  After the introduction of the ionization-injection scheme in Laser Wake Field
Acceleration and of related high-quality electron beam generation methods as
two-color or the Resonant Multi Pulse Ionization injection, the theory of
thermal emittance by C. Schroeder et al, has been used to predict the beam
normalised emittance obtainable with those schemes. In this manuscript we
recast and extend such a theory, including both higher order terms in the
polinomial laser field expansion and non polinomial corrections due to the
onset of saturation effects in a single cycle. Also, a very accurate model for
predicting the cycle-averaged $3D$ momentum distribution of the extracted
electrons, including saturation and multi-process events, is proposed and
tested. We show that our theory is very accurate for the selected processes of
Kr$^{8^+\rightarrow10^+}$ and Ar$^{8^+\rightarrow10^+}$, resulting in a a
maximum error below $1\%$ even in deep saturation regime. This highly accurate
prediction of the beam phase-space can be implemented e.g., in laser-envelope
Particle in Cell (PIC) or hybrid PIC-fluid codes, to correctly mimic the
cycle-averaged momentum distribution without the need of resolving the
intra-cycle dynamics. Finally, we introduce further spatial averaging with
Gaussian longitudinal and transverse laser profiles, obtaining expressions for
the whole-beam emittance that fits with Monte Carlo simulations in a saturated
regime, too.
