Open Access
Add to BibSonomy
Abstract
Energy transfer coefficients in thulium-doped silica fibers are key parameters for a reliable numerical modeling and optimization of high-power fiber lasers pumped at 790 nm and emitting around eye-safer wavelength of 2 µm. We report on determination of the energy transfer coefficients by comparing experimental values and rate-equation-based modeling of thulium fluorescence decays. The study was performed using two pump, 792 nm and 1620 nm, and two signal, 800 nm and 2 µm, wavelengths. The following values were obtained k3011 = (1.9 ± 0.2) × 10−22 m3s-1, k1130 = (1.4 ± 0.2) × 10−23 m3s-1 and k1120 = (3.5 ± 0.1) × 10−23 m3s-1. To the best of our knowledge, this study involves so far the most comprehensive comparison of various fluorescence decays in terms of excitation and detection wavelengths as well as excitation powers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thulium-doped fiber laser (TDFL) devices emitting in the 2 µm spectral range are a subject of increasing attention in current research thanks to a broad range of applications spread out from medicine [1,2], sensors [3], throughout defense [4] to material processing [5]. Optical communications [6] and nonlinear conversion of wavelength into mid-infrared spectral range [7] should not be omitted as well. TDFL sources operated in both, continuous-wave and short-pulse, regimes [8] as well as methods for coherent combination [9] are investigated. Despite intensive research, a scaling up to a kW output-power level still presents a considerable challenge as reviewed in [4,10,11].
TDFL based on silica glass are in the forefront of interest primarily for their expedient high-power operation. Owing to the energy level structure, thulium offers a particularly suitable way to generate two laser photons around 2 µm per one pump photon at 790 nm. This so-called two-for-one cross relaxation process thus offers a theoretical quantum efficiency up to 200% leading to a slope efficiency of 80% [12]. However, involvement of various possible interactions between nearby thulium ions leads to a complex theory of energy transfer (ET) processes that eventually lower the laser efficiency actually attained.
Despite general longtime interest in thulium ET processes in various host materials [13–16], silica-based fiber lasers profiting from them, and reaching quantum efficiency up to 180%, have been demonstrated only recently [10,17,18]. To achieve such high efficiency requires optimized fiber composition; in particular relatively high thulium concentration, 1.5–4 wt.%, is necessary [10,19]. Though being very promising and extensively studied, the cross relaxation process still has potential for improvement in practical exploitation. To push the limits, reliable theoretical models are essential [12]. Their main credibility limitation lies in a lack of spectroscopic input parameters, where the ET coefficients remain the main uncertainty [20].
The values of ET coefficients in silica-based glass were reviewed by Jackson [21] and by Smith [22]. However, both studies were not focused primarily on the coefficients. While Jackson used data for Tm,Ho co-doped YAG, the Smith’s k3011 estimations were based values reported in [23] and k1130 was guessed from thermodynamic equilibrium relations. Both references were in a fair agreement. The ET coefficients can be determined using a rate equation modeling. This approach involves solving a set of rate equations relating the populations of energy levels and comparing the modeling results with experimental spectroscopic data [24,25]. The desired coefficients are derived from fluorescence measurements by fitting the fluorescence decay curves with model equations. Such approach was demonstrated by Shi in the case of Er-doped YAG [26], and by Taher [27] and Albalawi [28] for Tm-doped tellurite glasses. This method was also used by our group for Tm-doped silica fibers [29], but too strong assumptions were used to simplify calculations, and therefore the published results were only approximative.
In this article, we used the numerical modeling of fluorescence decay curves for determination of ET coefficients in thulium-doped silica fibers. We applied a comprehensive approach involving two pump wavelengths, 792 nm and 1620 nm, and two signal wavelengths, 800 nm and 2 µm. Moreover, the modeling was applied for various pump powers up to 70 mW.
2. Theoretical model
The diagram of Tm3+ ion energy levels with all transitions involved in the model is shown in Fig. 1. Stimulated absorption and emission rates Wij account for amplified spontaneous emission. Spontaneous decay processes are described by Aij and ${A}_{i}^{{nr}}$ for the radiative and nonradiative decay rates respectively. The searched energy transfer coefficients are marked k3011, k1130 and k1120. The first coefficient describes the cross relaxation process 3H4,3H6 → 3F4,3F4; the second one is associated with an opposite process, energy transfer upconversion 3F4,3F4 → 3H4,3H6. The coefficient k1120 describes a process during which one electron is transferred from 3F4 to 3H6, and the other one is excited from 3F4 to 3H5. Because of a very fast nonradiative decay from 3H5 to 3F4, the net result of the k1120 process is one electron transferred from 3F4 to 3H6. In fact, k1120 is associated in broader sense with interactions of two ions at 3F4 that results in one ion at 3H6 and the other one at 3F4, i.e., it ends without populating any higher-lying level.
The population density of thulium ions N2 can be neglected because the nonradiative decay rate from 3H5 level is very high (> 105 s-1). Due to this fact, this level is also not suitable for fluorescence decay measurements. The emission from this level in silica-based glass is practically undetectable [24].
According to the energy level diagram in Fig. 1, we can write time-dependent differential rate equations for the population density in the relevant levels:
Parameters τi and ${\tau }_{i}^{r}$ describe fluorescence lifetimes and radiative lifetimes respectively, parameters βij represent branching ratios between levels i and j. Complete definition of variables and other details about the numerical model are to be found in [24,30,31]. Values of parameters used for the simulations are summarized in Table 1.
Table 1. Parameters used in numerical simulations
It should be noted that not all of the needed parameters can be obtained with one fiber. The fiber under study was of high dopant concentrations, while the 3H4 fluorescence lifetime (τ3) was measured by using a fiber with a very low (as low as possible) thulium concentration to avoid the 3H4 fluorescence lifetime shortening by the cross relaxation. The fiber under study is marked SG1598, and the lowly doped fiber is marked SG1604. The Al2O3 concentration was comparable in both fibers, see Table 2.
The calculations were done in two separate parts – pumping and decay. In the pumping stage, steady-state populations of energy levels were calculated applying pump rates W01 or W03 depending on the pump wavelength λ. In this regime, the pump rates (W01 or W03) dominate over all other Wij rates. The pump rates were calculated according to (6).
The definitions of variables are listed in Table 1, h is the Planck constant, c is the vacuum speed of light, P represents pump power and S is the fiber core area. The overlap factor Γλ represents the portion of light with wavelength λ that overlaps with the doped core area; it was calculated according to Eq. (6) in Ref. [30].
In the decay part, the pump is switched off, and all Wij terms can be omitted. Fluorescence decay curves were modeled according to equations (1-5) applying Wij = 0. Steady-state populations calculated in the first step were applied as initial level populations for the decay part. The desired coefficients k3011, k1130 and k1120 were fitted and the calculated curves were compared with the experimental data. Least squares minimization was used as a criterion.
3. Experimental
3.1 Preparation and characterization of fibers
Optical preforms were prepared by a modified chemical vapor deposition method in combination with a ceramic nanoparticle-doping method [33]. The preforms were doped with aluminum oxide and thulium ions. Dopant concentrations in preforms were analyzed using a Cameca SX-100 electron probe microanalyzer (EPMA); refractive index profiles were measured by a Photon Kinetics A2600 profiler. The fibers were drawn as single-mode ones and characterized given to their spectral absorption using a Nicolet 8700 FTIR spectrometer (1000–2500 nm) and an ANDO AQ-1425 optical spectrum analyzer (380–1600 nm). Details of the preparation and characterization can be found in [33]. Relevant preform and fiber parameters are listed in Table 2.
3.2 Fluorescence decay measurement
The fiber under study SG1598 was pumped at two wavelengths – 792 nm and 1620 nm. Fluorescence decay curves were collected at around 800 nm and 2 µm. All four combinations of pump and detected wavelength were examined for various pump powers up to 70 mW. The maximum pump power was limited by available optical components. In order to measure the 3H4 fluorescence lifetime of the lowly doped fiber SG1604, the fiber was pumped at 792 nm, and the fluorescence decays at 800 nm were detected.
The fluorescence decay curves were measured according to the setup depicted in Fig. 2. The active tested fiber was spliced with a standard single-mode fiber on both ends. Used components were as follows: a Lumics LU793M250 diode, a ThorLabs FPL1054S diode, an ILX Lightwave 3900 laser diode controller, an Agilent 33512B pulse generator, a Hamamatsu G12183-10 K InGaAs photodiode detector (1000–2200 nm), a ThorLabs PDA36A Si detector (450–1000 nm), a Teledyne LeCroy HDO6034 oscilloscope and a computer for data acquisition. The total response time of the used system was measured to be around 3 µs. Emitted light was detected from a side perpendicular to the stripped optical fiber. The tested fiber was about 1 mm long and the detector was placed to its close proximity and directly behind the splice with a standard single mode fiber to avoid any influence of reabsorption, amplified spontaneous emission and other disturbing phenomena to the fluorescence decay curves as was described in Refs. [34,35].
Measured fluorescence decay curves were analyzed and normalized. In the case of detection directly from a pumped manifold, the 1/e decay times were determined. Even though most of the fluorescence decay curves did not behave as single exponentials, the decay time was taken as the time at 1/e of the maximum fluorescence intensity. To determine the fluorescence lifetime, i.e., the decay time at zero pump, where the ET processes are negligible, the measurements were taken by using variable pump power which was extrapolated to a zero value.
In the case of the 3F4 fluorescence lifetime, the values were validated also from the decay tails. In the tail, where the population inversion is low, the decays behave as single exponentials, and fluorescence lifetime without the influence of energy transfers can be evaluated [36]. Both approaches were in a fair agreement. In the case of 3H4 fluorescence lifetime, the tail fitting cannot be used due to the effects of ET processes.
4. Results and discussion
4.1 Fibers under study
The refractive index profile together with the dopant concentration profiles of the preforms SG1598 and SG1604 are depicted in Fig. 3(a) and (b) respectively. The thulium concentration in the preform SG1604 was below the EPMA detection limit. For illustration, a step-index profile defined by FWHM and maximum of the SG1598 measured profile is shown in Fig. 3(a).
The parameters of both studied fibers are listed in Table 2. Concentrations of Al2O3 were obtained by EPMA, Tm3+ concentrations were calculated from absorption measurements. SG1598 was prepared as highly doped fiber; its composition was optimized according to Refs. [10,12]. SG1604 was prepared with thulium concentration as low as possible; a few tens of molar ppm are a reasonable value. While SG1598 was used to determine the energy transfer coefficients, SG1604 was used only to measure the fluorescence lifetime of the 3H4 energy level (parameter τ3 in Table 1).
Table 2. Parameters of the fibers under study
4.2 Fluorescence decay times and lifetimes
Figure 4 portrays the fluorescence decay times of fibers under study as a function of pump power. The 3F4 decay times of fiber SG1598 are depicted in Fig. 4(a), while the 3H4 decay times of fiber SG1604 are shown in Fig. 4(b). In both cases, the functions are extrapolated to the limit as the pump approaches zero and fluorescence lifetimes are highlighted. Fiber SG1598 was pumped at 1620 nm, and SG1604 was pumped at 792 nm.
Fig. 4. Fluorescence decay times as functions of pump power (a) SG1598 3F4 decay time, (b) SG1604 3H4 decay time.
The 3F4 decay time of fiber SG1598 declines rapidly with increasing pump power. This dependence is a result of significant ET processes between thulium ions. The correct value of the fluorescence lifetime is obtained by extrapolation to zero pump power where the ET processes are negligible.
Looking at Fig. 4(b), the SG1604 3H4 decay time is also affected by the pump power. The decrease with increasing pump indicates a presence of ET processes among thulium ions even for such thulium concentration as low as 40 ppm. Despite such low concentration, and effort undertaken to minimize clustering, some Tm3+ clusters might still be present.
4.3 Modeling of energy transfer coefficients
Experimental data obtained with fiber SG1598 under various pump powers were fitted with the model equations. The modeling was made for four combinations of pump and detected wavelength. The set of equations was solved numerically; initial value ranges of ET coefficients were taken from our previous work [29]. The ET coefficients of the best fit were searched by the least square minimization method, and they are listed in Table 3. The experimental data, together with the modeled curves calculated using the found ET coefficients, are depicted in Fig. 5.
Figures 5(a) and 5(b) portray the decay from 3F4 level. Pumped either directly or via the 3H4 level, it is evident that the effect of ET processes is pronounced especially at the beginning of the decay and mainly for higher pump power. After some time, approximately two times the 3F4 fluorescence lifetime, the decays start to behave as single exponentials driven by the 3F4 fluorescence lifetime. The modeled curves are in an excellent agreement with the experimental data across a wide range of time scale and fluorescence intensity.
Table 3. List of published values of ET coefficients in silica-based glass
The decays detected at 800 nm, shown in Figs. 5(c) and 5(d), are rather complicated. If pumped at 1620 nm, the fluorescence is quite weak and falls relatively fast to the level of noise. Especially the decay pumped with 15 mW is scattered considerably. In the case of higher pumps, the model is in a reasonable agreement with the data.
The decays detected at 800 nm, however, when pumped at 792 nm, are not satisfactorily described by the model. The decays obtained experimentally are faster than the calculated ones, especially at longer times. The deviations can be explained by several effects. At first, the presented model does not include higher-lying levels (1G4 and 1D2 for extended diagram see [20,30]) which may be involved in further ET processes. Even though the influence of ET involving 1G4 was described to be low [37], and we confirmed experimentally the 1G4 → 3H6 blue emission is more than 30 dB below intensity of the other studied emissions, it still may have some effect. The ET upconversion 3H4,3H4 → 1G4,3F4 would cause a faster depletion of 3H4 level, compared to our model, as was already observed. Because of a lack of credible spectroscopic data for the ET upconversion to energy levels above 3H4, involvement of such levels would make the search of ET coefficients far more complex without bringing reliable clarification.
The deviations can be explained also by the fact that real dopants profile is not a strict step-index one, the thulium concentration drops from the fiber core center towards the core edges slowly, see Fig. 3. The effect of the non-uniform doping profile on the cross relaxation was discussed by Shardlow [18] and by Ramírez-Martínez [17]. It was stated that variations in thulium concentration across the fiber core would lead to variations in the cross relaxation efficiency. Thus, the cross relaxation efficiency decreases towards the core edges where the thulium concentration drops. Similar variations can be assumed for the ET coefficients. In addition, portion of the thulium ions may rest in clusters.
The energy transfer coefficients, used for plotting the modeled curves in Fig. 5, are summarized in Table 3. The combination of excitation at 792 nm and detection at 800 nm was not involved in the search of the ET coefficients by the least square minimization, as there were much larger deviations than in the other three experimental arrangements. For comparison, Table 3. includes also previously published values of the ET coefficients.
The application of rate equations for the estimation of the coefficients was demonstrated already in our previous work [29]. Nevertheless, the calculations were simplified by using too strong assumptions which led to underestimated k1130 coefficient. Moreover, we did not involve the coefficient k1120. The obtained ET coefficients k3011 and k1130 are in a good agreement with those given in [21,22] although the authors have used different approaches. The only significant discrepancy lies in the k1120 coefficient. While Smith & Smith [22] did not consider this coefficient, Jackson & King [21] stated a value about and order of magnitude smaller than us. The main difference lies in fact that Jackson used data for Tm,Ho:YAG and taken from various sources. The variations in fiber composition among the references may also play a role. The interrelations of the upconversion processes in silica glass were discussed by Jackson in his later work in 2004 [12]. Based on the energy mismatches between involved manifolds, the ET (11→20) is exothermic and involves emission of at least two phonons, while the ET (11→30) is endothermic and requires at least one phonon to absorb. Thus, the rate of ET (11→20) would dominate over the rate of ET (11→30). Our result k1120 > k1130 is fully compliant with this statement.
From the point of view of laser properties, high k1120 depopulates the upper lasing level faster, and thus in principle increases laser threshold and decreases slope efficiency. On the other hand, the coefficient k3011 exceeds the other two coefficients and remains the dominant factor in energy transfer processes, especially when pumped at 790 nm. In addition, the effect of k1120 depends on a population inversion and signal gain saturation. Indeed, from Eq. (1) one can infer that for high power operation (i.e. high stimulated emission rate W10 and low N1 population) the exploitation of N1 by the laser signal dominates over the energy leakage process described by k1120. Thus, low population inversion shall be kept by proper selection of pump wavelength or by tailoring the pump absorption along the fiber, e. g., by special coiling and twisting techniques and fiber design [38–40]. In our preliminary simulations of thulium fiber laser operated in high power continuous-wave regime, no significant effect of k1120 was observed. The cooperative process described by k1120 becomes important in low power, high inversion setups, for example in low power fiber amplifiers, broadband ASE sources or during the fluorescence lifetime measurements, such as shown here. Detailed analysis of the energy transfer effects on high power fiber laser performance is out of the scope of this paper.
5. Conclusions
We have used a rate equation modeling to evaluate the energy transfer coefficients in a thulium-doped silica optical fiber. The fiber was pumped by a various pump power at two wavelengths, 792 nm and 1620 nm, while fluorescence decays were detected at 800 nm and 2 µm. The experimental decays were fitted with theoretical rate equations in order to find the energy transfer coefficients. The coefficients were found as follows k3011 = (1.9 ± 0.2) × 10−22 m3s-1, k1130 = (1.4 ± 0.2) × 10−23 m3s-1 and k1120 = (3.5 ± 0.1) × 10−23 m3s-1. The values correspond well to the data already published although the previous approaches were distinct. To the best of our knowledge, this is the first such complex study that involves comparison of various fluorescence decays in terms of excitation and detection wavelength as well as excitation powers. The values of energy transfer coefficients could be used in the design and modeling of high-power thulium-doped fiber lasers.
Funding
Grantová Agentura České Republiky (19-03141S).
Acknowledgements
The authors thank Vladimíra Sedláková for careful English proofreading and Jan Pokorný for a help with the error analysis.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. M. Michalska, W. Brojek, Z. Rybak, P. Sznelewski, M. Mamajek, and J. Swiderski, “Highly stable, efficient Tm-doped fiber laser—a potential scalpel for low invasive surgery,” Laser. Phys. Lett. 13(11), 115101 (2016). [CrossRef]
2. P. Kronenberg and O. Traxer, “The laser of the future: reality and expectations about the new thulium fiber laser—a systematic review,” Transl. Androl. Urol. 8(S4), S398–S417 (2019). [CrossRef]
3. J. Aubrecht, P. Peterka, P. Honzátko, O. Moravec, M. Kamrádek, and I. Kašík, “Broadband thulium-doped fiber ASE source,” Opt. Lett. 45(8), 2164–2167 (2020). [CrossRef]
4. S. D. Jackson, “Towards high-power mid-infrared emission from a fibre laser,” Nat. Photonics 6(7), 423–431 (2012). [CrossRef]
5. I. Mingareev, N. Gehlich, T. Bonhoff, A. Abdulfattah, A. M. Sincore, P. Kadwani, L. Shah, and M. Richardson, “Principles and applications of trans-wafer processing using a 2-µm thulium fiber laser,” Int. J. Adv. Manuf. Technol. 84(9-12), 2567–2578 (2016). [CrossRef]
6. A. M. Heidt, Z. Li, and D. J. Richardson, “High power diode-seeded fiber amplifiers at 2 µm—from architectures to applications,” IEEE J. Sel. Top. Quantum Electron. 20(5), 525–536 (2014). [CrossRef]
7. I. Moskalev, S. Mirov, M. Mirov, S. Vasilyev, V. Smolski, A. Zakrevskiy, and V. Gapontsev, “140 W Cr:ZnSe laser system,” Opt. Express 24(18), 21090–21104 (2016). [CrossRef]
8. C. Gaida, M. Gebhardt, T. Heuermann, F. Stutzki, C. Jauregui, and J. Limpert, “Ultrafast thulium fiber laser system emitting more than 1 kW of average power,” Opt. Lett. 43(23), 5853–5856 (2018). [CrossRef]
9. P. Honzatko, Y. Baravets, F. Todorov, P. Peterka, and M. Becker, “Coherently combined power of 20 W at 2000nm from a pair of thulium-doped fiber lasers,” Laser. Phys. Lett. 10(9), 095104 (2013). [CrossRef]
10. A. Sincore, J. D. Bradford, J. Cook, L. Shah, and M. C. Richardson, “High average power thulium-doped silica fiber lasers: review of systems and concepts,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–8 (2018). [CrossRef]
11. F. Todorov, J. Aubrecht, P. Peterka, O. Schreiber, A. A. Jasim, J. Mrazek, O. Podrazky, M. Kamradek, N. Kanagaraj, M. Grabner, Y. Baravets, J. Cajzl, P. Koska, A. Fisar, I. Kasik, and P. Honzatko, “Active optical fibers and components for fiber lasers emitting in the 2-(m spectral range,” Materials 13(22), 5177 (2020). [CrossRef]
12. S. D. Jackson, “Cross relaxation and energy transfer upconversion processes relevant to the functioning of 2 (m Tm3+-doped silica fibre lasers,” Opt. Commun. 230(1-3), 197–203 (2004). [CrossRef]
13. G. Armagan, B. M. Walsh, N. O. P. Barnes, E. A. Modlin, and A. M. Buoncristiani, “Determination of Tm-Ho rate coefficients from spectroscopic measurements,” in OSA Proceedings on Advanced Solid-State Lasers20, 141–145 (1994).
14. M. Falconieri, A. Lanzi, G. Salvetti, and A. Toncelli, “Fluorescence dynamics in an optically-excited Tm,Ho:YAG crystal,” Opt. Mater. 7(3), 135–143 (1997). [CrossRef]
15. R. R. Petrin, M. G. Jani, R. C. Powell, and M. Kokta, “Spectral dynamics of laser-pumped Y3Al5O12:Tm,Ho lasers,” Opt. Mater. 1(2), 111–124 (1992). [CrossRef]
16. G. Rustad and K. Stenersen, “Modeling of laser-pumped Tm and Ho lasers accounting for upconversion and ground-state depletion,” IEEE J. Quant. Electron. 32(9), 1645–1656 (1996). [CrossRef]
17. N. J. Ramirez-Martinez, M. Nunez-Velazquez, A. A. Umnikov, and J. K. Sahu, “Highly efficient thulium-doped high-power laser fibers fabricated by MCVD,” Opt. Express 27(1), 196–201 (2019). [CrossRef]
18. P. Shardlow, D. Jain, R. Parker, J. K. Sahu, and W. A. Clarkson, “Optimising Tm-doped silica fibres for high lasing efficiency,” in Proceedings of CLEO 2015, (2015).
19. S. D. Jackson and S. Mossman, “Efficiency dependence on the Tm3+ and Al3+ concentrations for Tm3+-doped silica double-clad fiber lasers,” Appl. Opt. 42(15), 2702–2707 (2003). [CrossRef]
20. M. Eichhorn, “Quasi-three-level solid-state lasers in the near and mid infrared based on trivalent rare earth ions,” Appl. Phys. B 93(2-3), 269–316 (2008). [CrossRef]
21. S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17(5), 948–956 (1999). [CrossRef]
22. A. V. Smith and J. J. Smith, “Mode instability thresholds for Tm-doped fiber amplifiers pumped at 790 nm,” Opt. Express 24(2), 975–992 (2016). [CrossRef]
23. P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15(1), 85–92 (2009). [CrossRef]
24. P. Peterka, I. Kasik, A. Dhar, B. Dussardier, and W. Blanc, “Theoretical modeling of fiber laser at 810 nm based on thulium-doped silica fibers with enhanced 3H4 level lifetime,” Opt. Express 19(3), 2773–2781 (2011). [CrossRef]
25. D. A. Simpson, G. W. Baxter, S. F. Collins, W. E. K. Gibbs, W. Blanc, B. Dussardier, and G. Monnom, “Energy transfer up-conversion in Tm3+-doped silica fiber,” J. Non-Cryst. Solids 352(2), 136–141 (2006). [CrossRef]
26. W. Q. Shi, M. Bass, and M. Birnbaum, “Effects of energy-transfer among Er3+ ions on the fluorescence decay and lasing properties of heavily doped Er-Y3Al5O12,” J. Opt. Soc. Am. B 7(8), 1456–1462 (1990). [CrossRef]
27. M. Taher, H. Gebavi, S. Taccheo, D. Milanese, and R. Balda, “Novel approach towards cross-relaxation energy transfer calculation applied on highly thulium doped tellurite glasses,” Opt. Express 19(27), 26269–26274 (2011). [CrossRef]
28. A. Albalawi, S. Varas, A. Chiasera, H. Gebavi, W. Albalawi, W. Blanc, R. Balda, A. Lukowiak, M. Ferrari, and S. Taccheo, “Determination of reverse cross-relaxation process constant in Tm-doped glass by 3H4 fluorescence decay tail fitting,” Opt. Mater. Express 7(10), 3760–3768 (2017). [CrossRef]
29. J. Cajzl, P. Peterka, M. Kowalczyk, J. Tarka, G. Sobon, J. Sotor, J. Aubrecht, P. Honzatko, and I. Kasik, “Thulium-doped silica fibers with enhanced fluorescence lifetime and their application in ultrafast fiber lasers,” Fibers 6(3), 66 (2018). [CrossRef]
30. P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quant. Electron. 36(1-3), 201–212 (2004). [CrossRef]
31. P. Peterka, W. Blanc, B. Dussardier, G. Monnom, D. Simpson, and G. Baxter, “Estimation of energy transfer parameters in thulium- and ytterbium-doped silica fibers,” Proc. SPIE 7138, 71381K (2008). [CrossRef]
32. B. M. Walsh and N. P. Barnes, “Comparison of Tm : ZBLAN and Tm : silica fiber lasers; Spectroscopy and tunable pulsed laser operation around 1.9 (m,” Appl. Phys. B 78(3-4), 325–333 (2004). [CrossRef]
33. M. Kamrádek, I. Kašík, J. Aubrecht, J. Mrázek, O. Podrazký, J. Cajzl, P. Vařák, V. Kubeček, P. Peterka, and P. Honzátko, “Nanoparticle and solution doping for efficient holmium fiber lasers,” IEEE Photonics J. 11(5), 1–10 (2019). [CrossRef]
34. S. Sujecki, L. Sojka, Z. Tang, D. Jayasuriya, D. Furniss, E. Barney, T. Benson, and A. Seddon, “Spatiotemporal modeling of mid-infrared photoluminescence from terbium(III) ion doped chalcogenide-selenide multimode fibers,” J. Rare Earths 37(11), 1157–1163 (2019). [CrossRef]
35. A. Schawuchow, S. Unger, S. Jetschke, and J. Kirchhof, “Advanced attenuation and fluorescence measurement methods in the investigation of photodarkening and related properties of ytterbium-doped fibers,” Appl. Opt. 53(7), 1466–1473 (2014). [CrossRef]
36. M. Digonnet, “Rare-earth-doped fiber lasers and amplifiers,” Inc. New York EUA, (2001).
37. S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express 14(1), 50–57 (2006). [CrossRef]
38. C. A. Codemard, A. Malinowski, and M. N. Zervas, “Numerical optimisation of pump absorption in doped double-clad fiber with transverse and longitudinal perturbation,” Proc. SPIE 10083, 1008315 (2017). [CrossRef]
39. P. Koška, P. Peterka, and V. Doya, “Numerical modeling of pump absorption in coiled and twisted double-clad fibers,” IEEE J. Sel. Top. Quantum Electron. 22(2), 55–62 (2016). [CrossRef]
40. M. Grábner, K. Nithyanandan, P. Peterka, P. Koška, A. A. Jasim, and P. Honzátko, “Simulations of pump absorption in tandem-pumped octagon double-clad fibers,” IEEE Photonics J. 13(2), 1–14 (2021). [CrossRef]
