Direct acceleration of electrons by a CO2 laser in a curved plasma waveguide

Abstract

Laser plasma interaction with micro-engineered targets at relativistic intensities has been greatly promoted by recent progress in the high contrast lasers and the manufacture of advanced micro- and nano-structures. This opens new possibilities for the physics of laser-matter interaction. Here we propose a novel approach that leverages the advantages of high-pressure CO₂ laser, laser-waveguide interaction, as well as micro-engineered plasma structure to accelerate electrons to peak energy greater than 1 GeV with narrow slice energy spread (~1%) and high overall efficiency. The acceleration gradient is 26 GV/m for a 1.3 TW CO₂ laser system. The micro-bunching of a long electron beam leads to the generation of a chain of ultrashort electron bunches with the duration roughly equal to half-laser-cycle. These results open a way for developing a compact and economic electron source for diverse applications.

OPEN

Direct acceleration of electrons by a CO laser in a curved plasma waveguide

LongqingYi,, Alexander Pukhov & Baifei Shen,

Laser plasma interaction with micro-engineered targets at relativistic intensities has been greatly promoted by recent progress in the high contrast lasers and the manufacture of advanced micro- and nano-structures. This opens new possibilities for the physics of laser-matter interaction. Here we propose a novel approach that leverages the advantages of high-pressure CO laser, laser-waveguide interaction, as well as micro-engineered plasma structure to accelerate electrons to peak energy laser system. The micro-bunching of a long electron beam leads to the generation of a chain of ultrashort electron bunches with the duration roughly equal to half-laser-cycle. These results open a way for developing a compact and economic electron source for diverse applications.

Laser wakeeld acceleration1 of particles to relativistic energies has been greatly promoted by the invention of chirped-pulse-amplication (CPA)2 more than twenty years ago. The ultrashort laser pulses with huge peak powers enabled by the CPA technique allowed for quasimonoenergetic electron bunches to be generated in underdense plasmas in so-called bubble regime38 of laser wakeeld. Pioneering experiments had reported that electrons with a few percent energy spread and sub-milliradian divergences beyond 4.2GeV can be produced9,

which demonstrates the impressive progress in plasma-based acceleration. However, a signicant drawback is the traditional CPA lasers use TiSa crystals which deliver an average power of a few Watts only at a low overall efficiency10. On the other hand, well-known for its industrial applications, the overall efficiency (520% from wall plug) of CO2 laser is among the highest of all lasers. Hence, it is the most economic choice when considering high energy physics applications, where high luminosities are usually required.

Nowadays, high-pressure CO2 laser has already reached multi-Terawatt-level11 and been successfully applied for a series of proton acceleration experiments12,13. However, it has been less progresses in CO2 laser-driven wakeeld acceleration14,15, mainly because of the difficulty with building ultra-short CO2 laser system. In general, it is well known that the longitudinal dimension of the driver of plasma wakeeld should be comparable to plasma wavelength in order to resonantly excite a bubble3,8,16, such that for the typical CO2 laser pulse duration (~10ps), an extremely low-density plasma (ne~1013cm3) is required, which is of little interest to the accelerator community since the maximum acceleration gradient (i.e. wave breaking eld) is on the same order of magnitude with the conventional RF accelerators.

In parallel, direct laser acceleration (DLA) oers an attractive alternative17,18, where no threshold intensity19 and no limitiations on the pulse duration. Normally, a waveguide can be used to guide laser pulses over distances much larger than the Rayleigh length

= 2 0 (w0 is the spot size and 0 is the laser wavelength), while simultaneously, transverse magnetic (TM) optical modes are excited in the channel. The co-propagating electrons in a proper phase can be accelerated with a peak longitudinal electric eld that can be estimated by2022

0

R m

State Collaborative Innovation Center of IFSA (CICIFSA), Shanghai

R

A

P

E GV cm a

[ / ] 8

[ ] , (1)

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Figure 1. Sketch of direct laser acceleration of electron in a CPW. (a) A linearly polarized CO2 laser and a relativistic electron beam are injected into a CPW from the le side. The electrons located in the right phase can be accelerated by the TM modes, resulting in the generation of a chain of energetic ultrashort electron bunches (right exit of CPW). (bd) Shows the longitudinal electric eld and electron motion at dierent propagating distances (marked by the red cube-frame in (a)), where a quarter of Ex eld are removed to avoid overlapping, and the electron energy is presented by the color. The gure on the back and le walls in (bd) presents theEx eld and electron position (black dots) at longitudinal slice z= 0 and transverse cross-section x=6.50, respectively. The waveguide is properly designed with a curvature in the polarization direction, detail dimension of half of the CPW period is shown in (e) for x-y plain (the waveguide is uniform along z-direction, and the size in z dimension is 120 in the presented 3D PIC simulation). The l1 is the half-CPW curvature period, and the ratio of l1 and l2 is optimized according to the simulation (l2=0.442l1). If l1 matches the dephasing length, the electron bunch can continuously gain energy until it overtakes the entire laser pulse.

where a0=eE0/mec0 is the normalized laser amplitude, c is the light velocity in vacuum me is the mass of an electron, e is the unit charge, 0 is the frequency of laser, and R is the radius of waveguide in m. For a 1.3TW CO2 laser pulse with a0= 5 and channel radius R=6064m, the peak acceleration gradient is roughly 0.64GV/cm, which compare favorably to laser wakeeld acceleration with ultra-short laser pulses at the same power level.

However, the slippage between laser phase velocity and the electron velocity (essentially c) forbids the electrons to stay in the acceleration phase, which sets a limit on the maximum energy that can be aquired. Historically, periodical grating surfaces23,24 and neutral gas lling25 have been proposed to solve the dephasing problem, but none of them can survive the relativistic intensities required for high acceleration gradient even for the pulse duration. In fully ionized plasma channel, the phase velocity of laser is superluminal. A corrugated plasma waveguide has been proposed as ultrahigh intensity optical slow-wave structure, where net energy gain can be achieved using a radially polarized laser propagating in a density-modulated gas jet19. More recently, owing to the advancements in laser pulse cleaning techniques26,27 and 3D direct laser writing (DLW) of materials28, laser interaction with ne plasma structure is drawing more and more attention. The micro- and nano-structured plasma targets have been introduced to manipulate laser matter interactions22,2931. Simulations suggest that the longitudinal electric eld in excess of 1TV/m can be achieved in an overdense micro-plasma-waveguide22. An taylored plasma microstructure that can overcome the phase slippage would allow for enormous acceleration gradients.

Results

In this letter, we propose a novel electron acceleration scheme using DLA in a curved plasma channel that is capable of generating energetic (>1GeV) ultra-short (duration ~ half-laser-cycle) electron bunch chain with slice energy spread ~1%. These high-quality electron beams can be widely applied in high energy physics, study of atomic and molecular dynamics and generating coherent x-rays. In the presented study, A CPW is used to overcome the phase slippage as shown in Fig.1. Inspired by the fact that the longitudinal electric eld in the CPW is anti-symmetric with respect to the propagation axis in the polarization direction (as shown in Fig.1(bd)), we properly design the spatial period of CPW to match the dephasing length of electrons. So that the relative displacement between

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Figure 2. Simulation results of the electron acceleration. (a) The on-axis longitudinal electric eld and witness electron density prole in the rst CPW period at propagation distance L= 0, 2, 4, 6, 8, and 10mm (b) peak energy of electron bunch at the end of each CPW period, the inset shows detail energy evolution in the rst CPW period (blue rectangular in (b)).

electrons and laser eld in the longitudinal and transverse directions keeps the electrons in the accelerating phase (for most of the time), enabling continuous energy gain of witness beam until it overtakes the entire laser pulse. In addition, a linearly polarized CO2 laser beam is employed as the driver, not only due to the high overall efficiency discussed above, but also because a long infrared wavelength enables a large acceleration bracket, which increases the number of particles per bunch. Nevertheless, the wavelength of drive laser is not mandatory, it should be carefully chosen according to the applications.

Electron acceleration process and the energy gain. Here we rst demonstrate the acceleration process with three dimensional (3D) particle-in-cell (PIC) simulations using the code VLPL32, the parameters can be found in Methods. The CPW is constituted with an overdense slab and an arc, the detailed dimensions for the structure are shown in Fig.1(e). Due to the up-and-down motion of laser pulse in CPW tends to push the electrons out of the waveguide, especially in the rst CPW period (since the gamma factor of electrons is smaller in the beginning), the initial energy of the electron bunch is chosen as 100MeV to ensure that an essential part of the injected electrons could be stably accelerated by The TM modes. Figure1(bd) show the relative motion of the electron bunch and longitudinal electric eld in half of the CPW period, which indicate the transverse motion of the guided laser beam perfectly compensates the phase slippage eect. As soon as the longitudinal phase slippage reaches half of laser cycle, electrons exactly fall into the acceleration phase on the opposite side of channel as expected. Moreover, since only the electrons with proper phase (Ex< 0) can be captured and accelerated within a long electron bunch, a chain of ultrashort electron bunches is generated. The duration of a single electron bunch is governed by the laser wavelength, an attosecond electron train can be generated if a short-wavelength laser is employed as the driver, which may be applied in the ultrafast electron diraction and 4D microscopy3335.

In order to obtain a deeper insight in the laser pulse propagation and electron acceleration, we perform 2D PIC simulation on the electron acceleration over 10 centimeters (10 CPW periods). A long laser pulse (duration ~1 ps) is employed to enable the acceleration over a long distance. The on-axis longitudinal electric eld and witness electron density prole are plotted in Fig.2(a) for one CPW period. One can see that the micro-bunching occurs simultaneously with the acceleration during the rst 2-mm propagation in the CPW. The generated micro bunches stay in the acceleration phase for most of time.

Figure2(b) presents the peak energy of electron bunch at the end of each CPW period, where the nal electron energy attained is 1.5GeV. The acceleration gradient decreases as the laser energy depletes in the CPW. Aer 10cm propagation, the laser loses 80% of its energy. For electrons with divergence smaller than 1mrad, the overall laser-to-electron energy efficiency is roughly 11%. The inset of Fig.2(b) shows the peak electron energy evolution in the rst CPW period. The slight energy decrease observed at L 3mm and 8mm is due to the brief passage of the witness bunch through the decelerating phase as the laser pulse is guided across the beam axis (see Figs1(c) and 2(a)). The net energy gain in the rst CPW period (1cm) is 260MeV that results in an average acceleration gradient of 26GV/m, or roughly 40% of the peak electric eld predicted in Eq.(1).

Theoretical analysis of the optical modes in plasma waveguide. In the following, we try to give an estimation on the electromagnetic eld and acquire basic results for dephasing distance by investigating laser

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propagation in a plane plasma waveguide. Further PIC simulation should be relied on to obtain optimum parameters for a real CPW. Considering a y-polarized laser pulse entering the plasma waveguide along x-axis, the waveguide has a rectangular cross-section in y-z plane (y= y0~y0, and z= z0~z0). Following the methods in refs 21,22, one can easily write the electric and magnetic elds in terms of two Hertz potentials e and h in Cartesian coordinate system:

=

x + =

2 e e

2 2

2

h h

x +

E x k H x k

, , (2)

2

2

2 2

e h

y

h e

E x y i z H x y i z

, ,

=

y

=

+

(3)

2 2

e h

z

h e

E x z i y H x z i y

=

+

z

=

, ,

(4)

where

< <

<

<

e

=

x

A k y k z e y y z z

B k z e y y z z

C k y e y y z z

sin( )cos( ) , ( , )

cos( ) , ( , )

sin( ) , ( , ) (5)

y z

ik x

e

e

z

x yp

x zp

0 0

0 0

0 0

ik x ik y

e

ik x ik z

y

A k y k z e y y z z

B k z e y y z z

C k y e y y z z

h

=

x

cos( )sin( ) , ( , )

sin( ) , ( , )

cos( ) , ( , ) (6)

y z

ik x

h

h

z

x yp

x zp

0 0

0 0

0 0

ik x ik y

h

< <

<

<

ik x ik z

y

2 2 2 is the total wave number in the vacuum core, and kx, ky, kz are the wave numbers in each direction, kyp, kzp are the transverse wave numbers inside the plasma channel walls. Apparently, since laser is properly guided in the x direction, kyp,

kzp are both imaginary.

At boundaries y=y0 and z=z0, the tangential components of both E and H should be continuous, which leads to the following eigenvalue equations,

here Ae, Be etc are coefficients determined by the incident laser amplitude, = + +

k k k k

x y z

p

2

2

p p

=

Y Y Y Y Y Y Y Y

1 sin( ) cos( ) cos( ) sin( ) 0,

(7)

+ =

Z Z Z Z Z Z Z Z

1 cos( ) sin( ) [ sin( ) cos( )] 0,

(8)

p

2

2

p p

where p is the plasma frequency, and Y = kyy0, Yp = ikypy0, Z = kzz0, Zp = ikzpz0, which satisfy

+ =

2

2 , and + =

2

2 .

Y Y k y

p

2 2 2

p

2

p

Z Z k z

p

2 2 2

0

2

0 0, the wave number kz is negligible. Let be the 1st root for Y in Eq.(7) ( = 1.54),

which corresponding to the lowest TM mode in the waveguide. One can write the longitudinal electric eld according to Eq.(2) as

In 2D limit, i.e.

0 z y

2

E m cy e a

yy e

0

sin

ik x

x

x

0

.

0

0

(9)

Discussion

A discussion on the energy spread. Apparently, the acceleration gradient is only uniform along the z direction, the nonuniformity in x- and y-direction tends to broaden energy spread of witness beam during the acceleration. The energy spread of witness bunch can be controlled by the duration of injected electron bunch as shown in Fig.3(a), the bunch duration above 5m is not considered because its beyond the longitudinal size of each electron bunch aer the micro-bunching of a long electron beam. The linear growth of energy spread is resulted from the longitudinal energy chirp owing to sinusoid distribution of acceleration gradient in Eq.(9). However, we emphasis that the slice energy spread is small (~1%) regardless the witness beam length, which suggest a high quality of the acceleration. When a short witness electron bunch (~1 m, occupies about 10% of the laser cycle) is injected into the CPW, the witness bunch gains little longitudinal energy chirp as shown in Fig.3(b). Also, the transverse phase space map (Fig.3(c)) illustrates that although the accelerating eld varies

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Figure 3. Energy spread of the accelerated electron bunch in CPW. (a) The r.m.s. energy spread obtainedfor dierent injected bunch durations. (b) Longitudinal and (c) transverse phase space of electrons in the rst CPW period for 1-m witness duration. (d) The relative energy spread evolution during the 10-cm acceleration of 1-m duration witness bunch. The red line in (b) shows the electron energy spectrum, the dierent colors in (c) corresponding to the phase space at corresponding distance, and the inset of (d) presents the phase space of electrons at highest energies.

along the y-axis, the integration in one period of CPW is uniform. So the electrons at dierent transverse position gains the same amount of energy in one CPW period. As a result, when a short (1m) electron bunch is injected into the proper accelerating phase, a quasi-monoenergetic electron beam can be obtained as shown in Fig.3(d). The relative r.m.s energy spread at highest energy (1.5 GeV) is about 2% for the whole beam, this result can be further optimized by employing shorter witness bunch. The absolute energy spread increases slightly owing to longitudinal energy chirp as shown by the inset phase space map, and the r.m.s slice energy spread is 0.83%.

A discussion on the matching condition. The above results also allow us to derive the matching condition which is crucial to the proposed scheme. In a sufficiently short propagation distance dx, the change in trans-verse size of CPW is negligible, the CPW can be treated as a plane waveguide, and the phase slippage between relativistic electrons (v c) and TM10 mode is

=

x dx h x dx

( ) /2 ( )

0

l l

1 2

= + =

2

2 2 2 2 , where h(x) is the CPW dimension along y-direction. The matching condition states that the phase slippage in half CPW period must be equal to 0/2, i.e.

l l

1 2

0

0

0

0 0

2

2

2

2

( x dx h x dx h x dx

) 2 2 ( ) 2 ( ) 2 , (10)

1

2

2

2

l

2

2

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Figure 4. Energy gain of injected electron bunch in CPW with dierent period.

where

= +

h x r r x

( ) 12

1 0 0 0

2 2 and

= + +

h x r r x r x l

( ) 24 2 ( )

2 0 0 0

2 2

0

2

1

2 are the expres-

sions for h(x) in the at and curve areas, respectively.

Equation(10) indicates the dephasing length is very sensitive to the transverse size of CPW, and it should be noted we have ignored the small transverse motion of laser pulse for simplicity. According to Eq.(10), we know that the longitudinal dimension of CPW should be chosen around the match condition l1 = Ld 4880, which roughly agrees with our numerical observations. By scanning over a range of CPW periods, we found the optimal acceleration is obtained for a slightly-shorter CPW with l1=4750 as shown in Fig.4. Apparently the violation of matching condition lead to early saturation which limits the energy gain. The maximum energy decreases 50% for 25 0 deviation from the matched cases.

In conclusion, a novel DLA scheme based on CPW at high laser intensities is proposed and tested by multi-dimension PIC simulations. Our results indicate that a CPW can be used as an electron accelerator when coupled with state-of-art CO2 laser beams. The proposed scheme demonstrates high acceleration gradient and beam quality, which makes it a promising candidate for future tabletop accelerator design. In addition, the overall efficiency of the scheme is high because the CO2 laser pulses have high wall-plug efficiencies. The underlying physics is discussed using PIC simulations and theoretical analysis, the matching condition is presented, which agrees with our numerical observation. The integration of longitudinal electric eld in one CPW period results in uniform accelerating structure in transverse direction. A quasi-monoenergetic electron bunch with mean energy 1.5 GeV, r.m.s. energy spread 2% can be obtained within 10-cm acceleration. Meanwhile, when a long electron beam is injected into the CPW, micro-bunching eect comes into play, which is capable of generating a chain of ultra-fast electron bunchs with the dimension of half-laser-cycle.

Methods

Due to the computational difficulty with simulating the realistic long CO2 laser pulse, we perform three dimensional (3D) simulation with an relatively small window

=

L L L 12 30 20

x y z 0 0 0 focused on the laser-electron interacting position over half of the CPW period (Lacc = 4750 0.5 cm) to examine the electron motion between two accelerating phases. The simulation resolution is dx = 0.040, dy = dz = 0.080 in each direction, where 0 = 10.6 m is the wavelength of CO2 laser. In 2D simulations, a bigger simulation window (LxLy=320260) and a ner resolution (dx = 0.020, dy = 0.050) are employed, while other parameters remain the same. A Moving window is used for both 2D and 3D simulations for computational efficiency. The particle per cell used in the simulation is 10 for 2D and 5 for 3D simulations. The absorbtion boundary for particles and periodic boundary for electric/ magnetic elds are employed. The laser pulse in the window is assumed to have a trapezoidal prole in time with normalized amplitude a0= 5, which propagates in the positive x direction. The plasma channel wall has a uniform density of n=3nc, where

=

n m e

/4

c e 0

2 2 is the critical density. It should be noted although the density of MPW is limited by computational efficiency, in real experiments, the laser can hardly penetrate into the area with n>3nc due to nite density gradients. In addition, the CPW contains a uniform low-density (1015cm3) plasma to provide necessary focusing for witness particles, it has little inuence to the accelerating eld and laser propagation. The witness electron bunch has a at density prole in x direction (duration40) and a Gaussian prole (FWHM20) in transverse direction.

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Acknowledgements

This work is supported by DFG Transregio TR18, EU FP7 EUCARD-2 projects and National Natural Science Foundation of China (No. 11505262, No. 11125526, and No. 11335013).

Author Contributions

L.Y. wrote the paper with contributions from A.P.; L.Y. conducted simulations and analysis; A.P. developed the code used for simulations (VLPL) and supervised the work; B.S. provided useful suggestions.

Additional Information

Competing nancial interests: The authors declare no competing nancial interests.

How to cite this article: Yi, L. et al. Direct acceleration of electrons by a CO2 laser in a curved plasma waveguide. Sci. Rep. 6, 28147; doi: 10.1038/srep28147 (2016).

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