We have proposed a method to predict the beam quality of a quasi-continuous-wave
(QCW)-pumped, passively Q-switched (PQS) monolithic YAG/Nd:YAG/Cr4+:YAG microchip. The boundary conditions
for the pulse width and optical intensity of the PQS microchip were
obtained by solving the rate equations with different initial
transmittances of the saturable absorber and reflectivities of the
output coupler. The optimal working point was selected with the
highest pulse energy within the boundary conditions. Fundamental
cavity mode diameters under different working conditions were
calculated according to the thermal lensing of the microchip. Critical
repetition rates were obtained with a pump waist diameter equal to the
calculated fundamental cavity mode diameter for different pump
focusing conditions. Good beam quality was maintained at repetition
rates below this critical threshold, but it gradually deteriorated
with higher repetition rates. Notably, the pulse energy, temporal
profile, and spectrum were also related to the beam quality owing to
the increased content of the higher-order mode. The experimental
results were in good agreement with the simulations, and therefore,
our investigation has paved a promising path towards the proper
operations of a monolithic QCW-pumped PQS microchip, especially for a
high beam quality output.