The characteristics of a convex-concave linear resonator under the thermal lensing effect are theoretically analyzed to find an analytical model for designing end-pumped solid-state lasers with flexible cavity lengths. By exploiting the design model, the power scaling for continuous-wave operation under strong thermal lensing can be easily achieved in the proposed resonator with different cavity lengths. Furthermore, the proposed resonator is applied to explore the exclusive influence of cavity length on the self-mode-locked (SML) operation. It is discovered that the lasing longitudinal modes will split into multiple groups in optical spectrum to lead to a multi-pulse mode-locked temporal state when the cavity length increases. Finally, a theoretical model is derived to reconstruct the experimental results of SML operation to deduce a simple relationship between the group number of lasing modes and the cavity length.