Physics of Self-Modulation Instability Konstantin Lotov Budker Institute of Nuclear Physics SB RAS, Novosibirsk, Russia Novosibirsk State University, Novosibirsk, Russia AWAKE Collaboration

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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. Beam portrait (2 nd half) Excited field (  ) AWAKE [PoP 18, ]

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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 (  r =1) is neither small, nor large Long beam (~150 wave periods), difficult to simulate, does not fit the screen p+,  =400 (long simulation time) Emittance driven divergence Units of measure: speed of light c for velocities, electron mass m for masses, initial plasma density n 0 for densities, inverse plasma frequency ω p −1 for times, plasma skin depth k p −1 = c/ω p for distances, wavebreaking field E 0 = mcω p /e for fields; also use ξ = z − ct. 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+,  =1000 (faster simulations, keeps pace with the light) Small angular spread Qualitative behavior is important, not the numbers

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

presented by K.Lotov at AWAKE collaboration meeting, Geneva Typical behavior of the maximum amplitude 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 Uniform plasma: 5 stages Here optimum step = steep increase of the plasma density by 8.5% at z = 360. Important characteristics: maximum wakefield Φ a, established wakefield Φ f [PRL 107, ]

presented by K.Lotov at AWAKE collaboration meeting, Geneva Efficiency of long beams In the uniform plasma, the longer the beam, the smaller is the ratio established/maximum field Long beams are inefficient in uniform plasmas L=160

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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  n

presented by K.Lotov at AWAKE collaboration meeting, Geneva The optimum step magnitude simulations of LHC beam (Phys. Plasmas 18, ) 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 the same average shift of the wave with respect to the beam for beams of different lengths

presented by K.Lotov at AWAKE collaboration meeting, Geneva Wave excitation, consequences of the linear theory Bunch contribution depends on bunch location in the potential well of the master wave Convenient: complex wave amplitude Bunch contribution in relation to the wave excited by previous bunches decelerated, no focusing amplitude growth, same phase decelerated and focused both amplitude growth and phase advance focused, no acceleration same amplitude, phase advance FjFj F j-1

presented by K.Lotov at AWAKE collaboration meeting, Geneva Motion of beam particles, how we look at: Important: r, ξ, p r Not important: ϕ, p ϕ, p z – not considered How to relate with potential wells? Separatrix: We look at −160 < ξ <−147.8 (23 rd, 24 th bunches) We plot beam particles in (r, ξ, p r )-space and look how they move

presented by K.Lotov at AWAKE collaboration meeting, Geneva Motion of beam particles, the uniform plasma Potential wells move backward with respect to the beam (already well known fact, v ph <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 v ph =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

presented by K.Lotov at AWAKE collaboration meeting, Geneva Motion of beam particles, the uniform plasma As the instability develops, potential wells must move with respect to the beam, ingest 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 (3 rd stage), since absence of incoming (accelerated) particles is favorable for wave growth z=

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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 uniform stepped-up note: the number of periods does not change uniformstepped-up

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

presented by K.Lotov at AWAKE collaboration meeting, Geneva Wave excitation, individual contributions of bunches Complex wave amplitude, uniform stepped-up Optimum spiral: 45 o to the radius-vector (it is between uniform and stepped up cases)

presented by K.Lotov at AWAKE collaboration meeting, Geneva Wave excitation, individual contributions of bunches Complex wave amplitude, uniform stepped-up optimum inefficient danger defocusing optimally working bunches, but small, as evolve through dangerous area not optimal bunches, but big main job of bunches is to push wave backward

presented by K.Lotov at AWAKE collaboration meeting, Geneva 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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