  In this paper, we investigate the quasi-particle (QP) relaxation of strongly
disordered superconducting resonators under optical illumination at different
bath temperatures with the Rothwarf and Taylor equations and the gap-broadening
theory described by the Usadal equation. The analysis is validated with various
single-photon responses of Titanium Nitride (TiN) microwave kinetic inductance
detectors (MKIDs) under pulsed 405~nm laser illumination. The QP relaxation in
TiN is dominated by QPs with energy below the energy gap smeared by the
disorder, and its duration is still inversely proportional to the QP density.
The QP lifetime versus temperature can be fitted. The relaxation of the
resonator can be further modeled with QP diffusion. The fitted QP diffusion
coefficient of TiN is significantly smaller than expected. Our result also
shows a significant increase in QP generation efficiency as the bath
temperature increases.
