1. Self-modulation of long particle beams
in plasma wakefield accelerators
Konstantin Lotov
Budker Institute of Nuclear Physics SB RAS, Novosibirsk, Russia
Novosibirsk State University, Novosibirsk, Russia
AWAKE Collaboration

2. Presented by K.Lotov at EAAC-2015, 16.09.2015
The self-modulation instability (SMI) is a cornerstone effect of the AWAKE experiment, so it is important to understand how does it work and why does it work better with the plasma density step.
AWAKE
[PoP 18, ]
Beam portrait (2nd half)
Excited field (F)

3. Presented by K.Lotov at EAAC-2015, 16.09.2015
In AWAKE, the self-modulation instability (SMI) is mixed with many other effects, so we introduce a simpler model
AWAKE:
Plasma of a finite radius
Ion motion is of importance
Half-Gaussian beam shape (longitudinal)
Nonlinear limitation of the wave growth
Radius (sr=1) is neither small, nor large
Long beam (~150 wave periods), difficult to simulate, does not fit the screen
p+, G=400 (long simulation time)
Emittance driven divergence
The studied case:
Infinite radially uniform plasma
Immobile ions
Constant current beam:
Linear plasma response,
“Small” radius (analytics available)
Look at first 25 periods (L=160)
e+, g=1000 (faster simulations, keeps pace with the light)
Small angular spread
Units of measure: speed of light c for velocities, electron mass m for masses, initial plasma density n0 for densities, inverse plasma frequency ωp−1 for times, plasma skin depth kp−1 = c/ωp for distances, wavebreaking field E0 = mcωp/e for fields; also use ξ = z − ct.
Qualitative behavior is important, not the numbers

4. The quantities to look at:
Presented by K.Lotov at EAAC-2015,
The quantities to look at:
(“ax” means on-axis value)
The dimensionless wakefield potential:
z=0
F = e(E+[ez, B]) = -F
Potential well = bunch (even if not formed yet)
Location ξj of the j-th potential well
(coordinate of the local maximum),
Amplitude Φj after j-th bunch (measured as half-amplitude at ξj )
are functions
of propagation distance z
Maximum amplitude Φm(z,L)
By default, L=160 (24 bunches)
z=1000

5. Typical behavior of the maximum amplitude
Presented by K.Lotov at EAAC-2015,
Typical behavior of the maximum amplitude
Uniform plasma: 5 stages
1 stage: wakefield structure changes from that of the seed perturbation to that of fastest instability growth.
2 stage: nearly exponential growth (analytically tractable)
3 stage: non-exponential growth
4 stage: fast field decrease
5 stage: almost constant wakefield
Here optimum step = steep increase of the plasma density by 8.5% at z = 360.
[PRL 107, ]
Important characteristics:
maximum wakefield Φa, established wakefield Φf

6. Efficiency of long beams in the uniform plasma
Presented by K.Lotov at EAAC-2015,
Efficiency of long beams in the uniform plasma
The longer the beam, the smaller is the ratio established/maximum field
L=160
Long beams are inefficient in uniform plasmas

7. The map of density steps
Presented by K.Lotov at EAAC-2015,
The map of density steps
The density step must happen at the exponential stage of the instability (not at full bunching)
Smooth density increase acts as a sharp step
-> length of the transition area is not important
There are local maxima at multiples of the optimum step magnitude dn

8. The optimum step magnitude
Presented by K.Lotov at EAAC-2015,
The optimum step magnitude
the optimum step makes the beam exactly one plasma period longer, if measured in local plasma wavelengths
first two bunches play a special role in the self-modulation
simulations of LHC beam (Phys. Plasmas 18, )

9. Wave excitation, consequences of the linear theory
Presented by K.Lotov at EAAC-2015,
Wave excitation, consequences of the linear theory
Bunch contribution depends on bunch location in the potential well of the master wave
decelerated, no focusing
amplitude growth, same phase
focused, no acceleration
same amplitude, phase advance
decelerated and focused
both amplitude growth and phase advance
Complex wave amplitude

10. Motion of beam particles, how we look at:
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, how we look at:
Important: r, ξ, pr
Not important: ϕ, pϕ, pz – not considered
How to relate with potential wells?
Separatrix:
We look at −160 < ξ <−147.8
(23rd, 24th bunches)
We plot beam particles in (r, ξ, pr)-space and look how they move

11. Motion of beam particles, the uniform plasma
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, the uniform plasma
Potential wells move backward with respect to the beam
(already well known fact, vph<c)
Why?
The wave can grow, only if potential wells contain more decelerated particles than accelerated ones.
The potential well attracts (radially) equal numbers of particles to decelerated and accelerated phases.
If vph=c, then the densest parts of the beam are at well bottoms, no wave drive.
If the well shifts back after attracting particles, then the densest part is decelerated.
This is the most efficient way of wave excitation, so it wins against other perturbations

12. Motion of beam particles, the uniform plasma
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, the uniform plasma
As the instability develops, potential wells must move with respect to the beam, trap beam particles at one side and release at the other.
This is the exponential stage of wave growth, but it finishes when the beam density is strongly disturbed.
What after that?
Potential well has no more particles “to eat”, but the wave continues to grow and move (3rd stage), since absence of incoming (accelerated) particles is favorable for wave growth z=

13. Motion of beam particles, the uniform plasma
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, the uniform plasma
At the field maximum, particle void area approaches the decelerating phase…
… but the well continues to move, since:
- defocused particles need time to leave,
- preceding bunches evolve slower.
z=1200

14. Motion of beam particles, the uniform plasma
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, the uniform plasma
At the established state, only a small fraction of beam particles remains in potential wells.
(Potential wells are where there are no particles).
z=10000

15. Motion of beam particles, the density step
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, the density step
Higher plasma density,
shorter plasma wavelength, =
a “force” resisting further elongation of the wave period,
Proper force => no motion of potential wells
note: the number of periods does not change
stepped-up
uniform
stepped-up
uniform

16. Motion of beam particles, the density step
Presented by K.Lotov at EAAC-2015,
Motion of beam particles, the density step
Asymmetric well population, how?
uniform
leading edges are defocused
trailing edges are focused
bunches move backward in ξ
and destroyed in zero Ieff region
focused - defocused
stepped-up
head bunches move like in uniform plasma
tail bunches fully survive, but are inefficient
middle bunches do the job
stepped-up

17. Instead of conclusion: understanding is good,
Presented by K.Lotov at EAAC-2015,
Instead of conclusion:
understanding is good,
but which findings are practically useful?
Magnitude of the step:
Location of the step: stage of the exponential wave growth
Beam behavior in the stepped-up plasma is far from the optimal, there is a room for improvements (e.g., with more sophisticated density profiles)

18. Thank you