  We study the hole and magnon spectral functions as a function of hole doping
in the two-dimensional (2D) t-J and t-t'-t"-J models working within the limits
of the spin-wave theory, by linearizing the hole-spin-deviation interaction and
by adapting the non-crossing approximation (NCA). We find that the staggered
magnetization decreases rather rapidly with doping and it goes to zero at a few
percent of hole concentration in both t-J and the t-t'-t"-J model. We find that
with doping, the residue of the quasiparticle peak at G=(pi/2,pi/2) decreases
rapidly with doping and the spectral function is in agreement with high
resolution angle-resolved photo-emission spectroscopy (ARPES) studies of the
copper-oxide superconductors. The observed large shift of the chemical
potential inside the Mott gap is found to be a result of broadening of the
quasiparticle peak. We find pockets centered at G, similar to those observed by
quantum oscillation measurements, (i) with an elliptical shape with large
eccentricity along the anti-nodal direction in the case of the t-J model, and
(ii) with an almost circular shape in the case of the t-t'-t"-J model. We also
find that the spectral intensity distribution in the doped antiferromagnet has
a waterfall-like patten along the nodal direction of the Brillouin zone, a
feature that is also seen in ARPES measurements.
