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1

Polarization control of electron injection and

acceleration in the plasma by a self-steepening laser

pulse

Jihoon Kim1, Tianhong Wang1, Vladimir Khudik2,Gennady Shvets1

1School of Applied and Engineering Physics, Cornell University, Ithaca, NY14850, USA.2Department of Physics and Institute for Fusion Studies, The University ofTexas at Austin, Austin, TX 78712, USA.

E-mail: jk2628@cornell.edu

November 2021

Abstract. We describe an interplay between two injection mechanism ofbackground electrons into an evolving plasma bubble behind an intense laserpulse: one due to the overall bubble expansion, and another due to its periodicundulation. The two mechanisms are found to occur simultaneously when anintense laser pulse propagating inside plasma forms a shock-like steepened front.Periodic undulations of the plasma bubble along the laser propagation pathcan either inhibit, or conspire with electron injection due to bubble expansion.We show that Carrier-Envelope-Phase (CEP) of the self-steepening laser pulseproduces a unique electron injector – Expanding Phase-controlled UndulatingBubble (EPUB). The longitudinal structure of the electron bunch trapped bythe EPUB can be controlled by laser polarization and power, resulting in high-charge (multiple nano-Coulombs) high-current (tens of kilo-Amperes) electronbeams with ultra-short (femtosecond-scale) temporal structure. Generation ofhigh-energy betatron radiation with polarization-controlled spectrum is analyzedas a promising application of EPUB-produced beams.

1. Introduction

An electron injector is an integral part of any accelerator, as it produces high-quality moderate energy particles for further acceleration. A remarkable feature ofa Laser Wakefield Accelerator(LWFA) [1, 2, 3] is the availability of an abundantreservoir of charged particles from the background plasma. Therefore, plasma cansimultaneously serve as an acceleration medium sustaining intense plasma waves,and an electron injector. While the key attraction of LWFAs is their compactnessowing to ultrahigh accelerating electric field – in excess of 100GV/m in many recentimplementations [4, 43, 6, 7, 8, 9] – of the plasma wave generated by intense laserpulses, its other advantage is the availability of large numbers of initially quiescentelectrons that can be injected into the plasma wave, capable of forming currentsexceeting 100kA[10]. If such injection can be carried out in a controllable way, it maybe possible to produce high-charge low-emittance beams in single compact device.

2

(a) (b)

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Figure 1: Schematic of Injection process for plasma bubble driven by (a) linearlypolarizaed laser pulse and (b) circularly polarized laser pulse. An expanding andtransversely undulating bubble is formed by a laser with steepened front. Accordingto laser polarization, periodically modulated or flat current beam can be generated.Laser intensity(red), plasma bubble shape at early(light blue) and later(dark blue)time, electron bunch density(yellow), plasma bubble back movement(green arrows)and injected current (white axes, yellow lines)

A number of promising approaches to injecting electrons into plasma wavesgenerated in the wake of a laser pulse, including the highly nonlinear “plasmabubbles” [20, 21], have been suggested and experimentally implemented. Thoseinclude injections due to ionization injection [33, 34, 35, 36], engineered densityramps [37, 38, 39, 40], or rapid variation of the bubble’s size along the laser’spath [42, 44, 43, 41, 45, 46, 47]. Recently, a new approach to electron injection based onplasma wake undulation controlled by carrier-envelope phase (CEP) has been proposedand experimentally demonstrated using single-cycle laser pulses [16, 24, 12, 13, 48, 49,14, 15]. Specifically, a near single cycle(NSC) laser pulse propagating in an underdenseplsma can generate a phase-controlled undulating bubble (PUB) with characteristicperiodicity TCEP = λL/(vph − vg) ≈ (λL/c)(ω

2L/ω

2p) [12, 13, 14, 16] and produce

periodic electron injections into the bubble [15]. CEP-based injection is a conceptualdeparture from the standard description of plasma wave generation by multi-cyclelaser pulses that relies on the phase-averaged (ponderomotive) approximation [11].Despite the promise of CEP-based injection to generating high-current ultra-shortelectron bunches [15], it has its limitations because it requires NSC pulses. Anothercircumstance under which CEP effect can manifest itself is when the pulse front islocally depleted due the etching by plasma. The front of such self-steepened laserpulse envelope can vary on a scale comparable to that of laser cycle [24, 25, 26],resulting in an Expanding Phase-controlled Undulating Bubble (EPUB) which is thesubject of this work.

In this paper, we examine the combined effect of expansion and undulation ofa plasma bubble on the injection, acceleration, and temporal shaping of an electronbunch produced by an EPUB, as shown in Fig 1. The paper is organized as follows.In Section 2, we set the stage by presenting the results of PIC simulations thatdemonstrate phase and polarization dependent injection of electrons into a plasmabubble produced by a self steepening few cycle laser pulse with cTFWHM ∼ 3λL,where TFWHM and λL are the pulse duration and wavelength, respectively. Theparameters of the laser pulse are chosen to be within reach of the BESTIA (BrookhavenExperimental Supra-Terawatt Infrared at ATF) laser system [28]. In Section 3 we

3

interpret these results by developing a simple single-particle model of electron trappingin a plasma bubble undergoing simultaneous expansion and undulation. This modelis used to demonstrate how laser polarization (i.e. linear versus circular) can be usedto generate the desired current profile (spiky versus smooth) of an injected electronbunch. Additional detailed PIC simulations presented in Section 4 demonstrate thathighly-charged (Q ∼ 10nC) bunches modulated on a temporal scale comparableto the laser period TL = λL/c can be formed, with promising implications forstructured X-ray generation. The temporal structure of the bunch can be controlledvia polarization and laser power. In Section 5, we discuss the efficiency of theproposed injection/acceleration scheme, and the possibility of controlling the spectrumof betatron radiation using laser polarization.

2. PIC Simulations

We use a 3D PIC code VLPL[22] to self-consistently model the propagation andeventual self-steepening of an intense laser pulse[25, 26], followed by self-injectionof some of the plasma electrons into the laser wakefield, acceleration of the injectedbunch, and subsequent plasma field depletion by the injected electrons [23]. Thefollowing laser parameters are used: peak power PL = 40TW , wavelength λL = 9.2µm, pulse duration TFWHM = 100fs, and the matched spot size σ⊥ = 8.5λL ≈ 78µm.Plasma is assumed to start with a linear density ramp of the length Lramp = 0.37mm,followed by a long plateau region with constant density n = 9.1 × 1016/cm3.A numerical grid used in the simulations was chosen to have the dimensions of∆x × ∆y × ∆z = 0.05λL × 0.25λL × 0.25λL, where x is the propagation directionof the laser pulse through the plasma.

We first consider a laser pulse linearly-polarized (LP) in the z−direction. Becausethe pulse front needs to steepen before CEP effect becomes visible, the plasma bubbledoes not execute transverse undulations immediately after the laser pulse entersthe plasma as shown in Fig 2 (a) and (d). Electrons are injected into the plasmabubble from the very beginning, but this initial population of injected electrons doesnot exhibit any transverse asymmetry in the z−direction. After ct = 400λL (orx = 3.7mm) of propagation through the plasma, the pulse front is depleted andsteepened as shown in Fig. 2(b), with further depletion at ct = 700λL(or x = 6.4mm)as apparent from Fig. 2(c).

The sharpness of the self-steepened front at ct = 400λL, as well as its depletion,are reflected in its spectrum plotted (red line) in Fig. 2(b). When compared withthe initial laser spectrum at ct = 100λL (blue-line), the spectrum of the steepenedpulse is red-shifted by approximately 25%, i.e. from ωL = ωL0 to ωL = 0.75ωL0 –a clear evidence of pulse depletion via plasma wake generation. Moreover, its largeFWHM spectral bandwidth ∆ω ∼ 0.5ωL is a clear evidence of pulse steepening onthe time scale of a laser period. This increase of the spectral bandwidth is a result ofstrongly-nonlinear interaction between the laser pulse and the plasma.

Laser pulse self-steepening on a wavelength scale produce transverse undulationsof the plasma bubble cavity along the laser polarization [16, 24, 15], which is analogousto the undulations produced using NSC laser pulses [16, 12, 13, 48, 49, 14, 15]. Bubbleoscillations in the z−direction are manifested as nonzero on-axis transverse wakefieldW⊥ ≡ (Wy,Wz) = E⊥ + ex × B⊥. For an LP pulse polarized in the z−direction,we find that Wy ≈ 0 and Wz 6= 0 as shown in Fig. 2(d). Here E⊥ and B⊥ arethe transverse electric and magnetic wakefields inside the plasma bubble, and W⊥ is

4

(a)(a) (b) (c)

(a)(d) (e) (f)

Figure 2: Three-dimensional particle-in-cell simulations of plasma bubble undulationsinduced by self-steepening of a laser pulse. (a-c) On-axis electric field of a z-polarizedlaser pulse (red line) and plasma density (color map) at (a) ct = 100λL , (b)ct = 400λL, and (c) ct = 700λL. (d-e) On-axis transverse wakefield W⊥ = (Wy,Wz)for (d) linearly- and (e) circularly-polarized pulse. Blue (orange) line: Wy (Wz) atξ ≡ x− ct = 35. Red dashed lines on (d): propagation distances corresponding to (a),(b), and (c). (f) Spectra of the laser pulse at ct = 100λL (blue line) and at ct = 400λL

(red line).

proportional to the transverse force exerted by the wakefield on a charge moving withthe speed of light in the x-direction inside the plasma bubble.

We now compare the effect of using a circularly polarized (CP) laser pulse ofthe same duration and intensity as the LP pulse considered above. As evidenced byFig. 2(d)-(e), plasma cavity undulations begin around the same propagation distancefor the LP and CP pulses because pulse steepening takes place after the same,polarization-independent, propagation distance through the plasma. After bubbleundulations start, they acquire a very different behavior for the CP laser pulse.Specifically, instead of executing undulations in the z− direction as in the case of theLP pulse (see Fig. 2(d)), the bubble executes helical motion in the y − z plane whenit is driven by a CP pulse (see Fig. 2(e)). As a result, the on-axis transverse wakefieldW⊥ of the bubble driven by the CP pulse has equal Wy and Wz components that areapproximately offset from each other by π/2 phase difference. For both polarizations,the size of the plasma bubble is continuously elongated. We note that the longitudinalbubble elongation is larger than bubble transverse expansion, as shown in Figs. 2(a)-(c). This reduces the bubble back velocity, having a direct impact on the bunching ofthe injected electrons, as will be seen below.

Electrons are injected into the bubble with distinct periodicity, close to TCEP /2,as shown in Figure 3 (a). Injected charge Q ≈ 10nC directly reflects the periodicinjection process via current bunching, as shown in Figure 3 (d). The current ismodulated at approximately ∆ξ/c = 1.3λL/c ≈ 40fs, close to the predicted value∆ξ ≈ cTCEP /4γ

2bb = 1, with γbb ≈ 5, cTCEP ≈ 100λL.

5

(d)

(e)

(f)

(a)

(��

(c)

Figure 3: Effect of laser ellipticity on injection and bunch formation. (a-c): Injectedelectron initial position on x0 − z0 plane(blue dots), Injection rate (red line). Currentprofile (d)-(f) after 6.4mm propagation. The red dashed lines mark the electrons thatform a monoenergetic peak. Linearly polarized (ǫ = 0) [(a),(d)], Elliptical (ǫ = 0.268)[(b),(e)], Circularly polarized(ǫ = 1)[(c),(f))], with ellipticity ǫ = Ey/Ez

2.1. Effect of laser polarization

Polarization dependence of bubble undulation is reflected on the injection dynamics.The bunch current modulation can be controlled via varying laser polarization.We note the huge injection spike located at x0 = 0.92mm(100λL), where densitytransitions from linear ramp to plateau [18, 17]. This injection is consistent in allsimulations with different power, phase, and polarization. After this initial injection,electrons are continuously injected from background plasma mostly independent oflaser polarization up to x0 ≈ 1.8mm(200λL), after which phase-controlled undulationamplitude grows.

After the onset of undulation, injection dynamics become polarization-dependentas can be seen from Fig 3 (a)-(c). While electrons injected from a bubble drivenby LP laser are injected in short bursts from alternating locations with z0 ≈ rb andrb ≈ 5, those injected from CP laser originate from a spiral-shaped initial positions.The injection rate shows that as laser polarization changes from LP to elliptical toCP, injection rate transitions from short periodic bursts into a mostly constant rate,with less interruptions. Note that even for CP injection, there is a dip in injectionrate at x0 = 4.8mm. This is because eventually, the huge injected charge reduces theaccelerating gradient, and can actually suppress electron injection[23].

This injection dynamics directly modifies the injected current profile, as seen in

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Fig 3 (d)-(f). Current modulation switches from highly modulated to less modulatedto almost flat, as the laser polarization changes from linear to elliptical to circular.Note the modulation in current profile for CP near ξ = 25, due to the beam loadingeffect modifying the injection process as mentioned before.

Finally, we note that despite the relatively long length of the entire bunch train(ora single bunch, in case of CP) of ∆ξ/c ≈ 370fs, an appreciable fraction(∼ 30%) ofearlier injected electrons [Fig 3(d)-(f)] collapse into a monoenergetic peak due to phasespace rotation, as will be shown in the later sections.

3. Expanding Phase-dependent Undulating Bubble (EPUB) injectionmechanism

To interpret this electron injection into an evolving plasma cavity, we use asimplified model of an positively-charged (devoid of electrons) spherical plasmabubble[42, 44, 19, 20]. The bubble has radius R(t) = R0(1 + ǫt) with initial radiusR0 expanding with rate ǫ propagating with uniform velocity vb. A Hamiltoniandescribing plasma electrons’ interaction with the bubble can be written as H(ρ, t) =√

1 + (P+A(t))2 − vbPx − φ(t), where ρ = (ξ, y, z − zosc), ξ = x− vbt, zosc(t) is thetransverse coordinate of the undulating bubble center, P is the canonical momentum,andA(t) (φ(t)) are the vector (scalar) potentials. Time, length, potential, and electronmomentum are normalized to ω−1

p , k−1p = c/ωp, mec

2/|e|, andmec, respectively, where

ωp =√

4πe2np/m is the electron plasma frequency and np is the plasma density.We use the Ax(t) = −φ(t) = Φ(t)/2 gauge, and assume that Φ(t) = (ρ(t)2 −

R(t)2)/4 inside and Φ(t) = 0 outside the bubble. Transverse plasma bubbleundulations zosc(t) ≡ zu cos(ωCEPt + φCEP) and bubble expansion, R(t), introducestime dependence of Hamiltonian. Here ωCEP ≡ 2π/TCEP is the CEP slip rate, zu isthe maximum bubble oscillation amplitude, and φCEP ≡ φCEP(t(x0), x0) is the initialCEP evaluated at the time t(x0) corresponding to electron’s entrance into the bubbleat x = x0.

To simplify the discussion, we consider the electron motion in the x-z plane. Fromthe Hamiltonian, the following equations of motion and Hamiltonian time-dependencecan be derived using dP/dt = −∂H/∂ρ, dρ/dt = −∂H/∂P:

dξ

dt=

pxγ

− vb, (1)

dpxdt

= −1

4

[

R(t)R(t) + ξ(1 + vb) + (vz − zosc)z]

(2)

dz

dt=

pzγ, (3)

dpzdt

= − (vx + 1)z

4(4)

dH

dt= −1 + vx

4

[

z(t)zosc(t) +R(t)R(t)]

(5)

3.1. Analytic estimate for injection using Hamiltonian model

Under specific conditions, electrons get injected into the bubble and are acceleratedto ultra relativistic energy[42, 44, 19]. It was shown that an electron can be trapped

7

when the condition H < 0 is fulfilled[42]; under this condition, electrons cannot escapethe bubble, even when they overtake the bubble. It was noted that there is anotherpopulation of electrons, the injected electrons, which can gain similar peak energy inthe bubble but can escape the bubble after it reaches the front end of the bubble[44].In this paper, we present an estimate for injection condition, providing a slightlyrelaxed condition for electrons to enter and overtake the bubble.

Electrons can get injected into a non-evolving moderately relativistic bubble(R ∼ γb) if

√2γb < R, where γb is the bubble relativistic factor [19]. The condition

was derived assuming that in case of a moderately relativistic bubble, electrons willbe able to catch up with the bubble if it can reflect off the bubble’s rear wall atleast once. Since this results in electron spending longer time in the bubble, the aboveinjection condition can determine if the electrons will be injected or not for moderatelyrelativistic bubble. In an expanding bubble, this condition can be relaxed becausethe Hamiltonian of the electron is altered[42]. In a non-evolving bubble, maximumexcursion of electron from bubble’s center axis is rm ≈ 4H + R2 − 2px/γ

2b − 2/px,

with H = 1. Assuming small expansion rate(ǫ ≪ 1), the maximum momentumgained by the electron is almost identical, so rm can be modified simply by pluggingin the modified H . Using the parameter free estimation for longitudinal momentumpx ≈ 1.1R2[19], one can obtain the modified injection condition by requiring rm < R,

γb/R < 1.1/√2H (6)

Electrons will be injected when the Hamiltonian decreases below Hthresh ≈0.6R2/γ2

b . For ultra-relativistic bubble where γb ≫ R, this will only hold whenH ≈ 0, but for a moderately relativistic bubble, this will hold for moderate valueof 0 < H < 1.

When such a bubble undergoes undulation, its Hamiltonian can both increase anddecrease, since the term zosc in dH/dt will change sign according to period TCEP [15].This further modification of electron Hamiltonian can suppress electron injection atsub-optimal undulation phases, φCEP = π/2 for z = −R and φCEP = −π/2 for z = Rby increasing the Hamiltonian above the injection threshold. The combined effect ofexpansion and undulation results in a periodic injection of electrons from backgroundplasma, which has direct effect on the injected electron current.

To illustrate this, we solve equations of motion (1)-(4) for three initially quiescentelectrons entering the bubble at y = 0, ξ = x − vbt = 0, z = −R. [Fig. 4 (a)].The bubble oscillates at period TCEP = 50, and also expands at different rateuntil texp = 20. The green trajectory interacts with an expanding bubble, but theexpansion rate ǫ = 0.002 is not fast enough, and the trajectory will not overtake thebubble. The blue trajectory interacts with a bubble with slightly faster expansion rateǫ = 0.0025, and overtake the bubble. The red trajectory interacts with an expanding(ǫ = 0.0025) and undulating bubble. Even though the expansion rate is identical tothe blue trajectory, the bubble undulates at a sub-optimal phase. Consequently, thered trajectory is not injected and passes through the bubble without gaining muchenergy.

The difference in behavior can also be seen from Hamiltonian evolution. Onlythe blue trajectory electron whose Hamiltonian decreases below the derived Hthresh

is injected, while the red and green trajectory electron pass through; in fact, theHamiltonian of red trajectory electron actually increases above H = 1.

To model injection of background particles, we give an estimate for finalHamiltonian for optimally positioned electrons(z = R)[42, 19] entering the bubble

8

Figure 4: Analytic estimates for injection. (a) Trajectory in ξ − z plane for injectedparticles. Bubble boundaries(black lines): initial(dashed) and final(solid). (b)Hamiltonian evolution versus time. Dashed line: Hthresh = 0.6γ2

b/R2. Three colored

lines: injected (blue) and passing particles due to slow expansion(green) and bubbleundulation(red). (c) Hamiltonian estimate for particles entering the bubble at differenttime(∆H(1)). −1 + Hthresh(red), −1 + Hexp(black dashed),−1 + H(black solid),Time during which electrons are injected(blue shadow). Parameters:(a-b) R = 6,zu = 0(blue, green) zu = 1.5(red), ǫ = 0.002(green),ǫ = 0.0025(blue,red), γb = 6,TCEP = 50, Bubble expansion halts at t = 20.(c): R = 6,γb = 6, ǫ = 0.0025, zu = 1.0,TCEP = 50.

as the bubble propagates. We compute Hamiltonian of electrons located on thesinusoidal trajectory defined by x0 = vbt, y0 = 0, z0 = R + zosc(t). Electrons locatedon this trajectory graze the bubble boundary, entering the bubble at its edge. Changein Hamiltonian of electrons entering bubble at different time can be estimated byintegrating

∆H =

∫

dt

[

pz(t)zosc(t)−1 + vx

4R(t) ˙R(t)

]

(7)

where the integral is calculated along the electron trajectory.To lowest order in bubble oscillation amplitude and expansion rate, we can

use the quantities from a non-evolving bubble(zu = 0, ǫ = 0) to estimate changein Hamiltonian. Assuming passage time of electron through the bubble, Tpass, ismuch smaller than oscillation period, TCEP , zosc ≈ −zuωCEP sin(ωCEP (tenter)),with tenter the time electron enters the bubble. Furthermore, if the bubble radiusvaries slowly (ǫ ≪ 1), R(t)R(t) ≈ R2ǫ. This simplifies the integral to ∆H ≈−[

zoscωCEP sin(φCEP )∆pz + (texit +∆x)(R2ǫ)/4]

where ∆pz is the exit transversemomentum of the electron, texit is the time at which electron exits the bubble, and∆x is the longitudinal distance electron travels during interaction with the bubble.

From the parameter free equations [19, 15], ∆x + texit = 5.4R,∆pz = 0.16R2,and the Hamiltonian increment is given by

∆H(1) = −1.35ǫR30 + 0.16zoscωCEP sin(ωCEP tenter)R

20 (8)

This Hamiltonian estimate consists of two terms with distinct behaviors, ∆Hexp =−1.35ǫR3

0 and ∆Hosc = 0.16zoscωCEP sin(ωCEP tenter)R20. The former depends on the

expansion of the bubble and is constant regardless of tenter . The latter depends on thebubble undulation and will oscillate in time with period TCEP . Undulation of bubblecan affect injection in two ways. (1) If the expansion rate ǫ is not large enough,

9

Hexp > Hthresh; without undulation, electrons cannot be injected. However, ∆Hosc

can reduce H further. If the amplitude of ∆Hosc is large enough, it can reduce theHamiltonian below Hthresh, and there will be periodic intervals during which electronscan be injected. In principle, these electron injections can be as short as possible, sincethe time during which H < Hthresh can become arbitrarily short. (2) Second, if ǫ islarge enough, Hexp < Hthresh, there will be continuous injection without the presenceof ∆Hosc. However, ∆Hosc will increase ∆H periodically, suppressing injection forshort duration. We note that these injections cannot become arbitrarily short, sincethe time during which injection is suppressed can get only as long as TCEP . The timedependent injection for scenario (2) is shown in Fig 3 (c). Both scenarios predict thatinjection process has periodicity TCEP .

Up to now, we have only considered ∆H in the case of a plasma bubble undulatingin one direction. Plasma bubble whose center moves along a helix can be also generatedby a circularly polarized laser pulse. In such case, the bubble center will always havea finite speed, only changing its direction azimuthally. This has a direct impact oninjection, since at any point in time, there will be angles at which injection is eithersuppressed or enhanced, as we shall see below. We note that it is possible to useelliptically polarized laser to drive the bubble, which can generate a bubble whosecenter follows a helix elongated in one transverse direction .

3.2. Result for particle swarm simulations

Our calculations demonstrate the feasibility of controlling electron injection viacombination of bubble expansion and undulation. However, the Hamiltonian modelrelied on several simplifying assumptions (parameter-free equations and slow bubbleevolution) and studied interaction with a single electron at a given time with aprescribed bubble.

In reality, a vast number of background electrons interact with an evolvingbubble and get injected into it. To understand this process, we seed test particleson a three dimensional volume spanning R < x0 < 80,−10.5 < y0, z0 < 10.5 andlaunch three different evolving spherical potentials that can capture and accelerateparticles: expanding, expanding and helically undulating, and expanding and linearlyundulating. In case of a helically undulating bubble, the center of the bubble movestransversely according to zosc(t) ≡ zu cos(ωCEPt + φCEP), yosc(t) ≡ zu sin(ωCEPt +φCEP), and in case of an expanding bubble, there is no transverse motion. The particleequations of motion and change in Hamiltonian can be derived from the Hamiltoniansimilarly to Eqns. (1-5)

As can be seen from the injection rate[Fig 5 (a)-(b)], only the expanding andlinearly undulating bubbles’ injection rate is periodically modulated. While theinjection rate of a helically undulating bubble does not exhibit modulation, thetransverse location from which they originate from does show periodic modulation[Fig 5 (a)].

We note the periodicity of the injected electron distribution in Fig 5 (a)-(b) andinjection rate in Fig 5 (e). Injected electrons’ initial location projected onto x-z planehas an approximate peridicity of cTCEP ≈ 50, in agreement with the time dependenceof ∆H(1). We note that injection rate for linearly undulating bubble has periodicitycTCEP /2, since the injection process happens twice, at z ≈ ±R, for each undulationperiod.

After the electrons are injected into the bubble, they quickly gain relativistic

10

(a) (b)

(d) (e) (f)

(c)

Figure 5: Injection modeling using particle swarm simulations. (a)-(b) Injectedparticles plotted in x0-z0 plane(blue dots) and injection rate (red line) and (d)-(e) Current profile of injected particles at t = 150 for helically undulating bubble[(a),(d)] and linearly undulating bubble [(b),(e)]. Injected particles’ distribution in

γ, p⊥ =√

p2y + p2z plane for expanding (c) and expanding and linearly undulating(f)

bubble at t=400. Parameters: R0 = 6, γb = 6, ǫ = 0.002, zu = 0[(a), (d)], zu =1.5[(b)− (c), (e)− (f)]

energy from the accelerating field and move at ultra relativistic velocity. Becausethe bubble phase velocity is slower than that of the injected electrons, electrons willadvance through the bubble after acceleration to ultra relativistic energy. This slippageof the back of the bubble from the injected bunches determines the longitudinalstructure of the injected bunch in case of a linearly undulating bubble. One canestimate the periodicity of the bunch modulation via converting injection periodicityto that in the moving-frame ξ = x−vbt. The rear of the bubble moves at vbb = vb−Rǫslower than bubble velocity because of bubble expansion. While the CEP phase slipsone cycle, the back of the bubble slips away from the ultra-relativistic particles bydistance ∆ξ = (c − vbb)TCEP . Because there are two injection per one oscillation,injected bunch forms a structure with longitudinal modulation ∆ξ = (c−vbb)TCEP /2[Figure 5(e)].

Since periodic transverse modulation is seen in the injection process, one mayexpect transverse coherent structure to form after electrons are injected, akin to thatseen in the longitudinal bunch structure. However, electrons injected at different timeshave finite energy spread between the different bunches, and the electrons executetransverse oscillation with different frequency. Due to incoherent transverse motion,the transverse structure of the bunch is not well estimated by one simple simpleformula.

We also note that electrons injected by an expanding and undulating bubble willgain more transverse momentum than that of an expanding bubble. We plot the

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electron distribution in (γ, p⊥) space, with p⊥ =√

p2y + p2z at further propagation

distance of t=400 for expanding [Fig 5 (c)] and expanding and linearly undulating[Fig 5(f)] bubbles. Electrons injected by an expanding bubble does not spread outin phase space, forming a line-like feature while those injected by an expanding andlinearly undulating bubble is spread out in phase space. This difference will have directconsequences in the radiated photon spectrum, since the photon energy emitted bythe electrons scales according to Ephoton ∝ γ3/2p⊥[50]

4. Control of injected bunch profile using CEP and laser power

In the previous section, we have established that in case of a linearly undulatingbubble, (1) injection is controlled by bubble undulation and (2) injected electronmodulation depends on CEP undulation period TCEP and bubble rear velocity γbb.We demonstrate via further PIC simulations that we can indeed control the injectedbunch profile via CEP control and power variation.

4.1. Effect of pulse absolute Carrier Envelope Phase on the injected bunch

(a) (b) (c)

(d) (e) (f)Δ� � ≈ 40f�Δ�/� ≈ 55��

Figure 6: Injection control using LP laser phase and power. Injection rate(a) andcurrent(b) for different CEP with same laser power(40TW). Injection rate(d) andcurrent(e) for different power with same CEP. Transverse wake(c), and bubble rearposition(f). 40TW φCEP = 0 laser pulse(black line), 40TW, φCEP = π/2 laserpulse(blue line), 50TW, φCEP = 0 laser pulse (orange line)

While bunches are injected into the bubble throughout multiple undulation, theabsolute Carrier-Envelope-Phase of the few-cycle pulse also has direct effect on theinjected electrons. To confirm this, we ran a PIC simulation using a 40TW LPlaser with φCEP = π/2. This leads to phase-shifting of the wake undulation byφCEP = π/2[Fig 6(c)]. Because the undulations control injection process, the trailingcurrent spikes’ position are shifted. Namely, the locations of the troughs in theφCEP = 0 become locations of peaks for φCEP = π/2[Fig 6 (a)-(b)]. We note that

12

earlier injected electrons’ positions are not as well controlled because the undulationamplitude is not so large, and other effects such as beam loading may have played amore dominant role in determining injection dynamics.

4.2. Effect of pulse power on bunch modulation

Modulation period of injected electrons are estimated by ∆ξ ≈ cTCEP/2γ2bb. TCEP ≈

λL/(vph − vg) = (λL/c)(ω2L/ω

2p), and only depends on laser wavelength and plasma

density. However, velocity of the back of the bubble depends on bubble expansion rateaccording to vbb = vb −Rǫ. This suggests that if one can decrease vbb, it will increasethe modulation period.

One way to achieve this is by simply increasing the power of the laser pulse. Whilethe phase velocity of the plasma wake is independent of laser power in blowout regime,bubble radius and bubble elongation rate increases for larger power, assuming one usesthe matched profile[6]. Conversely, one can also decrease the power, decreasing themodulation period of the injected bunches.

To show this, we run a PIC simulation with a 50TW LP pulse with matchedprofile, otherwise with same parameters. As shown in Fig 6 (c) , plasma bubbleundulation period is same regardless of laser power. However, the velocity of back ofthe bubble becomes slower when pulse power is larger due to faster expansion rate[Fig6 (f)]; the 50TW bubble rear has relativistic factor γbb ≈ 4.5, lower than that of a40 TW bubble(γbb ≈ 5), as shown by evolution of ξbb = x − vbbt. This results insame injection period [Fig 6(d)], but longer bunch modulation period [Fig 6(e)]; inthe range 20 < ξ < 33, there are 6 current peaks for the 50TW case with modulationperiodicity∆ξ/c ≈ 55fs, while there are 8 current peaks in the 40TW case withmodulation periodicity ∆ξ/c ≈ 40fs.

5. Discussions

5.1. Efficiency considerations

We comment on the high efficiency of the acceleration schemes. It has been shownthat few-cycle driven lasers can efficiently tranfer their energy into the injectedelectrons[27]. In our simulations, electrons are injected almost as soon as the pulseenters plasma, injecting high charge (∼10nC) into the wakefield. The large injectedcharge can efficiently convert wakefield energy excited by the pulse into electron energy.Furthermore, pulse is almost depleted by the time the monoenergetic peak reachesenergy of 200MeV, which means almost no pulse energy is wasted due to electronsentering the decelerating part of the bubble. This results in high efficiency( 50%) oflaser energy conversion into electron kinetic energy.

5.2. Bunch structure and Radiation generation

Laser-wakefield generated electrons can generate collimated, high-brightness X-raysvia betatron radiation[29]. Because the injected electrons form beams with O(fs)duration, the resulting radiation also generate ultrashort pulses of X-ray radiation[30].The resulting X-rays have been used to image various targets with fine detailssuch as irregular eutectic in the aluminum-silicon (Al-Si) system[31] and biologicalsamples[32].

13

(b)(a)

800

600

400

200

800

600

400

200

(d)(c)

Figure 7: Radiation from injected electrons. (a)-(b) Distribution in γ − p⊥ space ofelectrons over 100MeV energy for linearly polarized pulse(a) and circularly polarizedpulse(b) at 6.4mm propagation. Colormap: density in phase space, pink lines: energycorresponding to critical frequency. (c) Energy spectrum at 6.4mm propagation and(d) Distribution of emitted photons up to 6.4mm propagation for linearly polarizedpulse(red) and circularly polarized pulse(blue)

In our EPUB induced injection, the ability to control electron longitudinalbunching gives us further control of betatron radiation. The beam injected bya linearly polarized laser is highly modulated, generating a corresponding highlymodulated betatron radiation. In contrast, continuous current beam injected bycircularly polarized laser will generate a single longer pulse betatron radiation. Bychanging laser polarization from circular to linear, the duration and number of pulseof betatron radiation can alternate between a continuous 460fs radiation to a pulsed40fs radiation.

As previously stated, betatron radiation is characterized by a critical frequencyωc ∝ γ3/2p⊥. The energy spectra of CP and LP simulations are comparable with amonoenergetic peak at γ = 400[Fig 7(c)]. The monoenergetic peak formation is dueto bubble expansion and phase space rotation[42, 44], with slight difference in energyspread due to different current profiles leading to different beam loading effect[23].

However, EPUB driven by LP and CP laser exhibit distinct transverse undulation[Fig 2 (d),(e)]. As a result, injected electrons’ transverse momentum distribution in theγ−p⊥ space shows stark difference[Fig 7 (a)-(b)]. The CP case lower-energy electrons(γ < 400) have a higher transverse momentum, while there is a small population ofelectrons at γ ∼ 400 which gain higher transverse energy in the LP case.

Consequently, CP case has copious electrons with critical frequency at 200eV <

14

~ωC < 800eV [Fig 7(a)], while the LP case has a small number of electrons whosecritical frequency extends to higher energy (800eV < ~ωC)[Fig 7(b)]. This translatesto more radiation in CP case for photon energy Eph < 800eV , but a higher tail ofenergy in LP case for photon energy Eph > 800eV .

6. Conclusions

In this paper, we propose and theoretically demonstrate a versatile approach usingfew-cycle TW class laser and a preformed plasma to generate a spatially structuredbeam. The degree of bunching is controlled via changing laser polarization, alternatingbetween highly modulated high-current beam or a flat-current beam. An appreciablefraction of the injected beam form a highly mono energetic energy peak, and laserpulse energy is efficiently transferred (50%) to the injected electrons. The modulationperiod can be altered via changing laser power or density, and the precise location ofsome of the injected bunch can be controlled via changing laser CEP. Furthermore, theinjected electrons exhibit polarization dependent betatron radiation . We envision thatour injection and acceleration scheme will be a efficient way to generate bright betatronX-ray source in the soft X-ray regime switching between pulsed and continuousconfiguration, with controllable duration and time-delay.

7. Acknowledgments

This work was supported by the Department of Energy under a Grant No. DE-SC0019431 and by the National Science Foundation under a Grant No. PHY-2109087.The authors thank the Texas Advanced Computing Center (TACC) at The Universityof Texas at Austin for providing the HPC resources.

8. References

[1] V. Malka, J. Faure, Y. A. Gauduel, E. Lefebvre, A. Rousse, and K. T. Phuoc, “Principles andapplications of compact laser–plasma accelerators”, Nat. Phys, 4, 447-453 (2008).

[2] S. M. Hooker, “Developments in laser-driven plasma accelerators”, Nat. Photon, 7, 775–782(2013).

[3] E. Esarey, C. B. Schroeder, and W. P. Leemans, “Physics of laser-driven plasma-based electronaccelerators”, Review of Modern Physics, 81, 1229 (2009).

[4] K. Nakamura, B. Nagler, C. T´oth, C. G. R. Geddes, C. B. Schroeder, E. Esarey, and W.P. Leemans, “GeV electron beams from a centimeter-scale channel guided laser wakefieldaccelerator”, Phys. Plasmas, 14, 056708 (2007)

[5] X. Wang, R. Zgadzaj, N. Fazel, Z. Li, S. A. Yi, X. Zhang, W. Henderson, Y.-Y. Chang,R. Korzekwa, H.-E. Tsai, C.-H. Pai, H. Quevedo, G. Dyer, E. Gaul, M. Martinez, A. C.Bernstein, T. Borger, M. Spinks, M. Donovan,V. Khudik, G. Shvets, T. Ditmire, and M.C. Downer, “Quasi-monoenergetic laser-plasma acceleration of electrons to 2 GeV”, Nat.

Comms, 4,1988(2013)[6] W. Lu, M. Tzoufras, C. Joshi, F. S. Tsung, W. B. Mori, J. Vieira, R. A. Fonseca, and L. O.

Silva, “Generating multi-GeV electron bunches using single stage laser wakefield accelerationin a 3D nonlinear regime”, Phys. Rev. ST Accel. Beams, 10, 061301 (2007)

[7] W. P. Leemans, A. J. Gonsalves, H.-S. Mao, K. Nakamura, C. Benedetti, C. B. Schroeder, Cs.T´oth, J.Daniels, D. E. Mittelberger, S. S. Bulanov, J.-L. Vay,C. G. R. Geddes, and E. Esarey,“Multi-GeV electron beams from capillary-discharge-guided subpetawatt laser pulses in theself-trapping regime”, Phys. Rev. Lett, 113, 245002 (2014)

[8] H. T. Kim, V. B. Pathak, K. H. Pae, A. Lifschitz, F.Sylla, J. H. Shin, C. Hojbota, S. Ku. Lee, J.H. Sung, H. W. Lee, E. Guillaume, C. Thaury, K. Nakajima, J.Vieira, L. O. Silva, V. Malkaand C. H. Nam, “Stable multi-GeV electron accelerator driven by waveform-controlled PWlaser pulses”, Sci.Rep, 7,10203 (2017)

15

[9] A. J. Gonsalves, K. Nakamura, J. Daniels, C. Benedetti,C. Pieronek, T. C. H. de Raadt, S.Steinke, J. H. Bin,S. S. Bulanov, J. van Tilborg, C. G. R. Geddes, C. B.Schroeder, Cs. T´oth,E. Esarey, K. Swanson, L. Fan-Chiang, G. Bagdasarov, N. Bobrova, V. Gasilov, G.Korn, P.Sasorov, and W. P. Leemans, “Petawatt Laser Guiding and Electron Beam Acceleration to 8GeV in a Laser-Heated Capillary Discharge Waveguide”, Phys.Rev. Lett, 122, 084801 (2019).

[10] M.F. Gilljohann, H. Ding, A. Dopp, J. Gotzfried, S. Schindler, G. Schilling, S. Corde, A. Debus,T. Heinemann, B. Hidding, S.M. Hooker, A. Irman, O. Kononenko, T. Kurz, A. Martinez dela Ossa, U. Schramm, and S. Karsch Phys.Rev.X, 9, 011046 (2019).

[11] P. Mora, and T.M. Antonsen, Jr, “Kinetic modeling of intense, short laser pulses propagatingin tenuous plasmas”, Phys. Plasmas , 4,217 (1997).

[12] J. Huijts, I. Andriyash, L. Rovige, A. Vernier, and J. Faure, “Identifying observablecarrier-envelope phase effects in laser wakefield acceleration with near-single-cycle pulses”,Phys.Plasmas , 28, 043101,(2021)

[13] S. Xu, J. Zhang, N. Tang, S. Wang, W. Lu, and Z. Li, “Periodic self-injection of electrons in afewcycle laser driven oscillating plasma wake”, AIP Advances, 10,095310,(2020)

[14] F. Salehi, M. Le, L. Railing, M. Kolesik, and H. M. Milchberg, “Laser-accelerated, low divergence15 MeV quasi-monoenergetic electron bunches at 1 kHz”, Phys. Rev. X 11,021055.(2021)

[15] J. Kim, T. Wang, V. Khudik, and G. Shvets, ”Subfemtosecond wakefield injector and acceleratorbased on an undulating plasma bubble controlled by a laser phase” Phys. Rev. Lett 127,164801.(2021)

[16] E. N. Nerush, and I. Yu. Kostyukov, “Carrier-Envelope Phase Effects in Plasma-Based ElectronAcceleration with Few-Cycle Laser Pulses”, Phys. Rev. Lett., 103, 035001 (2009).

[17] M.K. Weikum, F.Y.Li, R.W. Assmann, Z.M. Sheng, and D. Jaroszynski , “Generation ofattosecond electron bunches in a laser-plasma accelerator using a plasma density upramp”,Nucl. Instrum. Methods Phys. Res. A, 829, 33-36 (2016).

[18] F. Y. Li, Z.M. Sheng, Y. Liu, J. Meyer-ter-Vehn, W.B.Mori, W.Lu, and J. Zhang, “DenseAttosecond Electron Sheets from Laser Wakefields Using an Up-Ramp Density Transition”,Phys. Rev. Lett., 110, 135002 (2013).

[19] I. Yu. Kostyukov, E. N. Nerush, A. Pukhov, and V. Seredov, “Electron Self-Injection inMultidimensional Relativistic-Plasma Wake Fields”, Phys. Rev. Lett., 103, 175003 (2009).

[20] I. Kostyukov, A. Pukhov, and S. Kiselev, “Phenomenological theory of laser-plasma interactionin “bubble” regime”, Phys. Plasmas, 11, 5256 (2004).

[21] J. B. Rosenzweig, B. Breizman, T. Katsouleas, and J. J. Su, “Acceleration and focusing ofelectrons in two-dimensional nonlinear plasma wake fields”, Phys.Rev.A, 44, R6189(R) (1991).

[22] A. Pukhov, “Three-dimensional electromagnetic relativistic particle-in-cell code VLPL (VirtualLaser Plasma Lab)”, J. Plasma Phys., 61, 425 (1999).

[23] M. Tzoufras, W. Lu, F. S. Tsung, C. Huang, W.B. Mori, T. Katsouleas, J. Vieira, R. A. Fonseca,and L. . Silva, “Beam loading by electrons in nonlinear plasma wakes”, Phys. Plasmas , 16,056705.(2009)

[24] Yong Ma, Liming Chen, Dazhang Li, Wenchao Yan, Kai Huang, Min Chen, Zhengming Sheng,Kazuhisa Nakajima, Toshiki Tajima, and Jie Zhang, “Generation of femtosecond gamma-raybursts stimulated by laser-driven hosing evolution”, Sc. Rep, 6, 6,30491 (2016).

[25] J Vieira et al, “Onset of self-steepening of intense laser pulses in plasmas”, New J. Phys 12,

045025 (2010)[26] C. D. Decker,W.B.Mori, K. -C Tzeng, and T. Katsouleas, “The evolution of ultra-intense, short-

pulse lasers in underdense plasmas”, Phys. Plasmas 3, 2047 (1992)[27] Daniel Papp et al “Highly efficient few-cycle laser wakefield electron accelerator” Plasma Phys.

Control. Fusion, 63 065019 (2021).[28] M.N. Polyanskiy, M. Babizen, and I.V. Pogorelsky, “BESTIA (Brookhaven Experimental Supra-

Terawatt Infrared at ATF) laser: A status report” AIP Conference Proceedings, 1812 110007(2017).

[29] F. Albert, and A G R Thomas “Applications of laser wakefield accelerator-based light sources”Plasma Phys. Control. Fusion 58 103001 (2016)

[30] A. Rousse, K T Phuoc, R Shah, A Pukhov, E Lefebvre, V Malka, S Kiselev, F Burgy, J Rousseau,D Umstadter, and D Hulin “Production of a keV X-Ray Beam from Synchrotron Radiationin Relativistic Laser-Plasma Interaction” Phys.Rev.Lett 93 135005(2004)

[31] A.E. Hussein, N. Senabulya, Y. Ma, et al. “Laser-wakefield accelerators for high-resolution X-rayimaging of complex microstructures” Sci Rep 9, 3249 (2019).

[32] S. Fourmaux, S. Corde, K. Ta Phuoc, P. Lassonde, G. Lebrun, S. Payeur, F. Martin, S. Sebban,V. Malka, A. Rousse, and J. C. Kieffer, “Single shot phase contrast imaging using laser-produced Betatron x-ray beams,” Opt. Lett. 36, 2426-2428 (2011)

16

[33] M. Chen, Z. M. Sheng, Y. Y. Ma, and J. Zhang, “Electron injection and trapping in a laserwakefield by field ionization to high-charge states of gases”, J. Appl. Phys., 99, 056109 (2006).

[34] A. Pak, K. A. Marsh, S. F. Martins, W. Lu, W. B. Mori, and C. Joshi, “Injection and Trapping ofTunnel-Ionized Electrons into Laser-Produced Wakes”, Phys. Rev. Lett., 104, 025003 (2010).

[35] C. McGuffey et.al., “Ionization Induced Trapping in a Laser Wakefield Accelerator”, Phys. Rev.Lett., 104, 025004 (2010).

[36] X. L. Xu, C.-H. Pai, C. J. Zhang, F. Li, Y. Wan, Y. P. Wu, J. F. Hua, W. Lu, W. An, P. Yu, C.Joshi, and W. B. Mori, “Nanoscale Electron Bunching in Laser-Triggered Ionization”, Phys.Rev. Lett., 117, 034801 (2016).

[37] C. G. R. Geddes, K. Nakamura, G. R. Plateau, Cs. Toth, E. Cormier-Michel, E. Esarey, C. B.Schroeder, J. R. Cary, and W. P. Leemans, Phys. Rev. Lett. 100, 215004 (2008).

[38] K. Schmid, A. Buck, C. M. S. Sears, J. M. Mikhailova, R. Tautz, D. Herrmann, M. Geissler, F.Krausz, and L. Veisz, Phys. Rev. Sp. Top. Acc. Beams 13, 091301 (2010).

[39] A. Buck, J. Wenz, J. Xu, K. Khrennikov, K. Schmid, M. Heigoldt, J. M. Mikhailova, M. Geissler,B. Shen, F. Krausz, S. Karsch, and L. Veisz, Phys. Rev. Lett. 110, 185006 (2013).

[40] A. J. Gonsalves, K. Nakamura, C. Lin, D. Panasenko1, S. Shiraishi, T. Sokollik, C. Benedetti,C. B. Schroeder, C. G. R. Geddes, J. Van Tilborg et al., Nature Physics 7, 862 (2011).

[41] S. A. Yi, V. Khudik, C. Siemon, and G. Shvets, Phys. Plasmas 20, 013108 (2013).[42] S. Kalmykov, S. A. Yi, V. Khudik, and G. Shvets, Phys. Rev. Lett. 103, 135004 (2009).[43] X. Wang, R. Zgadzaj, N. Fazel, Z. Li, S. A. Yi, X. Zhang, W. Henderson, Y. Chang, R. Korzekwa,

H. Tsai et al. Nature Comms. 4, 1988 (2013).[44] S. A. Yi, V. Khudik, S. Y. Kalmykov, and G. Shvets, Plasma Phys. Control. Fusion 53, 014012

(2011).[45] A. Pak, K. A. Marsh, S. F. Martins, W. Lu, W. B. Mori, and C. Joshi, Phys. Rev. Lett. 104,

025003 (2010).[46] H. Suk, N. Barov, J. B. Rosenzweig, Phys. Rev. Lett. 86, 1011 (2001).[47] R. Lehe, A. F. Lifschitz, X. Davoine, C. Thaury, and V. Malka, Phys. Rev. Lett. 111, 085005

(2013).[48] J. Faure et.al., “A review of recent progress on laser-plasma acceleration at kHz repetition rate”,

Plasma Phys. Control Fusion ,61, 014012 (2018).[49] S. Chou, J. Xu, D. Cardenas, D. Rivas, T. Wittmann, F. Krausz, S. Karsch, and L. Veisz,

“Sub-2-Cycle Laser-Driven Wakefield Electron Acceleration”, 2013 Conference on Lasers and

Electro-Optics - International Quantum Electronics Conference (2013)[50] Xi Zhang et al, “Effects of laser polarization and wavelength on hybrid laser wakefield and direct

acceleration” Plasma Phys. Control. Fusion ,60 105002, (2018)