We recently setup a high energy, tunable, injection seeded Ti:Sapphire
laser source whose implementation was reported in [Rev. Sci. Instrum.
78, 033102 (2007)]. The analysis of the spectral linewidth has demonstrated
that the pulses were time Fourier-transform limited. The present paper
reports on the modeling of this source with the aim of understanding
the energy conversion in the injected as well as in the free run regimes.
The experimental lasing threshold, output energy, slope efficiencies,
buildup time, and pulse duration are compared to the analytical solution
obtained by solving the laser rate equations. The developed model
deals with the spatial dependence of the photon and population inversion
densities, the physical properties of the Ti:Sa medium, and the unstable
resonator (based on a super-Gaussian output coupler). The energy-related
as well as the temporal features are discussed. The use of a squared
asymmetrical hyperbolic secant function to describe the temporal pulse
shapes is justified. The pulse durations exhibit a simple analytical
shape when they are plotted versus the output pulse energy. The model
is applied to experimental data obtained at different wavelengths
by tuning the laser amplifier previously described. It can readily
be applied to other pulsed Ti:Sapphire sources.