  Generation of highly collimated monoenergetic relativistic ion beams is one
of the most challenging and promising areas in ultra-intense laser-matter
interactions because of the numerous scientific and technological applications
that require such beams. We address this challenge by introducing the concept
of laser-ion lensing and acceleration (LILA). Using a simple analogy with a
gradient-index lens, we demonstrate that simultaneous focusing and acceleration
of ions is accomplished by illuminating a shaped solid-density target by an
intense laser pulse at ~10^22 W/cm^2 intensity, and using radiation pressure of
the laser to deform/focus the target into a cubic micron spot. We show that the
LILA process can be approximated using a simple deformable mirror model, and
then validate it using three-dimensional particle-in-cell simulations of a
two-species plasma target comprised of electrons and ions. Extensive scans of
the laser and target parameters identify the stable propagation regime where
the Rayleigh-Taylor (RT)-like instability is suppressed. Stable focusing are
found at different laser powers (from few- to multi-petawatt). Focused ion
beams with the focused density of order 10^23 cm^-3, energies in access of
750MeV, and energy density up to 2*10^13 J/cm^3 at the focal point are
predicted for future multi-petawatt laser systems.
