  We use particle-in-cell (PIC) simulations and simple analytic models to
investigate the laser-plasma interaction known as ponderomotive steepening.
When normally incident laser light reflects at the critical surface of a
plasma, the resulting standing electromagnetic wave modifies the electron
density profile via the ponderomotive force, which creates peaks in the
electron density separated by approximately half of the laser wavelength. What
is less well studied is how this charge imbalance accelerates ions towards the
electron density peaks, modifying the ion density profile of the plasma.
Idealized PIC simulations with an extended underdense plasma shelf are used to
isolate the dynamics of ion density peak growth for a 42 fs pulse from an 800
nm laser with an intensity of 10$^{18}$ W cm$^{-2}$. These simulations exhibit
sustained longitudinal electric fields of 200 GV m$^{-1}$, which produce
counter-steaming populations of ions reaching a few keV in energy. We compare
these simulations to theoretical models, and we explore how ion energy depends
on factors such as the plasma density and the laser wavelength, pulse duration,
and intensity. We also provide relations for the strength of longitudinal
electric fields and an approximate timescale for the density peaks to develop.
These conclusions may be useful investigating the phenomenon of ponderomotive
steepening as advances in laser technology allow shorter and more intense
pulses to be produced at various wavelengths. We also discuss the parallels
with other work studying the interference from two counter-propagating laser
pulses.
