## 2 Experimental Method and Results
### Anomalous Dispersion Region  
First, we focus on the anomalous dispersion region. When the net cavity dispersion \(D\) was between \(-0.319\) and \(-0.0197\) ps\({}^{2}\), stable soliton mode-locking was achieved. When the spectral filter was not used, sech\({}^{2}\) shaped pulse spectra with strong Kelly sidebands, which are characteristics of a soliton mode-locked pulse, were generated stably. Since the chromatic dispersion of SMF was large and negative at this wavelength range, the magnitudes of Kelly sidebands were large, compared to conventional Er and Yb-doped fiber lasers. The spectral width was increased as the net cavity dispersion was close to zero. Figure 3: Wavelength dependence of developed tunable filter, (**a**) comparison of numerical and experimental transmission spectra, and (**b**) variation of transmitted spectra during the wavelength tuning operation. Figure 4: Variation of optical spectra at fiber laser output when net dispersion was varied. (**a,b**) film A was used and spectral filter was (**a**) not used and (**b**) used. (**c,d**) film B was used and spectral filter was (**c**) not used and (**d**) used. When the spectral filter was used, passive mode-locking was achieved for the dispersion of \(-0.319\) to \(-0.0197\) ps\({}^{2}\). For \(D=-0.319\) and \(-0.144\) ps\({}^{2}\), sech\({}^{2}\) shaped pulses were obtained stably. Particularly when \(D=-0.144\) ps\({}^{2}\), the properties of the spectral filter fitted the laser performance well. A sech\({}^{2}\) shaped pulse with 7.2 nm spectral width and suppressed Kelly sidebands were obtained stably. It is interesting to note that, as shown by the blue and red curves in Figure 4, there were conditions in which mode-locking was obtained at two different center wavelengths. When \(D=-0.319\) ps\({}^{2}\), a continuous wavelength tuning operation was achieved by the control of the PC in the spectral filter. As shown by the green arrows in Figure 4b,d, the magnitudes of the tuning range were 5 to 16 nm. In addition, the switching of the center wavelength of the mode-locked pulse, which is shown in the blue and red curves, was demonstrated by the operation of the PC before the spectral filter. The shortest center wavelength was 1840 nm, which is useful to generate a high power soliton pulse at the 2.0 \(\upmu\)m range using a Tm-doped fiber amplifier [20]. For \(D=-0.05\) and \(-0.0197\) ps\({}^{2}\), narrow spectra of 2.8-2.9 nm were observed. It was considered that these narrow spectra were due to the bandwidth limit of the spectral filter. Figure 5 show the representative characteristics of output pulses for the net anomalous dispersion region when the spectral filter was not used. Here, the net cavity dispersion \(D=-0.144\) ps\({}^{2}\). A sech\({}^{2}\) shaped pulse with intense Kelly sidebands is shown in Figure 5a. Here, the spectral width was 7.1 nm at full-width half-maximum (FWHM). A clean autocorrelation trace without pedestal component is shown in Figure 5b. The temporal width was 999 fs, and the corresponding pulse width was 648 fs under the assumption of the sech\({}^{2}\) shaped pulse. The estimated time bandwidth product was 0.387, which was close to that of the Fourier transform limited sech\({}^{2}\) shaped pulse. In the radio frequency (RF) domain, clean RF spectra were observed. For the fundamental frequency shown in Figure 5d, the observed signal-to-noise ratio (SNR) was about 70 dB, and a low noise property was confirmed. In the RF spectra in the wide range shown in Figure 5e, clean RF spectra with almost constant intensity were observed up to 2 GHz, and stable mode-locking was confirmed. Figure 5: Characteristics of output pulses when \(D=-0.144\) ps\({}^{2}\) and spectral filter was not used, (**a**) optical spectra, (**b**) autocorrelation trace, (**c**) pulse train, (**d**,**e**) RF spectra of (**d**) fundamental frequency and (**e**) wide range. Figure 6 shows the characteristics of output pulses when \(D=-0.144\) ps\({}^{2}\) and the spectral filter was used.