Objective Mid-infrared fiber laser between 2 and 5 μm has many crucial applications, such as military confrontation, medicine, remote sensing, and spectroscopy, making them a research hotspot in the field of laser technology. Recently, researchers used fluoride glass fibers doped with various rare-earth ions as gain medium, and achieved 2.8, 3.2, 3.5, and 3.9 μm mid-infrared signal output. Among them, owing to the advantages, such as great transparency and unique emission spectrum at mid-infrared region, Er ³⁺-based ZBLAN (Er∶ZBLAN) glass fibers are the most widely used rare-earth doped active gain fiber. The transitions between the energy levels ⁴I_11/2 to ⁴I_13/2 and ⁴F_9/2 to ⁴I_9/2 can emission the signal photon covering 2.7 to 2.9 (2.8 μm) and 3.3 to 3.8 μm (3.5 μm), respectively. The experimental study of 3.5-μm-Er∶ZBLAN has been conducted since the 1990, from the early 655 nm red visible light pumping scheme to 976 and 1976 nm dual-wavelength pumping (DWP) scheme. The output performance has been significantly improved. Continuous wave (CW) laser with maximum output power of 5.6 W, Q-switch, and mode-locked pulse output has been reported. Compared with the experimental study, 3.5-μm-DWP-Er∶ZBLAN is also valuable for advanced research on the theoretical dynamics mechanism because of its complex energy level structure. For example, by exploring suitable fiber parameters and DWP scheme theoretically, the 3.5 μm signal output performance is optimized or the power ratio of dual wavelength signal is controlled. However, such advanced research not only requires repeated running of simulation programs and processes a large amount of simulation data, but always lacks the experimental results as a reference, which puts forward high requirements for convergence speed, convergence accuracy, and guessed value stability of boundary value problem (Bvp) algorithm. The existing theoretical research of 3.5-μm-DWP-Er∶ZBLAN is based on the exploration of the dynamics mechanism, and there is still no report of numerical algorithm. In this paper, we present a novel Bvp algorithm combined with two standard algorithms used in the fiber laser simulation: Relaxation method (RM) and shooting method (SM). We called the new algorithm relaxation-shooting method (RSM). The advantages of RSM are that it avoids the problem of RM's low convergence speed with an inaccurate guessed value, and the problem of diverging easily of SM, which make RSM possesses high convergence speed, convergence accuracy, and stability to guessed value. We believe that the research results of RSM will contribute to the development of the theory of mid-infrared fiber lasers, and guide the subsequent experiments of 2.8-μm-Er∶ZBLAN and 3.5-μm-DWP-Er∶ZBLAN.

Methods In this paper, first, we present a numerical model based on 3.5-μm-DWP-Er∶ZBLAN for steady-state analysis of the dynamic mechanism. Then, we investigate two standard Bvp analysis methods used in the field of fiber laser. Besides, we discuss their programs and mechanism of convergence judgment. We analyze the potential advantages of RSM and compile the program of RSM based on the characteristics and limitations of the two standard algorithms. Then, the simulated and experimental results in the classical literature are restored using RSM algorithm, which verifies the accuracy of the numerical model and Bvp algorithm. Finally, we discuss the convergence speed, computation time, and stability to guessed value of RSM. The results were compared with RM at the same pump scheme.

Results and Discussions Fig. 7 shows the stability comparison between RM and RSM for 3.44 μm signal guessed value at three DWP schemes. It is worth noting that the actual values of Ps2+(0) at three various pumping schemes can be obtained at around 0.7320, 1.2125, and 1.6336 W, and there is a minimum number of cycles of RM near the actual value. However, as the guessed value of Ps2+(0) drifts to the left or right, the number of cycles of RM increases rapidly, and it will increase to several times at the deviation of ±0.3 W. The total number of cycles of RSM is always lower than RM, and there is no obvious increase in number of cycles. Besides, the higher the deviation, the more obvious the advantage of RSM. There is no solution divergence in the whole computation process, indicating the stability of RSM to guessed value. Fig. 8 shows the comparison of convergence speed and accuracy between RM and RSM at three DWP schemes. The guessed value deviation was set at -0.3 W. We obtain that the RSM can converge at about ten cycles; absolute error of the last iteration is several orders of magnitude lower than the allowable error. The RM method requires 60 to 120 iterations to meet the given accuracy requirements. The result showed that RSM possesses higher convergence speed and convergence accuracy than the standard RM algorithm. Fig. 10 shows the test of total computational time when the initial guessed value was set at~0.3 W. We obtain that the total computational time of RSM is always lower than RM at the accuracy requirement range from 10 ^-4 to 10 ^-10, at three DWP schemes. When the maximum allowable error is set at 10 ^-10, the computational time is lower than 1/10 of using the RM method, which intuitively illustrates the RSM's actual calculation effect.

Conclusions In this paper, we present a novel Bvp algorithm called RSM, combined with two standard algorithms used in the field of steady-state analysis of fiber lasers, and based on the numerical model of 3.5-μm-DWP-Er∶ZBLAN. The results show that RSM has more obvious advantages in convergence speed, convergence accuracy, and stability to guessed value compared with RM. Besides, the RSM algorithm has the advantages of SM at high accuracy requirement, which can greatly reduce the number of cycles required for refining initial guessed value. The problem of diverging easily of SM when the solution of initial value problem (Ivp) is unstable, but it is can be solved by introducing a part of pre-convergence of RM. In addition, the stability of RSM to guessed value is much higher than RM; thus, it is no necessary to input a greatly accurate initial guessed value before simulation. However, the less accurate the guessed value, the more obvious the advantage of computation efficiency. In this study, the convergence speed of RSM can reach more than ten times of RM, making it suitable for the field of 3.5-μm-DWP-Er∶ZBLAN where reports of experimental and theoretical research are relatively rare. We believe that our research of RSM will contribute to the development of theoretical research of mid-infrared fiber lasers, and guide the subsequent experiments of 2.8-μm-Er∶ZBLAN and 3.5-μm-DWP-Er∶ZBLAN.