1. Introduction

Laser sources emitting at 2.3 µm has seen a gain of interest in recent years due to the applications of such radiation for spectroscopy in the short-wave infrared spectral range. These sources are important, in particular, in the context of LIDAR systems, as their laser wavelengths are ideally located in the K-band of the atmosphere [1]. They are also useful for the study of combustion processes [2]. Various 2.3 µm sources have already been developed, notably based on optical parametric oscillators / amplifiers [3], semiconductor lasers [4] or solid-state lasers based on transition-metal or rare-earth ions directly emitting at this wavelength. Indeed, Cr²⁺:ZnSe and Cr²⁺:ZnS lasers cover the spectral range around 2.3 µm [5], as well as thulium (Tm³⁺) based lasers operating on the ³H₄ → ³H₅ electronic transition [6].

Tm-lasers have shown the advantage of being directly diode-pumped, whereas Cr²⁺-based laser sources typically use Tm-lasers at 1.9 µm as pump sources. Direct laser emission at 2.3 µm from Tm³⁺-doped materials has been a topic of multiple studies in the past few years. Tm-lasers emitting at 2.3 µm (on the ³H₄→ ³H₅ transition) suffer from a strong competition with the high-gain ³F₄ → ³H₆ transition, leading to emission around 1.9 µm. Various strategies have been developed to avoid colasing on this transition. The most common one consists of making unfavorable the undesired 1.9 µm laser using dichroic mirrors, i.e., those coated for high losses for the spectral range of the ³F₄ → ³H₆ transition typically between 1.7 and 2.1 µm [7–9]. However, this strategy is not always the most successful one. In some cases, these losses are not sufficient to avoid lasing at the ³F₄ → ³H₆ transition. This is notably the case of fiber lasers where Fresnel reflections on both the input and output facets of the fiber are sufficient for lasing around 1.9 µm [10–12]. The same problem can be observed for waveguide lasers whose gain is high [13]. Similarly, when using crystals with a strong electron-phonon interaction where phonon-assisted laser emission can be observed such as Tm:Y₃Al₅O₁₂ (shortly Tm:YAG) or Tm:KLu(WO₄)₂ (Tm:KLuW), the emission wavelength of lasers operating on the ³F₄ → ³H₆ laser transition can be strongly shifted towards 2.1 to 2.2 µm [14,15]. Since the wavelength of this emission is very close to that of the purely electronic ³H₄→ ³H₅ transition, it is technologically difficult to produce correct dichroic mirrors that introduce high losses in the range of 1.7 - 2.2 µm when low losses are required at 2.3 µm.

Due to these difficulties, an alternative strategy consists in taking the opposite of this technique by benefiting from the co-laser at 2.3 µm and 1.9 µm since they operate in cascade. This type of cascade laser has already been used in Tm³⁺,Ho³⁺-codoped crystals to improve the efficiency of the 2.1 µm transition of holmium ions [16]. Publications using Tm³⁺-doped LiYF₄ [17] and CaGdAlO₄ (CALGO) [18] also point out the potential advantage of this strategy. They show in different configurations how the addition of the 1.9 µm transition allows a significant increase in the laser emission at 2.3 µm. The improvement in laser efficiency is mainly explained by recycling of thulium ions stored in the metastable intermediate ³F₄ level, which are forced down to the ground-state level. These two cascade lasers thus avoid the bottleneck effect in the metastable median electronic level being typical for 2.3 µm thulium lasers. On the other hand, the population reduction of the ³F₄ level by adding a laser transition at 1.9 µm also strongly reduces the energy transfer upconversion (ETU) from this level. The latter process contributes to improvement of the laser efficiency at 2.3 µm since it helps to populate the upper laser level [19]. Consequently, ETU between two thulium ions in the ³F₄ state allows to send one ion to the ³H₄ level and one - to the ³H₆ ground-state, and it modifies the 2.3 µm laser population rate as well as the absorption saturation process when Tm³⁺ ions are pumped at 790 nm. Since both the absorption and the ETU are strongly concentration-dependent [19], we expect that the effect of a laser cascade will not be always beneficial and will be doping level dependent.

In the present paper, we investigate this doping question for the first time to our best knowledge. First, the effect of cascade laser for different Tm³⁺ doping levels is studied experimentally for the well-known and largely-used Tm:LiYF₄ crystal. Several questions are addressed. Over which range of doping concentration is the cascade laser beneficial for 2.3 µm laser emission and for which is it detrimental and must be avoided? Is there an accessible optimal doping level? And what improvement in terms of laser efficiency can be expected? To answer these questions, we parallelly developed a simple analytical predictive model and confronted it to the experimental data obtained with Tm: LiYF₄. To conclude our study, we also extend the use of our model to evaluate the potential cascade laser benefice for other Tm³⁺-doped crystals.

2. Experimental setup

To study cascade laser emission from a Tm-laser, we construct a partially overlapped dual laser cavity allowing for simultaneous laser operation at 1.9 µm and 2.3 µm. The Tm³⁺-doped laser crystal is pumped by a continuous-wave Ti:Sapphire laser emitting up to 2.5 W around 780 nm. A focusing lens with an optimized focal length of 100 mm is used to focus the pump radiation into the crystal. The pump beam waist diameter is 43 µm in the crystal. To separate the emission on the ³H₄ → ³H₅ transition and that on the ³F₄→ ³H₆ one, we build two nested V-cavities of identical size so that the two waists of the cavities perfectly overlap in the crystal. They share the pump mirror, the laser crystal and a plano-concave mirror with a radius of curvature (RoC) of −200 mm. The two cavities are separated using a dichroic mirror coated for high reflection (HR, R > 99.95% at 2.15 −2.4 µm) and high transmission (HT, T > 99.5% at 1.7-2.1 µm). An additional mirror with the same features is used as a filter to avoid any 2.3 µm laser transition in the “1.9 µm – laser” arm. The two cavities are closed by two independent output couplers (OCs). The pump mirror is coated for HR at 2.2-2.4 µm and has a transmission of 5% at 1.9 µm. The distance between the pump mirror and the concave plane mirror is experimentally optimized around 105 mm. The distance between the latter mirror and the two output couplers is 350 mm for both cavities. For the cavity arm supporting the ³F₄ → ³H₆ laser transition, the OC provides a transmission of 2% at 1.70 - 1.95 µm, and for the second cavity arm supporting the ³F₄ → ³H₆ transition, the output coupling is 3% at 2.1 - 2.4 µm.

3. Tm:LiYF₄ cascade laser

3.1 Laser crystals

We select the Tm:LiYF₄ crystal for experimental demonstration of the cascade Tm-laser as this crystal is currently one of the most commonly used materials for lasers at 2.3 µm due to its good spectroscopic and thermal properties, as well as commercial availability.

Let us briefly describe the spectroscopic properties of Tm³⁺ ions in this material. For the ³H₆ → ³H₄ pump transition, the maximum absorption cross-section σ_GSA is 0.64 × 10⁻²⁰ cm² at 781 nm for π-polarization [20]. For the considered laser transitions, the stimulated-emission cross-sections σ_SE are 0.42 × 10⁻²⁰ cm² at 1877 nm (the ³F₄ → ³H₆ transition) [21] and 0.52 × 10⁻²⁰ cm² at 2303 nm (the ³H₄ → ³H₅ transition) for π-polarized light [22]. The ion density corresponding to a Tm³⁺ doping level of 1 at.% is 1.38 × 10²⁰ cm⁻³. The intrinsic lifetimes of the ³F₄ and ³H₄ levels (for low Tm³⁺ doping levels) are 11 ms and 2.3 ms, respectively.

For our study, three Tm:LiYF₄ crystals with different doping levels are used and their parameters are listed in Table 1. The crystal thicknesses are chosen to provide pump absorption at 781 nm around 80%. All crystals are antireflection coated for both the pump and laser wavelengths to avoid Fresnel losses.

Table 1. Parameters Used in the Simulation of the Cascade Laser Performance

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3.2 Experimental result

The Tm:LiYF₄ crystal is positioned in the double cavity described in Fig. 1(b). To observe the influence of the 1.9 µm laser emission on the 2.3 µm one, the input-output dependences (i.e., the output power at 2.3 µm vs. the pump power at 780 nm) are measured for all the studied laser crystals. The 1.9 µm laser emission is allowed or not by using a pinhole depending on whether the cascade laser is desired or not without changing any alignments.

 

Fig. 1. (a) Energy-level scheme for Tm³⁺ ions in LiYF₄ showing processes relevant for the cascade laser study: GSA – ground-state absorption, CR – cross-relaxation, ETU – energy-transfer up-converison, NR – multiphonon non-radiative relaxation; (b) Scheme of the laser set-up used for the cascade laser study in Tm:LiYF₄.

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The experimental results presented in Fig. 2 show different cases in which the presence of the cascade laser involving the emission at 1.9 µm is favorable or unfavorable for the laser at 2.3 µm, as a function of different doping level. For the highest studied concentration of Tm³⁺ ions of 6 at.%, by allowing the laser to operate on the ³F₄ → ³H₆ transition, no laser emission at 2.3 µm is observed. Without the cascade laser scheme, the output power amounts to 90 mW at 2.3 µm but it drops to zero when the 1.9 µm laser operates. As a result, the cascade laser scheme is clearly in disfavor of the 2.3 µm emission. In contrast, for a doping level of 2.5 at.%, the addition of the laser cascade at 1.9 µm increases the output power at 2.3 µm from 90 mW to 180 mW, i.e., representing a two-fold increase. For an intermediate Tm³⁺ concentration of 3 at.%, by allowing the laser to operate at 1.9 µm, the increase of the output power at 2.3 µm is less significant, namely from 200 mW to 248 mW (24% variation).

 

Fig. 2. Cascade Tm:LiYF₄ lasers: (a-c) Output power at 2.3 µm and 1.9 µm and (d-f) pump absorption efficiency at 781 nm. The Tm³⁺ doping level is (a,d) 6 at.%, (b,e) 3 at.%, and (c,f) 2.5 at.%.

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To understand the involved phenomena, the pump absorption efficiency at 781 nm is measured in three different regimes, namely, first, when the laser operates on the cascade scheme, second, when only the laser emission at 2.3 µm is supported and third, without any laser operation. Compared to the case of purely 2.3 µm laser emission, by allowing the laser to operate on the ³F₄ → ³H₆ transition, the pump absorption increases by 45%, 22% and 19% for 2.5 at.%, 3 at.%, and 6 at.% Tm³⁺ doping, respectively. The 2.3 µm laser emission also leads to an increase of the pump absorption compared to the case without any laser action but in a less extent. The observed increase in the pump absorption under cascade laser operation is the principal reason to explain the increase in the output power at 2.3 µm simply due to the increased pumping efficiency. However, this explanation is not sufficient since similar increases in absorption are measured for 3 at.% and 6 at.% Tm³⁺ doping levels but the effects on the 2.3 µm laser emission are radically different. The differences in the impact of the 1.9 µm laser channel on the 2.3 µm one can nevertheless be explained by the fact that adding the emission at 1.9 µm empties the ³F₄ level population. Emptying this manifold also reduces the influence of the ETU which helps to fill the upper laser level for the ³H₄ → ³H₅ transition.

3.3 Modeling of laser performance

To further understand the effect of Tm³⁺ doping concentration on the laser performance and the other physical quantities involved, an analytical model is proposed. The aim is to analyze the variation of the laser power at 2.3 µm due to variations in the pump absorption and ETU rates. The notations and values used in the present work are summarized in Table 1.

In this model, the variation of the laser output at 2.3 µm is assumed to be due to two main reasons: first, the variation of the pump absorption efficiency and second, the variation of the ETU efficiency refilling the upper laser level (³H₄): ΔP_2.3µm = ΔP_2.3µm,abs + ΔP_2.3µm,ETU. The increase in the output power due to increased pump absorption is quantified as follows: ΔP_2.3µm,abs = P_p(η_cA_c – η_ncA_nc). Assuming that the cascade laser operation mainly induces mainly a variation of pump absorption with low perturbation of the laser slope efficiency, η_c and η_nc can be considered to be equal: ΔP_2.3µm,abs = P_pη_cΔA. This simplifying assumption (which is valuable from the predictive point of view) is of course not valid when the cascade laser scheme prevents oscillations at 2.3 µm laser so that η_c = 0 and the model is clearly less precise in such a case (which corresponds to the case of 6 at.% Tm:LiYF₄ laser for example). The change in absorption ΔA = A_c – A_nc can either be directly determined from the previously described absorption measurements or simply calculated from the Beer-Lambert law. For such a calculation, a simplification assumption is made that all the Tm³⁺ ions are either in the fundamental ³H₆ level, or they are accumulated in the metastable ³F₄ level. Considering the long lifetime of this level compared to the fluorescence lifetime of the ³H₄ manifold quenched by the cross-relaxation process [23–24], this assumption is in general valid. In our case, since the laser at 2.3 µm empties furthermore the ³H₄ level, it become acceptable. In addition, the ³H₅ level is rapidly thermally relaxed to the lower-lying ³F₄ level and its population is nearly negligible, so that the absorption can be calculated as ${A_x} = 1 - {e^{ - {\sigma _a}lN({1 - {n_{{.^3}{F_4}\; x}}} )}}$ with $x \in \{{c,nc} \}$ (with c for “cascade” and nc for “no cascade”).

For the power variation at 2.3 µm due to the ETU, a quantification can be defined as follows, based on the rate equations [23]:

 

Fig. 3. The energy-transfer upconversion parameter K_ETU vs. Tm³⁺ doping level in LiYF₄. Triangles – literature data [19,25–27], squares – the K_ETU values used in the present work.

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Note that the power variation in this equation intrinsically depends on the cavity architecture through the parameters η_c and η_opt$.$ To limit the number of free parameters in the model, we therefore focus our interest on the normalized power variation, i.e., by dividing the power variation by the output power in the presence of the cascade laser operation:

The present model is then compared with the experimental measurements made with the Tm:LiYF₄ crystals to qualify it and verify that it could be useful to fairly predict the variations in laser efficiencies of cascade Tm-lasers based on crystals with different doping levels. The $n_{{.^3}{F_4}\; x\; }^2$ values are estimated using the absorption curves. Since the different tested crystals have different pump absorption efficiencies and / or thicknesses, this would imply a model with different input parameters for each one. It is particularly the $n_{{.^3}{F_4}\; x\; }^2$ parameter that is dependent on the doping level because the energy-transfer upconversion tends to empty the ³F₄ level for high doping levels (see Table 1).

Since the goal of the present study is to provide an estimation of the tendency from the point of view of having a semi-empirical predictive model, we are plotting this curve using only one variable parameter, i.e., the doping concentration. Because of this, we decide to take the average values for the other parameter, acknowledging that these ones could varying in a limited extend, especially $n_{{.^3}{F_4}\; x\; }^2$. We then use the set of values of 83%, 50%, 8 mm for the parameters $n_{{.^3}{F_4}\; nc\; }^2,\; n_{{.^3}{F_4}\; c\; }^2 $ and l, respectively, to obtain a unique trend curve that could have been predicted. The K_ETU value is obtained using both a linear model (with a concentration-independent coefficient of 0.8 × 10⁻⁴⁰ cm⁶s⁻¹$)$ or the interpolation based on the literature data, as shown in Fig. 3. Considering these values and the different hypotheses, the model gives good evidence of cases when the cascade laser emission is beneficial but remains overestimating for high doping levels. Indeed, the K_ETU parameter increases significantly for constant $n_{{.^3}{F_4}\; x\; }^2$ whereas this population may decrease and compensate the increase of K_ETU. Anyway, the model allows to have a good estimate especially for the interesting points at low doping levels and a physically coherent tendency for higher doping. Furthermore, changing of the $n_{{.^3}{F_4}\; x\; }^2$ parameter has a very low influence on the crossroads of the normalized power variation $\frac{{\mathrm{\Delta }{P_{2.3\; \mu m}}}}{{{P_{2.3\; \mu m\; c}}}}$ with the zero axis, that is the keystone to know whether a cascade laser scheme is beneficial or not.

Despite a simple model based on several assumptions described above, we obtained a semi-empirical equation accurately reproducing the trend observed experimentally. To answer the question whether the model could be more precise considering the data measured for each particular crystal and not the average values, we plot in Fig. 4 the simulation results for the 2.5 at.% and 3 at.% Tm³⁺-doped LiYF₄ crystals. As expected, the agreement becomes more precise. Moreover, one can observe that the limit where the cascade laser is advantageous does not vary for the different curves. To be able to test this model with more experimental points, we also plot for the two above-mentioned crystals the same normalized power variations with the cascade laser for various pump powers as shown in Fig. 4(c,d). For these curves, the variables $n_{{.^3}{F_4}\; nc\; }^2,\; n_{{.^3}{F_4}\; c\; }^2 $ and K_ETU are the ones corresponding to each crystal. The values of K_ETU for both crystals have been added to Fig. 3.

 

Fig. 4. Normalized power variation for laser emission at 2.3 µm from cascade Tm:LiYF₄ lasers: (a,b) effect of the doping concentration, symbols – experimental data, curves - theoretical modeling: calculations using (a) averaged or (b) exact experimental data for each crystal; (c,d) effect of the pump power at 781 nm: symbols – experimental data, curves - theoretical modeling for (c) 3 at.% Tm:LiYF₄ and (d) 2.5 at.%. Tm:LiYF₄.

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It can be seen that for the 3 at.% Tm³⁺-doped crystal, the value $\mathrm{\Delta }{P_{2.3\; \mu m}}/{P_{2.3\; \mu m\; c\; }} $ reaches 19.5% while the relative absorption difference $\mathrm{\Delta }A/{A_{\; c\; }}$ approaches 25%. This result corresponds well to those expected by our model, as $\mathrm{\Delta }{P_{2.3\; \mu m}}/{P_{2.3\; \mu m\; c\; }}$ tends towards the relative absorption difference when the pump power becomes sufficiently high. For the 2.5 at.% Tm³⁺-doped crystal, the measured value of $\mathrm{\Delta }{P_{2.3\; \mu m}}/{P_{2.3\; \mu m\; c\; }}$ reaches 49.2% while the relative variation in absorption is 53%. One may observe that, for the pump powers closer to the threshold, the model becomes less accurate. It can be explained by the fact that the assumption ${\eta _c}\sim {\eta _{nc}} $ is no longer valid in this case.

4. Extension of the analytical model to diode-pumping

Using fiber-coupled laser diodes for pumping Tm³⁺-doped crystals instead of Ti:Sapphire lasers allows for design of power-scalable and compact systems. The main difference consists of the available power and beam quality (brightness) of pumps. As a consequence, it is interesting to adapt our model and to trace the impact of this change on the predicted cascade laser effect since it is mainly affects the term related to the ETU process through the available pump power and the pump volume such as:

 

Fig. 5. Normalized power variation for laser emission at 2.3 µm from cascade diode-pumped Tm-lasers: (a) For diode-pumped Tm:LiYF₄ lasers already described in the literature, theoretical modeling of $\mathrm{\Delta }{P_{2.3\; \mu m}}/{P_{2.3\; \mu m\; c\; }}$ vs. the doping level corresponds to parameters described in the text and the experimental point is coming from Huang et al. [17]; (b) For several Ti:Sapphir pumped Tm³⁺-doped crystals, experimental (symbols) and theoretical modeling (curves) data are also shown. (c) For diode-pumped Tm:YVO₄, theoretical modeling of $\mathrm{\Delta }{P_{2.3\; \mu m}}/{P_{2.3\; \mu m\; c\; }}$ vs. the doping level corresponds to parameters described in the text. The experimental point is coming from Yu et al. [30,31];

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In Fig. 5, simulations for this diode emitting up to 25 W of output power focused into a spot of 200 µm [17] is also presented. For comparison, we plot simulated curves for two previously reported diode-pumping schemes with Tm:YLF crystal : 90 W output, focused into a waist of 260 µm [28] and 450 W output, 500 µm spot (M² ∼ 150) [29] as shown in Fig. 5(a). The conclusion is that the optimization of cascade implies slightly lower doping levels for the diode pumping but remains relatively equivalent, and the model can be extended easily to the diode-pumped configurations.

5. Confrontation of the analytical model with other crystals

Other crystals than Tm:LiYF₄ also demonstrated cascade laser emission at 2.3 µm and 1.9 µm, such as Tm:CALGO [18] and appeared promising for emission at 2.3 µm for different reasons (e.g., broadband emission properties). It is interesting to see if our model can be extended to other Tm³⁺-doped materials. From the previous results for Tm:CALGO, the relative variation of the power at 2.3 µm vs. the doping level can be plotted to compare the experimental and simulated results. An output power at 2.3 µm of 332 mW (with cascade laser emission) and only 264 mW (without cascade laser) was reported for an incident pump power of 2.5 W. Concerning the modelling for Tm:CALGO, the population of the ³F₄ level is fixed at 55% with cascade laser against 90% without it, according to the measured pump absorption efficiency (75% with cascade laser and 63% without it) given in [18].

Yu et al., demonstrated diode pumped cascade lasing in Tm:YVO₄ [30,31] and Tm:GdVO₄ with an output power of 2.88 W at 2.3 µm [32]. Refence [30,31] presented by the authors show a 1.5 at. % Tm:YVO₄ crystal in the same configuration changing only the coating of the output coupler to have or no cascade laser at 1.9 µm. With an incident pump power of 25 W and an output coupler of 1% at 2.3 µm, the output power is 1.1 W with cascade laser [30] and 1.3 W without cascade laser [31]. The calculation showed a lower efficient cascade laser in these crystals : indeed the absorption is still high under no lasing condition, so that the gain that consist of increasing the absorption is low. For the model, the population of the ³F₄ level is estimated to be 45% with the cascade laser and 70% without it.

To provide another comparison, we also look at the Tm:KLuW crystal which is particular since the long-wave phonon sideband of the ³F₄→ ³H₆ electronic transition extends up to at least 2.2 µm. To our best knowledge, the influence of the cascade laser scheme on the 2.3 µm laser emission of this crystal has not been reported so far. The crystal used in the present study is 3.0 at.% Tm³⁺-doped and 5 mm long. For the laser cavity, a CaF₂ prism is added into the arm supporting laser oscillations at 2.3 µm to avoid any vibronic cascade laser emission in the spectral range of 2.1-2.2 µm. The pump polarization is chosen to be E || N_p in the crystal corresponding to an absorption cross-section of σ_GSA = 9.96 × 10⁻²⁰ cm² at 793 nm. The power obtained at 2.29 µm is 10 mW (in the presence of cascade laser emission), compare with 30 mW without cascade laser at 1.9 µm, making the cascade laser scheme unfavorable in this case. It can be explained by a small difference of pump absorption between the two configurations (95% without the cascade laser effect, 99% with it), making the improvement in the absorption efficiency marginal. For the model, the population of the ³F₄ level is estimated to be 40% with the cascade laser and 75% without it. The difference between measurement and calculation can be explained by the fact that the laser operates close to the laser threshold. The assumption ${\eta _c}\sim {\eta _{nc}} $ is no longer valid in this case.

We decided to compare these experimental results with three different crystals with our analytical model. However, as the coefficient K_ETU (as well as its concentration dependence) is not well referenced for other crystals as it is done for Tm:LiYF₄, we have to find a solution to evaluate these parameters for our model. The cross-relaxation coefficients are usually better referenced in literature. Considering that ETU and cross-relaxation are related effects (they are antagonistic effects), we assume that the ratio between their coefficients can be considered equivalent for Tm:LiYF₄ and other crystals used in our model. It is then possible to compare the experiment and the analytical model, as shown in Fig. 5(b). It is interesting to note that even with some lacune for certain spectroscopic data for the considered crystals, the numerical simulations are in relatively good agreement with the experimental results, indicating a low dispersion of the missing parameters between crystals and validating the extension of the model to other Tm³⁺-doped materials. It can be explained by the fact that the optimal laser crystals are often designed to have similar pump absorption, lengths within a typical range of doping levels. Being confident in our model, we also plot in Fig. 5(b) the prediction for Tm:YAG and Tm:YAP which is also a crystal of interest.

6. Conclusion

In conclusion, we studied positive effect of the cascade laser involving the ³H₄ → ³H₅ and ³F₄ → ³H₆ consecutive laser transitions in Tm:LiYF₄ and answered the question how the addition of laser emission at 1.9 µm can improve the output at 2.3 µm. This improvement strongly depends on the thulium doping concentration, and it can mainly be explained by a significant increase in the pump absorption, counterbalanced by the negative effect of ETU. An analytical model to evaluate this phenomenon is developed. Corroborated with experimental studies on three LiYF₄ crystals doped with 2.5 - 6 at. %, this model shows that, in some configurations, the addition of a laser cascade can double the output at 2.3 µm. The cascade laser technique is shown to be an alternative way of pump recycling to improve the efficiency of thulium-based 2.3 µm lasers. It has the advantage of being implemented simply by ensuring that the used laser mirrors provide high reflectivity additionally around 1.9 µm, given that they must already be reflective at 2.3 µm. These mirrors are often easier and less constrained to produce than dichroic mirrors that are currently used. Moreover, in the case of Tm:YAG and Tm:KLuW crystals where emission on the ³F₄ → ³H₆ transition is possible between 2.0 and 2.2 µm using the phonon assisted mechanism, using the cascade laser is even more critical. After validation, the simple presented analytical model was extended, using several fair hypotheses, to be a predictive model for different practical configurations such as diode pumping and other Tm³⁺-doped materials. The numerical results have been confronted with various host matrices based on the results available in the literature, and they are in line with the experiments. Thus, the model allows one to estimate the range of Tm³⁺ doping levels for which cascade laser is advantageous for boosting the laser efficiency at 2.3 µm.