November 2, 2018

Magnetically assisted self-injection and radiation generation for plasma basedacceleration

J. Vieira1,∗ J. L. Martins1, V.B. Pathak1, R. A. Fonseca1,2, W.B. Mori3, and L. O. Silva1†1GoLP/Instituto de Plasmas e Fusao Nuclear-Laboratorio Associado, Instituto Superior Tecnico, Lisboa, Portugal

2DCTI/ISCTE Lisbon University Institute, 1649-026 Lisbon, Portugal and3Department of Physics and Astronomy, UCLA, Los Angeles, California 90095, USA

It is shown through analytical modeling and numerical simulations that external magnetic fieldscan relax the self-trapping thresholds in plasma based accelerators. In addition, the transverselocation where self-trapping occurs can be selected by adequate choice of the spatial profile of theexternal magnetic field. We also find that magnetic-field assisted self-injection can lead to theemission of betatron radiation at well defined frequencies. This controlled injection technique couldbe explored using state-of-the-art magnetic fields in current/next generation plasma/laser wakefieldaccelerator experiments.

PACS numbers: 52.38.Kd, 41.75.Jv, 52.25.Xz, 52.65.Rr

INTRODUCTION

Plasma based accelerators (PBA) use high intensity

laser pulses [1], with intensities above I ∼ 1018 W/cm2,

or highly charged particle bunch drivers [2], with morethan 1010 charged particles, to excite ultra-relativisticplasma waves. The ideal plasma density to maximizecharge and energy gain depends on the nature of thedriver (i.e. lepton, hadron or laser pulse), typically rang-ing between n0 = 1014 − 1019 cm−3. At these plasmadensities, charged particle bunches can be accelerated byplasma wakefields to 1-100 GeV in 1-100 cm [3].

The proof-of-principle of PBA is firmly demon-strated [4, 5]. Presently, connection with applications [6]is an essential step to further improve this technology.To this end, fine control over the properties of the ac-celerated electrons is required. Several techniques wereproposed to control self-trapping. Control over chargeand energy of accelerated bunches can be reached usingplasma ramps [7], counter propagating lasers [8], to ion-ization mechanisms [9], and resorting to non-linear opti-cal effects such as self-focusing [10].

A novel technique using transverse magnetic fieldsto relax self-injection thresholds has been recently pro-posed [11]. The use of external magnetic fields in plasmaacceleration was first proposed to extend the accelera-tion distances in plasma accelerators in the surfatronmodel [12]. The role of external magnetic fields in PBAwas also explored in [13], and the use of longitudinal mag-netic fields to enhance the self-injected charge in laserwakefield acceleration was investigated in [14].

This paper presents a detailed derivation of the self-trapping threshold condition in the presence of externalfields. Using the particle-in-cell (PIC) code Osiris [15],it is shown that magnetic injection can be used to gener-ate single or multiple off-axis self-injected bunches withwell defined radial injection positions. Using the post-

processing radiation code JRad [16] it is demonstratedthat these electrons may emit betatron radiation at welldefined frequencies close to the undulator regime. Thispaper is structured as follows. In Sec. describes ananalytical trapping condition in the presence of externalfields. In Sec. , 3D PIC simulation results are employedto analyze the relevant physical mechanisms of magneticfield assisted self-injection. The use of different B-fieldgeometries to control transverse features of magneticallyinjected electrons is described in Sec. . Section showsthat magnetically assisted injection can lead to the emis-sion of clearly defined betatron radiation harmonics forthe first time in PBAs. Finally, conclusions are stated inSec. .

TRAPPING CONDITIONS IN THE PRESENCEOF EXTERNAL FIELDS

The dynamics of the electrons in the fields createdby an intense laser in the blowout regime can be de-scribed using Hamiltonian dynamics [9]. A general trap-ping condition in the presence of external fields can befound by examining the evolution of the Hamiltonian ofplasma electrons in the co-moving frame, (x = x, y =y, ξ = vφt − z, s = z), given by H = H − vφP‖, where(x, y) are the transverse coordinates, z the distance,vφ the wake phase velocity (determined by the drivergroup velocity), P‖ the longitudinal canonical momen-

tum, H =√m2ec

4 + (P + eA)2 − eφ the Hamiltonian ofa charged particle in the presence of electric and magneticfields, me and e the electron mass and charge, c the speedof light, A and φ the vector and scalar potentials, andP = p− eA, where P and p are the canonical and linearmomentum respectively. Normalized units will be usedhenceforth unless explicitly stated. Mass and charges arenormalized to me and e, respectively, velocity v to c, time

arX

iv:1

210.

1947

v1 [

phys

ics.

plas

m-p

h] 6

Oct

201

2

2

to ωp =√

4πn0e2/me, momentum to mec, and densityto the background plasma density n0. Vector and scalarpotentials are normalized to e/mec

2 and e/mec respec-tively. Magnetic fields (B) are normalized to ωc/ωp whereωc = eB/me is the cyclotron frequency.

In order to derive a trapping threshold condition in thepresence of external fields, we consider first the expres-sion for the temporal evolution of H:

dHdt

= (vφ − v‖)dHdξ

=∂H

∂s=

[v · ∂A

∂s− ∂φ

∂s

], (1)

Integration over the particle trajectory yields:

Hf −Hi =

∫dt

dHdt

=

∫dξ

vφ − v‖dHdt, (2)

where the subscripts ’i‘ and ’f ‘ refer to the initial andfinal (trapped) electron positions. The integration is per-formed along the electron trajectory, and dt = dξ/(vφ −vz). Combining Eq. (1) with Eq. (2) gives:

Hf −Hi =

∫dξ

dHdξ

=

∫dξ

[v · ∂A

∂s− ∂φ

∂s

], (3)

Using the definition for H, and considering that initiallyelectrons are at rest (i.e. pi = 0):

Hf−Hi = γf−vφpf‖−1−(φf − vφAf‖

)−(φi − vφAi‖

),

(4)

with φ = φpl + φext, A‖ = Apl‖ + Aext

‖ , and where the

superscripts ’pl‘ and ’ext‘ refer to the plasma and externalfields respectively, and γ = (1−v2)−1/2 is the relativisticfactor. Defining the wake potential ψ = φ− vφA‖, ∆ψ =ψf −ψi, and assuming that for trapping the longitudinalvelocity of the electron must reach the velocity of thewake i.e. vz = vφ, Eq. (4) readily becomes:

Hf −Hi =γ

γ2φ

− 1−∆ψpl −∆ψext. (5)

where γφ = (1 − v2φ)−1/2 is the gamma factor of the

plasma wave. Using Eq. (3) to express Hf − Hi leadsto the trapping condition [11]:

1 + ∆ψpl =γ

γ2φ

−∫

dHdξ

dξ −∆ψext. (6)

Equation (6) is a general trapping condition in the pres-ence of external fields, and valid beyond the range of va-lidity of the quasi-static approximation [17]. Analyticalsolutions to Eq. (6), however, are not yet known becausethe calculation of ∆ψpl and

∫dξHdξ requires accurate

prediction of the particle trajectories and field structuresat the back of the bubble, where the applicability of thestandard analytical models [18] is limited.

To retrieve a general trapping threshold in the absenceof external fields, and in the conditions where the quasi-static approximation is valid, it should be considered

∆ψext = 0, and∫

dξHdξ = 0 in Eq. (6). For an ultra-relativistic plasma wave with γφ → ∞ trapping occurswhen 1 + ∆ψpl = 0, or, equivalently, ∆ψpl = −1. Gen-erally, this condition can only be met at the back of theplasma wave, in regions of maximum accelerating fields,where ψpl is minimum and approaches ψpl = −1 [9]. Inthe presence of static external fields the trapping condi-tion becomes 1 + ∆ψpl = −∆ψext. The trapping thresh-olds are relaxed because they can be met when ∆ψpl islarger than −1 provided that ∆ψext > 0. In other words,trapping may occur for lower values of peak accelerat-ing gradients. Moreover, trapping may be suppressed if∆ψext < 0.

Trapping can also be relaxed (or suppressed) when theexternal fields vary spatially in z because of the contri-bution of finite

∫dξHdξ 6= 0 to Eq. (6). If the pro-

file of the external fields profile leads to∫

dξHdξ > 0(∫

dξHdξ < 0) along the electron trajectory then trap-ping is facilitated (suppressed) [10, 11]. Physically, thefact that

∫dξHdξ > 0 is typically associated with the re-

duction of the wake phase velocity through the accordioneffect, thus facilitating self-injection [10, 11, 19].

MAGNETICALLY CONTROLLEDSELF-INJECTION IN LWFAS AND PWFAS

To investigate controlled self-trapping in the presenceof external static magnetic fields we present in this sec-tion 3D particle-in-cell simulations of laser (LWFA) andplasma (PWFA) wakefield accelerators using the particle-in-cell code Osiris. Figure 1 illustrates the evolution,and highlights the key mechanisms of injection assistedby external B-fields in the LWFA. The simulation win-dow moves at the speed of light, with dimensions of24 × 24 × 12 (c/ωp)

3, divided into 480 × 480 × 1200cells with 1 × 1 × 2 electrons per cell in the (x, y, z)directions respectively. The plasma ions are immobile.A linearly polarized laser pulse with central frequencyω0/ωp = 20 was used, with a peak vector potential ofa0 = 3, a duration ωpτFWHM = 2

√a0, and a trans-

verse spot size matched to the pulse duration such thatW0 = cτFWHM [3]. The plasma density is of the formn = n0(z)

(1 + ∆nr2

)for r <

√10 c/ωp and n = 0 for

r >√

10c/ωp with ∆n = ∆nc = 4/W 40 (i.e. the nor-

malized matching condition given by ∆nc = 4/(πreW20 ),

where re = e2/mec2 is the classical electron radius) be-

ing the linear guiding condition in the normalized units,and where n0(z) is a linear function of z which increasesfrom n0 = 0 to n0 = 1 for 50 c/ωp to ensure a smoothvacuum-plasma transition. The channel guides the frontof the laser thereby minimizing the evolution of the bub-ble. A static external B-field pointing in the positivey-direction was used. At the point where the plasma den-sity reaches its maximum value, the external field riseswith Bext

y = ωc/ωp = 0.6 sin2[πz/(2Lramp) + Φ1], with

3

Lramp = 10c/ωp. It is constant and equal to Bexty = 0.6

for Lflat = 40 c/ωp and then drops back to zero withBexty = 0.6 sin2[πz/(2Lramp) + Φ2]. Moreover, Φ1 and

Φ2 are phases chosen to guarantee the continuity of theexternal B-field profile. Qualitatively, the longitudinalprofile of the magnetic field thus resembles that of [20].

0 4 10 12

[c/ ]p

0

40

80

120

p [m

c]

eII

x [c/

] p

-4

0

4

8

0

4

8

Den

sity

[n ] 0

0

60

30

E

[m

c

e

]-1

pe

lase

r

Den

sity

[A

rb. U

nit.]

0.01

10.0

(a) (c) (e)

(b) (d) (f)

Bext

Bext

Bext

-8

B profileext

|Den

sity

| �[A

rb.U

nits]

0

50

10-10-10

10

px [m

e c]

py [me c]

FIG. 1: 3D osiris simulation results illustrating the mag-netic self-injection mechanisms. (a), (c), and (e) show theelectron plasma density in gray, the self-trapped particles inblue, and the laser pulse envelope in red colors at t = 110/ωp,t = 126/ωp and t = 159/ωp. (b), (d), and (f) show the cor-responding p‖ − ξ phase-space. The magnetic field leads tooff-axis self-injection in a narrow angular region. The insetin (f) represents the transverse momentum phase-space of theself-injected electron bunch residing within the bubble. TheB-field profile is schematically represented on the top of thefigure. The laser driver moves from left to right as indicatedby the arrow.

Self-injection is absent from the regions where the B-field rises. In these regions, electrons traveling backwards(vz < 0) near the back of the bubble feel an increas-ing v × Bext force that rotates electrons anti-clockwisethereby locally decreasing (increasing) the blowout ra-dius for x > 0 (x < 0). Then, as the B-field rises, thelocal wake phase velocity at the back of the bubble in-creases (decreases) for x > 0 (x < 0). For x > 0, vφis superluminal,

∫dξHdξ < 0, and self-injection can not

occur. For x < 0 trapping is precluded because electronsreach the axis in regions where the plasma focusing andaccelerating fields are unable to focus and trap electrons.Thus, although for x < 0

∫dξHdξ > 0, we have that∫

dξHdξ + ∆ψext < 0.Self-trapping occurs in the uniform regions of the ex-

ternal magnetic field where x > 0. For x > 0, elec-trons rotating anti-clockwise reach the axis in regions ofmaximum focusing and accelerating fields with larger p‖and can be trapped. For x < 0, electrons reach regionsof the axis (where focusing and accelerating fields arelower) with lower p‖, and are are lost to the surround-ing plasma. A threshold B-field for injection may beretrieved in the limit where γφ → ∞. Neglecting theplasma fields (∆ψpl = 0), and noting that the exter-nal longitudinal vector potential Aext

‖ = −Byx is consis-tent with the considered magnetic field, leads to a sim-

plified trapping condition ∆ψext = −By∆x = 1, where∆x = xf − xi ' −rb, where rb is the blowout radius andwhere it was considered that the initial (final) trappedelectron radial position is x = rb (x = 0). It shows thatinjection is facilitated in the region where ∆x < 0 is min-imum. As a consequence, injection occurs off-axis (forx > 0), and in a well defined azimuthal region defined by−Byrb sin θ = 1, where θ is the angle between the planeof the electron trajectory with the B field [11]. Note,however, that this trapping threshold condition overes-timates the threshold B-field for self-injection because itneglects the plasma fields.

There is an upper By value, given by ωc/ωp . 1, be-yond which injection may be suppressed in regions wherethe B field is flat. The later condition ensures that theplasma wakefields are nearly undisturbed by the exter-nal fields. Simulations then showed that when ωc/ωp � 1there is a suppression of the wakefields that prevents in-jection. Hence trapping can be relaxed in the regions ofuniform B-fields provided that 1/rb . By . 1 or, equiv-

alently 170/rb[10µm] . B[T] . 32√n0[1016cm−3].

The above-mentioned upper B-field limit for injectionis absent from the downramp regions, where a strongerself-injection burst occurs for x > 0. Injection occurswithin the same angular and radial region as in the uni-form B-field section (Fig. 1c). For x > 0, when the B-field lowers rb increases, vφ lowers and

∫dξHdξ > 0,

facilitating injection. For x < 0, vφ > 1, and trapping issuppressed. The resulting phase space at t = 126c/ωp isshown in Fig. 1d.

After the magnetized plasma region, the magneticallyinjected electron bunch is clearly detached from the backof the bubble, leading to the generation of a quasi-mono-energetic electron bunch. The magnetic injected electronbunch right after the B-field is shown at t = 159c/ωpin Fig. 1e, and the corresponding phase-space in Fig. 1f.The inset of Fig. 1f shows the transverse phase space ofthe magnetically injected electron bunch residing withinthe blowout region. The asymmetrical distribution re-sults from the fact that the injection process occurs off-axis. At this location, the emittance of the beam is onthe order of 1π mm mrad in both transverse directions.Although comparison of beam emittance with a similarscenario without the B-field is not meaninfull becausewithout the field the amount of self-injected charge ismuch smaller. However, the measured beam emittanceis at the same values or lower than typical emittances ofLWFAs.

External magnetic fields also relax the self-trappingthresholds in the PWFA. Figure 2 shows results from a3D simulation of a magnetized PWFA. A 30 GeV elec-tron bunch was considered with density profile given

by nb = nb0 exp(−x2⊥/(2σ

2⊥))

exp(−ξ2/(2σ2

ξ ))

, with

σ⊥ = 0.17 c/ωp, σξ = 1.95 c/ωp, and nb/n0 = 19. Theseparameters ensure that rb is similar to the magnetized

4

LWFA investigated above. The simulation window di-mensions are 24 × 24 × 24 (c/ωp)

3, and it is divided in480× 480× 640 cells with 1× 1× 2 electrons per cell inthe (x, y, z) directions respectively. The magnetic fieldprofile is similar to the LWFA case.

255 260 265 270 275

ξ[cωp]

6

4

2

0

-2

-4

-6

[cω

p]

x

7

6

5

4

3

2

1

0

Den

sity

[A

.U.]

101

10-4

# [A

.U.]

0 150 300Energy [MeV]

0.0

0.2

0.6

B =0.0ext

B =0.2ext

B =0.6ext

Bext

Bext

(a) (b) (c)

FIG. 2: 3D osiris simulation results illustrating magnetic self-injection in the plasma wakefield accelerator using (a) Bext =0, (b) Bext = 0.2, and (c) Bext = 0.6 at t = 275/ωp. In thisscenario the B-field injection mechanism is dominated by theplasma bubble dynamics in the B-field down-ramp. It is clearthat the trapped charge increases with the amplitude of theexternal field. The driver moves from left to right as indicatedby the arrow in (c).

The accelerating structures are similar for the LWFAand PWFA parameters described above since the blowoutradius are similar for both cases. However, as shown byEq. (6), self-injection thresholds are harder to meet in thePWFA than in the LWFA because γPWFA

φ ' 60000 �γLWFAφ ' 20. In contrast to the LWFA scenario, injec-

tion is then absent in the PWFA in the uniform regionsof the B-field, where the larger pz at the back of the bub-ble for x > 0, associated with the additional electronsv×B anti-clockwise rotation, is still below that requiredfor injection. Magnetic self-injection occurs only in theB-field downramp (Fig. 2b-c), where the injection mecha-nism is similar to that ascribed to the LWFA. In general,stronger self-injection bursts occur in the B-field down-ramp for both LWFA and PWFA.

The amount of self-injected charge can be tuned bychanging the B-field amplitude. The inset of Fig. 2cshows the spectra of the self-injected charge in the firstplasma bucket using Bext = 0.6 (red curve), Bext = 0.2(green curve), and Bext = 0.0 (blue curve). The amountof trapped charge is negligible in the un-magnetized sce-nario, and it is roughly 8 times larger for Bext = 0.6than for Bext = 0.2 (notice that the plot is logarithmic inthe vertical y-direction). These results show that higherB-field amplitudes increase the total amount of injectedcharge.

Because of beam-loading [21], higher amounts of self-injected charge lead to lower accelerating gradients. Con-sequently, the maximum energy that can be achieved islower for self-injected bunches with higher charges. Thisis consistent with the inset of Fig. 2c which shows thatself-injected bunches with lower charges reach higher en-ergies.

The inset of Fig. 2c also shows that the energy spreadsof the magnetically injected electrons are on the orderof 100%. Due to the short duration of the self-injectedbunch in comparison to the plasma wavelength, whichguarantees uniform acceleration throughout the entirebunch length, the relative energy spread would decreaseas the beam accelerates. Moreover, the energy spreadwould further narrow down near the dephasing lengthdue to the bunch phase-space rotation [22].

For these parameters the threshold magnetic field forself-injection is Bext

y & 0.2. To connect these sim-ulations with actual experimental conditions, we taken0 = 1017 cm−3 for which the electron beam and plasmaparameters match those available at SLAC [5] with σ⊥ =50.4 µm, σz = 84 µm and a total number of 3×1010 elec-trons. For these parameters, Bext

y = 0.2 corresponds to20 T. These magnetic fields could be produced with state-of-the-art magnetic field generation techniques [20, 25].By tuning further the plasma parameters controlled in-jection with magnetic fields as low as 5 T can also beachieved (cf. Sec. ).

SIMULTANEOUS GENERATION OF MULTIPLESELF-INJECTED ELECTRON BUNCHES

The transverse location where self-trapping is relaxedcan be selected by adequate choice of the profile of theexternal magnetic field. As an example, Fig. 3 shows theresults from a 2D slab geometry simulation using a mag-netic field which reverses sign at x = 0 (this is equivalentto an azimuthal B-field profile in cylindrical symmetry).In this case, the magnetic field points outside (inside)the simulation plane for x > 0 (x < 0). The 2D sim-ulations use a simulation box that moves at c with di-mensions 12× 32 (c/ωp)

2, and is divided into 640× 3000cells with 3× 3 electrons per cell in the (x, ξ) directionsrespectively. The laser pulse and plasma channel param-eters are similar to those of the 3D LWFA simulation(cf. Fig. 1). The amplitude of the external B-field isBexty = ωc/ωp = 0.6 sin2[πz/(2Lramp) + Φ1]x/|x|, with

Lramp = 10c/ωp, it is constant and equal to Bexty = 0.6

for Lflat = 50 c/ωp and drops back to zero with Bexty =

0.6 sin2[πz/(2Lramp) + Φ2]x/|x|, where the choice of Φ1

and Φ2 ensures the continuity of the B-field longitudinalprofile.

Figure 3a shows the magnetically injected electrons inthe regions where the B-field is uniform. Two off-axis in-jection bursts occur at well defined transverse positionsin the flat B-field regions. The two bunches are then in-jected symmetrically close to x = 0. An additional andstronger self-injection burst occurs at the B-field down-ramp (Fig. 3b). After the magnetized plasma region, thetwo self-injected electron bunches continuously acceler-ate in the wakefield (Fig. 3c). Note that Fig. 3c refersto the early propagation of the electron bunch, much

5

shorter than the dephasing length. Similarly, the propa-gation distance is much smaller than the betatron periodof oscillation. The physical mechanisms under which self-injection occurs in the present configuration are identicalto those presented in Sections and .

Interestingly, Fig. 3 reveals that injection occurs in ahighly spatially localized region. Off axis injection fromwell defined radial and azimuthal regions was observed inSec. in 3D simulations. Generally, however, this effectis more noticeable in 2D slab geometry simulations thanin 3D. These results also suggest that ring like electronbunches could be obtained in 3D. This could be advan-tageous for radiation generation purposes because bunchparticles would perform betatron oscillations with similaramplitudes.

x [c/

] p

-8

0

4

8(a)

0

2

0 4 8 [c/ ]p

4D

ensi

ty [n ] 0

E [m

c

e ]

-1p

ela

ser

0

50

100

150(b) (c)

-4

12

Bext

Bext

Bext

B profileext

FIG. 3: 2D Osiris simulation of a magnetized LWFA using anazimuthal-like magnetic field. In the upper (lower) simulationmid-plane the B-field is directed outside (inside) the simula-tion plane. (a) is taken in the B-field flat region at t = 79/ωp,(b) in the B-field down-ramp at t = 92/ωp, and (c) after theB-field at t = 102/ωp. The B-field profile is schematicallyrepresented on the top of the figure. This configuration leadsto identical magnetically injected electron bunches in the up-per and lower simulation mid-plane. The laser driver movesfrom left to right as indicated by the arrow. The dark pointsrepresent self-injected particles.

EMISSION OF BETATRON RADIATION ATWELL DEFINED FREQUENCIES

Typical synchrotron radiation experiments in plasmaaccelerators reveal that radiation emission occurs in thewiggler regime. The wiggler regime enables emission ofx-rays with broad spectra [23]. This contrasts with theundulator regime, where radiation is emitted at well de-fined harmonics. Although not yet attained experimen-tally, the undulator regime provides ideal conditions forradiation amplification, being critical for the realizationof a ion-channel plasma based laser [24]. This section il-lustrates how could magnetically injected electrons emitbetatron radiation at well defined frequencies, closer tothe undulator regime.

The PWFA beam and plasma simulation parameters,presented in Sec. , were tuned in order to lower the re-quired magnetic field for injection, such that it could bemore easily reached experimentally, and in order to lower

the amplitude of the betatron oscillations in comparisonto the plasma skin depth, such that distinguishable be-tatron radiation harmonics could be emitted. System-atic 3D parameter scans then showed that the thresholdmagnetic field for injection is 5.5 T at n0 = 1015 cm−3.At this plasma density, Lflat = 40c/ωp = 6.8 mm, andLramp = 10 c/ωp = 1.68 mm, and the maximum B-fieldamplitude is Bext

y = 0.55 ωc/ωp. These parameters arewithin current technological reach [20, 25]. Simulationsused a simulation box with 12× 12× 16 (c/ωp)

3, dividedinto 480× 480× 640 cells with 2× 2× 1 particles per cellfor the electron beam and background plasma.

450350350250

Propagation Distance p [c/ω ]

0.2

0

-0.2

px

[c/ω

]

0 325Energy [MeV]

track

s/

[A.U

.]

0 3

60

1

x [c/ ] ωp

6040200-20-40

Energy [arb.units] 200

(a) (b)

ω [

ωp]

0.8

0.6

0.4

0.2

y [c/ ] ωp

6040200-20-400.0

x104

(c)

Energy [arb.units] 400

y

x

y

x

Detector plane Detector plane

FIG. 4: (a) B-injected electron bunch trajectories in theplasma wakefield accelerator. The trajectories gentle rise inthe x2 direction is due to the subtle transverse velocity shiftimprinted on the particle beam driver in the B-field region.The inset represents the distribution of the radiation strengthparameter αβ . A significant fraction (83%) of the trajectoriesare characterized by αβ � 1. (b)-(c) Corresponding radiationprofile lineouts passing through the center of a virtual detectorlocated 5100c/ωp from the end of the plasma. The whitedashed lines corresponds to the predictions of Eq. (7) usingγ = 400 and rβ = 0.06.The location where the spectra weretaken in the detector plane is also shown.

Self-trapping occurs off-axis at the B-field downramp.Injection is localized radially and azimuthally, enablingthe bunch to perform synchronized betatron oscillations(Fig. 4a). The transverse x-axis shift of the electron tra-jectories result from the electron beam driver deflectionwhen traversing the magnetized plasma region. The de-flection angle is small and could be corrected by addingadditional magnetized plasma regions with alternatingB-fields along the propagation direction [11].

The small blowout radius (rb ' 1.5c/ωp) ensuresthat the betatron amplitudes of oscillation rβ are muchsmaller than the plasma skin-depth (〈rβ〉 ' 0.06c/ωp).The corresponding radiation strength parameter (αβ =γKβrβkp) distribution, where Kβ = 1/

√2γ is the nor-

malized betatron frequency, and kp the plasma wavenum-ber, is shown in the inset of Fig. 4a. It shows that asignificant portion (83%) of the electrons radiate withαβ < 1, an indication that single harmonics could bedistinguishable in the emitted radiation spectrum. Toretrieve the radiation spectrum, a random sample of theself-injected electrons was post-processed using the radi-ation code JRad [16]. Figure 4b-c shows the radiationspectrum in the transverse central lines of a virtual de-

6

tector placed at a distance 5100 c/ωp from the exit of theplasma. The detector lies on the x− y plane.

The asymmetries in the x direction depicted in Fig. 4bresult from the tilt of the magnetically injected electrontrajectories. Figure 4b-c reveal that radiation is emittedat well defined frequencies, which are particularly clearat larger angles, i.e. for larger |x|. The width of eachharmonic present in Fig. 4b is larger than that expectedin an idealized scenario, where radiation would be purelyemitted in the undulator regime. This widening is dueto the spread on the αβ distribution (through γ and rβspreads) and also because some electrons radiate withstrength parameters which are larger than unity αβ & 1.

For an electron bunch with constant relativistic factorγ, and constant rβ in a pure ion-channel, the frequencyof the betatron radiation harmonics emitted in the un-dulator regime are given by [26]:

ωnωp

=2nγ2Kβ(

1 + α2β/2)

cos θ + 2γ2 (1− cos θ), (7)

where n corresponds to the nth emitted harmonic, and θto the angle between the velocity vector of the electronand the point in the detector. Radiation collected on-axisonly exhibits odd-harmonics. To compare the predictionsof Eq. (7) with simulation results we computed the parti-cles trajectories average 〈γ〉 = 400 and 〈rβ〉 = 0.06. Thisyields αβ ' 0.7, consistent with the inset of Fig. 4. Theanalytical prediction Eq. (7) are shown by the dashedlines in Fig. 4b-c. Eq. (7) is in good agreement with thesimulation results specially for larger values of |x|. Dis-crepancies are due to the fact that the beam trajectoriesare tilted, and that αβ , rβ , and γ vary in time and foreach electron.

CONCLUSIONS

In conclusion, we explored further a recent controlledinjection technique that uses transverse, static mag-netic fields to tailor transverse properties of self-injection.This scheme leads to off-axis self-injection in well de-fined radial and azimuthal regions. A configuration con-sisting of a section of transversely uniform magnetizedplasma yielding off-axis self-injection was investigated.It was shown that simultaneous self-injection of electronbunches could be achieved by using transversely non-uniform fields. This work also suggests that a series ofmagnetized regions could be used to produce a temporalsequence of electron bunches. Moreover, multiple spa-tially separated electrons could be produced simultane-ously with transversely non-uniform B-fields. We showedthat this technique could be used to produce electronbunches capable to emit betatron radiation at well de-fined frequencies with current technology.

ACKNOWLEDGMENTS

Work partially supported by FCT (Portugal)through the grants SFRH/BPD/71166/2010, andPTDC/FIS/111720/2009, by the European communitythrough LaserLab-Europe/Charpac EC FP7 ContractNo. 228464. The simulations were performed at the ISTCluster, at Jaguar supercomputer under INCITE andon the JuGENE supercomputer.

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∗ Electronic address: jorge.vieira@ist.utl.pt† Electronic address: luis.silva@ist.utl.pt

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