Effective Electron Beam Injection With BroadEnergy Initial Beam

J. H. Cooley1,2,3, R. F. Hubbard3, D. F. Gordon3, A. Zigler2, A. Ting3 andP. Sprangle3

1 University of Maryland, College Park, MD2 Icaras Research Inc., Bethesda, MD

3 Plasma Physics Division, Naval Research Laboratories, Washington, DC.

Abstract. Laser Wakefield Accelerators (LWFA), in the resonant regime, require use of aninjected electron beam. Several optical methods for generating electron bunches exist e.g., LaserIonization and Ponderomotive Acceleration (LIPA) and Self-Modulated LWFA among others.Each of these schemes produces an electron bunch with a characteristic energy distribution. Weexamine the trapping characteristics in a resonant LWFA for an injection electron beam with abroad energy spread that can be characterized using a Boltzmann distribution with an "effectivetemperature". We present results of both analytic calculations and simulations which provide amethodology for optimizing the resulting accelerated electron bunch characteristics i.e., energyand energy spread, for a given LWFA configuration.

INTRODUCTION

Laser Wakefield Accelerators (LWFA) [1] have the potential for providing high-energy electron beams at a fraction of the current cost for a standard linear accelerator.Several potential schemes for LWFA have been proposed and studied. One method isthe resonant LWFA [1], in which a single laser pulse that is resonant with the plasmawave (ω pτ laser ≅ π , and

�

ω p = 4πq2ne /me is the plasma frequency) excites an

accelerating electric field. The last element required to demonstrate LWFA in theresonant case is an initial electron beam to inject into the plasma wakefield.

Ideally, the injected electrons will be produced in such a way that they are mono-energetic and placed in the accelerating phase of the plasma wave. One difficulty inproviding these electrons is the need for precise timing between the electron bunchand the laser pulse used to provide the plasma wave wake. Effects of poor timing, un-phased injection, have been presented elsewhere [2,3] and will not be discussed.Several methods for creating a phased electron bunch have been proposed. Theseinclude Laser Ionization and Ponderomotive Acceleration (LIPA) [4], use of wiretargets [5], use of SM-LWFA [6], Laser Injection Laser Acceleration [7] (LiLAc), andcolliding pulse injection [8] (CPI). Each of these methods has a characteristic electrondistribution in both energy and bunch duration. However, one common trait for mostof these methods is a broad electron energy. In this paper, we examine the propertiesof the acceleration process for an electron distribution that initially has a broad energy

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spectrum and demonstrate that it is possible to utilize these electrons and still producean adequate accelerated electron bunch.

HAMILTONIAN FORMALISM

Detailed discussion of the one-dimensional Hamiltonian formalism is presentedelsewhere [9]. Therefore, the current discussion will be limited to specific resultsnecessary for understanding the following analysis. In particular, in one-dimension,the Hamiltonian function given by

H (γ ,ψ ) = γ 1− βgβ( )mec2 + qφ sinψ (1)

is constant for an electron in the frame of the pulse,

�

ξ = z /vg − t . Here φ = qΦ / mec2

is the normalized plasma wake-field electrostatic potential,

�

ψ = ω pξ , and

�

me and q are

the mass and charge for an electron. The quantity

�

γ is the electron Lorentz factor and

is related to

�

β = ve /c by

�

γ =1/ 1−β 2 . Finally,

�

βg =1− λ2 /2λp2 − λ2 /2π 2ro

2

characterizes the laser pulse group velocity and includes the effects due to finite laserspot size [12], λp = 2πc /ω p is the plasma wave wavelength and

�

vg = cβg .

This Hamiltonian function neglects the transverse wake field affects thatcontribute a radial force to the dynamics of the electrons. In uniform plasma, thisradial force is out of phase with the accelerating force in the wake by π/2. We caninclude these radial effects in the current case by assuming that only electrons thatremain in the region of phase with a focusing electric field gradient in the radialdirection will remain in the accelerating wake. Based on this assumption only phasesfrom 0 to π need be examined in detail.

With this Hamiltonian, we can obtain the equations of motion for the normalizedmomentum, p = p / mec , and phase,

�

ψ , as the laser pulse propagates in the laboratoryframe coordinate, z. These equations are

d pdz

= −qφ0ω p

cβg

1+ 1p2

⎛⎝⎜

⎞⎠⎟

1/2

cosψ , (2a)

and

dψdz

=ω p1vg

−1c1+ 1p2

⎛⎝⎜

⎞⎠⎟

1/2⎡

⎣⎢⎢

⎤

⎦⎥⎥

. (2b)

Figure 1 shows the phase space trajectories of several particles with initialnormalized momentum around four and initial phase from π/2 to 2 integratedaccording to Eqs. 2. The conditions for the wake amplitude, φ=0.1, correspondapproximately to the plasma wake stimulated by a 10x1012 W laser focused into anelectron density ne = 5x10

17 with a vacuum spot size of 30 µm. The solid linerepresents the orbit that encompasses the maximum region of phase space that remainsin a focusing electric field. The electron that follows this trajectory has a normalized

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momentum of 4.3 at phase of π/2. This electron represents the lowest momentum thatcan be trapped by the plasma wakefield.

FIGURE 1. Phase space trajectories for particles evolved according to Eqs. 2 with initial conditionslocated as marked. The solid lines at y=0 and y=π demark the transition into de-focusing regions whereelectrons would be expected to be lost due to radial de-focusing.

SIMULATION RESULTS

We have performed a number of two-dimensional simulations using the codeWAKE [10] with a test particle electron bunch. These simulations correspond toparameters for projected experiments to be performed at the Naval ResearchLaboratory (NRL) in the near future. The laser pulse parameters are a peak power ofapproximately 10x1012 W and a laser spot size of 30 µm, peak intensity approximately7.1x1017 W/cm2.

Figure 2 shows the phase-space for the injected electrons. Figure 2a shows theinitial bunch distribution. Additionally, figure 2a includes both the trajectory thatcorresponds to the minimum-trapping orbit for the one-dimensional Hamiltoniandiscussed above and the Hamiltonian separatrix, which defines the largest closed orbitfor the Hamiltonian. Notice that the test particles have momentum below the orbit forminimum trapping in uniform plasma. In the simulation, the accelerating and focusingfields in the channel were found to have an increase in useful phase up to−π / 6 <ψ useful < π / 2 , consistent with the findings of Andreev [13]. Figure 2b shows

the phase space trajectories for the initial bunch and five evenly spaced intervalsduring the acceleration process, approximately every 1.3 cm.

Figure 3 shows results of integrating Eqs. 2 for a set of test particles chosen tocorrespond to a bounding box around the test particles in the above simulation. Noticethat the relative momentum spread, the solid line, initially decreases until at about 3.5cm propagation the relative spread is around 2 percent. At this acceleration length themean momentum of the electron bunch is about 550. As the electrons continue toaccelerate the relative energy spread rapidly increases. These results agree with thesimulation discussed above. In particular the results of both the Hamiltonian analysisand the simulation indicate that the relative momentum spread and maximum

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FIGURE 2. a) Initial particle distribution for this simulation. Also included are both the Hamiltonianorbit that corresponds the minimum momentum for trapping in a uniform plasma and the separatrix, thelargest closed orbit for the Hamiltonian. Notice that none of the initial particles should be capturedaccording to one-dimensional theory. b) Phase-space plot for the entire simulation. Notice that aftermore then 6 cm of propagation about 1% of the test-particles are trapped and accelerated. Also noticethat the minimum energy spread, ~2%, occurs after approximately 3.8 cm of acceleration but onlyresults in momentum gains of 600, e.g. 300 MeV electrons.

momentum gain are not independent and that the optimal beam quality will require atrade-off with total energy gain. This result is most clearly seen in Figure 2b whichshows that a high quality electron bunch is present at half the acceleration lengthapproximately 3.8 cm. The normalized momentum is over 600 while the relativemomentum spread is 2.1%. As the electron bunch continues to accelerate the rear ofelectron bunch can be seen to gain energy relative to the head of the bunch, thusincreasing the relative spread in momentum.

The final issue that we explore in two-dimensions is the maximum charge that canbe trapped from an electron beam with a large energy spread. There are two issuesinvolved in this beam-loading problem. The first involves the depletion of the

FIGURE 3. Plot of the Relative momentum spread (solid) and mean normalized momentum (dashed)as the electron bunch is integrated according to Eqs. 2. The initial particles were chosen to correspondto a six-point distribution containing the lowest energy electrons that will be trapped in the wakefieldbased on the WAKE simulation results. Notice that the minimum spread in momentum does notcorrespond to the maximum energy gain for the accelerator.

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wakefield amplitude as the plasma wave energy is converted into energy in theaccelerated electrons. This problem has been described in detail [14]. The beam-loading issue that is of concern for the present work is associated with the effect of theelectrons that have energy to low to be trapped, yet can disrupt the plasma wakefieldinitially such that the trapping and acceleration of higher energy electrons is reduced.

To explore this issue in two-dimensions we utilize the code TurboWave [11] intwo-dimensional slab geometry. We treat the laser field in the ponderomotive limit.We have performed a series of simulations with self-consistent beam-loading effects.Figure 4 presents the charge in the accelerated bunch as a function of the total chargein the initial beam for several values of total charge. The electron bunch was initiallyphased with a drift momentum of 0.1 and momentum spread of 1. The laser pulse waschosen to provide a normalized wakefield amplitude of 0.1, with a laser spot size 30µm consistent with the early results from WAKE.

We see in Figure 4 that approximately one percent of the electrons are trapped andaccelerated up to a total charge of approximately 1x1011 electrons at which time theplasma wake is disrupted by the large electron charge present at the start of thesimulations. This disruption leads to a maximum total trapped charge ofapproximately 1.7x109 electrons for a total charge of 1.2x1011 electrons. Furtherincreasing the total charge disrupts the trapping process and leads to a decrease in totalcaptured charge.

FIGURE 4. This figure shows the effect of beam loading on the trapping and acceleration process forinitially thermal electron bunches. Notice that the charge of trapped electrons is linear in terms of totalcharge in the bunch up to a total charge of approximately 1x1011 electrons. At this point the efficiencyof trapping electrons by the plasma wake decreases until at 1.2x1011 electrons the total trapped andaccelerated electrons reaches a maximum and further increasing the total charge is disruptive to thetrapping process.

CONCLUSIONS

Based on the results of the above analysis and simulations, electron bunches withbroad energy spread can be used for injection into resonant LWFA. However, it is notsufficient to specify the maximum energy gain for an accelerator system, rather one

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must determine the proposed application for the accelerated electron bunch and thusdefine the optimal final parameters. For instance, if maximum total energy gain isimportant and energy spread of the electron bunch is not important, then the LWFAlength should correspond to the de-phasing length of the plasma wave. However, ifthe energy spread is critical then a reduced accelerator length and thus energy gainshould be used.

ACKNOWLEDGMENTS

JHC and AZ would like to thank DOE for the SBIR-phase II grant that funded thiswork. Work at the Naval Research Laboratories was supported by DOE and ONR.

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