  The electron residual energy originated from the stochastic heating in
under-dense 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 counter-propagating wave for
triggering the stochastic electron motion. On the other hand, the analyses of
intensity spectrum reveal this fact that the dominant scattering mechanism
tends to Thomson rather Raman scattering by increasing the pulse length. A
covariant formalism is used to describe the plasma heating so that it enables
us to measure electron temperature inside the pulse region.
