  A laser-boosted relativistic solid-density paraboloidal foil is known to
efficiently reflect and focus a counterpropagating laser pulse. Here we show
that in the case of an ultrarelativistic counterpropagating pulse, a
high-energy and ultrahigh intensity reflected pulse can be more effectively
generated by a relatively slow and heavy foil than by a fast and light one.
This counterintuitive result is explained with the larger reflectivity of a
heavy foil, which compensates for its lower relativistic Doppler factor.
Moreover, since the counterpropagating pulse is ultrarelativistic, the foil is
abruptly dispersed and only the first few cycles of the counterpropagating
pulse are reflected. Our multi-dimensional particle-in-cell simulations show
that even few-cycle counterpropagating laser pulses can be further shortened
(both temporally and in the number of laser cycles) with pulse amplification. A
single few-cycle, multi-petawatt laser pulse with several joule of energy and
with peak intensity exceeding 10^23 W/cm^2 can be generated already employing
next-generation high-power laser systems. In addition, the carrier-envelope
phase of the generated few-cycle pulse can be tuned provided that the
carrier-envelope phase of the initial counterpropagating pulse is controlled.
