Numerical Modeling of Dual-Wavelength Pumped Heavily-Ho3+-Doped Fluoroindate Fiber Lasers With Efficient Output at 3.92 μm

In this article, laser emission at 3.92 μm in heavily-holmium-doped fluoride fibers using 888 nm and 1660 nm laser as pumping sources is investigated through numerical modeling. The dual-wavelength pumping system has lower threshold and higher slope efficiency than the single-wavelength pumping system at 888 nm. In addition, the system can decrease the heat load due to the 1660 nm input influencing the energy transfer up-conversion and cross-relaxation processes effectively, which indicates the possibility of raising the laser heat-damage threshold and providing high-power 3.92 μm laser generation. This research represents a novel method for improving the performance of 3.92 μm fiber lasers.

have attracted more and more attention in research.In 1992, H. Többen et al. generated 8 mW of laser output at 3.5μm in erbium-doped ZBLAN fiber using a 655 nm laser as the pump laser [8].In 2014, O. Henderson-Sapir et al. developed a dual-wavelength (985 nm and 1973 nm) pumping (DWP) system and generated a 3.5μm laser in Er: ZBLAN fiber with an output power of 260 mW [9].Using this new pumping scheme, the system's efficiency showed an improvement of 16%, by taking advantage of the long-lived 4 I 11/2 state.Two years later, they improved the performance of a 3.5μm laser with output power at the Watt level [5].In 2021, M. Lemieux-Tanguay.et al. reported a DWP 3.55μm fiber laser with a record output power of 14.9 W [10], demonstrating the advantages of DWP systems compared to single wavelength pumping systems.ZBLAN fiber has been paramount in the successes in fiber laser development in the 3-3.5μm range.However, when the operating wavelength is around 3.9μm, the maximum output power decreases dramatically to 11 mW owing to the relatively high phonon energy of ZBLAN glass, thus limiting the operation of Ho: ZBLAN fiber lasers to cryogenic temperatures, and this is a significant disadvantage that needs to be overcome.
Compared to ZBLAN glass, InF 3 -based glass has a lower phonon energy and a longer multiphonon absorption edge.In 2018, Maes et al. demonstrated a 3.92μm fiber laser operating at room temperature with a record output power of 197 mW. by employing Ho 3+ -doped InF 3 -based fiber as the gain medium and an 888 nm laser as the pump.Despite the progress in midinfrared fiber lasers, the performance of 3.9μm fiber lasers is still far behind that of 3-3.5μm lasers.One fundamental problem is that the 3.9μm transition (Ho 3+ : 5 I 5 → 5 I 6 ) is a self-terminating process since the lifetime of the lower state ( 5 I 6 ) is much longer than that of the higher state ( 5 I 5 ).Therefore, exploring a more efficient scheme for generating 3.92μm fiber lasers is crucial.
So far, some efforts have been made to suppress the selfterminating process, which acts as a bottleneck for developing 3.92μm lasers.In 2021, F. Zhou et al. theoretically investigated a new method based on a DWP scheme [11].The results revealed that a DWP system using 888 nm and 962 nm pump lasers contributed to reducing the system's threshold and effectively enhancing the laser output.In the same year, Loconsole et al. developed a gain-switched pulsed laser using an 8 cm-long section of fluoroindate glass fiber heavily doped with holmium (N Ho = 100000 ppm).The laser was designed to operate with an input peak power of P p = 10 W and a pump repetition rate of f = 100 kHz, resulting in an output signal peak power of approximately P s = 14.62 W at a wavelength of λ = 3.92μm [12].Subsequently, J. Cao et.al.demonstrated cascade lasing at 3.92μm and 2.92μm, enhancing slope efficiency while reducing the laser threshold and heat accumulation [13].Meanwhile, the utilization of co-doped fluoroindate glasses has been regarded as a more effective approach for the production of 3.92μm lasers [14], [15], [16].In a study conducted in 2023, Loconsole et al. employed a nine-level system to simulate the variation of a Ho: Nd-codoped fluoroindate glass fiber.They achieved laser emission at 3.92μm by employing an 808 nm pump laser, resulting in a slope efficiency (η) of 16.67% and an input power threshold (P th ) of 0.2 W [17].
In this work, we demonstrate a new DWP method for a 3.92μm fiber laser, using 888 nm and 1660 nm lasers as the pumps.A selfcirculatory system is formed in the 5 I 5 , 5 I 6 , and 5 I 7 levels due to the 1660 nm pumping.This then transports the ions in the 5 I 7 state to the 5 I 5 state so that the occupation number of the higher state can be maintained at a high level.Since the pump source of the system is a 1660 nm laser, commercially available highpower lasers can be used.Consequently, this system reduces the threshold and heat accumulation while enhancing the slope efficiency of a 3.92μm fiber laser.Numerical simulations are used to explore the feasibility of this scheme, and the optimal fiber parameters are determined through analysis.

II. NUMERICAL MODEL
The transition process under consideration in the heavily holmium-doped InF 3 fiber laser is presented in Fig. 1 for a DWP system.The essential processes, including cross-relaxation (CR), nonradiative multiphonon decay (NR), energy transfer upconversion (ETU), ground state absorption (GSA), excited state absorption (ESA) as well as stimulated emission (SE), are shown.
The Ho 3+ -doped fiber laser can be modeled as a five-energylevel system.Due to large rates of multiphonon relaxation, the population of state 5 I 4 can be ignored.In the study conducted by [12], the authors also examined the ESA process ( 5 I 6 → 5 S 2 ) in the gain-switched pulsed laser.However, due to the limitations imposed by the fiber damage threshold, it is not feasible to employ such high pumping powers for the continuous wave (CW) pumping laser.Consequently, the low rate of this ESA process renders it negligible in the overall analysis.The rate equations of the five-energy-level system can be listed as follows [18], [19], [20]: (2) where N i is the population density of ions in level i, β ij is the luminescence branching ratio, and τ i is the radiative lifetime of the excited states of Ho 3+ (labeled as i = 1, 2, 3, and 4).The rates of NR processes are W NR43 = 58234 s −1 and W NR32 = 7260 s −1 , respectively [18].The ETU rates R ETU1 and R ETU2 can be defined as where W 11 and W 22 represent the parameters of the ETU 1 and ETU 2 processes.The cross-relaxation rate of R CR30 is given by where W 30 is the cross-relaxation parameter.Meanwhile, the GSA, ESA 1 and ESA 2 pump rates R GSA , R ESA1 and R ESA2 can be defined as follows [21]: where λ p1 and λ p2 are the 888 nm and 1660 nm pump wavelengths, respectively.Γ p1 denotes the power filling factor for 888 nm light and can be estimated as the ratio of the core crosssectional area A core to the cladding cross-sectional area A clad .h is Planck's constant; c is the speed of light in a vacuum.σ GSA03 , σ ESA14 and σ ESA13 are the GSA and ESA 1 and ESA 2 absorption cross-sections, respectively.σ EMI30 , σ EMI41 and σ EMI31 are the GSA, ESA 1 and ESA 2 corresponding emission cross-sections, respectively.P + p1 (x, t) and P − p1 (x, t) denote the forward and backward propagating pump power at 888 nm, while P + p2 (x, t) and P − p2 (x, t) represent the forward and backward pump power at 1660 nm.The rate of stimulated emission R SE32 for the 3.92μm emission is where λ s is the signal wavelength for the 5 I 5 → 5 I 6 transition.σ EMI32 and σ ABS23 are the effective cross-sections of emission and absorption for the 5 I 5 → 5 I 6 transition, respectively, and can be calculated by: where σ SE32 is the emission cross-section, b 2 and b 3 are the Boltzmann factors, and g 2 , g 3 are the higher and lower laser sublevel degeneracies.P + s (x, t) and P − s (x, t) denote the forward and backward propagating signal powers for the 3.92μm emission.Γ s represents the power filling factor for the 3.92μm laser propagating in the fiber core.The light is assumed to have a Gaussian distribution; thus it can be calculated from [19]: where d core is the fiber core radius, and ω 0 is the mode field radius.ω 0 is estimated for single-mode and step-index fibers using the Marcuse empirical formula [18]: where the normalized frequency V can be calculated by 2πr core NA/λ SE32 and NA is the numerical aperture of the fiber.A schematic of the setup of the DWP system is shown in Fig. 2. The evolution of optical power is defined by the following equations [11]: where α P1 and α P2 are the background loss coefficients for 888 nm and 1660 nm pump light, respectively, and α s is the background loss coefficient at the signal wavelength.The boundary conditions for the pump power and laser power at both ends of the fiber can be defined as follows: where R 03f and R 03b represent the input and output mirror reflectivity for the 888 nm pump source, respectively.R 13f and R 13b are the input and output mirror reflectivity for the 1660 nm pump source.L is the fiber length.P P1, Launch , and P P2, Launch denote the launched pump power at 888 nm and 1660 nm, respectively.R 32f and R 32b represent the reflectivity of the front mirror and output mirror for the 3.92μm signal.

III. SIMULATION PARAMETERS
Most of the values of the parameters used in this work are from published works.Only a few articles provide experimental values for the spectral parameters of Ho 3+ -doped fluoride glass fibers, and some parameters have yet to be experimentally measured.In this case, we determine them by speculation, approximation, or fitting in this work.
The intrinsic lifetimes and the corresponding branching ratios of the investigated states are listed in Table I.The values of the lifetimes of levels 1-4 and their corresponding branching ratios are taken from [1].The pump and signal light parameters are listed in Table II, which contains the wavelengths of a pump (888 nm, 1660 nm) and signal (3.92μm) lights, as well as the effective absorption and emission cross-sections used in the simulation.The GSA cross-section σ GSA03 and ESA 1 cross-section σ ESA14 of P 1 and the emission cross-section σ EMI32 have been presented in previous works [11].The unknown emission cross-sections are determined by measuring them for our custom-fabricated Ho 3+doped fluoride glass.The cross-section of the corresponding absorption process of ESA 2 can be calculated from the emission cross-section using the McCumber theory [18]: where σ EMI and σ ABS are the cross-sections of emission and absorption at wavenumber ν for the transition between a lower level i and a higher level j, k B is Boltzmann's constant, T is the temperature that is assumed to be 300 K (k B T = 207 cm −1 ).ω is the emission photon energy (in cm −1 ), and N l is the equilibrium population at T = 300 K for the initial state (i.e., 5 I 7 at E 1 = 5116 cm −1 ) and N u is the final state (i.e., 5 I 5 at E 2 = 11165 cm −1 ).The term g i is the i-level degeneracy

TABLE III SUMMARY OF THE FIBER PARAMETERS
with values g l = 15 for the 5 I 7 state and g u = 11 for the 5 I 5 state.The spectra of the calculated cross-sections are shown in Fig. 3.The Ho 3+ ions concentration of the glass used in this work is N Ho = 15000 ppm, so the value of the cross-section may deviate from the true values of heavily doped Ho 3+ glass.
The parameters used in our simulations are shown in Table III.These are mainly from the previous experiment using the single 888 nm pumping [22].The radius of the core and cladding of the fiber is 8μm and 45μm, respectively.The fiber has a numerical aperture (NA) of 0.2.The fiber losses are 0.2 dB/m at 3.92μm and 0.42 dB/m at 888 nm.The Ho 3+ doping concentration in InF 3 fiber is 100000 ppm, and the corresponding density is 2 × 10 27 m −3 .The input and output mirror reflectivities at 1660 nm pump light R P2_1 and R P2_2 are assumed to be 4%.

IV. VALIDATION
Firstly, the simulation results in this work are supposed to be compared with the experimental results of previous literature.Reference [23] calculated the ETU and CR parameters W ETU1 , Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.W ETU2 , and W CR with the unit of s −1 .There is a method from [24] to transfer the interionic parameters unit from s −1 to m 3 s −1 .Using W 11 = W ETU1 /N S1 and W 22 = W ETU2 /N S2 , the ETU and CR parameters can be obtained, where N S1 and N S2 are their critical excited densities.However, the values were measured in 10 mol% Ho 3+ -doped InF 3 glass.As is known from the previous reports about Er 3+ -doped and Ho 3+ -doped ZBLAN fiber lasers [25], the ETU and CR parameters in the fiber are usually lower than the bulk glass test results.These two sets of parameters are called weakly interacting (WI) and strong interacting (SI) parameters, and there is a certain ratio between them.The SI parameters values of W ETU1 , W ETU2 and W CR were calculated to be 1.27 × 10 −22 m 3 s −1 , 2.1 × 10 −22 m 3 s −1 , and 3.48 × 10 −23 m 3 s −1 , respectively.The calculated WI parameters used in this simulation are shown in Table IV (marked as Cal. in the table) to get the best agreement between the simulation and experimental results.

TABLE IV SUMMARY OF THE INTERIONIC PARAMETERS
Fig. 4 presents the experimental data linear fitting and simulation result.The simulated laser threshold and slope efficiency are 4.3 W and 10.3%, respectively.Both are close to the experimental results reported in reference [11], in which the threshold is 4.3 W, and the slope efficiency is 10.2%, respectively.
The consistency with the previous results of a single 888 nm wavelength pump experiment proves the validity of the simulation in this work.On this basis, this simulation model can be used to predict the performance of 3.92μm fiber laser.

V. SIMULATION RESULTS
Since a larger pump power requires a smaller step size, the operation time will be greatly increased, when P 2 reaches 6 W, a single operation takes several days, so we only take P 2 = 5W at most in this work.Fig. 5(a) shows that the output power of a 3.92μm fiber laser varies with pump power at 888 nm P 1 when the pump power at 1660 nm P 2 is fixed at 1-5 W. The output power when P 2 is switched off is also given.Increasing the 1660 nm pump power can significantly decrease the laser threshold.Moreover, the threshold approaches zero when the pump power at 1660 nm is set as 5 W.This phenomenon can be ascribed to the ESA 2 process, which transports 5 I 7 state ions to the 5 I 5 state and consequently decreases the rate of the ETU 1 process.In addition, the population density of the 5 I 6 state decreases while the population density of the 5 I 5 state increases.This process finally leads to a more significant inversion population and increases output power.In the following preliminary assumption, the 2.92μm cascade system formed by Fig. 6.Simulated result in the dual-wavelength pumping system.(a).Signal output as a function of fiber length when P 1 is fixed at 6.5 W, R 32b is fixed at 84% and P 2 is varied from 0 TO 5 W. (b) Slope efficiency as a function of fiber length when P 1 is fixed at 6.5 W, P 2 is fixed at 5 W, and R 32b is varied from 60%-90%.(c).Threshold as a function of fiber length when P 1 is fixed at 6.5 W, P 2 is fixed at 5 W, and R 32b is varied from 60%-90%.(d).Laser output as a function of P 1 when P 2 is fixed at 5 W, fiber length is fixed at 0.2 m and R 32b is varied from 60%-90%. the 2.92μm mirror or FBGs may enhance the 1660 nm pumped self-circulation system.This topic will be studied in the future.
Based on this, we performed a series of optimization simulations on the DWP system to improve the efficiency and output power of the 3.92μm fiber laser.Fig. 5(b) shows that the output power of the 3.92μm fiber laser varies with P 2 when P 1 is fixed at 0 W, 2 W, 4 W, and 6 W. We observe that the influence of the 1660 nm laser increases with increasing 888 nm laser pump power.Meanwhile, we see that, with an increasing pump power of the 1660 nm laser, the slope efficiency first increases and then decreases to zero gradually.The reason for this is that the rate at which the 5 I 7 state accumulates is lower than the rate at which the 1660 nm laser dissipates.If P 1 and P 2 are set to 6 W, the 3.92μm laser output power will be 0.54 W.
In Fig. 6(a), we see that when the pump power of the 888 nm laser is fixed at 4 W, the optimal fiber length increases with increasing pump power of the 1660 nm laser.When the power of the 1660 nm pump reaches 5 W, the maximum output power of the 3.92μm fiber laser reaches 0.54 W with a fiber length of 0.175 m.As we can see from Fig. 6(a), in the case of the same pump power at 888 nm P 1 , when pump power at 1660 nm P 2 is turned on, the dependence of the 3.92μm laser concerning fiber length becomes more evident.This phenomenon is because the 1660 nm pump laser helps to enhance the absorption of the 888 nm pump light.Simulations for different conditions were performed to verify the influence of the 1660 nm pump laser on the 3.92μm fiber laser.In one, we switched off the ESA 2 process by fixing P 2 at 0 W, termed "ESA 2 off".On the other, we switched on the ESA 2 process by fixing P 2 at 5 W, termed "ESA 2 on".We found that the unabsorbed power of the 888 nm laser for "ESA 2 on" is less than that for "ESA 2 off".
Based on this, the fiber length of the DWP system is optimized when the 1660 nm pump power P 2 is fixed at 5 W. Fig. 6(b) and (c) show how the laser output threshold and slope efficiency vary with the fiber length when the output coupling reflectance is 60%-90%, and 84%, respectively.Due to the high doping concentration, the optimal length of the fiber is 20-30 cm.The threshold value decreases with increased output coupling reflectivity while the slope efficiency decreases.To further discuss the effect of output reflectivity on the output power of the 3.92μm laser, the relationship between P 1 and P 2 was studied with the mirror's reflectance at the end of the fiber R 32b fixed.The 3.92μm output power as a function of 888 nm pump power is shown in Fig. 6(d).The output power reaches 0.526 W when the output coupling reflectance is 84%, pump power P 1 = 6.5 W and P 2 = 5 W, respectively.From this curve, we conclude that a higher reflectance contributes to a lower threshold but results in low slope efficiency.The optimal reflectance range is between 80-90% when the input power is lower than 10 W. A lower reflectance enables high-power laser output in the high-power pumping system when the pump power of 888 nm is more significant than 6 W.
It was mentioned in [26] that the fiber cavity failed under a high-power input condition because of too much heat load at the butt-coupling between the fiber and the entrance DM1.To explore whether our DWP system helps reduce the heat load on the fiber, we studied the relationship between the fiber heat load and the 1660 nm pump power.The heat load was calculated by considering the nonradiative decay from the excited states and the exothermic or endothermic heat generated by Energy Transfer (ET) processes according to [13]: (27) where q is the heat load generated per unit length (W/m), A core is the area of the fiber core (m 2 ), N i is the population density in level i (m -3 ), W NRij is the nonradiative decay rate of level i. δE ij and E i are the exothermic or endothermic energy of ET W ij and energy of level i, respectively.The ET processes in this work are ETU 1 , ETU 2 , and CR, W ij represents the rate of ET process.Heat loads at x = 0 as a function of 1660 nm pump power when the reflectance of the mirror in the end of the fiber is fixed at 84%, fiber length L = 0.2 m, and pump power at 888 nm P 1 is set as 6.2 W are shown in Fig. 7.The heat load in the fiber is effectively reduced when the 1660 nm pump power is increased.This may be due to the decrease of the energy-endothermic of the ET process rates with the increase to pump power at 1660 nm P 2 .To confirm this conjecture, the ET processes as a function of 1660 nm input power are shown in Fig. 8.The ET rates decrease with an increase in the launched 1660 nm pump power.
However, the 3.9μm laser output generated by the designed pumping system still cannot exceed 1W.As can be hypothesized from Fig. 5(a), this may be due to the accumulation of too many 5 I 6 level particles, resulting in the continuous increase of 1660 nm laser without more 5 I 7 level particles being promoted to 5 I 5 level.The resulting higher 1660nm pump power also does not enable laser enhancement.If the new co-doping scheme is adopted [17] in the subsequent work and the cascade system [11] is used, the efficiency will be improved even more.Fig. 7. Calculated heat load in fiber at x = 0 as a function of pump powers at 1660 nm when R 32b = 84%, L = 0.2 m, and P 1 is fixed as 6.2 W. Fig. 8. ET rates as a function of input power at 1660 nm when R 32b = 84%, L = 0.2 m, and P 1 is fixed as 6.2 W.

VI. CONCLUSION
In this work, we demonstrate a new DWP system to generate a 3.92μm laser with at most greater efficiency, lower threshold of about 2 W, and higher slope efficiency of about 12.1% than the conventional 888 nm single-wavelength pumping system.This is due to the self-circulatory system produced by the 1660 nm ESA 2 process, which plays a significant part in maintaining the population inversion of the 3.92μm transition.In this system, the nonradiative process effectively influences the ESA 2 process, which decreases the rate of the ETU 1 process.A decrease in the population of the 5 I 7 state due to the ESA 1 avoids the accumulation of population in the 5 I 6 state.Correspondingly, this also increases the rate of the ETU 2 process.In highly Ho 3+ doped glass, the ETU 2 process strongly transports ions from 5 I 6 to 5 F 5 and 5 I 8 , thereby perfectly solving the problem of particle accumulation in the 5 I 6 state [27].Compared to single-wavelength pumping, the DWP system is more efficient at maintaining the population inversion.In addition, we showed that the 1660 nm laser could effectively reduce the heat load of Ho 3+ : InF 3 fiber.One possible limitation is that the cross-relaxation process will prevent the formation of a higher population in the 5 I 5 state.This is especially noticeable in glasses with a higher concentration of dopants, specifically those exceeding 10 mol%.In order to solve this problem, cascaded systems are recommended.Our results will be confirmed through experiments in the future.This new scheme could provide significant practical guidance for designing a DWP system for 3.92μm laser output.

Fig. 4 .
Fig. 4. Simulated and experimental laser output power as functions of launched pump power.

Fig. 5 .
Fig. 5. Simulated 3.92μm signal output in the dual-wavelength pumping system when fiber length is fixed at 0.2 m and back lens reflection index at 3.92μm is fixed at 0.84.(a) Laser output as a function of P 2 when P 1 is fixed at 0, 2, 4, 6 W. (b) Laser output as a function of P 1 when P 2 is fixed at 0-5 W.

TABLE I SUMMARY
OF THE VALUES OF THE EXCITED-STATE LIFETIMES AND BRANCHING RATIOS OF HO 3+ IN INF 3 HOST TABLE II SUMMARY OF THE SPECTRUM PARAMETERS