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

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, 103101] Beam portrait (2nd half) Excited field (F)

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

The quantities to look at: Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Typical behavior of the maximum amplitude Presented by K.Lotov at EAAC-2015, 16.09.2015 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, 145003] Important characteristics: maximum wakefield Φa, established wakefield Φf

Efficiency of long beams in the uniform plasma Presented by K.Lotov at EAAC-2015, 16.09.2015 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

The map of density steps Presented by K.Lotov at EAAC-2015, 16.09.2015 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

The optimum step magnitude Presented by K.Lotov at EAAC-2015, 16.09.2015 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, 103101)

Wave excitation, consequences of the linear theory Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Motion of beam particles, how we look at: Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Motion of beam particles, the uniform plasma Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Motion of beam particles, the uniform plasma Presented by K.Lotov at EAAC-2015, 16.09.2015 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=0..1000

Motion of beam particles, the uniform plasma Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Motion of beam particles, the uniform plasma Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Motion of beam particles, the density step Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Motion of beam particles, the density step Presented by K.Lotov at EAAC-2015, 16.09.2015 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

Instead of conclusion: understanding is good, Presented by K.Lotov at EAAC-2015, 16.09.2015 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)

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