.0035 | 0.0035 |

T (eV) _{e} | 100 | 100 | 50 | 100 | 100 | 100 |

f-number | 30 | 30 | 40 | 15 | 13 | 10 |

Z (_{R}μm) | 3204 | 3462 | 7193 | 801 | 602 | 356 |

t (fs) _{pulse} | 30 | 30 | 100 | 30 | 30 | 30 |

λ/2_{p}c (fs) | 30 | 14.5 | 8 | 30 | 30 | 30 |

γ_{0}/ω _{pe} | 0.016 | 0.0126 | 0.009 | 0.016 | 0.016 | 0.016 |

a_{0} | 0.0091 | 0.0091 | 0.011 | 0.0091 | 0.0091 | 0.0091 |

a_{1} | 0.205 | 0.205 | 0.011 | 0.205 | 0.205 | 0.205 |

### A. Wake generation

The focus of our work is on short nonrelativistic pulses with a pulse duration, *t _{pulse}* [full-half-width-maximum (FWHM)], comparable in spatial extent to the plasma wavelength

*ct*∼

_{pulse}*λ*(with

_{p}*λ*= 2

_{p}*π*/

*k*and

_{p}*k*=

_{p}*ω*/

_{pe}*c*, where

*ω*is the plasma frequency and

_{pe}*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

*λ*and phase velocity

_{p}*v*≈

_{p}*c*. We denote in the following such pulses as “seed” pulses with the field amplitude

*a*

_{1}. In the linear approximation, the plasma electron density perturbation associated with the wake,

*δn*, is given by

_{w}^{1}

where *ζ* = *x* – *v _{g}*

_{1}

*t*denotes the coordinate in the frame where the seed pulse moves with the group velocity

*v*

_{g}_{1}in the

*x*-direction,

*n*

_{0}denotes the background plasma electron density, the seed electric field amplitude is given by $a1=eE1mec\omega 1=8.55\xd710\u221210(I1\lambda 12[W\u2009cm\u22122\u2009\mu m2])1/2$ in terms of the intensity of the short seed laser pulse

*I*

_{1}and its wavelength

*λ*

_{1}(

*λ*

_{1}= 1.13

*μ*m, when

*n*= 0.0035

_{e}*n*, where

_{c}*n*is the critical electron density for this laser frequency), and $k\u2032p=\omega pe/vg1$. The operator $\u2207\zeta \u2032,y,z2=\u2202\zeta \u203222+\u2202y22+\u2202z22$ 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

_{c}*t*= 30 fs in time (and

_{pulse}*l*=

_{s}*ct*≈ 9

_{pulse}*μ*m in length) and the peak seed pulse of amplitude

*a*

_{1}= 0.205 in a plasma with density

*n*/

_{e}*n*= 0.0035 (cf. Ref. 25). This simulation corresponds to case “1” (see Table I for all simulation cases discussed).

_{c}The generation of the wake is not a resonant process, but Eq. (1) predicts the maximum response for *l _{s}* =

*λ*/2. Alternatively, longer pulses

_{p}*l*>

_{s}*λ*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.

_{p}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, *n _{e}*/

*n*= 0.015, resulting in a shorter plasma wave length

_{c}*λ*, case “2” has the same conditions as case “1.”

_{p}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 *Z _{R}* determined by

*Z*=

_{R}*πf*

^{2}

*λ*, 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

*r*

_{0}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 *λ _{p}*, 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 $vph=c1\u2212ne/nc$. This results in the dephasing constraint $(c\u2212vph)td=\lambda p/2$. Defining the dephasing length as

*L*=

_{d}*ct*results in $\lambda p/2=Ld[1\u2212(1\u2212\lambda 2/\lambda p2)1/2]$ being $\u2248Ld\lambda 2/2\lambda p2$ for

_{d}*n*/

_{e}*n*≪ 1, so that the dephasing length can be approximated as

_{c}^{1}$Ld\u2248\lambda p3/\lambda 2$. 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.

### B. Backward Raman amplifier

Backward Raman Amplification (BRA) involves the use of a long pump pulse (*ω*_{0}, *k*_{0}) counterpropagating with respect to the short seed pulse (*ω*_{1} = *ω*_{0} – *ω _{pe}* and

*k*

_{1}=

*k*–

_{p}*k*

_{0}). When the two lasers overlap, they beat at the plasma frequency, exciting a Langmuir wave (

*ω*=

_{L}*ω*and

_{pe}*k*≈ 2

_{L}*k*

_{0}–

*k*) that resonantly couples to the seed. Energy flows from the pump to the seed, resulting in amplification and compression of the seed.

_{p}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 *a*_{0}, the seed *a*_{1}, and the plasma (Langmuir) wave *a _{L}*.

^{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}

where *a*_{0,1} ≡ *eE*_{0}/(*m _{e}cω*

_{0,1}) are the normalized amplitudes of the electromagnetic fields

*E*

_{0,1}of the pump (0) and the seed (1) wave, with $a\u03021=(\omega 0/\omega 1)1/2a1$ and with ($\omega 0,1,k\u21920,1$) as the corresponding frequencies and wave vectors; $aL=(\omega L/\omega 0)1/2eEL/(mec\omega p)$ denotes the normalized amplitude of the Langmuir wave electric field,

*E*, with ($\omega L,k\u2192L$), and the coupling constant reads $\gamma 0=(kLc/4)\omega p/(\omega L\omega 1)1/2$. The equations are written in the stationary frame of the seed pulse, using the variables

_{L}*ζ*=

*x*–

*v*

_{g}_{1}

*t*and

*τ*=

*t*, where

*v*

_{g}_{0}and

*v*

_{g}_{1}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 $\nu L=(\pi \omega pe3/2k2)\u2202fM/\u2202v|v=\omega L/k$ evaluated at the Maxwellian distribution function,

*f*. On the long time scale,

_{M}*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.

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 solution^{10} of Stimulated Raman Scattering (SRS). The seed pulse amplitude at time *t* = 3.57ps has already experienced amplification with respect to its initial amplitude *a*_{1} = 0.011. Amplification is due to the pump laser—propagating from the right to the left—initially at *a*_{0} = 0.011 through coupling with the Langmuir wave. The seed pulse duration *t _{pulse}* is 100 fs at FWHM, and the plasma density and electron temperature are

*n*/

_{e}*n*= 0.05 and

_{c}*T*= 50 eV, respectively. The example of case “3,” and previous studies based on PIC simulation with similar parameters,

_{e}^{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 solutions

^{10}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 *k*_{0}, the wake at *k _{p}* =

*ω*/

_{pe}*c*, the seed at

*k*

_{1}=

*k*

_{0}–

*k*, and the Langmuir waves from Raman coupling

_{p}*k*≈ 2

_{L}*k*

_{0}–

*k*. The one dimensional Fourier transform is taken in space and is presented as “FFT(

_{p}*E*)” and “FFT(

_{y}*E*)” for the Fourier transform of

_{x}*E*and

_{y}*E*, respectively, into

_{x}*k*.

_{x}## III. BACKWARD RAMAN AMPLIFIER FOR LASER WAKEFIELD GENERATION

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 × 10^{16} W/cm^{2} at best focus and a flat top plane wave pump of intensity 1 × 10^{14} W/cm^{2}, in a plasma of density *n _{e}* = 0.0035

*n*. The parameters are chosen such that the seed efficiently excites plasma waves of the wake (

_{c}*ct*=

_{seed}*λ*/2) in the non-relativistic regime (

_{p}*a*

_{1}= 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, $G=Lx\gamma 02|a0|2/|\nu Lvg1|$, requires the length

*L*≈ 70 mm to reach

_{x}*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

*k*≈ 0.46

_{L}*k*. Recent work

_{D}^{26}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 *k _{L}*/

*k*≈ 0.46 of the BRA Langmuir wave characterizes the kinetic regime of the SRS

_{D}^{29,30}and of the Langmuir wave nonlinear evolution.

^{31}This leads to strong linear Landau damping,

*ν*≈ 0.14

_{L}*ω*, and in the nonlinear regime to electron trapping and to trapped particle modulational instability.

_{pe}^{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,

*k*≈ 0.32

_{L}*k*and 200 eV, and

_{D}*k*≈ 0.65

_{L}*k*) show only small variations in the resulting seed and wake amplitudes [see Figs. 9(a) and 9(b)].

_{D}Figure 5 shows the results of a 2D plane wave seed and pump simulation with parameters of case “1,” the transverse dimension of 10 *μ*m, 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 *k _{L}* ≈ 0.46

*k*and the wake at

_{D}*k*=

_{p}*ω*/

_{pe}*c*≈ 0.014

*k*. Superimposed is the 1D cut at

_{D}*y*= 5

*μ*m 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

*eE*/

_{L}*mcω*≈ 0.02 (see Fig. 6), the bounce frequency $\omega b=(kLeEL/m)1/2\u22489.3\xd710131/s$, and therefore, the characteristic trapping time

_{L}*τ*= 2

_{b}*π*/

*ω*= 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.

_{b}^{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(

*E*)” and “fft(

_{x}*E*)” for the Fourier transform of

_{y}*E*and

_{x}*E*, into

_{y}*k*and

_{x}*k*, 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 (

_{y}*k*=

_{x}*k*= 0.059

_{p}*k*

_{0}and

*k*≈ 0) and the SRS Langmuir wave (

_{y}*k*=

_{x}*k*= 1.94

_{L}*k*

_{0}and

*k*≈ 0), one can see broad continua of transverse components, resulting from the trapped particle modulational instability

_{y}^{29–31}and the second harmonic of the

*k*-wave. The detailed analysis of the spectra about

_{L}*k*≈ 2

_{x}*k*

_{0}has revealed additional components at

*k*±

_{L}*nk*and

_{p}*ω*±

_{L}*nω*[cf. inset in Fig. 5(b)] which result from the coupling between the strong wakefield wave (

_{p}*k*,

_{p}*ω*) 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.

_{p}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 *k _{L}* ±

*nk*and

_{p}*ω*±

_{L}*nω*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.

_{p}The characteristic feature of the electrostatic field spectra is large separation between the wake at *k _{p}* and the SRS Langmuir wave at

*k*(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

_{L}*n*/

_{e}*n*= 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,

_{c}*ν*=

*ν*= 0.14

_{L}*ω*. 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

_{p}*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, *ct _{pulse}* ∼

*λ*/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.

_{p}^{32}The effect of the broad seed pulse spectrum was examined in PIC simulations,

^{33}showing Brillouin coupling

^{34}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.

### A. Resonant BRA

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, *E _{x}* in Fig. 7(a) and

*E*

_{y}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, Δ

*E*

_{x}_{(}

_{y}

_{)}=

*E*

_{x}_{(}

_{y}

_{)}(with BRA)–

*E*

_{x}_{(}

_{y}

_{)}(without BRA). From the quantity Δ

*E*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

_{y}^{−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.

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.

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).

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 regime^{9,35,36} ($\omega pe\u22642\omega 0a0a1$) 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 $dE1=\u2212dE0=2\alpha I0A1(t)dt$. Where *A*_{1}(*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

where *d* indicates the spatial dimension of the solution, i.e., *d* = 2 or = 3 for 2D or 3D respectively, *l _{s}* =

*ct*, and

_{pulse}*Z*is the Rayleigh length of the seed pulse. The solution to Eq. (5) with

_{R}*α*= 0.45 efficiency and constant pump intensity

*I*

_{0}= 1 × 10

^{14}W/cm

^{2}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 $\u2202I1\u2202t=0$ to get the solution

One can verify this result by substituting Eq. (6) into Eq. (5) to confirm *I*_{1}(*t*) = *I*_{1}(0). Since diffraction results in a dynamic loss in seed intensity, the pump requirements *I*_{0}(*t*) (6) are not constant in time but can be evaluated near the seed's Rayleigh length (*t *≈* Z _{R}*/

*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 × 10

^{14}W/cm

^{2}, while our BRA pump intensity is 1 × 10

^{14}W/cm

^{2}, 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 *I*_{0} = 1 × 10^{14} W/cm^{2} is applied to seed lasers with the best-focus intensity *I*_{1}(0) = 5 × 10^{16} W/cm^{2}, 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.

### B. Loss of the BRA resonance due to the wake effects

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.

It has been shown^{37} that a short laser pulse that propagates in plasma with a density profile due to the wake's Langmuir wave, *δn _{w}*/

*n*

_{0}, will develop frequency shifts that can be described by

with *δk*_{1} = *∂ _{x}*Φ =

*∂*Φ and

_{ζ}*δω*

_{1}= –

*∂*Φ, where $\Phi (\zeta ,\tau )=\u2212\omega p2\delta nw\tau /(2\omega 1n0)$. The characteristic linear dependence of

_{t}*δk*

_{1}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

where we assumed *k _{p}l_{s}* =

*π*and

*v*

_{g}_{1}≃

*c*.

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

where $\delta \omega =\omega pe22\omega 1\delta nwn0$. 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.

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, $\omega 0=\omega 1+\omega L,\u2009k\u21920=k\u2192L+k\u21921$ (*k*_{0} = –*k*_{1} + *k _{L}* for the backscatter, with $ka=|k\u2192a|,a=0,1,L$). When we describe the application of BRA to wake enhancement, we will consider short pulses on the order of the plasma wavelength

*λ*. Any application of the wave coupling model [Eqs. (2)–(4)] in the description of BRA must include a nonlinearity of the wake where $\delta nw\u223c|a1|2$ in the phase shift of Eq. (7).

_{p}Figure 13(a) shows the results of SCPIC simulations at *t* = 40 ps for the seed pulse amplitude *a*_{1} and the pump amplitude *a*_{0} and for the normalized amplitude of the electrostatic field, *a _{L}*, 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

*n*/

_{e}*n*= 0.0035, the initial seed pulse duration is

_{c}*τ*= 30 fs measured as the FWHM of a Gaussian envelope and the left-propagating laser pump amplitude

_{L}*a*

_{0}= 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:

where the numerical constant *g*_{1} = 1/(12*π*) was calculated for a $sin\u2009$ 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 *a*_{0} and the seed $a\u03021$ 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.

### C. Chirping the pump

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 pump^{32} 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 rate^{9,12} by substituting *a*_{1}(0) with *a*_{1}(*t*) = *a*_{1}(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:

The pump frequency satisfies the usual matching conditions for coupling when $\omega 0(t)=\omega 0(0)+c\u2009\u27e8\delta k1\u27e9(t)$, corresponding to a chirped pump with *ω*_{0}(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, *E _{w}* ∼ 4.5 × 10

^{10}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 *ω _{L}* may deviate from resonance. However, since our parameters correspond to the superradiant regime

^{9,35,36}($\omega pe\u22642\omega 0a0a1$) where there is an increased bandwidth in the Langmuir waves, the interaction is still effective. To satisfy the wavenumber matching conditions $k\u21920=k\u2192L+k\u21921$, 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: $\Phi 0(x,t)=\omega 0(t+x/c)+\alpha (t+x/c)2/2$ with $\alpha =\u2212\omega 0\omega pe12\pi 2nenc|a1(0)2|$, where

*α*is the linear chirp coefficient and has a value of –2.3 × 10

^{23}s

^{−2}for our standard parameters (“1,” “4,” “5,” and “6”). Linear chirp coefficients up to 4.47 × 10

^{23}s

^{−2}have been obtained in experiments

^{32}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 tapering^{42} 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.

## IV. CONCLUSIONS

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 *n _{e}* = 0.0035

*n*, where it is possible for a 30 fs laser seed pulse to have a FWHM comparable with half of a plasma wavelength

_{c}*λ*/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,

_{p}^{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 relevant

^{28}choice of

*T*= 100 eV. By varying electron temperature, we could alter the properties of the resonant Langmuir wave driven by the SRS. At

_{e}*T*= 100 eV and

_{e}*n*= 0.0035

_{e}*n*, the k-vector of the plasma wave,

_{c}*k*≈ 0.46

_{L}*k*, and BRA operates in the kinetic regime of SRS.

_{D}^{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.

## ACKNOWLEDGMENTS

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.