Analytical and numerical investigation of the reflection and transmission of a counter-propagating relativistically strong laser pulse from a relativistically flying dense plasma double-sided mirror is studied. We assume that the incident laser pulse is short, so that we can neglect the slow ion dynamics and consider the electron motion only. Numerical results of the amplitudes of the reflected/transmitted electric fields from a uniformly moving mirror, accelerated mirror, and oscillating mirror are obtained. Fourier spectrum of the reflected intensity from the moving mirror shows that the intensity decreases with increase in the frequency. The reflected pulse has an up-shifted frequency and increased intensity. It is seen that the first few cycles of the reflected radiation exhibit presence of high harmonics, while the later cycles are compressed together with harmonics in comparison with the earlier cycles. The variation of the reflection coefficient for a uniformly moving mirror as a function of the thin foil plasma-density parameter is numerically studied.
