Dual-Wavelength Cascade Pumping for Low-Threshold and High-Efficiency 4.4 μm Emission in Dy³⁺:InF₃ Fiber Laser: A Numerical Investigation

Abstract

Dy³⁺:InF₃ fiber shows promise for 4.4 μm mid-infrared lasing, but the much shorter lifetime of its upper laser level compared to the lower level causes inevitable self-termination. While cascade 4.4 μm/3 μm lasing has been proposed as a potential solution, this method faces complex configuration and an extremely high pump threshold (>30 W under continuous-wave operation), rendering it impractical for high-power use, especially given InF₃’s soft-glass nature. To address the self-termination challenge and enable the low-threshold, high-efficiency lasing, this study proposes, for the first time to our knowledge, a dual-wavelength cascade-pumping scheme utilizing 2.8 μm and 2.4 μm pumps. Numerical simulations demonstrate that the dual-wavelength cascade-pumped Dy³⁺:InF₃ fiber laser exhibits an optical-to-optical efficiency of up to 18.4% and a maximum slope efficiency of 38.5%. The total pump threshold is as low as 5.4 W, remarkably lower than that required by the cascade lasing approach. This work provides a viable solution and design guidelines for the development of 4 μm-class mid-infrared fiber lasers.

1. Introduction

Mid-infrared (mid-IR) lasers in the 3–5 μm wavelength range, which lie within the atmospheric transparency window and cover the fingerprint regions of numerous gas molecules, have significant application value and demand in fields such as environmental monitoring, biomedicine, and molecular spectroscopy [1,2,3,4,5,6]. Typically, such lasers can be obtained via optical parametric oscillations (OPOs) [7], quantum cascade lasers (QCLs) [8], and fiber lasers [2]. However, OPO laser sources show complex structures, high cost, and high susceptibility to environmental perturbations, while QCLs suffer from extremely limited output power. Among these approaches, fiber lasers, characterized by excellent beam quality, long-term stability, and power scalability, are drawing growing attention.

Currently, with the development of rare-earth doping and fluoride fiber fabrication technologies, 3 μm-class Er³⁺-doped, Ho³⁺-doped, and Dy³⁺-doped ZBLAN fiber lasers have achieved considerable progress and moved towards commercialization. Specifically, the output power of Er³⁺-doped ZBLAN fiber lasers has reached 70 W around 2.8 μm [9] and 15 W around 3.5 μm [10], while the output power of Ho³⁺-doped and Dy³⁺-doped ZBLAN fiber lasers around 3 μm has achieved 41 W [11] and 10 W [12], respectively. Another key focus lies in extending the emission wavelength of these systems to the 4 μm range. However, further extending the emission wavelength presents significant challenges, as the multiphonon absorption of ZBLAN fibers exhibits exponential growth with wavelengths exceeding 3.5 μm [13]. This necessitates the use of host materials with lower phonon energy. Chalcogenide fibers and InF₃ fibers have been demonstrated as promising host materials, featuring lower phonon energy and an enhanced infrared transparency range compared to ZBLAN fibers [14,15,16,17,18,19]. Among these, achieving direct mid-IR oscillation in rare-earth-doped chalcogenide fibers remains challenging, primarily attributed to limited doping concentration and homogeneity of rare-earth ions. In contrast, InF₃ fibers permit heavily doping with good homogeneity, paving the way for direct generation of 4 μm-class lasers [20]. Additionally, InF₃ fibers possess a broad transparency range spanning from the visible to mid-IR region [5,21], offering greater flexibility in the pump source selection compared to most chalcogenides. Presently, a record output power of 200 mW has been achieved from a Ho³⁺-doped InF₃ fiber laser operating near 3.9 μm [22], which represents the longest wavelength directly generated by a fluoride fiber laser in experimentally reported results to date.

Despite these advancements, extending laser emission beyond 4 μm remains a substantial challenge. In 2018, Majewski et al. observed ~4.4 μm fluorescence emission from Dy³⁺:InF₃ fiber, originating from the ⁶H_11/2 → ⁶H_13/2 transition [23]. However, laser operation has yet to be achieved, primarily due to the inherent self-termination effect stemming from the significantly shorter lifetime of the upper energy level (⁶H_11/2, 1 μs [24]) compared to the lower level (⁶H_13/2, 500 μs [24]). While cascade lasing (simultaneous 4.4 μm and 3 μm emission) has been proposed as a potential solution, this approach suffers from a complex laser configuration and an extremely high pump threshold (>30 W under continuous-wave operation) [13]. Such characteristics render it impractical for high-power applications, particularly considering the soft-glass nature of InF₃ materials. Additionally, theoretical analyses suggest that dual-wavelength pumping (2.85 μm and 4.09 μm) of a dysprosium-doped fiber laser could be feasible [25]. Nevertheless, the output power of this in-band pumping scheme is severely constrained by the inherently small absorption cross-section at 4.09 μm [26]. Furthermore, the scarcity of high-power pump sources at 4.09 μm also limits its practical implementation.

To address the self-termination challenge and enable the low-threshold, high-efficiency lasing from the Dy³⁺:InF₃ fiber, this study proposes, for the first time to our knowledge, a dual-wavelength cascade-pumping scheme utilizing 2.8 μm and 2.4 μm pumps. Simulation results demonstrate that the dual-wavelength cascade-pumped Dy³⁺:InF₃ fiber laser exhibits an optical-to-optical efficiency of up to 18.4% and a maximum slope efficiency of 38.5%. The total pump threshold is as low as 5.4 W, less than one fifth of that required by the cascade lasing approach under continuous-wave operation.

2. Numerical Model

Figure 1a displays the schematic of the energy level structure and energy transfer processes relevant to the proposed dual-wavelength cascade-pumped Dy³⁺:InF₃ fiber laser at 4.4 μm. The 2.8 μm pump (pump 1) excites the Dy³⁺ ions from ground state level ⁶H_15/2 to level ⁶H_13/2. The ⁶H_13/2 level is characterized by a long lifetime of 500 μs [25], thus resulting in a substantial accumulation of Dy³⁺ population on this level under the continuous pumping of a 2.8 μm laser source. Consequently, the level ⁶H_13/2 can be regarded as a virtual ground state (VGS) for higher energy level transitions. With the pumping of a 2.4 μm laser source (pump 2), the Dy³⁺ ions on level ⁶H_13/2 are excited to ⁶H_9/2 + ⁶F_11/2 level through virtual ground state absorption (VGSA). Given that ⁶H_9/2 + ⁶F_11/2 level has an extremely short lifetime of 0.05 μs, the ions on it are unstable and transit to the upper laser level ⁶H_11/2 through rapid multiphonon relaxation (MPR). In a population inversion state, the transition between the ⁶H_11/2 and the lower laser level ⁶H_13/2 emits radiation at ~4.4 μm. Notably, benefiting from the energy recycling mechanism from level ⁶H_13/2 to ⁶H_11/2, the self-terminating effect faced by single pumping can be overcome.

Figure 1. (a) Energy level structure and energy transfer processes of the dual-wavelength cascade- pumped Dy³⁺:InF₃ fiber laser at 4.4 μm. GSA: ground state absorption, VGSA: virtual ground state absorption, M: non-radiative decay rate, W: radiative decay rate. (b) Schematic diagram of the dual-wavelength cascade-pumped Dy³⁺:InF₃ fiber laser. HT: high transmission, HR: high reflectivity, FBG: fiber Bragg grating.

The designed dual-wavelength cascade-pumped Dy³⁺:InF₃ fiber laser is illustrated in Figure 1b. The pump lasers at 2.8 μm (pump 1) and 2.4 μm (pump 2) are focused into the core of the Dy³⁺:InF₃ fiber via a convex lens L₁ after passing through a dichroic mirror M₁ (HT at 2.8 μm, HR at 2.4 μm) and a gold mirror M₃ (HR at 2.4 μm), respectively. The cavity consists of fiber Bragg gratings FBG₁ and FBG₂, where FBG₂ acts as an output coupler. To guarantee the optimal absorption of the 2.4 μm pump by the Dy³⁺:InF₃ fiber, thereby lowering the pump threshold, FBG₃ and M₁ are employed to confine the 2.4 μm pump within the cavity. After filtering the unabsorbed 2.8 μm pump laser with a dichroic mirror M₂ (HT at 4.4 μm, HR at 2.8 μm) placed at a 45° angle, the 4.4 μm laser is output. In practical applications of this laser configuration, an AlF₃ endcap can be spliced to the fiber at the pump end, which enhances the fiber’s damage threshold.

Based on the energy level structure and energy transfer mechanisms of Dy³⁺ ions, the rate equations for the 4.4 μm dual-wavelength cascade-pumped Dy³⁺:InF₃ fiber laser are established as follows:

$$\begin{array}{ll}{\displaystyle \frac{d{N}_1(z,t)}{dt}}& =-(W_{10}+M_{10})N_1(z,t)+R_{\mathrm{GSA}}(z,t)-R_{\mathrm{VGSA}}(z,t)+R_{\mathrm{SE}}(z,t)+(M_{21}+W_{21})N_2(z,t)\\ & +W_{31}{N}_3(z,t)\end{array}$$

(1)

$${\displaystyle \frac{d{N}_2(z,t)}{dt}}=-(W_{20}+W_{21}+M_{21})N_2(z,t)-R_{\mathrm{SE}}(z,t)+(M_{32}+W_{32})N_3(z,t)$$

(2)

$${\displaystyle \frac{d{N}_3(z,t)}{dt}}=-(W_{30}+W_{31}+W_{32}+M_{32})N_3(z,t)+R_{\mathrm{VGSA}}(z,t)$$

(3)

$$N_{\mathrm{Dy}}={\displaystyle \sum _{i=0,\dots ,3}{N}_i}$$

(4)

In Equations (1)–(4), N_i represents the population density of Dy³⁺ ions at the i-th energy level. N_Dy stands for the doping concentration of Dy³⁺ ions in the InF₃ fiber. W_ij denotes the radiative decay rate from the upper energy level i to the lower energy level j, and M_ij is the non-radiative decay rate for the transition of Dy³⁺ ions from the upper energy level i to the lower level j. R_GSA and R_VGSA respectively represent the rates of GSA at 2.8 μm and VGSA at 2.4 μm, while R_SE denotes the stimulated emission rate of the 4.4 μm signal light, which can be given by the following:

$$R_{\mathrm{GSA}}(z,t)={\displaystyle \frac{{\mathsf{\Gamma }}_{\mathrm{p}1}{\lambda }_{\mathrm{p}1}}{hc{A}_{\mathrm{core}}}}({\sigma }_{01}{N}_0(z,t)-{\sigma }_{10}{N}_1(z,t))(P_{\mathrm{p}1}^{+}+P_{\mathrm{p}1}^{-})$$

(5)

$$R_{\mathrm{VGSA}}(z,t)={\displaystyle \frac{{\mathsf{\Gamma }}_{\mathrm{p}2}{\lambda }_{\mathrm{p}2}}{hc{A}_{\mathrm{core}}}}({\sigma }_{13}{N}_1(z,t)-{\sigma }_{31}{N}_3(z,t))(P_{\mathrm{p}2}^{+}+P_{\mathrm{p}2}^{-})$$

(6)

$$R_{\mathrm{SE}}(z,t)={\displaystyle \frac{{\mathsf{\Gamma }}_{\mathrm{s}}{\lambda }_{\mathrm{s}}}{hc{A}_{\mathrm{core}}}}({\sigma }_{21}{N}_2(z,t)-{\sigma }_{12}{N}_1(z,t))(P_{\mathrm{s}}^{+}+P_{\mathrm{s}}^{-})$$

(7)

where λ_p1 and λ_p2 denote the wavelengths of pump 1 and pump 2, respectively, and λ_s represents the laser wavelength. h stands for the Planck constant and c represents the speed of light in a vacuum. Γ_p1, Γ_p2, and Γ_s denote the power filling factors for the 2.8 μm pump, 2.4 μm pump, and 4.4 μm laser, respectively, where the filling factor is defined as Γ_x = 1 − exp[−2(r_core/ω)] with x = p1, p2, s. The signal is assumed to follow a Gaussian distribution with a mode radius given by ω = r_core (0.65 + 1.619V^−1.5 + 2.876V⁻⁶), where r_core denotes the fiber core radius. V = 2πr_coreNA/λ_x is the normalized frequency for the pump or laser, where NA (0.16) denotes the numerical aperture, and A_core = π$r_{core}^2$ refers to the cross-sectional area of the Dy³⁺:InF₃ fiber core. The term σ_ij represents the absorption cross-section for the transitions from lower energy level i to upper energy level j (i < j), or the emission cross-section for the transitions from upper energy level i to lower energy level j (i > j). Specifically, σ₁₀ and σ₃₁ represent the emission cross-sections at pump 1 and pump 2 wavelengths, respectively, while σ₀₁ and σ₁₃ denote the corresponding absorption cross-sections. For the laser signal, σ₂₁ and σ₁₂ are the emission cross-section and absorption cross-section at the signal wavelength, respectively. P_p1, P_p2, and P_s denote the longitudinally varying powers of pump 1, pump 2, and the signal light along the fiber.

The positive and negative signs of these power values indicate the forward and backward propagation directions along the fiber, respectively. The propagation equations for the power evolution of the pump lasers and the laser light along the propagation can be obtained as follows:

$${\displaystyle \frac{d{P}_{\mathrm{p}1}^{\pm }}{dz}}=\mp {\mathsf{\Gamma }}_{\mathrm{p}1}({\sigma }_{01}{N}_0(z,t)-{\sigma }_{10}{N}_1(z,t))P_{\mathrm{p}1}^{\pm }\mp {\alpha }_{\mathrm{p}1}{P}_{\mathrm{p}1}^{\pm }$$

(8)

$${\displaystyle \frac{d{P}_{\mathrm{p}2}^{\pm }}{dz}}=\mp {\mathsf{\Gamma }}_{\mathrm{p}2}({\sigma }_{13}{N}_1(z,t)-{\sigma }_{31}{N}_3(z,t))P_{\mathrm{p}2}^{\pm }\mp {\alpha }_{\mathrm{p}2}{P}_{\mathrm{p}2}^{\pm }$$

(9)

$${\displaystyle \frac{d{P}_{\mathrm{s}}^{\pm }}{dz}}=\pm {\mathsf{\Gamma }}_{\mathrm{s}}({\sigma }_{21}{N}_2(z,t)-{\sigma }_{12}{N}_1(z,t))P_{\mathrm{s}}^{\pm }\mp {\alpha }_{\mathrm{s}}{P}_{\mathrm{s}}^{\pm }\pm P_{\mathrm{s}}^{SP}$$

(10)

where α_p1, α_p2, and α_s denote the background loss coefficients of the InF₃ fiber at pump and signal wavelengths. $P_{\mathrm{s}}^{\mathrm{SP}}$ represents the spontaneous emission power, which is defined as the following:

$$P_{\mathrm{s}}^{\mathrm{SP}}=2{\displaystyle \frac{c}{{\lambda }_{\mathrm{s}}}}{\sigma }_{21}{N}_2(z,t){\Delta }_{\mathrm{ASE}}$$

(11)

where Δ_ASE is the noise bandwidth of the amplified spontaneous emission.

The powers of the pump lasers and laser light at both ends of the fiber are subjected to the following boundary conditions:

$$P_{\mathrm{p}1}^{+}(0)=P_{\mathrm{input}\text{\_}\mathrm{p}1}+R_{1,\mathrm{p}1}{P}_{\mathrm{p}1}^{-}(0)$$

(12)

$$P_{\mathrm{p}1}^{-}(L)=(R_{2,\mathrm{p}1}+R_{3,\mathrm{p}1})P_{\mathrm{p}1}^{+}(L)$$

(13)

$$P_{\mathrm{p}2}^{+}(0)=P_{\mathrm{input}\text{\_}\mathrm{p}2}+(R_{1,\mathrm{p}2}+R_{0,\mathrm{p}2})P_{\mathrm{p}2}^{-}(0)$$

(14)

$$P_{\mathrm{p}2}^{-}(L)=(R_{2,\mathrm{p}2}+R_{3,\mathrm{p}2})P_{\mathrm{p}2}^{+}(L)$$

(15)

$$P_{\mathrm{s}}^{+}(0)=R_{1,\mathrm{s}}{P}_{\mathrm{s}}^{-}(0)$$

(16)

$$P_{\mathrm{s}}^{-}(L)=(R_{2,\mathrm{s}}+R_{3,\mathrm{s}})P_{\mathrm{s}}^{+}(L)$$

(17)

where P_{input_p1} and P_{input_p2} are the powers launched into the fiber core for pump 1 and pump 2, respectively. R_1,p1 and R_1,p2 represent the reflectivities of FBG₁ at pump 1 and pump 2 wavelengths, respectively. R_0,p2 represents the reflectivity of M₁ at pump 2 wavelength. Similarly, R_2,p1 and R_2,p2 represent the reflectivities of FBG₂ at pump 1 and pump 2 wavelengths, respectively. R_3,p1 and R_3,p2 represent the reflectivities of FBG₃ at pump 1 and pump 2 wavelengths, respectively. R_1,s, R_2,s, and R_3,s are the reflectivities of FBG₁, FBG₂, and FBG₃ at the laser wavelength.

For the laser system under study, the output power P_output of the fiber laser is given by the following equation:

$$P_{\mathrm{output}}=(1-R_{2,\mathrm{s}}-R_{3,\mathrm{s}})P_{\mathrm{s}}^{+}(L)$$

(18)

Based on the defined boundary conditions, the rate equations and propagation equations are numerically solved using the fourth-order Runge–Kutta algorithm [27]. In the simulation, the fiber length L is uniformly discretized, with transient solutions computed for each spatial segment until reaching steady state. Subsequently, the power distributions of the pump lasers and the signal laser light for each segment are calculated by applying the finite difference method [28]. It should be noted that unless otherwise specified, all simulations utilize the parameters in Table 1.

Table 1. Parameters used in the simulations.

Parameter	Description	Value	Ref.
M10	Non-radiative decay rate: 6H13/2 → 6H15/2	1.98 × 103 s−1	[29]
M21	Non-radiative decay rate: 6H11/2 → 6H13/2	1 × 106 s−1	[29]
M32	Non-radiative decay rate: 6H9/2 → 6H11/2	2 × 107 s−1	[30]
λp1, λp2, λs	Wavelengths of pump and signal lasers	2.8 μm, 2.4 μm, 4.4 μm	—
W10	Radiative decay rate: 6H13/2 → 6H15/2	17.83 s−1	[31]
W21	Radiative decay rate: 6H11/2 → 6H13/2	4.01 s−1	[31]
W20	Radiative decay rate: 6H11/2 → 6H15/2	64.80 s−1	[31]
W32	Radiative decay rate: 6H9/2 → 6H11/2	4.62 s−1	[31]
W31	Radiative decay rate: 6H9/2 → 6H13/2	50.79 s−1	[31]
W30	Radiative decay rate: 6H9/2 → 6H15/2	315.77 s−1	[31]
σ01	Absorption cross-section: 6H15/2 → 6H13/2	3.1 × 10−25 m2	[32]
σ12	Absorption cross-section: 6H13/2 → 6H11/2	1.51 × 10−25 m2	[13]
σ10	Emission cross-section: 6H13/2 → 6H15/2	2.05 × 10−25 m2	[32]
σ21	Emission cross-section: 6H13/2 → 6H11/2	2.06 × 10−25 m2	[13]
NDy	Dy3+ doping concentration	0.5 mol% (9.13 × 1025 m−3)	[13]
L	Fiber length	0.06 m	Optimized
rcore	Fiber core radius	6 μm	[13]
Γp1, Γp2	Power filling factor for 2.8 μm, 2.4 μm	0.7560, 0.8270	Calculated
Γs	Power filling factor for 4.4 μm	0.3671	Calculated
αp1, αp2	Fiber loss at 2.8 μm, 2.4 μm	0.2 dB/m, 0.2 dB/m	[13]
αs	Fiber loss at 4.4 μm	0.2 dB/m	[13]
R0,p2	Reflectivity of M1 at 2.4 μm	99%	—
R1,p1, R1,p2	Reflectivities of FBG1 at 2.8 μm, 2.4 μm	0, 0	—
R1,s	Reflectivities of FBG1 at 4.4 μm	99%	—
R2,p1, R2,p2	Reflectivities of FBG2 at 2.8 μm, 2.4 μm	0, 0	—
R2,s	Reflectivities of FBG2 at 4.4 μm	95%	—
R3,p1, R3,p2	Reflectivities of FBG3 at 2.8 μm, 2.4 μm,	0, 99%	—
R3,s	Reflectivities of FBG3 at 4.4 μm	0	—
ΔASE	Noise bandwidth of the amplified spontaneous emission	1 nm	[13]

For σ₁₃ and σ₃₁, there are no direct experimental measurement data for reference. However, approximate values can be scaled from the absorption and emission cross-sections of Dy³⁺-doped chalcogenide glass [13]. Studies indicate that the fluorescence spectral shapes of Dy³⁺-doped fluoride glass exhibit minimal deviation from the chalcogenide glass host, primarily showing reduced cross-section intensity [13]. Using σ₁₂ and σ₂₁ scaling factors as a reference (chalcogenide host: σ₁₂ = 8.5 × 10⁻²⁵ m², σ₂₁ = 9.8 × 10⁻²⁵ m² [26]; fluoride host: σ₁₂ = 1.51 × 10⁻²⁵ m², σ₂₁ = 2.06 × 10⁻²⁵ m²), we obtain scaling factors of 5.63 (absorption cross-section) and 4.76 (emission cross-section). Applying these to chalcogenide glass values at 2.4 μm (absorption cross-section: 9.79 × 10⁻²⁵ m², emission cross-section: 9.14 × 10⁻²⁵ m² [26]), we derive fluoride glass cross-sections of σ₁₃ = 1.74 × 10⁻²⁵ m² and σ₃₁ = 1.92 × 10⁻²⁵ m² at 2.4 μm. These estimated cross-sections at 2.4 μm are comparable to those at 2.8 μm, which is consistent with these chalcogenide fibers.

3. Simulation Results and Discussion

In the proposed dual-wavelength cascade-pumped mid-infrared fiber laser system, the gain of the 4.4 μm laser is primarily driven by the 2.4 μm pump. However, the 2.8 μm pump intensity significantly impacts the VGSA of 2.4 μm. Figure 2a depicts the output power of the 4.4 μm laser versus 2.8 μm pump power P₁ at different fiber lengths, with 2.4 μm pump power P₂ fixed at 20 W. For all the curves, laser oscillation begins at a low pump power level (P1th = ~1 mW), with the output power initially increasing with the 2.8 μm pump power P₁. After reaching their respective peaks, the output powers of all curves decline as the 2.8 μm pump power P₁ continues to increase. This decrease can be attributed to the excessive 2.8 μm pump power, which leads to particle accumulation in the lower energy level and thus diminishes the population inversion between the upper and lower energy levels. With increasing fiber length, the laser output power declines more rapidly due to the higher fiber loss and stronger signal light reabsorption.

Figure 2. Simulated output power of the Dy³⁺:InF₃ fiber laser for different fiber lengths: L = 0.04, 0.06, 0.08, 0.1 m. (a) Output power vs. 2.8 μm pump power P₁. (b) Output power vs. 2.4 μm pump power P₂ with 2.8 μm pump power P₁ fixed at 5 mW. (c) Output power vs. 2.4 μm pump power P₂ with 2.8 μm pump power P₁ fixed at 20 mW.

Figure 2b,c illustrate the relationship between the output power and the 2.4 μm pump power P₂ for different fiber lengths, with the 2.8 μm pump power P₁ fixed at 5 mW and 20 mW, respectively. As shown in Figure 2b, the output power increases linearly with 2.4 μm pump power P₂. Relatively shorter fiber lengths correspond to lower pump thresholds P2th, achieving a minimum threshold of 8 W at a fiber length of 0.04 m. However, longer fiber lengths enable a more rapid rise in output power. In Figure 2c, when 2.8 μm pump power P₁ is increased to 20 mW, the pump threshold P2th increases accordingly, with distinct magnitudes of growth. The longer the fiber, the more pronounced the threshold increase. Additionally, the slope efficiency of the output power is almost the same across all the fiber lengths.

Figure 3a shows the output power of the Dy³⁺:InF₃ fiber laser versus 2.4 μm pump power P₂ at 2.8 μm pump power P₁ = 8, 12, 16, and 20 mW. Increasing 2.8 μm pump power P₁ would elevate the pump threshold P2th, as higher P₁ causes particle accumulation in the lower laser level, requiring a higher 2.4 μm pump power to achieve population inversion. Figure 3b depicts the output power versus 2.8 μm pump power P₁ for 2.4 μm pump power P₂ = 8, 12, 16, and 20 W. The results demonstrate that the influence of 2.8 μm pump power P₁ becomes more pronounced with higher 2.4 μm pump power P₂. No laser emission occurs at 2.4 μm pump power P₂ = 8 W due to insufficient population inversion. Under optimized pumping (P₁ = 12 mW, P₂ = 20 W), the fiber laser yields a maximum output power of 3.69 W, with the corresponding overall optical-to-optical efficiency of 18.4%.

Figure 3. Simulated output power of the Dy³⁺:InF₃ fiber laser for fiber length L = 0.06 m and output coupler reflectivity R_2,s = 95%. (a) Output power vs. 2.4 μm pump power P₂ for 2.8 μm pump power P₁ = 8, 12, 16, and 20 mW. (b) Output power vs. 2.8 μm pump power P₁ at 2.4 μm pump power P₂ = 8, 12, 16, and 20 W.

Figure 4a,b show the slope efficiency (with respect to 2.8 μm pump power P₁) and pump threshold (P2th) of the Dy³⁺:InF₃ fiber laser versus fiber length at 2.8 μm pump power P₁ = 12 mW, with output coupler reflectivity R_2,s = 90–98%. The maximum slope efficiency of 38.5% is achieved at the minimum output coupler reflectivity R_2,s = 90% and fiber length L = 0.08 m, while the minimum P2th of 5.4 W is achieved at the maximum output coupler reflectivity R_2,s = 98% and fiber length L = 0.03 m. The total pump threshold is markedly lower than the >30 W threshold required for the single-wavelength pumped cascade lasing approach under continuous-wave operation [13]. Figure 4c depicts the output power versus 2.8 μm pump power P₁ at 2.4 μm pump power P₂ = 20 W, fiber length L = 0.06 m, and output coupler reflectivity R_2,s = 90–98%. It can be observed that 2.8 μm pump power P₁ = 12 mW is identified as the optimal value for achieving high output power. Furthermore, when 2.8 μm pump power P₁ exceeds 8 mW, output coupler reflectivity, R_2,s = 95%, exhibits the optimal output power performance.

Figure 4. Simulated output characteristics of the Dy³⁺:InF₃ fiber laser. (a) Slope efficiency (with respect to 2.8 μm pump power P₁) and (b) pump threshold (P2th) vs. fiber length at 2.8 μm pump power P₁ =12 mW, and output coupler reflectivity R_2,s = 90–98%. (c) Output power vs. 2.8 μm pump power P₁ at 2.4 μm pump power P₂ = 20 W, fiber length L = 0.06 m, and output coupler reflectivity R_2,s = 90–98%.

Under optimized pumping conditions (P₁ = 12 mW, P₂ = 20 W) and an output coupler reflectivity R_2,s = 95%, further analysis is conducted on the effect of Dy³⁺ doping concentration on the laser performance, as depicted in Figure 5. When the dopant concentration of Dy³⁺ rises from 0.3 mol% to 0.7 mol%, the optimal fiber length for peak output power decreases from 0.06 m to 0.03 m, and the corresponding peak output power increases from 3.49 W to 4 W. These results highlight that enhanced efficiency can be achieved by employing gain fibers with higher Dy³⁺ doping concentrations.

Figure 5. Simulated output power of the Dy³⁺:InF₃ fiber laser as a function of the fiber length for different doping concentrations.

The above simulation results are based on an estimated value of σ₁₃ = 1.74 × 10⁻²⁵ m². Considering potential discrepancies in the practical scenarios, we further investigate the influence of σ₁₃ on optical-to-optical conversion efficiency under optimized pumping conditions (P₁ = 12 mW, P₂ = 20 W) and an output coupler reflectivity R_2,s = 95%, as shown in Figure 6a. As observed, the optical-to-optical conversion efficiency increases from 12.8% at σ₁₃ = 1 × 10⁻²⁵ m² to 21.7% at σ₁₃ = 3 × 10⁻²⁵ m². Based on these results, we believe much higher optical-to-optical conversion efficiency can be achieved in Dy³⁺-doped chalcogenide fiber lasers theoretically, since such fibers exhibit an absorption cross-section of up to 9.79 × 10⁻²⁵ m² [26].

Figure 6. (a) Influence of σ₁₃ on optical-to-optical conversion efficiency under optimized pumping conditions (P₁ = 12 mW, P₂ = 20 W) and an output coupler reflectivity R_2,s = 95%. (b) Dependence of pumping threshold (under conditions: P₁ = 12 mW, R_2,s = 98%, L = 0.03 m) and optical-to-optical conversion efficiency (under conditions: P₁ = 12 mW, P₂ = 20 W, R_2,s = 95%, L = 0.06 m) on 2.4 μm pump laser escape loss.

Furthermore, we assume that the 2.4 μm pump laser undergoes reflection between M₁ and FBG₃ with 1% escape loss, which might be higher in practical implementations. Thus, we investigate the impacts of such loss on the pumping threshold (under conditions: P₁ = 12 mW, R_2,s = 98%, L = 0.03 m) and optical-to-optical conversion efficiency (under conditions: P₁ = 12 mW, P₂ = 20 W, R_2,s = 95%, L = 0.06 m) by varying R_0,p2 from 95% to 99%. As illustrated in Figure 6b, with a decrease in R_0,p2 (i.e., an increase in loss), the pump threshold increases from 5.4 W to 7.4 W, while the optical-to-optical conversion efficiency decrease from 18.4% to 11.2%. Therefore, in the practical scenarios, to achieve low-threshold and high-efficiency operation of the laser, such loss should be minimized. For example, an antireflection coating can be used to reduce the reflection loss of the lens, and an aspherical lens can be employed to reduce aberrations of the focused spot. We would like to mention that Majewski et al. numerically demonstrated that pulse pumping can significantly lower the lasing threshold compared to continuous-wave pumping [13]. Thus, employing a 2.4 μm pulsed laser as the pumping source is expected to further reduce the threshold of the dual-wavelength cascade-pumped laser.

4. Conclusions

In summary, this work presents a dual-wavelength cascade-pumping strategy to tackle the self-termination limitation in Dy³⁺:InF₃ fiber lasers, enabling low-threshold, high-efficiency 4.4 μm emission. Comprehensive simulation results demonstrate that this dual-wavelength cascade configuration achieves an overall optical-to-optical efficiency of up to 18.4% and a maximum slope efficiency of 38.5%. The total pump threshold is as low as 5.4 W, which is remarkably lower than the >30 W threshold required for single-wavelength pumped cascade lasing under continuous-wave operation. Furthermore, we identify that further efficiency enhancement can be realized by appropriately increasing the Dy³⁺ doping concentration. This study not only provides a viable technical pathway for developing low-threshold, high-efficiency 4.4 μm Dy³⁺:InF₃ fiber lasers, but also paves the way for their experimental implementation in 4 μm-class mid-infrared fiber laser systems.