  The propagation of the laser pulses in the underdense plasma is a very
crucial aspect of laser-plasma interaction process. In this work, we explored
the two regimes of laser propagation in plasma, one with $a_0 < 1$ and other
with $a_0 \gtrsim 10$. For $a_0<1$ case, we used a cold relativistic fluid
model, wherein apart from immobile ions no further approximations are made. The
effect of the laser pulse amplitude, pulse duration, and plasma density is
studied using the fluid model and compared with the expected scaling laws and
also with the PIC simulations. The agreement between the fluid model and the
PIC simulations are found to be excellent. Furthermore, for $a_0 \gtrsim 10$
case, we used the PIC simulations alone. The delicate interplay between the
conversion from the electromagnetic field energy to the longitudinal
electrostatic fields results in the dispersion and so the red-shift of the pump
laser pulse. We also studied the interaction of the dispersed pulse (after the
propagation in underdense plasma) with the sub-wavelength two-layer composite
target. The ions from the thin, low-density second layer are found to be
efficiently accelerated to $\sim 70$ MeV, which is not found to be the case
without dispersion.
