Self-consistent Plasma Wake Field Interaction and Ring Solitons in

paraxial Beam Transport

R. Fedele*, F. Tanjia*, S. De Nicola*†,

D. Jovanovic‡, and P.K. Shukla§

XCVIII Congresso Nazionale Napoli, 17-21 Settembre 2012

*Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and INFN Sezione di Napol † INO-CNR, Istituto di Cibernetica “Eduardo Caianiello” del CNR ‡ Institute of Physics Belgrade, Serbia § Center of Advanced Studies in Physical Sciences, Fakultät für Physik und Astronomie Ruhr-Universität Bochum, Bochum, Germany

Outline

• Plasma wake field excitation

• The “plasma + beam” system – Lorentz-Maxwell system

– Single particle Hamiltonian

– Hartree’s mean field approximation

• The Governing system of Equations – Nonlinear case

Ring solitons in local regime – Numerical analysis

– Vortices

• Conclusions

Plasma Wake Field Excitation

• Large amplitude EM waves due to collective interaction of charged particles beam with plasma through wake fields

• Due to the very strong nonlinear and collective beam-plasma interaction, the beam becomes the driver of a large amplitude plasma wave that follows the beam with almost its own speed (plasma wake) and carries out both longitudinal and transverse electric fields (plasma wake fields), e.g. in PWFA

• The entire beam experiences the effects of the PWF that itself has produced (self interaction)

The Sub Systems: Plasma + Charged Particle Beam

• Plasma

– Collisionless

– Magnetized: strong constant and uniform external magnetic field (B0 = B0 ez)

– Overdense regime: n0 >> nb

n0 = unperturbed plasma density

nb = unperturbed beam density

– The ions are supposed infinitely massive and constitute a background of positive charge with density n0

• Electron/positron Beam

– Relativistic, travelling along the magnetic field

– The beam length is much greater than the plasma wavelength (long beam limit)

Plasma Model: The Lorentz-Maxwell System

Beam Dynamics: Single Beam-Particle Hamiltonian

• The relativistic single-particle Hamiltonian associated with the perturbed transverse dynamics of the beam including external magnetic field + interaction with plasma (the longitudinal beam dynamics is ignored), can be expressed in terms of the four-potential (A,f):

• We linearize the system “plasma + beam” introducing small perturbations

[P. Chen, Part. Accel. 20, 171 (1987), P. Chen, J. J. Su, T. Katsouleas, S.

Wilks, and J. M. Dawson, IEEE Transactions on Plasma Science 15, 218

(1987)]

• We transform all the system of equations to the beam co-moving frame ξ = z-βct ≃ z-ct (β ≃ 1)

• We split p and A into the longitudinal and transverse components, viz., p = zpz + p, A = zAz + A

• We take the expansion of the Hamiltonian in p - (q/c)A up to second order to get an effective dimensionless Hamilonian

• We take into account the individual quantum nature of the beam particles via PWF excitation By using the Hartree's mean field approximation (quantum paraxial diffraction).

The Governing Equations

[R. Fedele, F. Tanjia, S. De Nicola, D. Jovanovic, and P. K. Shukla, Quantum ring solitons and nonlocal effects in plasma wake field excitations, to be published in Phys. Plasmas (2012)] [R. Fedele, F. Tanjia, S. De Nicola, D. Jovanovic, and P. K. Shukla, in Proc. Intl. Topical Conf. on Plasma Sc., Lisbon Portugal, AIP Conference Series, 142, 212 (2012)]

The Envelope Equation

• We use the virial equation to get the envelope equation

Nonlinear Case: Ring Solitons in the Local Regime

• The transverse beam size much greater than the plasma wavelength: kpsr >>1

Self equilibrium:

Numerical Analysis

• We introduce the following dimensionless quantities:

– ξ ξ / s02, r r /s0, 𝜓m 𝜓m , sm sm / s0

and Am (m!s02/ ) , s0 is the initial transverse beam spot size

– For the initial BWF:

2

0!sm !m

3D plots of | 𝜓m |2 as a function of x and y for different values of

dimensionless time ξ for m = 0, Kb = 1.5, δ = 1.0, Am = 1.0

3D plots of | 𝜓m |2 as a function of x and y for different values of

dimensionless time ξ for m = 0, Kb = 0.75, δ = 0.5, Am =0.75

(self equilibrium)

Vortices

3D and density plots of| 𝜓m |2 as a function of x/s and y/s for different

values of dimensionless time ξ for m = 1, Kb = 1.5, δ = 3.5, Am =2.0625

3D and density plots of| 𝜓m |2 as a function of x/s and y/s for different

values of dimensionless time ξ for m = 2, Kb = 1.5, δ = 5.0, Am =5.625

Conclusions

• The nonlocal effects on a cold relativistic electron/positron beam travelling through magnetized plasma in overdense regime via PWF interaction as well as the Hartree's approach in quantum paraxial approximation.

• We have demonstrated the possibility of coherent, stable hollow solutions that are rotating (m 0), in nonlinear regime, which possess such topology are often called the ring solitons.

• In nonlinear strictly local regime, the existence of quantum beam vortices in the form of ring solitons has been shown numerically. These beam vortex states are introduced by the orbital angular momentum.

• Due to the interplay between the strong transverse effects of the PWF (collective and nonlinear effects) and the external magnetic field, the envelope oscillations with weak and strong focusing and defocusing have been observed for different values of m.