  We report on the implementation of degenerate Raman sideband cooling of
$^{40}$K atoms. The scheme incorporates a 3D optical lattice, which confines
the atoms and drives the Raman transitions. The optical cooling cycle is closed
by two optical pumping beams. The wavelength of the laser beams forming the
lattice is close to the D$_2$ atomic transition, while the optical pumping is
operated near the D$_1$ transition. With this cooling method, we achieve
temperature of $\sim$$1\mu$K of a cloud with $\sim$$10^7$ atoms. This
corresponds to a phase space density of $\ge$$10^{-3}$. Moreover, the fermionic
ensemble is spin polarized to conditions which are favorable for subsequent
evaporative cooling. We study the dependence of the cooling scheme on several
parameters, including the applied magnetic field, the detuning, duration, and
intensity profile of the optical pumping beams. Adding this optical cooling
stage to current Fermi gas experiments can improve the final conditions and
increase the data rate.
