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• Investigated the stability region of a four-mirror ring resonator using the ABCD matrix model and thermal simulations. • Analyzed Ti:sapphire laser output under varying pump spot size, output coupler transmittance, and pump power. • Identified a 'double peak' power effect caused by pump beam focusing variations along x and y axes in the crystal. • Demonstrated correlation between output coupler transmittance and pump power, offering supporting evidence. With the in-depth research on light-atom interaction, there is a demand for improved performance in output power, noise, beam quality, stability and tunability of continuous-wave Ti:sapphire lasers. Based on the ABCD matrix model and combine with the latest numerical simulation results of Ti: sapphire crystal, the effect of cavity parameters on the stable region of the four-mirror ring resonant cavity is analyzed, and the output characteristics of the Ti:sapphire laser under different pump spot sizes, transmittance of output coupler mirrors, and pump power are measured and discussed. The experimental discovery of a "double-peak" power phenomenon is attributed to the movement of the pump focusing mirror, which is caused by variations in the focusing positions of the pump beam along the x and y directions within the crystal. This work further confirms the correlation between the optimal transmittance of output couplers and the applied pump power. By applying the "3σ" criterion and assessing class A uncertainty in our measurements, we ensure the reliability of our findings and offer valuable insights for the design optimization of tunable Ti:sapphire lasers, contributing to the advancement of laser technology in precision research. [ABSTRACT FROM AUTHOR]