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