The recent two decades have seen great progress of the rare-earth-doped fluoride fiber lasers in the mid-infrared (MIR) region of >2.5 μm, motivated by a lot of potential applications such as polymer processing [1–3], remote sensing [4–6], and laser surgery [7–9]. As the pioneer, the 980 nm diode pumped Er³⁺-doped ZrF₄ fiber laser based on the ⁴I_11/2→⁴I_13/2 transition has enabled a record power value of 41.6 W at 2.824 μm [10] and 30.5 W at the longer 2938 nm wavelength [11] at room temperature. Based on the red-shifted emission band of the ⁵I₆→⁵I₇ transition, the 1150 nm diode pumped Ho³⁺-doped ZrF₄ fiber laser could even operate at 3021 nm [12]. Nevertheless, most early reports were clamped at 2.7 μm–3.0 μm [13]. Until 2014, Henderson-Sapir et al. introduced the concept of dual-wavelength pumping at 985 nm and 1973 nm into an Er³⁺-doped ZrF₄ fiber system to excite the higher ⁴F_9/2→⁴I_9/2 transition, achieving ~260 mW laser power at ~3.5 μm, far beyond 3.0 μm [14]. Along this way, wavelength tuning within the range of 3.33 μm–3.78 μm [15] and significantly scaled power of 14.9 W at 3.55 μm [16] have been demonstrated afterwards. To bridge the gap between 3.0 μm and 3.3 μm, hence unleashing the application potential of this region, the ultrawide ⁶H_13/2→⁶H_15/2 transition band of Dy³⁺ around 3.0 μm has been exploited by Majewski and Jackson [17,18]. In-band pumping of a Dy³⁺-doped ZrF₄ fiber at ~2.8 μm has enabled 10.1 W power at 3.24 μm [19], and pumping at ~1.7 μm has realized entire wavelength coverage of 2.7 μm–3.4 μm [20]. In fact, Ho³⁺ also owns an emission band (i.e., ⁵S₂,⁵F₄→⁵F₅ [21]) in this region that is even stronger than Dy³⁺ [22], in addition to the well-known emission bands at ~2.1 μm (i.e., ⁵I₇→⁵I₈) and ~2.9 μm (i.e., ⁵I₆→⁵I₇), and the recent high-profile 3.9 μm band (i.e., ⁵I₅→⁵I₆). As early as 1998, direct exciting of a Ho³⁺-doped ZrF₄ fiber at 532 nm has successfully yielded a 3.22 μm laser at room temperature [21]. Limited by the poor pumping source and long-lived metastable states ⁵I₆ and ⁵I₇, however, only 11 mW power with slope efficiency of 2.8% was obtained. Since then, there has been little attention paid to the transition. This strongly motivates us to explore its potential in efficient lasing. With the improved fiber drawing technique, the recent reports imply that rare-earth ions behave better when doped into lower-phonon-energy InF₃ (509 cm^–1) compared with common ZrF₄ glass (574 cm^–1) [23–26]. For example, a Dy³⁺-doped InF₃ fiber has led to record slope efficiency of 82% beyond 3 μm [23]; the longest laser wavelength of 3.91 μm has been theoretically predicted from an Er³⁺-doped InF₃ fiber [24]; a Ho³⁺-doped InF₃ fiber has yielded unprecedented 197 mW laser power at 3.92 μm at room temperature [25]; wideband emission around 4.3 μm has been observed from a Dy³⁺-doped InF₃ fiber [26]. With respect to the ~3.2 μm transition of Ho³⁺, the upper- and lower-level lifetimes in InF₃ glass are higher and lower than that in ZrF₄, respectively, beneficial for population inversion formation.

In this work, we propose to adopt dual-wavelength pumping at 1150 nm and 980 nm to solve the ions bottleneck issue existed, and numerically explore the potential of the Ho³⁺-doped InF₃ fiber in getting efficient ~3.2 μm laser. Also, the laser performance after activating the cascaded ~3.9 μm transition simultaneously has been studied, where the heat load has been taken into account to evaluate the power scaling potential of this ~3.2 μm and ~3.9 μm dual-wavelength system.

Fig. 1 presents the schematic diagram of our designed dual-wavelength pumped double-clad Ho³⁺-doped InF₃ fiber laser at ~3.2 μm and ~3.9 μm, of which the cavity is formed by two dichroic mirrors (i.e., front cavity mirror (FCM) and rear cavity mirror (RCM)). Note that the first pumping at 1150 nm, available from the commercial laser diode (LD) [27], is coupled into the fiber inner clad, and the second pumping at 980 nm, achievable from high-brightness Yb³⁺-doped silica fiber laser [28], is coupled into the fiber core instead.

The relative energy level diagram is plotted in Fig. 2, where the ~3.2 μm emission and ~3.9 μm emission are stemmed from the ⁵F₄,⁵S₂→⁵F₅ and ⁵I₅→⁵I₆ transitions, respectively, where the upper level of the ~3.2 μm laser is populated through the cascaded ground state absorption (GSA) and visual ground state absorption (VGSA) processes. While the excited state absorption (ESA) and interionic processes of energy transfer upconversion (ETU) (i.e., ETU1 and ETU2 in Fig. 2) and cross relaxation (i.e., CR1 and CR2 in Fig. 2) also participate in the population evolution. Accordingly, the relevant numerical model including rate and power evolution equations has been shown as below. Note that we failed to match this model with the experiment at ~3.2 μm due to the absent fiber length information in the unique experimental report [21]. Nevertheless, we believe the model is valid since it is modified from our prior ~3.9 μm Ho³⁺-doped InF₃ fiber laser model [29] that could reproduce the experiment.

$ \frac{{{\text{d}}{N_5}}}{{{\text{d}}t}} = - \tau _5^{ - 1}{N_5} + {R_{{\text{VGSA}}}} - {R_{{\text{CR}}2}} - {R_{{\text{S}}{{\text{E}}_1}}} $ (1)

$ \frac{{{\text{d}}{N_4}}}{{{\text{d}}t}} = - \tau _4^{ - 1}{N_4} + {\beta _{5{\mathrm{,}}4}}\tau _5^{ - 1}{N_5} + {R_{{\text{S}}{{\text{E}}_1}}} + {R_{{\text{ESA}}}} + {R_{{\text{ETU2}}}} $ (2)

$ \frac{{{\text{d}}{N_3}}}{{{\text{d}}t}} = - \tau _3^{ - 1}{N_3} + \sum\limits_{i = 4{\mathrm{,}}5}^{} {{\beta _{i{\mathrm{,}}3}}\tau _i^{ - 1}{N_i}} + {R_{{\text{CR2}}}} - {R_{{\text{CR1}}}} - {R_{{\text{S}}{{\text{E}}_2}}} $ (3)

$ \frac{{{\text{d}}{N_2}}}{{{\text{d}}t}} = - \tau _2^{ - 1}{N_2} + \sum\limits_{i = 3{\mathrm{,}}4{\mathrm{,}}5} {{\beta _{i{\mathrm{,}}2}}\tau _i^{ - 1}{N_i}} - {R_{{\text{VGSA}}}} - 2{R_{{\text{ETU}}2}} + {R_{{\text{ETU}}1}} + {R_{{\text{GSA}}}} + {R_{{\text{S}}{{\text{E}}_{\text{2}}}}} $ (4)

$ \frac{{{\text{d}}{N_1}}}{{{\text{d}}t}} = - \tau _1^{ - 1}{N_1} + \sum\limits_{i = 2{\mathrm{,}}3{\mathrm{,}}4{\mathrm{,}}5} {{\beta _{i{\mathrm{,}}1}}\tau _i^{ - 1}{N_i}} - 2{R_{{\text{ETU}}1}} + 2{R_{{\text{CR}}1}} + {R_{{\text{CR}}2}} - {R_{{\text{ESA}}}} $ (5)

$ {N_{{\text{Ho}}}} = \sum\limits_{i = 0{\mathrm{,}} 1{\mathrm{,}} 2{\mathrm{,}} 3{\mathrm{,}} 4{\mathrm{,}} 5} {{N_i}} $ (6)

where N_i represents the population density on the energy level i, τ_i represents the intrinsic lifetime of the energy level i involving the contributions of the radiative transition and multi-phonon relaxation (MPR). β_i,j represents the branching ratio for the decay from the higher level i to the lower level j. If $i=j+1 $, β_i,j involves the contributions of both the radiative transition and MPR, otherwise, only that of the radiative transition. Considering the large MPR rate for ⁵I₄→⁵I₅, the population density on the ⁵I₄ level has been ignored. R_ETU1 and R_ETU2 are the rates of the ETU1 and ETU2 processes, respectively, defined as

$ {R_{{\text{ETU1}}}} = {W_{1{\mathrm{,}}1}}N_1^2 $ (7)

$ {R_{{\text{ETU2}}}} = {W_{2{\mathrm{,}}2}}N_2^2 $ (8)

where W_1,1 and W_2,2 represent the parameters of ETU1 and ETU2, respectively. R_CR1 and R_CR2 represent the rates of the CR1 and CR2 processes, respectively, defined as

$ {R_{{\text{CR}}1}} = {W_{3{\mathrm{,}}0}}{N_3}{N_0} $ (9)

$ {R_{{\text{CR2}}}} = {W_{5{\mathrm{,}}0}}{N_5}{N_0} $ (10)

where W_3,0 and W_5,0 stand for the parameters of CR1 and CR2, respectively. The units of these parameters (i.e., W_1,1, W_2,2, W_3,0, and W_5,0) are all m³/s. R_GSA, R_VGSA, and R_ESA are the rates of the GSA, VGSA, and ESA processes and given by

$ {R_{{\text{GSA}}}} = \frac{{{\lambda _{{p_1}}}{\Gamma _{{p_1}}}}}{{hc{A_{{\text{core}}}}}}\left( {{\sigma _{{\text{ab}}{{\text{s}}_{{\text{02}}}}}}{N_0} - {\sigma _{{\text{em}}{{\text{i}}_{{\text{20}}}}}}{N_2}} \right)\left( {P_{{p_1}}^ + + P_{{p_1}}^ - } \right) $ (11)

$ {R_{{\text{VGSA}}}} = \frac{{{\lambda _{{p_2}}}{\Gamma _{{p_2}}}}}{{hc{A_{{\text{core}}}}}}\left( {{\sigma _{{\text{ab}}{{\text{s}}_{{\text{25}}}}}}{N_2} - {\sigma _{{\text{em}}{{\text{i}}_{{\text{52}}}}}}{N_5}} \right)\left( {P_{{p_2}}^ + + P_{{p_2}}^ - } \right) $ (12)

$ {R_{{\text{ESA}}}} = \frac{{{\lambda _{{p_2}}}{\Gamma _{{p_2}}}}}{{hc{A_{{\text{core}}}}}}\left( {{\sigma _{{\text{ab}}{{\text{s}}_{{\text{14}}}}}}{N_1} - {\sigma _{{\text{em}}{{\text{i}}_{{\text{41}}}}}}{N_4}} \right)\left( {P_{{p_2}}^ + + P_{{p_2}}^ - } \right) $ (13)

where $ {\lambda _{{p_1}}} $ and $ {\lambda _{{p_2}}} $ are the two pumping wavelengths setting at 1150 nm and 980 nm, respectively; $ {\Gamma _{{p_1}}} $ and $ {\Gamma _{{p_2}}} $ are the corresponding power filling factors. For the clad-coupled 1150 nm pumping, $ {\Gamma _{{p_1}}} $ can be simply estimated as the ratio of the cross-section area of the fiber core to that of the inner clad. For the core-coupled 980 nm pumping, the normalized frequency of 10 for our selected fiber is far from the cut-off frequency, and thus the multi-mode nature has been considered as that in our recent work [30] when calculating $ {\Gamma _{{p_2}}} $. h is Planck’s constant and c is the speed of light in vacuum. A_core denotes the fiber core cross-section area. $ {\sigma _{{\text{ab}}{{\text{s}}_{{\text{02}}}}}} $, $ {\sigma _{{\text{ab}}{{\text{s}}_{{\text{14}}}}}} $, and $ {\sigma _{{\text{ab}}{{\text{s}}_{{\text{25}}}}}} $ represent the cross sections of the GSA, ESA, and VGSA processes, respectively, and $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{20}}}}}} $, $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{41}}}}}} $, and $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{52}}}}}} $ are the corresponding cross sections of the reverse processes. $ P_{{p_1}}^ + $ ($ P_{{p_1}}^ - $) and $ P_{{p_2}}^ + $ ($ P_{{p_2}}^ - $) represent the intra-cavity forward (backward) propagating 1150 nm and 980 nm pumping power, respectively.

Similarly, $ {R_{{\text{S}}{{\text{E}}_1}}} $ and $ {R_{{\text{S}}{{\text{E}}_2}}} $ represent the rates of the stimulated emission at ~3.2 μm and ~3.9 μm, respectively, and are given by

$ {R_{{\text{S}}{{\text{E}}_{\text{1}}}}} = \frac{{{\lambda _{{s_1}}}{\Gamma _{{s_1}}}}}{{hc{A_{{\text{core}}}}}}\left( {{\sigma _{{\text{em}}{{\text{i}}_{{\text{54}}}}}}{N_5} - {\sigma _{{\text{ab}}{{\text{s}}_{45}}}}{N_4}} \right)\left( {P_{{s_1}}^ + + P_{{s_1}}^ - } \right) $ (14)

$ {R_{{\text{S}}{{\text{E}}_2}}} = \frac{{{\lambda _{{s_2}}}{\Gamma _{{s_2}}}}}{{hc{A_{{\text{core}}}}}}\left( {{\sigma _{{\text{em}}{{\text{i}}_{32}}}}{N_3} - {\sigma _{{\text{ab}}{{\text{s}}_{{\text{23}}}}}}{N_2}} \right)\left( {P_{{s_2}}^ + + P_{{s_2}}^ - } \right) $ (15)

where $\lambda _{{s_1}} $ and $\lambda _{{s_2}} $ are the laser wavelengths setting at 3260 nm and 2920 nm, respectively; $\Gamma _{{s_1}} $ and $\Gamma _{{s_2}} $ are the corresponding power filling factors; $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{54}}}}}} $ and $ {\sigma _{{\text{em}}{{\text{i}}_{32}}}} $ are the stimulated emission cross sections at 3260 nm and 3920 nm, respectively; $ {\sigma _{{\text{ab}}{{\text{s}}_{45}}}} $ and $ {\sigma _{{\text{ab}}{{\text{s}}_{{\text{23}}}}}} $ are the corresponding cross sections of their reverse processes. Considering their normalized frequencies of 3.03 and 2.52 are slightly larger than the cut-off frequency, single-mode operations with the Gaussian intensity distribution are assumed. Thus, $\Gamma _{{s_1}} $ and $\Gamma _{{s_2}} $ can be estimated by

$ {\Gamma _{{s_1}}} \approx 1 - \exp \left( { - 2\frac{{r_{{\text{core}}}^2}}{{\omega _1^2}}} \right) $ (16)

$ {\Gamma _{{s_2}}} \approx 1 - \exp \left( { - 2\frac{{r_{{\text{core}}}^2}}{{\omega _2^2}}} \right) $ (17)

where r_core is the radius of the fiber core, and ω₁ and ω₂ are the mode field radii at 3260 nm and 3920 nm, respectively, which can be estimated based on the Marcuse empirical formula for the step-index fiber [31]:

$ {\omega _1} = {r_{{\text{core}}}}(0.65 + 1.619V_1^{ - 1.5} + 2.879V_1^{ - 6}) $ (18)

$ {\omega _2} = {r_{{\text{core}}}}(0.65 + 1.619V_2^{ - 1.5} + 2.879V_2^{ - 6}) $ (19)

where V₁ and V₂ are the corresponding normalized frequencies and given by

$ {V_1} = \frac{{2\pi {r_{{\text{core}}}}{\text{NA}}}}{{{\lambda _{{s_1}}}}} $ (20)

$ {V_2} = \frac{{2\pi {r_{{\text{core}}}}{\text{NA}}}}{{{\lambda _{{s_2}}}}} $ (21)

where NA is the numerical aperture. $P_{{s_1}}^ + $ ($P_{{s_1}}^ - $) and $P_{{s_2}}^ + $ ($P_{{s_2}}^ - $) represent the intra-cavity forward (backward) propagating 3260 nm and 3920 nm pumping power, respectively.

The relative power evolution equations of the pumping and laser are given below:

$ \frac{{{\text{d}}P_{{p_1}}^ \pm }}{{{\text{d}}z}} = - {\Gamma _{{p_1}}}\left( {{\sigma _{{\text{ab}}{{\text{s}}_{02}}}}{N_0} - {\sigma _{{\text{em}}{{\text{i}}_{{\text{20}}}}}}{N_2}} \right)P_{{p_1}}^ \pm - {\alpha _{{p_1}}}P_{{p_1}}^ \pm $ (22)

$ \frac{{{\text{d}}P_{{p_2}}^ \pm }}{{{\text{d}}z}} = - {\Gamma _{{p_2}}}\left( {{\sigma _{{\text{ab}}{{\text{s}}_{{\text{25}}}}}}{N_2} - {\sigma _{{\text{em}}{{\text{i}}_{{\text{52}}}}}}{N_5} + {\sigma _{{\text{ab}}{{\text{s}}_{{\text{14}}}}}}{N_1} - {\sigma _{{\text{em}}{{\text{i}}_{{\text{41}}}}}}{N_4}} \right)P_{{p_2}}^ \pm - {\alpha _{{p_2}}}P_{{p_2}}^ \pm $ (23)

$ \frac{{{\text{d}}P_{{s_1}}^ \pm }}{{{\text{d}}z}} = {\Gamma _{{s_1}}}\left( {{\sigma _{{\text{em}}{{\text{i}}_{{\text{54}}}}}}{N_5} - {\sigma _{{\text{ab}}{{\text{s}}_{45}}}}{N_4}} \right)P_{{s_1}}^ \pm - {\alpha _{{s_1}}}P_{{s_1}}^ \pm $ (24)

$ \frac{{{\text{d}}P_{{s_2}}^ \pm }}{{{\text{d}}z}} = {\Gamma _{{s_2}}}\left( {{\sigma _{{\text{em}}{{\text{i}}_{32}}}}{N_3} - {\sigma _{{\text{ab}}{{\text{s}}_{23}}}}{N_2}} \right)P_{{s_2}}^ \pm - {\alpha _{{s_2}}}P_{{s_2}}^ \pm $ (25)

where $ {\alpha _{{p_1}}} $ and $ {\alpha _{{p_2}}} $ represent the background loss coefficients at 1150 nm and 980 nm, respectively; $ {\alpha _{{s_1}}} $ and $ {\alpha _{{s_2}}} $ are the background loss coefficients at 3260 nm and 3920 nm, respectively. At the two ends, the boundary conditions of the pumping power and the laser power are given as follows:

$ P_{{p_1}}^ + (0) = {R_{{p_{11}}}}P_{{p_1}}^ - (0) + {P_1} $ (26)

$ P_{{p_1}}^ - (L) = {R_{{p_{12}}}}P_{{p_1}}^ + (L) $ (27)

$ P_{{p_2}}^ + (0) = {R_{{p_{21}}}}P_{{p_2}}^ - (0) + {P_2} $ (28)

$ P_{{p_2}}^ - (L) = {R_{{p_{22}}}}P_{{p_2}}^ + (L) $ (29)

$ P_{{s_1}}^ + (0) = {R_{{s_{11}}}}P_{{s_1}}^ - (0) $ (30)

$ P_{{s_1}}^ - (L) = {R_{{s_{12}}}}P_{{s_1}}^ + (L) $ (31)

$ P_{{s_2}}^ + (0) = {R_{{s_{21}}}}P_{{s_2}}^ - (0) $ (32)

$ P_{{s_2}}^ - (L) = {R_{{s_{22}}}}P_{{s_2}}^ + (L) $ (33)

where $ {R_{{p_{11}}}} $ ($ {R_{{p_{12}}}} $) and $ {R_{{p_{21}}}} $ ($ {R_{{p_{22}}}} $) are the reflectivity of FCM (BCM) at 1150 nm and 980 nm, respectively; L is the fiber length; P₁ and P₂ are the 1150 nm pumping power and 980 nm pumping power coupled into the fiber, respectively; $ {R_{{s_{11}}}} $ ($ {R_{{s_{12}}}} $) and $ {R_{{s_{21}}}} $ ($ {R_{{s_{22}}}} $) are the reflectivity of FCM (BCM) at 3260 nm and 3920 nm, respectively. Note that the two pumping lights propagate in the cavity with one round to ensure sufficient absorption. The laser is output in a forward direction. The solution method is the same as our recent work [29,30].

In the simulation, all the parameters we used are listed in the following Tables 1 [22,25,32–34] and 2.

The used InF₃ fiber size (i.e., r_core/r_clad=5.5 μm/50.0 μm, NA=0.3) is commercially available from Le Verre Fluore, and the low Ho³⁺ dopant concentration of 1 mol.% is selected in this case in order to relieve the heat load. ${\sigma _{{\text{ab}}{{\text{s}}_{{\text{02}}}}}}$, ${\sigma _{{\text{ab}}{{\text{s}}_{25}}}}$, $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{52}}}}}} $, and $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{32}}}}}} $ are directly derived from the previous studies related to InF₃ glass [25,32,33]. $ {\sigma _{{\text{em}}{{\text{i}}_{{\text{54}}}}}} $ is derived from Ho³⁺-doped ZrF₄ glass instead [22] due to the similar emission spectra of the ⁵I₆→⁵I₇ transition at ~2.9 μm in InF₃ and ZrF₄ glass. ${\sigma _{{\text{ab}}{{\text{s}}_{14}}}}$ is also taken from ZrF₄ glass [34] due to lack of the direct measurement in InF₃ glass. Their corresponding $ {\sigma _{{\text{em}}{{\text{i}}_{20}}}} $, $ {\sigma _{{\text{em}}{{\text{i}}_{41}}}} $, ${\sigma _{{\text{ab}}{{\text{s}}_{45}}}}$, and ${\sigma _{{\text{ab}}{{\text{s}}_{23}}}}$ are calculated by the common McCumber relation [35], in which the relevant energy level degeneracy and locations can be found in Refs. [36] and [37], respectively. W_1,1, W_2,2, W_3,0, and W_5,0 are obtained as follows: First, the rate parameters K_1,1, K_2,2, K_3,0, and K_5,0 in 1 mol.% doped bulk InF₃ glass are achieved by fitting in a power low function [33]. Divided by the critical concentration of excited Ho³⁺ ions [38], which is equal to 7×10²⁵ m^–3 for K_1,1, 2×10²⁵ m^–3 for K_2,2, and N_Ho for W_3,0 and W_5,0 [29,32], the corresponding W_1,1, W_2,2, W_3,0, and W_5,0 can be deduced. Considering the weakly interacting (WI) parameters are more accurate to reproduce the Ho³⁺-doped InF₃ fiber system [25] indicated by the previous simulations [29,39,40], the deduced strongly interacting (SI) parameters in bulk InF₃ glass above are then transformed to the WI parameters listed in Table 1 based on the ratios (i.e., 2.0, 3.2, 30.0, and 30.0) validated in Ref. [29].

τ_i and β_i,j are calculated using the radiative lifetimes, rates of MPR (labeled as ${W_{{\text{MP}}{{\text{R}}_{i{\mathrm{,}}j}}}}$), and luminescence branching ratios [33] based on the relation addressed in Ref. [41].

The heat load is an important metric that can be used to assess the potential of a laser system in power scaling. Same as the prior reports [30,39,40,42], the heat load of the pumped fiber end that has the greatest value is selected to characterize the whole system in our case, where the non-radiative decay (i.e., MPR) and exothermic/endothermic heat from the interionic processes (i.e., ETU1, ETU2, CR1, and CR2) have been considered. Thus the total heat load can be calculated by the following formula [30,42]:

$ $$\begin{array}{rl}Q=& A_{\text{core }}(\sum _{\begin{array}{c}i=1\\ j=i-1\end{array}}^4{N}_i{W}_{\mathrm{MPR}i,j}{E}_{i,j}+\sum _{i=1,2}{W}_{i,i}{N}_i^2\delta E_{i,i}+\\ & \sum _{j=3,5}{W}_{j,0}{N}_j{N}_0\delta E_{j,0})\end{array}$$ $ (34)

where Q is the heat load per unit length [W/m], E_i,j is the energy gap between the levels i and j, and δE is the exothermic energy associated with the interionic processes, where $\delta E_{1{\mathrm{,}}1}=E_{1{\mathrm{,}}0}-E_{2{\mathrm{,}}1} $, $\delta E_{2{\mathrm{,}}2}=E_{2{\mathrm{,}}0}-E_{4{\mathrm{,}}2} $, $\delta E_{3{\mathrm{,}}0}=E_{3{\mathrm{,}}1}-E_{1{\mathrm{,}}0} $, and $\delta E_{5{\mathrm{,}}0}=E_{5{\mathrm{,}}3}-E_{1{\mathrm{,}}0} $ and all are in J. The parameters are listed in Table 3.

First, the ~3.9 μm laser transition process is switched off in the model, and the laser performance under the single transition operation state at ~3.2 μm has been exhibited.

Fig. 3 plots the ~3.2 μm power distribution with the varied ${R_{{s_{12}}}}$ and L at the fixed pumping power, i.e., P₁ = 5 W and P₂ = 20 W which are technically available for the commercial LD [27] and state-of-the-art diffraction-limited Yb³⁺-doped fiber laser [28], respectively. Under the optimal ${R_{{s_{12}}}}$ of 0.11 and L of 1.85 m, output power of 3.6 W is theoretically predicted, with optical-to-optical efficiency of 14.4%.

The corresponding power evolutions with the pumping power are displayed in Fig. 4. Fig. 4 (a) plots that with P₁, where the output power increases first and then saturates rapidly with the increased P₁ owing to the excessive ions in the ⁵I₆ state which cannot be consumed by fixed P₂, thus converting to the laser. Particularly, the higher P₂, the higher P₁ needed to saturate the output, and the higher final output, identical to our familiar dual-wavelength pumped Er³⁺-doped ZrF₄ fiber laser at ~3.5 μm [42,43]. When varying P₂ under fixed P₁, the saturation behavior is observed as shown in Fig. 4 (b). This is different from the dual-wavelength pumped ~3.5 μm systems, where the laser quenching exists [16,43,44], stemming from the pumping resonant ESA process targeting the laser upper level [44].

To elucidate the effects of the interionic processes and ESA on the laser performance (e.g., slope efficiency, threshold, and saturation power), we re-calculated the outputs by switching off these processes, separately, in the model as presented in Fig. 5. Compared with the other interionic processes, the ETU2 process has a relatively obvious impact on the laser performance due to its large process parameter indicated in Table 1. Once switching off the ETU2 process, higher output power can be achieved, implying its detrimental effect on the laser output. The reason is that it not only consumes the ions that should be employed for populating the laser upper level ⁵F₄,⁵S₂ by pumping at 980 nm, but also feeds half of them to the laser lower level ⁵F₅, thus preventing the population inversion. In contrast to ETU2, the ESA process has a more remarkable impact, since the available output power almost halves, once switching off it, highlighting its dominated role in the laser operation. This can be explained by the fact that the ESA process efficiently recycles the ions in the long-lived metastable state ⁵I₇ (mainly originating from the direct radiative transition of the laser upper level, verified by us, owing to the large β₅₁) by bringing them back to the population circulation between the ⁵F₄,⁵S₂ and ⁵I₆ levels.

According to the above results, we can see that the saturation with P₂ is the main limitation for power scaling, although it can be delayed by increasing P₁. The cause behind it is that the excited ions after releasing ~3.2 μm photons cannot rapidly go back to the ⁵I₆ level, hence participating in the next excitation. Thus, the relatively long lifetime of the ⁵I₅ level (i.e., 135 μs) catches our attention. To validate this point, the simulation is performed by shortening its lifetime to 16.3 μs, same as that of the ⁵F₅ level as shown in Fig. 6, where the saturation has been alleviated to an extent, leading to an increased output, indicating the positive effect of the shorter lifetime of the ⁵I₅ level.

From a realistic perspective, cascading the ⁵I₅→⁵I₆ transition [25] to lase simultaneously is one of the considerable solutions, since it not only can accelerate depopulating of the ⁵I₅ level by bringing the ions back to the ⁵I₆ level, but also targets the ~3.9 μm emission, an attractive band for practical applications (e.g., infrared countermeasures and free-space communications [45,46]).

Accordingly, the ~3.9 μm transition is switched on in the model. In order to show the laser performance variation more clearly after activating the ~3.9 μm transition, ${R_{{s_{22}}}}$ is optimized only under the optimal ~3.2 μm cavity condition indicated by Fig. 3 (i.e., ${R_{{s_{12}}}}$ = 0.11 and L = 1.85 m). As a result, the relatively small ${R_{{s_{22}}}}$ of 0.1 yields the maximum ~3.9 μm power as shown in Fig. 7.

Fig. 8 presents the corresponding dual-wavelength power and heat load evolutions with P₂. It is seen that the available maximum ~3.2 μm laser power has been raised by ~1 W as a result of the delayed P₂ corresponding to saturation. Whilst the ~3.9 μm laser is efficiently operated with a threshold of 8.9 W and slope efficiency of 20.2% (relative to P₂). The sudden change of the ~3.2 μm power around the ~3.9 μm threshold indicates the remarkably varied population distribution once the ~3.9 μm transition is activated. When P₁ = 5 W and P₂ =25 W, the output power values of 4.68 W at ~3.2 μm and 2.76 W at ~3.9 μm are theoretically achievable, with optical-to-optical efficiency of 15.6% and 9.2%, respectively (relative to P₁ + P₂). In contrast to the prior 532 nm pumped ~3.2 μm Ho³⁺-doped fiber laser based on the same transition [21], our results exhibit the significant performance improvement and the capability of this transition in efficient laser generation at ~3.2 μm. Although the current advanced Dy³⁺-doped ZrF₄/InF₃ fiber laser, excited by the diode-pumped Er³⁺-doped ZrF₄ fiber laser at ~2.8 μm, behaves better at ~3.2 μm alone [19,23], our scheme is versatile since it can emit at ~3.2 μm and ~3.9 μm simultaneously, hence providing the new opportunities for a series of potential applications (e.g., difference frequency generation, dual-channel free-space communications, and composites processing). In addition, the predicted ~3.9 μm laser performance is also quite leading. In contrast to the state-of-the-art theoretically and experimentally reported Ho³⁺-doped fiber systems [29,40], the power has been enhanced by a factor of 2.2 (i.e., 197 mW) and 14 (i.e., 1.26 W), respectively. More importantly, our estimated heat load of 5.17 W/m is only ~40% of them, attributable to the used low dopant concentration, highlighting the great potential of our proposed cascaded scheme in high-power ~3.9 μm laser generation. Note that the heat load in Ref. [25] is estimated to be 12.6 W/m using the same model in our case and parameters in Ref. [29], which could reproduce the ~3.9 μm laser.

In this work, we theoretically unlock the potential of the ⁵F₄,⁵S₂→⁵F₅ transition of Ho³⁺, to which rare attention was paid in the past, in efficient lasing at ~3.2 μm via the 1150 nm and 980 nm dual-wavelength pumping, for the first time. Based on the cascaded excitation way (i.e., ⁵I₈→⁵I₆→⁵F₄,⁵S₂), the efficient laser at ~3.2 μm, giving output power of 3.6 W with optical-to-optical efficiency of 14.4%, is theoretically obtainable from a lightly-Ho³⁺-doped InF₃ fiber. In addition, we find that the ESA process (i.e., ⁵I₇→⁵F₅), resonating with the 980 nm pumping, plays a critical role in the efficient ~3.2 μm lasing, by bringing the excited ions, escaping from the circulation between the ⁵F₄,⁵S₂ and ⁵I₆ levels, back to it. To accelerate the ions circulation thus alleviating the power saturation behavior with the 980 nm pumping power, the cascaded ⁵I₅→⁵I₆ transition at ~3.9 μm has been activated simultaneously. Such an operation results in increased output power of 4.68 W with higher optical-to-optical efficiency of 15.6% under the same 1150 nm pumping power. More importantly, the cascaded ~3.9 μm laser can also efficiently operate, yielding output power of 2.76 W with optical-to-optical efficiency of 9.2% under a moderate heat load. In terms of output power and heat load, its performance is even better than all the prior experimentally and theoretically reported ~3.9 μm Ho³⁺-doped InF₃ fiber systems. Our proposed cascaded pumping scheme not only provides an alternative way for ~3.2 μm laser, but also exhibits a new tack for the dual-wavelength laser design at ~3.2 μm and ~3.9 μm, thus offering the new opportunities for many potential applications, e.g., polymer processing, infrared countermeasures, and free-space communications.

The authors declare no conflicts of interest.