# Widely Wavelength-Tunable High Power Single-Longitudinal-Mode Fiber Laser in Mid-Infrared Waveband
## 4 Discussion  
Next, we discuss the mechanism of SLM operation in Case 2. To achieve SLM operation, the bandwidth of the grating needs to be narrow enough so that it can limit the number of longitudinal modes that can oscillate during the round-trip transmission. The bandwidth of the grating can be estimated by Equation (1) [20]:  
\[\delta\lambda=\frac{\lambda}{|m|\mathrm{N}}=\frac{\lambda d\cos\theta}{a} \tag{1}\]  
where \(\lambda\) represents the central wavelength, \(m=1\) is the first diffraction order, where most of the energy is concentrated, \(N\) is the number of grating lines illuminated, \(a\) is the diameter of the collimated laser beam, \(\theta\) is the incident angle of the beam and \(\mathrm{d}\) is the interval between two grating lines. In our system, \(\lambda=2780\) nm, \(d=2.2\) um, \(a=10\) mm, and \(\theta=32^{\circ}\), and thus \(\delta\lambda\) is calculated to be 0.52 nm (\(\sim\)21 GHz). As the longitudinal mode interval of the fiber FP cavity (consist of the two perpendicularly cleaved fiber facets) is roughly 1.8 pm (\(\sim\)71 MHz) and the interval of the external FP cavity (30 cm for example) is roughly 12.9 pm (\(\sim\)500 MHz), more than one longitudinal mode should be able to oscillate, as shown in Figure 7. Thus, other suppression mechanisms for longitudinal modes must exist. Due to the interference effect occurring between the longitudinal modes caused by the three-mirror cavity [21] and losses caused by the mirrors, lenses, coupling efficiency and the surrounding environment, longitudinal modes in the cavity will compete with each other and finally only those longitudinal modes which conform to the least common multiple of the frequency intervals for all individual FP cavities would survive. After this screening mechanism, some surviving longitudinal modes which are far away from the central longitudinal mode will suffer more losses and less gain due to the narrowband filter effect of the grating and finally are completely suppressed. As a result, only one longitudinal mode can be selected to oscillate in the composite cavity.  
In order to verify the longitudinal mode suppression mechanism we have discussed above, another comparison experiment was designed. Here, the right fiber end-face was cleaved with an angle of \(6^{\circ}\) to decrease the Fresnel reflection greatly. In this situation, the laser structure without grating failed to oscillate when the pump power was below 5 W due to the extremely low reflection provided by the \(6^{\circ}\) fiber end-face. Once the grating was added, the laser started to oscillate with a threshold pump power of 1.2 W. However, since the compound cavity no longer existed due to the lack of reflection from the right fiber end-face, the beat frequency signals always exist at the RF spectrum, as shown in Figure 8a, no matter how the distance between the grating and the fiber end-face was adjusted. Furthermore, the measured optical spectrum bandwidth was also broader than  
Figure 7: Spectral responses centered at the same wavelength under the influence of all frequency tuning elements (fiber and external cavity modes and grating reflectance spectral characteristics).  
that in Case 2 at the same lasing wavelength as shown in Figure 8b. Since we kept all the other experimental conditions unchanged, it is obvious that the angle of the right fiber end-face plays a key role in the formation of the SLM operation. Hence, the compound cavity consisting of the external cavity and the cavity formed by the two perpendicularly cleaved fiber end-faces is a decisive factor in the formation of the SLM operation.