The intensity noise in a 1560 nm single frequency fiber laser after passing through an optical mode cleaner is analyzed both theoretically and experimentally. Experimental measurement shows that in addition to the evident suppression of intensity noise by the mode cleaner, there exist induced observable periodic fluctuations in the analyzing frequency range of 2 to 12 MHz, as well as the amplification of the intensity noise at low frequencies. The above results cannot be explained by the present mode cleaner model for noise suppression. In this paper, we propose a new theoretical model, in which the mode cleaner is considered equivalent to a delay line and through it the phase-noise of the fiber laser is partially converted to the intensity noise. The phase-induced relative intensity noise (RIN) amplitude is jointly determined by the laser linewidth, the mode cleaner linewidth, and the analyzing frequency. The theoretical analysis shows a very good agreement with the experimental results. The noise suppression effect of the acoustic optical modulator is further analyzed by inserting it into the setup and providing a frequency modulation for it. We have observed an evident improvement of the mode cleaner locking, while the bandwidth of the laser is slightly suppressed from 26 to 16kHz, and the degradation of the measured intensity noise after the mode cleaner is also moderate. The theoretical analysis according to our proposed model fits well with this result. This result further confirms that the phase-induced intensity noise has no direct connection to the mode cleaner locking quality. Through the above analysis, a complete theoretical mode for analyzing the noise suppression by a mode cleaner is built.