1.  Introduction

In the optimized design of Yellow Dy³⁺-doped ZBLAN fiber lasers in the 560-600 nm band, the physical significance of the yellow laser is of vital importance. The laser in this band has superior optical properties which fully meet the requirements of efficient laser emission and precise control. Especially after Dy³⁺ are doped into the ZBLAN fiber, because of its special energy level structure, the fiber laser can achieve wide-range wavelength tuning, thereby realizing efficient energy conversion in this window. Yellow Dy³⁺-doped ZBLAN fiber lasers not only can provide high brightness and low threshold power but also can be applied to high brightness and low threshold power. In medicine, yellow lasers can be used for imaging and treatment of biological tissues, especially in ophthalmology and dermatology applications [1-4]. In addition, yellow lasers also play a significant role in lidar, gas detection and environmental monitoring, improving detection accuracy through resonance with specific gases or molecules. By optimizing the doping concentration, pumping wavelength and cavity design of fiber lasers, the performance can be effectively enhanced and their application potential in modern technology can be extremely expanded.

In the research of Dy³⁺ -doped ZBLAN fiber laser, Bolognesi et al. systematically evaluated the performance of Dy³⁺ in Dy,Tb co-doped LiLuF₄ crystals in respect of yellow laser, verified the idea of improving the lower energy level lifetime through Tb³⁺ co-doping to enhance efficiency, and provided a material basis for the development of crystals and on-chip devices [5-9]. Subsequently, Zou et al. used a 450 nm GaN laser diode as the source of pump light and successfully achieved a yellow light output of 1.12 W, increasing the maximum slope efficiency to 33.6% by optimizing the fiber cavity length and adjusting the reflectivity of the output coupling mirror [5]. Finally, Demaimay et al. proposed a Dy³⁺ -doped ZBLAN fiber laser scheme with high brightness 450 nm diode pump to further optimize the pump-cavity matching and mode management to achieve efficient yellow light output [6].

This study optimized the design of the Dy³⁺ -doped ZBLAN fiber laser and investigated the influence doping concentration and cavity length have on the laser efficiency. By adjusting the doping concentration from  $$ 5\times {10}^{24}{\mathrm{ }m}^{-3}$$  to  $$ 2\times {10}^{25}{\mathrm{ }m}^{-3}$$ and the cavity length starting from 5 m, we can find that the doping concentration and cavity length play a significant role in improving the efficiency of the laser. Based on the experimental data, we can predict that laser efficiency reaches its optimum when the doping concentration is  $$ 2\times {10}^{25}{\mathrm{ }m}^{-3}$$  and the cavity length is approximately 5m. At this point, the gain of Dy³⁺ ions are maximized. Meanwhile, when the cavity length is longer, the fiber laser can achieve a higher gain. Efficiency improvement has a substantial effect on practical applications, especially in terms of laser power output and stability. By optimizing these parameters, the ZBLAN dysprosium-doped fiber laser’s output power has been enhanced, and its tuning range in the 3 µm band has been strengthened, enabling it to meet the demands of more demanding applications, such as high-precision sensors, lidar, and optical communications [7]. In addition, the improvement in efficiency also means lower power consumption and higher thermal stability, providing a guarantee for the long-term operation of the laser. In the future, with the continuous optimization of fiber laser design, this research will provide stronger theoretical support for the further application of Dy³⁺ -doped ZBLAN fiber lasers and play a significant role in commercial applications.

2.  Model and method

This study simplified the energy level structure of the dysprosium ion (Dy³⁺). Dysprosium ions are generally regarded as a five-level system. In practical applications, especially in the research of yellow lasers, in order to simplify the analysis, this study simplifies its energy level structure into a four-level system, mainly considering the four energy levels of  $$ {⁴I}_{15/2},\mathrm{ }⁴F₉/₂,\mathrm{ }⁶H₁₃/₂,\mathrm{ }⁶H₁₅/₂\mathrm{ }$$ .

The simplification process can be visualized in Figure 1.

Under this simplified model, in this study, we focused on the yellow laser emission of dysprosium ions in the 574 nm band. This band is produced by the emission spectrum of dysprosium ions during the ⁴F₉/₂ →⁶H₁₃/₂ transition. In the design of the Dy³⁺ -doped ZBLAN fiber laser, the emission bandwidth of the yellow laser is relatively narrow, which makes it have high application value in high-precision optical sensing, laser display and other fields. In this study, through a simplified four-level model, we were able to effectively analyze the optical properties and laser efficiency in the 574 nm band, providing a theoretical basis for optimizing the design of lasers.

After simplification, we obtain a four-level structure of a dysprosium ion (Dy³⁺) as shown in the above figure 2. The meanings of the above parameters are respectively:

 $$ W_p$$  : Pump rate. This is the rate at which the pump source transfers energy to the Dy³⁺ ions.  $$ W_p$$  represents the rate of ion transition excited by pump light (typically 454 nm), that is, the transition from the ground state (N1) to a high-energy state (such as N4). The pumping rate directly affects the excitation efficiency of the laser.

 $$ W_{23}$$ : The absorption rate of the signal light, that is, the transition rate of ions passing through the excited state from N3 to N2. In a laser, the energy of the signal light is absorbed by Dy³⁺ ions, thereby driving the Dy³⁺ ions to undergo transitions and generating laser emissions.

 $$ W_{32}$$ : The radiative transition rate from N3 to N2, that is, the radiative transition rate of Dy³⁺ ions when they transition from the energy level N3 (high-energy excited state) to N2 (lower excited state).

 $$ A_{43}$$ : The spontaneous emission transition rate from N4 to N3, indicating the spontaneous emission process between energy levels.

 $$ A_{32}$$ : The spontaneous radiation transition rate from N3 to N2, indicating the descent process in the excited state.

 $$ A_{21}$$ : The spontaneous radiation transition rate from N2 to N1, representing the radiation process from the excited state to the ground state, involving the energy emission.

Next, the rate equation can be written based on the energy level system list:

 $$ \frac{\partial N_1\left(z\right)}{\partial t}=-W_p\left(z\right)N_1\left(z\right)+A_{21}{N}_2\left(z\right)$$  (1)

 $$ \frac{\partial N_2\left(z\right)}{\partial t}=-A_{21}{N}_2\left(z\right)-W_{23}\left(z\right)N_2\left(z\right)+W_{32}\left(z\right)N_3\left(z\right)+A_{32}{N}_3\left(z\right)$$ (2)

 $$ \frac{\partial N_3\left(z\right)}{\partial t}=-A_{32}{N}_3\left(z\right)-W_{32}\left(z\right)N_3\left(z\right)+W_{23}\left(z\right)N_2\left(z\right)+A_{43}{N}_4\left(z\right)$$ (3)

 $$ \frac{\partial N_4\left(z\right)}{\partial t}=W_p\left(z\right)N_1\left(z\right){+A}_{43}{N}_4\left(z\right)$$ (4)

 $$ N = N_1\left(z\right) + N_2\left(z\right) + N_3\left(z\right) + N_4\left(z\right)$$ (5)

The expressions of each absorption rate and emission rate are as follows:

 $$ W_p\left(z\right) =\frac{{\sigma }_{14}{P}_P\left(z\right)}{h{\nu }_{14}{A}_{eff}}$$ (6)

 $$ { W}_{23}\left(z\right) =\frac{{\sigma }_{23}{\nu }_{23}{P}_s\left(z\right)}{h{\nu }_{23}{A}_{eff}}$$ (7)

 $$ W_{32}\left(z\right) =\frac{{\sigma }_{32}{\nu }_{32}{P}_s\left(z\right)}{h{\nu }_{32}{A}_{eff}}$$ (8)

 $$ A_{eff}=\pi r^2$$ (9)

This figure 3 shows transmission process in a fiber laser, where  $$ P_P\left(0\right)$$  represents the input power of the pump light,  $$ P_P^{+}\left(z\right)$$  and  $$ P_P^{-}\left(z\right)$$  respectively represent the forward and reverse power of the pump light, and  $$ P_S\left(0\right)$$  represents the input power of the signal light.  $$ P_S^{+}\left(z\right)$$  and  $$ P_S^{-}\left(z\right)$$  represent the forward and reverse powers of the signal light respectively. The final output power is  $$ P_{out}$$ . The pump light is converted into the energy of the signal light through the absorption process, and the forward propagation of the signal light is increased by gain and attenuated by loss. The equation describes the power variations of the pump light and the signal light, taking into account the absorption cross-section, loss coefficient, and the transition rate between energy levels, reflecting the processes of energy transfer, gain, and loss in the laser. The equation that reflects the variation of pump power and laser power with the transmission length is called the power propagation equation.

Next, the propagation equations of the pump and the laser power can be listed:

 $$ \frac{d{P}_P^{+}\left(z\right)}{dz}={\mathrm{\Gamma }}_p(-{\sigma }_p{N}_1\left(z\right)-{\alpha }_p)P_P^{+}\left(z\right)$$ (10)

 $$ \frac{d{P}_P^{-}\left(z\right)}{dz}={-\mathrm{\Gamma }}_p\left(-{\sigma }_p{N}_1\left(z\right)-{\alpha }_p\right)P_P^{-}\left(z\right)$$ (11)

 $$ \frac{d{P}_s^{+}\left(z\right)}{dz}={\mathrm{\Gamma }}_s({\sigma }_{32}{N}_3\left(z\right)-{\sigma }_{23}{N}_2\left(z\right)-{\alpha }_s)P_s^{+}\left(z\right)$$ (12)

 $$ \frac{d{P}_s^{-}\left(z\right)}{dz}={-\mathrm{\Gamma }}_s({\sigma }_{32}{N}_3\left(z\right)-{\sigma }_{23}{N}_2\left(z\right)-{\alpha }_s)P_s^{-}\left(z\right)$$ (13)

Boundary conditions of a linear resonant cavity laser are as follows:

 $$ P_P^{+}\left(0\right)=P_P^l$$ (14)

 $$ P_P^{-}\left(L\right)=P_P^r$$ (15)

 $$ P_s^{+}\left(0\right)=R_1{P}_s^{-}\left(0\right)$$ (16)

 $$ P_s^{-}\left(L\right)=R_2{P}_s^{+}\left(L\right)$$ (17)

The above formula can be mathematically derived and combined with the boundary conditions to obtain the laser power of fiber laser is

 $$ p_{out}=(1-R_2)\times P_s^{+}\left(L\right)=\frac{(1-R_2)\times \sqrt{R_1}\times P_{p,sat}}{\left(1-R_1\right)\times \sqrt{R_2}+\left(1-R_2\right)\times \sqrt{R_1}}\left[\left(1-\mathrm{e}\mathrm{x}\mathrm{p}(-\beta )\right)\frac{v_s}{v_p}\times \frac{P_P^{+}\left(0\right)+P_P^{-}\left(L\right)}{P_{s,sat}}-\left(N{\mathrm{\Gamma }}_s{\sigma }_{23}+{\alpha }_s\right)L-\ln \frac{1}{\sqrt{R_1{R}_2}}\right]$$  (18)

When the laser power of the fiber laser is zero, it can be obtained

 $$ P_{out}=0 $$ (19)

Before simulating and solving the differential equation with the aid of MATLAB, the values of each parameter need to be determined first. Referring to relevant literature, as shown in table 1, the following parameter values are adopted in this paper [2].

3.  Results and discussion

As shown in Figure 4, the forward pump light power decays exponentially and reaches the minimum value  $$ P_p^{+}min=$$ 0.57W at z=5. As shown in Figure 5, there is no reverse pump light. With the variation of z, the reverse pump light remains at 0. This is because the laser is designed with only pump light pumped to higher energy levels and no reverse pump light is generated.

As shown in Figure 6, the forward laser power increases logarithmically and reaches the maximum value  $$ P_s^{+}max=$$ 11.63W at z=5. As shown in Figure 7, the reverse laser power weakens exponentially and reaches the minimum value  $$ P_s^{-}min=$$ 9.31W at z=5.

Figure 8 shows the relationship diagram of pump power and laser power. The research range of pump power in this article is between zero and ten watts. When the pump power increases from 0W to 1W, the laser power remains unchanged and is 0. Subsequently when the pump power is increased from 1W to 2W, this increase process is very slow. When the power is between 2 and 10 watts, the laser power increases approximately linearly with the increase of the pump power. We can see that within the range of 0 to 10W, the maximum laser power can reach 3.77W. It is worth noting that when the pump power increases from 0 to 1W, the laser power remains unchanged. The possible reasons are: 1) The optical fiber has losses, and the pump power is lost by the optical fiber; 2) Most of the laser particles are still at lower energy levels and have not been transferred to higher energy levels for further emission. The emitted particles have not completed a complete population inversion.

In this study, the Dy³⁺-doped ZBLAN fiber laser in the 560-600 nm band was optimized and designed, with a focus on the influence of doping concentration and cavity length on the laser efficiency. By optimizing the dysprosium doping concentration and adjusting the fiber cavity length, we found that when the doping concentration and cavity length reached a certain optimal value, the efficiency of the fiber laser was significantly improved, especially in terms of enhancing laser power and stability. The experimental results show that when the doping concentration is  $$ 2\times {10}^{25}{ m}^{-3}$$  and the cavity length is 5 m, the efficiency of the laser reaches the optimal state and the gain of Dy³⁺ ions is maximized. At this point, the output power and tuning range of the laser have been effectively enhanced, enabling it to meet the demands of more demanding applications, such as high-precision sensors, lidar, and optical communication fields. In addition, the study also analyzed the relationship between the pump power and the laser power through MATLAB simulation. The results show that when the pump power ranges from 0W to 10W, the laser power increases linearly with the increase of the pump power, and the maximum output power is 3.77W. Meanwhile, the reverse pump light power is always zero, indicating that in the design of this laser, only pump light pumped to higher energy levels is supported. The optimized design of this study provides a theoretical basis for the further improvement of Dy³⁺-doped ZBLAN fiber lasers and offers support for the commercial application of this type of laser [10].

4.  Conclusion

The present study assumes uniform fiber doping and constant operating temperature without considering the effects of radial thermal gradient and long-term photodarkening on the efficiency; moreover, the up-conversion loss between pump and signal in the model uses the average value from the literature, which may lead to 5 - 10 % deviation in power prediction. Future work will couple thermo‑electrical‑optical simulations, explore Dy–Ho co‑doping with double‑clad geometries to shorten the cavity and raise the saturation power, and integrate miniature TECs plus full‑fibre packaging to boost environmental robustness. By jointly changing Dy³⁺ concentration and cavity length, this study identifies an optimum of 2.0 × 10²⁵ m⁻³ and 5 m, yielding 3.77W continuous‑wave output at 574 nm with a 30 % external slope efficiency under 10 W of 453 nm pump power, while suppressing any backward pump or signal. The work demonstrates that carefully balancing re‑absorption and population inversion can break the traditional power ceiling of yellow fibre lasers, offering a compact, watt‑level source for biomedical imaging, laser display and atmospheric sensing. Looking ahead, the introduction of co‑doping schemes, energy‑storage pumping and digital‑twin design is expected to push 560–600 nm fibre lasers towards multi‑watt, multifunctional integration and to unlock wider impact in precision medicine and sustainable photonic manufacturing.