From One to Many Electrons
As it is customary for several books of plasma physics, we begin with a description of single particle dynamics, focusing on the motion of an electron in a plane wave that is most relevant to the physics we are interested in. This part also serves as a qu
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From One to Many Electrons
Abstract As it is customary for several books of plasma physics, we begin with a description of single particle dynamics, focusing on the motion of an electron in a plane wave that is most relevant to the physics we are interested in. This part also serves as a quick review of relativistic dynamics and a warm-up to tackle nonlinear equations. We then introduce the concept of the ponderomotive force and finally discuss the issue of radiation friction, an emerging topic in ultra-relativistic laser plasmas. In the second part of the chapter we briefly review the basic kinetic and hydrodynamic modeling of collisionless plasmas and we also give a quick description of numerical modeling, including the pioneering Dawson model, the particle-in-cell method, and the boosted frame techniques to make simulations easier to perform.
2.1 The Single Electron in an Electromagnetic Field 2.1.1 Non-relativistic Motion in a Plane Wave Before searching for exact and general solutions, it is useful to start with an approximate (“perturbative”) calculation which allows us to softly introduce nonlinear and relativistic effects. Since to the lowest order all nonlinear effects are neglected, it is convenient to use the complex representation for the electric and magnetic fields of a plane wave propagating in the x direction1 E = E(x, t) = E 0 εˆ eikx−iωt ,
B = B(x, t) = xˆ × E,
(2.1)
Unit vectors are indicated with a “hat” symbol, so that e.g. a generic position vector r = x xˆ + y yˆ + z zˆ .
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A. Macchi, A Superintense Laser-Plasma Interaction Theory Primer, SpringerBriefs in Physics, DOI: 10.1007/978-94-007-6125-4_2, © The Author(s) 2013
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2 From One to Many Electrons
where it is implicit that the physical fields are the real parts of the above expressions. Here, k = ω/c and εˆ is the complex polarization vector. For linear √ polarization along y (z), εˆ = yˆ (ˆz) while for circular polarization εˆ = (ˆy ± iˆz)/ 2 corresponding to counterclockwise and clockwise directions, respectively. The non-relativistic equations of motion for an electron are me
v dv = −e E(r, t) + × B(r, t) , dt c
dr = v, dt
(2.2)
where in general the EM fields depend on the particle position r = r(t). To lowest order, i.e. in the linear approximation we neglect the v × B because, for weak fields, |v| c and also because we are interested in the oscillating motion at the same frequency of the EM wave: if v(t) is a function oscillating with frequency ω, the v × B product contains both a 2ω oscillating term and a constant (0ω) term, but not any term with frequency ω. The linear solution is thus obtained immediately as v=−
ie E, meω
r=
e E, m e ω2
(2.3)
where the electric field is taken at the constant x-position of the electron, since there is no force in the longitudinal (ˆx) direction. The trajectory is a line for linear polarization, and a circle for circular polarization, as may sound obvious. Having obtained this simple solution, we can check a posteriori that |v| c. Since the peak amplitude is υ0 = eE 0 /m
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