Dark current and beam loading in blowout

regime of laser-plasma acceleration:

A test-bed for reduced simulation methods

2014 US-Japan JIFT Workshop on Progress in kinetic plasma simulations,

New Orleans, Louisiana, November 1, 2014

Serge Kalmykov & Brad Shadwick

Department of Physics & Astronomy, University of Nebraska – Lincoln, Lincoln, NE 68588-0299 USA

Xavier Davoine

CEA, DAM, DIF, Arpajon 91287, France

Arnaud Beck

LLR, IN2P3–CNRS–École Polytechnique UMR 7638, Palaiseau 91128, France

1

Accelerating structure is no more a periodic quasi-1D nonlinear wake. It is electron density BUBBLE!

P. Mora and T. M. Antonsen, Jr., Phys. Rev. E 53, R2068 (1996)

A. Pukhov and J. Meyer-ter-Vehn, Appl. Phys. B: Laser Opt. 74, 355 (2002)

50 – 500 fs

Projectile in gas Laser pulse in plasma

Modern laser-plasma accelerators work in the blowout (“bubble”) regime

2

Why blowout regime?

1. Accelerating gradient strongly exceeds the cold wave breaking limit

2. Beam loading limitations are greatly relaxed [M. Tzoufras et al., Phys. Plasmas

16, 056705 (2009) ]

Efficiency is enhanced

3. In any transverse cross-section of the bucket, accelerating gradient is uniform

4. The entire bucket is focusing; focusing gradient is linear with radius: N, is

thus preserved

Electron beam quality is expected to be preserved

5. Electron are self-injected Expected technical simplicity

3

Electrons forming the dense shell surrounding the bubble are trapping

candidates

They follow the bubble for a long time, and become relativistic

Plasma electrons in the blowout regime

Bubble is essentially a quasi-static structure:

Majority of plasma electrons,

expelled by the radiation pressure and

attracted by the charge separation force,

are passing

Electron cavity ( “bubble” ) travels over the positive ion background at a relativistic speed

S. Y. Kalmykov, S. A. Yi, A. Beck et al., New J. Phys. 12, 046019 (2010) 4

Bubble is observable in the lab

P. Dong et al., PRL 104, 134801 (2010) M. Helle et al., PRL 105, 105001 (2010) Z. Li et al., PRL 113, 085001 (2014)

Experimental correlation:

Electron self-injection & generation of monoenergetic beams

Formation of the optical bullet – probe light trapped and focused by the bubble

5

Two major physical processes critical for the electron beam quality:

• Beam loading:

Compensation of the plasma wakefield with the own wakefield of

the accelerated electron beam, leading to the reduction of the accelerating gradient

• Dark current:

Self-injection of unwanted electrons through the acceleration process,

leading to the emittance dilution and formation of the massive polychromatic electron energy tail

6

There is a number of confusions caused by the abuse of fully kinetic codes:

1. Beam loading terminates injection and leads to the formation of the QM beam [C. G. R. Geddes et al., Nature 431, 538 (2004); S. P. D. Mangles et al., Phys. Plasmas 14, 056702 (2007); … ]

2. Beam loading causes elongation of the bubble and enforces continuous injection (dark current) [V. Malka et al., Phys. Plasmas 12, 056702 (2005); F. S. Tsung et al., Phys. Plasmas 13, 056708 (2006); … ]

7

Clearing up these confusions

calls for the reduced models

capable of “turning off” the electron beam own wakes, while

preserving accurate dynamics of the laser pulse and quasistatic bulk

electron response.

Correct theory resolves the confusion as to the role of beam loading in termination of injection. To saturate injection, the charge/current density must be at least an order of magnitude higher than in a quasi-monoenergetic bunch seen in experiments/simulations. The physical origin of bubble expansion/continuous injection remains unanswered! 8

First step: Accurate theory of beam loading for the fully cavitated wake

Conceptual framework for the self-injection/beam loading/dark current problem can be built using

exceptionally simple, efficient, and physically correct numerical model

Fully relativistic quasi-static cylindrical PIC code WAKE [1]

1. Ponderomotive guiding center (e.g. time averaged over the laser cycle)

2. Extended paraxial (correct group velocity dispersion in a broad frequency range, no side-scatter; accurate description of pulse depletion and self-compression due to the wake excitation)

3. Plasma macroparticles are quasistatic and are pushed by the time-averaged ponderomotive force of the laser.

+ built-in test electron tracking module (full 3D, dynamic, non-averaged) [2,3]

[1] P. Mora and T. M. Antonsen, Jr., Phys. Plasmas 4, 217 (1997)

[2] V. Malka … A. Solodov, Phys. Plasmas 8, 2605 (2001);

[3] S. Y. Kalmykov et al., Phys. Plasmas 18, 056704 (2011)

Further step: Reduced model – quasistatic PIC code with test particles

9

How good is the test-particle toolbox?

WAKE vs. CALDER-Circ*

* S. Y. Kalmykov, S. A. Yi, A. Beck et al., NJP 12, 045019 (2010)

S. Y. Kalmykov, S. A. Yi, A. Beck et al., Plasma Phys. Control. Fusion 53, 014006 (2011) S. Y. Kalmykov, A. Beck, S. A. Yi et al., Phys. Plasmas 18, 056704 (2011) S. Y. Kalmykov, A. Beck, X. Davoine et al., NJP 14, 033025 (2012)

CALDER-Circ quasi-cylindrical, fully explicit 3D PIC code with a poloidal mode decomposition of fields and currents; with 3rd order macroparticles, numerical Cherenkov-free solver

computational efficiency of a fully explicit electromagnetic 2D PIC code [A. F. Lifschitz et al., J. Comput. Phys. 228 , 1803 (2009)]

COMPARATIVE EFFICIENCY:

WAKE: 20 hours on a single-core 2.13 GHz Intel processor, 1 GB RAM

CALDER-Circ: 21 500 CPU hours (86 h on 250 cores) on “Titane” cluster at CEA/DIF/DAM, Bruyères-le-Châtel, France

WAKE runs > 103 times faster than CALDER-Circ and correctly captures all relevant

physics of plasma wake evolution and dynamics of electron self-injection

10

Early stage of acceleration:

Physical cause of injection and

formation of quasi-monoenergetic bunch

11

Parameters of numerical demonstration

(Texas Petawatt laser*)

Pulse power 1.33 PW

Wavelength 1.057 m

Density 2.5 1017 cm-3 g = p /0 = 63.25 (P/Pcr = 20)

Spot size 27.4 m r0 2.6 c/pe

Pulse FWHM 150 fs L = 4.2 /pe

Plasma length 1.5 cm

Laser is strongly mismatched for self-guiding:

kpr0 2.6 << 2(a0)1/2 6.2

* E. W. Gaul et al., Appl. Optics 49, 1676 (2010)

S. Y. Kalmykov et al., High Energy Density Phys. 6, 200 (2010)

12

Laser diffracts Bubble expands

Electrons injected

Laser self-guides Bubble steadily contracts

Electrons accelerated

In both WAKE and CALDER-Circ runs:

a b: Pulse diffracts as in vacuum.

Expanding bubble traps ~1010 electrons

(CALDER-Circ)

b c: Laser self-guides.

Contraction of the bubble extinguishes injection and truncates the tail of electron bunch

Injection begins and terminates at the same positions in both test-particle and full PIC simulation

beam loading has no effect on initiation and termination of injection

WAKE

C.-Circ

W.

C.-C.

Beam loading in CALDER-Circ reduces acceleration gradient and slows down phase space rotation

WAKE simulation

W.

C.-C.

W.

C.-C.

a

a

b

c

b

c

Electron density Longitudinal phase space

13

Beam loading

Electron bunch in CALDER-Circ modeling:

Q = 1.3 nC r.m.s. = 5 m Lb = 15 m

E1

E2

rt Rb

C.-c.

W.

Sheath electrons cross the axis

bubble remains closed until

R4b/(8r2

t) > 0 (σ2 r.m.s./2)(nb0/n0)

[M. Tzoufras et al., Phys. Plasmas 16, 056705 (2009)]

R4b/(8r2

t0) 5.3

Beam loading alone is unable to extinguish self-injection

Tip of the bunch: E1 0

CALDER macro-particles and WAKE test particles have the same energy

Tail: E2 > 0

Transverse fields of the bunch (C.-c.) perturb the sheath and reduce accelerating gradient

14

Uncontrolled acceleration through dephasing:

Physical origin of dark current and

manifestation of beam loading

15

Stage I

The pulse self-guides with minimal oscillations in the waist size, which cause injection and rapid formation of the QM bunch;

the pulse leading edge accumulates large red-shift.

Stage II

Self-steepening of the leading edge transforms the pulse into a single-cycle relativistic optical shock.

The bubble constantly elongates, causing continuous self-injection, contaminating electron spectrum with a polychromatic tail.

Beam loading is unable to saturate injection before dephasing.

Bucket expansion is mostly unrelated to the beam loading.

B. M. Cowan, S. Y. Kalmykov, A. Beck, X. Davoine et al., J. Plasma Phys. 78, 469 (2012)

Stage I Stage II CALDER-Circ code:

VORPAL-PD code:

t = 3.1 ps t = 4.03 ps t = 7.88 ps

Wave-length, 0

Power/ energy

TLD (FWHM in intensity)

Spot size, r0

Plasma density, n0

0.8 m 70 TW /

2.1 J 30 fs 13.6 m 6.51018 cm-3

Parameters are standard for experiments with sub-100 TW laser facilities

Uncontrolled acceleration through dephasing:

Two stages of the process

16

Stage I (formation of QME bunch)

Beam loading enhances bubble expansion

by ~ 30%, but is unable to prevent/delay

stabilization and contraction

BL slows down phase space rotation and

reduces energy gain by ~20%

Collection volume is unaltered (electrons are injected from the sheath only)

The bubble thus remains predominantly quasi-static.

Pulse self-focuses, its head remains guided, tail flops inside the bubble.

Bubble size varies

Electrons are injected

from the sheath

CALDER-Circ WAKE

Collection volume: electrons are injected from sheath only!

“Radius” of the bubble (length of accelerating phase on axis)

vs propagation distance

Effect of beam loading

Bubble expands: injection is continuous with broad spectrum

Bubble contracts: injection terminates; phase space rotation produces monochromatic energy spectrum

WAKE

CALDER -Circ

Q 215 pC

17

Stage II: acceleration through dephasing

Collection volume is unaltered (electrons are still injected from the sheath only) Near dephasing, beam loading enhances

bubble expansion by merely ~ 40%

Thus, it is not the dominant cause of expansion!

Laser pulse self-compresses; bubble continuously expands; continuous injection produces

polychromatic tail: QQME / Qtail 6.5

WAKE

CALDER

-Circ

C.-C.

W.

215 pC 1.4 nC

215 pC CALDER-Circ

WAKE

Collection phase space

Bubble size vs propagation distance

Collection volume: electrons are still injected from sheath only!

The bubble thus remains predominantly quasi-static through dephasing.

Beam loading as a cause of continuous injection is ruled out!

18

S. Y. Kalmykov et al., Phys. Plasmas 18, 056704 (2011); New J. Phys. 14, 033025 (2012)

Figures 5 and 4 from S. Y. Kalmykov et al., New J. Phys. 14, 033025 (2012) Simulation code: WAKE

As the relativistic optical shock builds up,

the bubble slows down

the dephasing length shortens

the energy gain drops

Pulse self-compression limits electron energy gain in the blowout regime

Negative charge builds up inside the optical shock (“snow-plow” effect)

sheath electrons – injection candidates – become exposed to higher longitudinal electric field due to stronger charge separation

sheath electrons receive stronger backward push their return to axis is delayed

bubble expands: massive continuous self-injection ensues

Ez

Ez

Ez along the trajectory of a sheath electron

pz along the trajectory of a sheath electron

GeV c

m-1

Half-way through dephasing At dephasing

Half-way through dephasing At dephasing

Buildup of the relativistic optical shock: The origin of the dark current

vg

19

A proper choice of the laser pulse phase, compensating for the nonlinear red-shift, shall delay formation of the optical shock through electron dephasing, (i) keeping the beam clean of the low-energy background, (ii) increasing the dephasing length, boosting electron energy.

1. Bandwidth requirements: The nonlinear red-shift due to wake excitation is of order 0. To compensate, the pulse bandwidth must approach the carrier frequency!

2. Phase profile: The red-shift is localized at the pulse leading edge. Advance higher frequencies in time – introduce the negative frequency chirp!

Additional measures:

3. Propagating the pulse in a channel

(a) suppresses diffraction of the pulse leading edge, further delaying self-steepening, and

(b) causes controllable periodic injection and multi-bunching in a phase space, generating comb-like electron beams

[1] S. Y. Kalmykov, “Optical control of electron phase space in laser-plasma accelerators,” invited talk QI2 .2, 56th Annual Meeting of the APS Division of Plasma Physics, Bull. Am. Phys. Soc. 59(15), 281 (2014)

The dark current is tied to the optical process.

Good news! it can be controlled by purely optical means [1].

20

Wigner transform: quasi-probability that helps estimate the photon density in phase

space (,), providing the measure for the local frequency shift

dezazazrW ci )/( ),2

1(),

2

1(

2

1),0,,(

z (mm) z (mm)

z – ct (m)

Transform-limited Gaussian pulse

z – ct (m)

Negatively chirped Gaussian pulse

vg

|W(,)| |W(,)|

Pulse leading edge is blue-shifted

Initial conditions:

21

Local frequency shifts, pulse front etching, suppression of bubble expansion (WAKE)

(2)

(1)

z – ct (m)

Pulse speeds up

Electron dephasing is delayed

Energy gain of the QME bunch (test electrons) nearly doubles

Flux in the tail drops nearly twice

z – ct (m) z – ct (m) z – ct (m) z – ct (m)

(3)

)exp(),( iaza

/)/(// 0

1

00 ct

When incident pulse is negatively chirped,

optical shock does not form;

pulse average frequency remains high;

quasistatic bubble expansion is greatly reduced

z – ct (m)

(1)

(3)

22

CALDER-Circ simulation:

z (mm)

Chirp & taper No chirp & no taper

Bubble size vs propagation distance

Collection volume

Lorentz factor (tail of the bubble)

“Collection phase space” (pz fin vs zin)

z (mm)

ne (c

m-3)

z (mm)

Unchirped pulse Negatively chirped pulse

< E > = 558 MeV

E, RMS = 60 MeV < E >

= 1004 MeV

E, RMS

= 45 MeV

Polychromatic background:

i. Charge drops nearly by half

ii. Average flux (charge per MeV) drops by a factor of

3.5

QME bunch:

i. Energy doubles, reaching

1 GeV

ii. Relative energy spread drops

by a factor 2.5

iii. Brightness increases by 60%

Effect of the chirp

23

GeV GeV

z 4.05 mm z 5.75 mm

Dark current control in the strongly nonlinear interaction regime

(200 TW, 55 fs pulse, ne0=5.71018cm-3)

Chirp

No chirp

No chirp

Chirp

Dephasing point for electrons in the chirped-pulse simulation

P/Pcr 40, peFWHM 7.4

z = 5.75 mm

Qqm

nC

E

MeV

E

MeV E/E

N, x

mm mrad

N, y

mm mrad

<B>qm

Chirp 3.4 640 80 0.125 38.4 38.4 1.75

1011

No chirp

3.1 750 250 0.3 60 60 0.25

1011

Negatively chirped driver makes even continuous injection much quieter!

Electron beams far beyond dephasing:

24

Quasistatic response of bulk plasma electrons to the steadily varying ponderomotive force of the evolving laser pulse is the cause of electron self-injection and the main physical phenomenon behind the electron beam phase space evolution in the blowout regime of laser wakefield acceleration.

Using reduced computational models allowing to “turn off” the fields and currents of injected electrons (viz. quasistatic WAKE code with fully dynamic test particles), in combination with full 3D PIC codes, makes it possible to find that

a) The bubble evolution (and, hence, dynamics of self-injection) remains almost unaffected by the beam loading through electron dephasing.

b) Early termination of self-injection is thus not associated with the beam loading.

c) The beam loading slows down phase space rotation and early formation of the quasi-monoenergetic beam.

d) Continuous expansion and massive self-injection, contaminating the electron spectrum with a high-charge, polychromatic tail, is a consequence of laser pulse self-compression, rather than of beam loading.

e) Negative chirp of the pulse (giving the pulse bandwidth equivalent to < 1.5 optical-cycle transform-limited duration) avoids formation of the optical shock and drastically reduces continuous injection.

Summary

25

Supplements

26

27

Standard results of acceleration until dephasing at ne0 61018 cm-3 with 50-60 fs, 60-200TW laser pulses: poor quality electron spectra with large polychromatic background

Common two-stage scenario of electron injection:

All simulations presented in this slide were made with fully explicit, 3D PIC codes

27

28 28

Misinterpretation due to exclusive use of a single model: PIC code

29 29

Accurate physical picture leading to the correct theory of beam loading in the blowout regime

Misinterpretation

30

Statement impossible to support or refute on the basis of fully kinetic simulations.

31

Statements impossible to support or refute on the basis of fully kinetic simulations.

Progress of the bubble and electron beam through the Stage II (CALDER-Circ simulation)

No chirp

Chirp

No chirp

Chirp

E (GeV) E (GeV) E (GeV)

Chirp No chirp

32

Red – CALDER-Circ; black – WAKE [S. Y. Kalmykov et al., New J. Phys. (2012)]

Black – CALDER-Circ; color – VORPAL with perfect dispersion [B. M. Cowan et al., J. Plasma Phys. (2012); ]

33