1. Introduction

Thulium (Tm³⁺) is an excellent candidate for infrared laser applications thanks to its broad emission spectrum at around 1.8 micron, which makes it very appealing for several applications from precise cut and ablation of biological tissues to sensing applications [1–6]. Furthermore, Tm³⁺ has a significant property, which is cross-relaxation process (³H₄, ³H₆ → ³F₄, ³F₄) where two ions are promoted in the upper level of laser by every single pumping photon. This process significantly improves pumping quantum efficiency and lasing at 1.8 μm. Yet most of laser simulations do not consider, or do not directly measure, the reverse cross-relaxation process (³F₄,³F₄,→³H₆,³H₄) that reduces the efficacy of cross-relaxation process.

Various glasses including silica, fluoride, germanate and tellurite have been used as laser host material [7–11]. Amongst of all oxide glasses, tellurite glasses have the lowest phonon energies (~750 cm⁻¹) and provide high rare earth ions solubility. Independently from the type of glass, laser development and design optimization rely on the accurate modeling and comparison of different doping levels [12,13]. While lifetimes and cross-sections are well known, the ion-ion related parameters, namely cross-relaxation and the reverse transfer process [12–17] are more difficult to obtain. In particular, these usually are not available as parameters validated over a large interval of doping level and therefore suitable for doping level optimization.

In previous papers, we obtained the cross-relaxation parameter for a class of tellurite glasses with a composition 75TeO₂:20ZnO:5Na₂O (mol%) with doping level ranging from 0.36mol% to 10mol% (corresponding to doping levels ranging from 0.82 to 22*10²⁰ ion/cm⁻³) [14,17]. In this paper, we numerically investigate the reverse transfer process in Tm-doped tellurite glasses and compare it with experiments. We demonstrate that we are able to fit a set of samples with doping level variations by a factor of 30 (0.36 mol% to 10 mol%) using the same parameters for all samples. We also demonstrate information can be obtained on the reverse cross-relaxation process parameter when we fit the slow decaying fluorescence tails, which is emitted when pump level is almost depopulated. This is an alternative approach to the one based on Kushida as in Ref [18]. This paper calculates the ratio between reverse and cross relaxation process as of 0.03. We also show measurements may vary from 0.01 to 0.09, if pump intensity is not perfectly known, due to low inversion regime. We, indeed, demonstrate in this paper that a more accurate measurement of the reverse process parameter can be obtained only in a regime of significant ground state depletion. In this paper we also evaluate possible impact of the reverse cross relaxation parameter on the amount of pump power required to achieve a given population inversion.

2. Theoretical modeling

We used the energy levels scheme and considered transitions shown in Fig. 1 [13,19].

 

Fig. 1 Energy level scheme of Thulium

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Figure 1 shows the lowest four energy manifolds of Tm³⁺ ion. In the figure, the laser transition, the pump transition, and direct and reverse cross-relaxation processes are indicated, together with spontaneous decay paths. The corresponding set of rate equations is as follows:

where N₁, N₂, N₃ and N₄ are the population of the energy levels ³H₆ (ground level), ³F₄ (upper laser level), ³H₅ and ³H₄ (pump level), respectively; W₁₄, W₄₁ are the pump rates, ${\tau }_i$ the lifetime of the i-level, and ${\beta }_{ij}$ are branch ratios from the $i$- to $j$-level [17]. The coefficients $P_{ij}$ describe the energy transfer processes: P₄₁ (³H₄, ³H_6$\to$³F₄, ³F₄) is the cross relaxation constant, which is proportional to doping level [14,17], and P₂₂ (³F₄,³F₄, $\to$³H₆, ³H₄) is the reverse cross-relaxation process constant, the investigation of which is the main aim of this study.

3. Simulation and fittings

We use the model to fit the fluoresce decay curves of a set on Tellurite glass samples with doping levels ranging from 0.36 mol% to 10 mol%. The experimental data were reported in details in Ref [17]. Used numerical values are listed in Table 1. The pump cross section at 790 nm was taken from Ref [17] while the emission cross-section was calculated using Ref [20]. To simulate the experimental pumping condition with non-instantaneous pump power decay, the pump intensity was used on and set to 1.3*10³ W/cm² for all samples, assuring a low excitation regime; and the pump power decrease from the peak value was simulated by using an exponential decay curve with a 2 μs time constant. Figure 2 shows for all samples, both the ³H₄ and ³F₄ experimental fluorescence decay curves and corresponding fitting using the data of Table 1.

Table 1. List of parameters used in the modeling

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Fig. 2 Normalized Fluorescence decay of ³H₄ (a) and ³F₄ (b) levels: Experimental (dots) and theoretical fitting (line). The number that identifies the samples (T#) corresponds to the doping level in mol%.

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The calculated lifetimes are reported in Fig. 3, with fitting using the formula proposed by Auzel et al. [21]. The obtained quenching concentration for ³F₄ and ³H₄ levels were 2 mol% and ~1.1 mol%, respectively. The data suggested that we are in the classical homogeneous distribution where there was no chemical clustering and the physical clustering was reflected by the decreasing of the observed lifetime, increasing the Tm content. The lifetime of isolated ions ${\tau }_0$ for ³H₄ was 0.57 ms, whereas that for ³F₄ is equal to 3 ms. Note that data slightly differ from Ref [17]. due to a different approach in considering the cross-relaxation contribution to ³H₄ lifetime fitting.

 

Fig. 3 Calculated lifetime values for a) ³H₄ level, and b) ³F₄ level versus Tm concentration, solid line is fitting using Ref [21].

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4. Determination of the reverse cross-relaxation process parameter

While many parameters were either easily recalculated as lifetimes or taken by previous publications, as cross-relaxation constant or branching ratios, we had to investigate the reverse cross-relaxation process in order to complete the set of parameters and be able to use the set of Eq. (1-4) to simulate and compare Tm-doped laser over a wide range of concentrations. To measure the P₂₂ parameter, we looked to the tail of ³H₄ fluorescence [11]. This is an alternative method to Kushida’s model [18], but is more directly related to rate-equation parameters. Figure 4 shows the same data of Fig. 2 but down to a very low signal level. To be able to properly taking into account the low signal tail, we removed the photodiode background noise from experimental data. We notice that when the ³H₄ population is almost depleted, and almost all ions are back to ground state, a second exponential constant kick in, this constant is not arbitrary but exactly half the ³F₄ lifetime due to the fact that ³H₄ level is populated by the slow reverse cross-relaxation process and depopulated by fast radiative decay. Therefore instantaneous population is determined by the P₂₂N₂² term in Eq. (1), and the quadratic dependence on N₂ gives a second exponential constant equal to half τ₂. The fact that the second exponential is about half τ₂ is a demonstration that we are not looking at measurement background noise. Once the pump power intensity is fixed, the time at which the slope change depends strongly on P₂₂. Note all the fitting curves in Fig. 4, used the same P₂₂ = 0.03*P₄₁ parameter. Figure 4a shows the T1.08 sample; in this case experimental data are mainly noise, hiding the second exponential. This is because the effect of reverse cross-relaxation is evident only at low inversion, i.e. very low signal level for low doped samples. Investigation of low doped samples would therefore require higher sensitivity detectors.

 

Fig. 4 Theory (blue line) and Experiment (red dots): Fluorescence decay for ³H₄ for samples T1.08 (a). T4 (b), T6 (c) and T7 (d).

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To understand the margin of error in our fitting, we ran further simulations. Simulations show that while the effect of each rate-equation parameters affects the ³H₄ fluorescence curve, this does not happen, at low pump level, if we keep constant the product of pump intensity times P₂₂. In Fig. 5(a), we show how the ³H₄ fluorescence varies as a function of parameter a (defined as the ratio a = P₂₂ / P₄₁) and intensity. We see that by dividing a (the ratio P₂₂ / P₄₁) by 3 and multiplying the power by 3 (so product is a constant value) we have about the same curves. Figure 5(a) shows that if during a measure at a low pump rate, i.e. low inversion as usually occur for a spectroscopic measurement, the pumping rate is over or under estimated, we still have similar decay curves, so fitting is less sensitive to errors. However, Fig. 5(b) shows that in case of higher pump rate leading to a higher inversion, an error of a factor of 2 in pump intensity is easily spotted (compare curve i = original*100 and i = original*200). Figure 5 suggests that a separate measurement should be done at a higher pump power to reach at least 20% inversion, to determine parameter a without risking that inaccurate pump rate will generate errors. This indicates that the value we found and we propose in this paper may be subject to an error due to pump power level not having properly taken into account. On the base of those simulations, we will plan a new investigation of Tm-doped glasses using low pump power level for standard spectroscopy and a higher pump level to determine the P₂₂ constant.

 

Fig. 5 Simulation of fluorescence decay from level ³H₄ for sample T4. Original refers to a' curve using values used in Fig. 2 and Fig. 4: a' is the ratio = 0.03 and I' is pump intensity = 1.3*10³ W/cm².

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5. Impact on device operation

To evaluate in first instance the impact on the device performance of the reverse cross-relaxation parameter, we calculated the required pump intensity to reach a given value of inversion. This is only a first indication since in laser or amplifier operation other pehomena may interplay with reverse cross-relaxation, like photodarkening, since pathways involve population of Tm levels affected by reverse cross-relaxation [22–26]. Figure 6 shows, for sample T4, the results for 20% and 40% inversion. All values have been normalized to the value at a = 0.03 (so that both curves are equal to 1 for a = 0.03); the inversion is defined as the difference between N₂ and N₁ (N₂-N₁) normalized to the doping concentration.

 

Fig. 6 Normalized threshold intensity versus parameter a. The values are normalized with respect to the values for a = 0.03. Calculation were done for sample T4

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We noticed that the dependence is not linear and becomes increasingly quadratic for a higher concentration. Indeed, Eq. (1) shows that P₂₂ is determined by a, and has a quadratic dependence on N₂ while the dependence on the pump power is linear, so the equation balance may justify such a quadratic dependence. A further comment is that if we underestimate the value of a by a factor of 3 (referring to discussion in Fig. 5) we significantly underestimate the required intensity. Finally, we noticed that a low value of a may be of a higher benefit in the case of a higher inversion, like in the case of amplifiers. This may suggest that a choice of the most appropriate glass may depend on the specific application and should take into account the whole set of parameters, including reverse cross-relaxation. From Fig. 6 we can also see that impact of the reverse cross-relaxation parameter is larger when high inversion are required, such as in amplifier devices, this may suggest that the choice of the most effective glass may depend on the specific application. For example, in Ref [12], the silica glass shows a longer (30%) ³H₄ lifetime at around 5.75 mol% doping level, but a = 0.08. This may indicate that the choice of glass may depend on the specific application. Other glasses show even lower a, Ref [27]. calculated, using Kushida’s model [18], a parameter below 0.01 for fluoroindogallate glass. This indicates that a comparable measurement of reverse cross-relaxation parameter across different types of glasses may be useful for laser scientist to choose the appropriate glass host for each specific application.

6. Conclusion

We have been able to define a set of spectroscopic parameters for Tm-doped tellurite glasses able to predict fluorescence decay over a wide doping level interval. To achieve this, we completed the available parameter with the inverse cross-relaxation process constant. In this paper, we proposed a new method to calculate the reverse cross-relaxation process and we assessed possible margin of error. This method can be applied to any type of glass, and we demonstrate that while a low inversion regime is optimum for lifetime, branching ratios, cross sections and cross-relaxation parameter calculations a second round of measurement, using higher pump level, should be performed to precisely characterize the reverse cross-relaxation process. Within the aim of this paper, to use existing experimental data, we found that the reverse process parameter is 3% of the cross-relaxation parameter (P₂₂ = 0.03*P₄₁) with a maximum error interval of 0.01-0.09. This error interval can be reduced to a factor of 2 with about 100 times higher pump intensity. Overall we were able to use a single set of parameters and fit samples with doping level varying by a factor of about 30 (from 0.36 mol% to 10 mol%). This set will allow laser engineering to appropriately simulate active device and to find the optimum doping level. We also show that the appropriate calculation of reverse cross-relaxation parameter may have a significant effect on the simulation of laser and amplifier devices.