The relationship between the longitudinal-mode structure (LMS) of the laser and the relative intensity noise (RIN) properties was investigated in this paper after an all-solid-state continuous-wave (CW) single-frequency 1064 nm laser with output power of 50.3 W was achieved. The LMS of the laser was manipulated by controlling the temperature of the deliberately introduced nonlinear lithium triborate (LBO) crystal in the resonator. When the laser worked with single-longitudinal-mode (SLM) operation, the stable RIN spectrum was observed and measured. Moreover, with the decrease of the nonlinear conversion efficiency (NCE), the frequency and amplitude of the resonant-relaxation oscillation (RRO) peak regularly shifted to the higher frequencies and increased, respectively. However, with further decrease of the NCE, the laser began to work with the multi-longitudinalmode (MLM) or mode-hopping operation and the unstable RIN spectra of the laser were both observed not only at low frequencies but also at high frequencies. Once the NCE was moved away, the MLM or mode-hopping can only enhance the fluctuation of the laser RIN spectrum below the frequency of RRO. The experimental results definitely revealed that the key to achieve a stable high power laser with low intensity noise was to realize single-frequency operation of the laser with free MLM and mode-hopping.