In this work, we demonstrate a single-walled carbon nanotubes-based wavelength multiplexed fiber laser, which generates dual-comb pulse in the train of soliton rain. The fiber laser cavity is manipulated in repetition frequency of 16.58 MHz, 3 dB spectral bandwidth of 8.4 nm. Two asynchronous pulses constitute the soliton rain pulse sequences, which the intensity difference is about 5.72 dB between the dual frequencies. A piece of graded-index multi-mode fiber as a filter based on the multi-mode interference effect is introduced into cavity to improving the signal to noise ratio to ~62 dB, and locate the central wavelength of the dual-comb at 1556.7 nm and 1561.5 nm. The repetition rate difference of the dual-frequency is about 169 Hz with the resolution bandwidth of 1 Hz. The time delay of the dual-frequency pulse detected by cross-correlation method is 5.78 ms, which is well matched with the results in radio frequency spectrum. Different from the stable period of the general cross-correlation signal, our experimental results show several different sub-periods due to the existence of the drifting solitons in the soliton rain sequences. Meanwhile, the number of different sub-periods in the correlation decreases from six to three as the pump power reduced from 100 mA to 97.3 mA. Our work provides a new sight into the quasi-steady multi-soliton dynamics process in fiber lasers, and will be promising solutions for interference ranging, and synchronization and timing.
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