**Introduction**

*Q*-switched and mode-locked [3-6]. However, few papers report narrow-linewidth and single longitudinal-mode fiber lasers around 2 μm [7], and these lasers are very important in many applications involving optical communications and optical fiber sensors [8]. As we all know, TDF can be used a homogeneous broadening gain medium at room temperature, and its feature of strong mode competition results in mode hopping and unstable output [9]. A lot of methods have been used to attempt to obtain stable narrow-linewidth fiber lasers in C-band or L-band, such as using a fiber Bragg grating (FBG)-based Fabry-Perot (F-P) filter to increase longitudinal-mode spacing and achieve single frequency oscillation [10,11], adopting polarization incoherent technology [12] and constructing a ultrashort cavity for several centimeters as well as a high concentration gain fiber to realize the single frequency output [13].

^{3+}doped narrow linewidth fiber lasers, which have important applications such as gas sensors, wavelength-divisionβmultiplexing (WDM) optical communication technology and liquid water sensing [17]. In this paper, we presented the theory of saturable absorber as a transient FBG based on standing-wave saturation effects, as well as analyzed the corresponding factors of parameters of a transient FBG.

**Principle and methods**

*P. V.*is the principal value of the integral derived and

*ω*

_{1}-

*ω*

_{2}is the spectral range where the absorption is nonnegligible,$\mathrm{\Delta}\mathit{\alpha}(\mathit{z},{\omega}^{\prime})$is the change in absorption,

*c*is the speed of light. The variation of the absorption coefficient will result in periodical spatial variation of refraction index, which forms the transient Bragg reflection grating. For simplicity, the maximum change of refractive index that can be obtained in doped fiber can be described as [22]

*N*is the doping concentration of the unpumped fiber, ${\sigma}_{e}$is the emission cross-section of the doped particles. The period of the FBG is given by

*β*is the mode propagation constant, $\kappa ={\displaystyle \frac{\pi \gamma \mathrm{\Delta}n}{\lambda}}$ is the coupling coefficient of the FBG, $\delta =\sqrt{{\kappa}^{\mathrm{2}}-{\left({\displaystyle \frac{\mathrm{\Delta}\beta}{\mathrm{2}}}\right)}^{\mathrm{2}}}$is the detuning, $l$ is the length of the doped fiber. The full width at half maximum (FWHM) of the FBG can be denoted as [24]

^{3+}as a two-level system of

^{3}H

_{6}and

^{3}F

_{4}, the TDF will absorb the signal wave (1570 nm) and produce population inversion, forming a saturable absorber.

**Simulation results**

^{2}, $N=\mathrm{3.5}\times {\mathrm{10}}^{\mathrm{25}}$ Tm

^{3+}/cm

^{3}, $\lambda =2000\mathrm{nm}$in Eq. (2), we can estimate $\mathrm{\Delta}n$ as $\mathrm{1.257}\times {\mathrm{10}}^{\mathrm{-}\mathrm{6}}$, and then we bring $\mathrm{\Delta}n$ into Eq. (4).

^{3+}and the length of TDF. From Fig. 1, it can be observed that the simulative central Bragg wavelength is 2000 nm, and the saturable absorber has the biggest reflection in the central wavelength. We can also find in Fig. 1 that for stable change of refraction index $\mathrm{\Delta}n$, the reflection of FBG became higher with the longer length of TDF (0.3, 0.5 and 0.7 m), while the FWHM became narrower. The relation between $\mathrm{\Delta}f$ and fiber length $l$ is shown in Fig. 2, $\mathrm{\Delta}f$decreases nonlinearly as $l$increases. We set $l$varying from 1 to 10 m, and we can get the linewidth of 14.45-130.4 MHz, or 0.1927-1.739 pm, which means the transient FBG can act as an ultra-narrow filter. Moreover, if the value of $\mathrm{\Delta}n$ decreases from 1.257×10

^{-6}to 3×10

^{-8}(see Fig. 3),the value of linewidth can be a few tenths of MHz and it can be seen as a single longitudinal-mode fiber laser if the linewidth is smaller than the free mode spacing. In theory, the FBG’s filtering effect is more excellent with narrower $\mathrm{\Delta}f$because less resonant modes are existed. If $\mathrm{\Delta}f$is less than the longitudinal-mode spacing, one can achieve a single longitudinal-mode fiber laser. When it comes to the unpumped TDF, stable standing-wave saturation effects cannot form with short length of doped fiber, so the longer length can achieve better performance. Even though, the threshold power will be higher, while the slope efficiency and the output power will get lower if we make the length of TDF too long because only the first half of it acts as an saturable absorber while the latter half become the attenuated fiber.