In the context of kinetic theory an expression for the growth rate of a free-electron laser, under the weak resonance instability condition, for full dispersion relation has been obtained. The space-charge potential is included in the analysis and the expression for growth rate reduces to that of the Compton regime under the low density condition. With the assumption of a spread in the longitudinal momentum in the form of a Gaussian distribution function, the effect of the thermal electron beam on the growth rate is studied. The results are compared to another linear theory, a computer simulation, and an experiment.

The electron residual energy originated from the stochastic heating in underdense field-ionized plasma is here investigated. The optical response of plasma is initially modeled by using the concept of two counter-propagating electromagnetic waves. The solution of motion equation of a single electron indicates that by including the ionization, the electron with higher residual energy compared to the case without ionization could be obtained. In agreement with chaotic nature of the motion, it is found that the electron residual energy will significantly be changed by applying a minor change to the initial conditions. Extensive kinetic 1D-3V particle-in-cell (PIC) simulations have been performed in order to resolve full plasma reactions. In this way, two different regimes of plasma behavior are observed by varying the pulse length. The results indicate that the amplitude of scattered fields in sufficient long pulse length is high enough to act as a second

The stochastic behavior of electrons during the interaction of an intense short laser pulse with under-dense plasma is investigated by employing a fully kinetic 1D-3V particle-in-cell (PIC) simulation. The development of chaos in the involved nonlinear regime and in the presence of plasma space charge is examined. Though the electron Lagrangian is extremely complicated in this case, our analyses suggest some potential ways for chaos development. In this regard, our simulation results show that chaotic motion can develop in three different ways. When the space charge field is weak, the scattered fields can provide the necessary condition for chaos to occur. When a strong space charge field is presented, the creation of chaos is initiated by wave breaking. The third procedure for creating chaos originates from the inhomogeneity of the density on the vacuum-plasma surface. In this case, a new electrostatic mode without any phase relation with the space charge electrostatic mode is generated.

In this paper, the electrons energy spectrum produced by stochastic acceleration in the interaction of an intense laser pulse with the underdense plasma is described by employing the fully kinetic 1D-3 V particle-in-cell simulation. In this way, two finite laser pulses with the same length 200 fs and with two different rise times 30 and 60 fs are typically selected. It is shown that the maximum energy of electrons in the laser pulse with the short rise time (30 fs) is about eight times greater than the maximum energy of the electrons with the long rise time (60 fs). Furthermore, unlike the pulse with the short rise time, the shape of energy spectrum and the electrons temperature in the long rise time laser pulse are approximately unchanged over the time. These results originated from the fact that in the case of long rise time laser pulse, all electrons are accelerated by the one chaotic mechanism because of the scattered fields generated in the plasma, but in the case of short rise time laser pulse, three different mechanisms accelerate the electrons: first, the stochastic acceleration because of the nonlinear wave breaking via plasma-vacuum boundary effect; second, the stochastic acceleration initiated by the wave breaking; and third, the direct laser acceleration of the released electrons.