Self-pulsing in passively Q-switched microchip lasers is investigated in de
tail. The typical range of values of the parameters motivates a new analysi
s of the laser rate equations. We determine basic properties of the laser i
ntensity oscillations such as threshold conditions, repetition rate, pulsew
idth, peak power, and pulse energy. We show that these oscillations appear
through a quasi-vertical Hopf bifurcation located slightly above the lasing
threshold. Our bifurcation results are verified numerically by modeling a
microchip laser experiment with Nd:YAG as the gain medium and Cr:YAG as sat
urable absorber. Our results agree with the experiment to within 10%.