  Full Poincar\'e (FP) beams, a special class of fully correlated beams
generated through the coaxial superposition of a Laguerre-Gauss and fundamental
Gaussian modes of orthogonal polarizations, contain all possible polarization
states on the surface of the Poincar\'e sphere in a single beam. While the
presence of all unconventional polarization states makes the FP beams useful
for various applications, the dependence of the refractive index on the
polarization restricts the efficient generation of FP beams across the
electromagnetic spectrum through nonlinear frequency processes. To avoid such
difficulty, we use two contiguous BIBO crystals with orthogonal optic axes and
generate an ultrafast FP beam at 405 nm with average power as high as 18.3 mW
at a single-pass conversion efficiency of 2.19%. Using Stokes parameters and
Stokes phases, we observed the doubling of C-points and L-lines singularities
and the orbital angular momentum in the SHG process. We have also devised a new
technique to estimate the polarization coverage of the pump and SHG FP beams
and observed the variation in polarization coverage with the intensity
weightage of the constituting beams. Interestingly, the SHG beam has the
highest polarization coverage for the FP beam of equal intensity weightage of
the superposed beams. We validated our experimental results in close agreement
with the theoretical results.
