By means of the emerging Dispersive Fourier transformation technique, we captured the pulse-resolved spectral evolution dynamics of the double-soliton (DS) states in a single-walled carbon nanotube based Er-doped fiber laser from the initial fluctuations, monitoring the evolution process up to 10 seconds (corresponding to ~260 million roundtrips) discontinuously. Two distinctly different evolutionary types of DS states have been investigated in detail: splitting from one pulse and forming simultaneously. Relaxation oscillations, beating, transient bound state, spectral broadening and pulse interval dynamics have been observed in the evolution process of the DS operation. Our study will be helpful for the further research of mode-locking operation.
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