Fractional thermal load in cryogenically cooled Yb:YLF and Yb:YAG lasers

: We present a method for the direct measurement of the fractional thermal load (FTL) in cryogenically cooled laser crystals. The experimental methodology involves characterizing the liquid nitrogen evaporation rate in a dewar containing the laser crystals, allowing for the accurate determination of FTL. The FTL is measured to be 1.7 × quantum defect (QD) for Yb:YLF and 1.5 × QD for Yb:YAG under continuous wave lasing conditions. The measured FTL values are then used to calculate the temperature distribution inside the crystals as a function of pump power, and the simulation results are found to be in very good agreement with the in-situ temperature measurements using contactless optical luminescence thermometry. The method and findings presented in this work hold great potential to benefit laser engineers and scientists working with cryogenic lasers to address and overcome temperature-dependent handicaps.


Introduction
The impact of pump-induced heating on laser gain media is a critical consideration that can significantly limit the performance of laser/amplifier systems.This is due to temperaturedependent dynamics such as thermal quenching of fluorescence lifetime [1][2][3], thermal lensing [4][5][6], thermally induced stress [7][8][9][10], output-mode instabilities [11][12][13], etc.Hence, an accurate determination of the temperature distribution within laser crystals becomes quite important to overcome these thermal limitations.A key parameter in this context is the fractional thermal load (FTL), representing the ratio of heat load generated to absorbed pump power in the laser gain medium.
Existing approaches to estimate FTL indirectly often rely on observation of the variation of crystal temperature or thermal lensing as a function of absorbed pump power, but these methods require precise knowledge of additional system parameters, such as thermal conductivity, thermo-optic coefficient, and photo-elastic coefficients [14][15][16][17][18]. Direct measurements, including laser calorimetry [19][20][21] and monitoring the current of the heat sink thermoelectric module [22], offer accurate alternative approaches.However, it is important to emphasize that these methods are only compatible with room temperature measurements.
The amount of fractional thermal load can typically be estimated by studying electron transfer mechanisms between energy levels, such as quantum defect, excited state absorption, auger up-conversion, and thermal quenching of fluorescence [23][24][25][26].However, these mechanisms are not always well-known for all gain media, and specific cases require detailed spectroscopic studies.Even with complete knowledge of energy transitions, it remains challenging to determine which mechanism predominates and to what extent.Consequently, determining the fractional thermal load remains essential.
In this paper, we introduce a novel method for direct measurement of fractional thermal load in cryogenically cooled lasers.Moreover, we report the measured FTL values for Yb:YLF and Yb:YAG materials under lasing conditions at cryogenic temperature for the first time.

Experimental methodology
Our approach to determining the fractional thermal load is based on the measurement of the liquid nitrogen evaporation rate in the dewar containing the laser crystal.First, the liquid nitrogen evaporation rate of the dewar is characterized as a function of a known heat load, created by a calibrated resistive heating element.Then, during continuous wave (CW) lasing operation, the evaporation rate of the dewar is measured as a function of absorbed pump power for both Yb:YLF and Yb:YAG crystals.One can then directly infer the heat load created by the laser crystals by comparing the measured liquid nitrogen evaporation rate with the earlier curve obtained using the resistive heating element.The variation of the heat load with absorbed pump power is then used to calculate the fractional thermal load induced inside the crystal.
Figure 1 shows the schematic diagram of the setup used for FTL measurements.The dewar used for the experiments is designed and fabricated in-house.For the characterization of the evaporation rate of the dewar as a function of heat load, a 200 W cartridge resistive heating element (RH, RS PRO 860-7075, Fig. 2(b, inset)) is placed into a copper block.To accurately mimic the crystals' heat load, the resistive heater is mounted at the position where we usually place the laser gain medium.For better visualization, Fig. 2 shows photos of Yb:YLF crystal indium soldered to the copper heat sink and the mounted resistive heating block, respectively.They are both thermally connected to the cold head of the dewar.The heating element is operated by an adjustable power supply (PS) via a vacuum feedthrough.The voltage applied and resistance response of the heating element are measured simultaneously by a digital multimeter (DMM, Fluke 116) that is connected across the heating element.By using the basic electric power formula (P = V 2 /R), the power that turns into heat energy is calculated (cable connection losses are negligible).While the heating element is powered, it is actively cooled by liquid nitrogen (LN 2 ).Subsequently, LN 2 evaporates and leaves the dewar over an exhaust pipe in gas form.The exhaust of the dewar is connected to a mass flow meter (MFM, Aalborg Instruments DPM47) measuring the gas flow (evaporation rate).Since the evaporation temperature of LN 2 is 77 K, it freezes the MFM.Therefore, a heating bath circulator (HBC, Thermo Scientific 1561071) is placed between the mass flow meter and dewar to increase and stabilize the temperature of the gas that reaches MFM around room temperature.
The setup used for the CW lasing experiments is also shown schematically in Fig. 1.The Yb-doped YLF crystal has a 1% doping concentration, and it is 20 mm long, 15 mm wide, and 10 mm thick.Similarly, the Yb-doped YAG crystal also has 1% doping concentration, and it is 23 mm long, 15 mm wide, and 5 mm thick.The Yb:YLF crystal has 3 mm long undoped endcaps on both ends and hence the total crystal length is 26 mm.The Yb:YAG crystal has a 3 mm undoped cap on its front side only, and therefore has a total length of 26 mm.Temperature stabilized laser diode modules (Laserline GmbH) providing up to 3 kW of output power are used as the pump source.The diodes emit un-polarized laser light at the center wavelength of 960 nm (Yb:YLF) and 940 nm (Yb:YAG) with a full-width half maximum (FWHM) of 2.5 nm.The diode modules are fiber coupled into 600 µm core diameter fibers, resulting in an M 2 of around 220.To focus the pump beam into the crystals, we utilize a f1-f2 telescope to image the fiber output.However, due to the small working range of the f1-f2 telescope, another 300 mm focal length lens (f3) is necessary to re-image the pump beam with a beam diameter of 2.1 mm inside  As we are interested in measuring the primary heat load that is responsible for the internal heating of the laser gain element, we tried to minimize all secondary heat sources.First of all, the inner walls of the dewar are coated with a highly absorptive material that absorbs more than 98% of light in the range of 400-1100 nm.The aim is to avoid any stray light (spontaneous emission or scattered pump light) reflecting from the inner walls and hitting back to the cold head of the LN 2 tank.If stray light hits back to the cold head, it might create an extra heat load and consume extra LN 2 , and this could result in an error in measuring the real fractional thermal load of the system.Additionally, the absorbed energy in the dewar walls is removed by a water chiller connected to the outer chamber of the dewar.The cold head used is made of copper which has a reflectivity above 80% in the infrared wavelength range [27].The setup is similar to what was used in our previous lasing/and amplification studies [28][29][30], and enables us to understand thermal load limitations for further power scaling of rod-based cryogenic Yb-doped YLF and YAG systems.As a side note, the absorber coatings on the inner walls and reflective coatings on the cold head are not perfect (100%), so the measurements give an upper limit for the fractional thermal load (FTL).The actual FTL may be slightly lower than what is measured, but not higher.In the next section, we show that the thermo-mechanical simulations indicate a very good agreement with in-situ temperature measurements, demonstrating that the error bar in the FTL measurement is rather small.

Experimental results and discussion
Figure 3 shows the measured liquid nitrogen (LN 2 ) evaporation rate of the dewar as a function of the heat load applied to the crystal via the resistive heater.The measurement has been replicated three times at different dates to confirm the repeatability of the measurement.As can be seen, measurements agree very well with each other.As expected, the liquid nitrogen evaporation rate is linearly increasing with heat load for at least up to 170 W. This confirms that the system is still rather far away from Leidenfrost effects starting at the surface [31].Leidenfrost effect occurs when a liquid encounters a surface that is considerably hotter than its boiling point, forming a vapor layer that insulates it from direct contact with the surface.Since the system is not reaching temperatures high enough to induce this phenomenon, effective cooling can be achieved at the dewar liquid nitrogen contact point.Note that the dewar consumes about 3 standard liters per minute (SL/min) of liquid nitrogen, without any heat load, and the liquid nitrogen consumption increases around 0.23 SL/min per W. Figure 4 shows the summary of continuous-wave laser performance of the cryogenically cooled Yb:YLF and Yb:YAG lasers, respectively.Since our previous studies demonstrated that the best continuous-wave performance is achieved with output couplers of 40% for Yb:YLF and 25% for Yb:YAG, we have chosen to utilize them in this study as well [29,30].With these output coupling values, we observed a lasing threshold of around 30 W and 60 W for Yb:YLF and Yb:YAG, respectively.For Yb:YLF, the laser produced a maximum CW output power of 330 W around 1019 nm at an absorbed pump power of 530 W, and the estimated slope efficiency is 69%.As a side note we have observed lasing at 995 nm up to 100 W of absorbed power.Above that level, the lasing wavelength shifts to 1019 nm due to temperature-induced changes in the gain spectrum (a detailed discussion of this effect can be found in our previous studies [29,30,32]).For the case of Yb:YAG, the laser produced a maximum CW output power of 140 W around 1030 nm at an absorbed pump power of 270 W, and the estimated slope efficiency is 59%.The reason for the lower absorbed power levels compared to the Yb:YLF system is that the thermal lens becomes too dynamic at absorbed powers above 250 W and stable continuous wave lasing was not feasible.The thermal lensing effects are visibly evident in the insets of Fig. 4, by the displayed near-field output beam profiles at each data point.For Yb:YAG, the center of the output beam profile becomes brighter, and the beam gets smaller due to thermal lensing, and instability arises above 250 W of absorbed pump power.On the other hand, the Yb:YLF output becomes multi-mode at increased power level, but the laser was still stably operating thanks to the much smaller overall thermal lensing in Yb:YLF [6,30].Hence, we could even apply higher pump power levels to Yb:YLF [29] but without any additional benefit for the FTL measurements since Yb:YLF demonstrated a linear output power dependency on absorbed power.Employing the same methodology utilized in the dewar characterization with the heating element, the evaporation rate of the liquid nitrogen (LN 2 ) as a function of the absorbed pump power for continuous wave (CW) Yb:YLF and Yb:YAG lasers are measured and shown in Fig. 5(a).Even though there are small fluctuations in the evaporation rates for both lasers, we observe that they both have a linear trend with respect to the absorbed pump power.The common point in the laser measurements and the reference heating element measurements is the evaporation rate.Therefore, we can directly infer the heat load deposited in the crystals as a function of absorbed pump power as shown in Fig. 5(b).Then the variation of the heat load with absorbed pump power could be used to calculate the fractional thermal load induced inside the crystals.For instance, to estimate the heat load for Yb:YLF at an absorbed power of 300 W, we first follow the curve representing Yb:YLF and find the evaporation rate corresponding to the 300 W absorbed power, which is approximately 10 SL/min.Then, by using the calibration curve we check the heat load that corresponds to a 10 SL/min evaporation rate on the right-hand side of the graph.This procedure results in an estimated heat load for Yb:YLF of approximately 35 W at an absorbed power of 300 W. This principle is followed to calculate the estimated heat load as a function of absorbed power for both lasers and the final characteristic is shown in Fig. 5(c).Similar to the previous curves, the estimated heat load also has a linear dependence on absorbed pump power.Finally, by linear fitting, the fractional thermal loads (slopes of the trends) are calculated as 10% for Yb:YLF and 13% for Yb:YAG under lasing conditions at cryogenic temperature.There are different contributions to FTL, but the basic contribution in Yb-systems is the quantum defect (QD).In our lasing cavity, the quantum defect for Yb:YLF (pumped at 960 nm, lasing at 1019 nm) and Yb:YAG (pumped at 940 nm, lasing at 1030 nm) is around 5.8% and 8.7%, respectively.Hence the estimated FTL is 1.7 and 1.5 times the quantum defect for Yb:YLF and Yb:YAG, respectively.The measured FTL for Yb:YAG in this work is in relatively good agreement with the earlier data presented by T. Y. Fan at room temperature [21].We believe that the extra heat load that is generated beyond the quantum defect is a result of undesired effects such as radiation trapping and impurity-induced nonradiative decay [33,34].A laser photon might be absorbed within the gain media and re-emitted randomly, potentially at a longer wavelength, a process known as radiation trapping.This sets off multiple absorption and re-emission cycles, creating thermal load in the gain medium through non-radiative transitions.The impact is more pronounced when there's a significant overlap between the absorption and emission spectra of the gain media like Yb-doped YLF and YAG [30,32,[35][36][37][38][39][40].It is not straightforward to calculate the strength of the radiation trapping since it is crystal geometry dependent due to the presence of effects such as total internal reflection (TIR) [41].TIR plays a significant role in determining the path of photons within the crystal, causing photons to be reflected back into the crystal rather than escaping, influencing the absorption and re-emission processes.Moreover, the existence of trace amounts of rare earth impurities (e.g., Neodymium, Erbium, Thulium, Holmium) and defects is shown to create radiation-quenching channels [40,[42][43][44].Due to spatial dependency and crystal-to-crystal differences of impurities and defects, it is challenging to generalize and determine their effect on thermal loading.
Furthermore, it is important to note that the fractional thermal load might be influenced by several factors, including pumping/ lasing wavelength, doping concentration and efficiency of the laser system.Pump and output laser wavelengths determine the efficiency of energy absorption and conversion (quantum defect), one of the main heat load mechanisms in lasers.Ideally, the choice of pumping and lasing wavelength as close as possible can maximize the efficiency and minimize heat load.Moreover, an increase in the doping concentration may lead to a higher effect of radiation trapping, potentially elevating the thermal load.Similarly, inefficient energy extraction from the gain medium, often attributed to mode-matching issues, may result in unextracted energy storage, leading to additional radiation trapping, and increasing the thermal load.Therefore, a comprehensive understanding of these and similar factors is crucial for minimizing the FTL and optimizing laser performance.
Moreover, to confirm our FTL findings, we have compared the estimated and calculated average temperatures of the crystals.First, the optical contactless temperature probing method based on variation of fluorescence spectra with temperature [45,46] is used to estimate the average temperature of the crystals at different absorbed power levels for cryogenic Yb:YAG and Yb:YLF lasers.Then, we calculated the average temperature in both crystals as a function of the absorbed power at cryogenic temperatures.Finally, Fig. 6 compares the estimated and calculated outcomes.Open marks are the estimated temperature by the contactless optical temperature probing methods while solid lines represent the calculations by assigning FTL as 1.7 × QD and 1.5 × QD for Yb:YLF and Yb:YAG, respectively.Figure 6 shows that our detailed thermomechanical calculations of the crystal temperature using the measured FTL values are in very good agreement with the estimated temperatures of the crystals, confirming the validity of the measured FTLs for cryogenically cooled Yb:YLF and Yb:YAG crystals under continuous wave extraction.We believe that the small difference observed between experimental data and simulations might be due to difficulties in accurately determining critical material parameters such as thermal conductivity.As a side note, for the temperature calculations, the model described in one of our previous studies [6] is utilized, with additional details provided in the appendix below.Fig. 6.Measured (open marks) and calculated (solid lines) temperature of cryogenically cooled Yb:YLF and Yb:YAG lasers as a function of absorbed pump power during CW lasing operation.The calculation has been performed for a fractional thermal load (FTL) of 1.7 and 1.5 times the quantum defect (QD) for Yb:YLF and Yb:YAG, respectively.

Conclusion
We have presented a method for the direct measurement of the fractional thermal load under lasing conditions at cryogenic temperatures.Our method starts with careful measurement of the liquid nitrogen evaporation rate of a dewar system at different heat loads generated by a resistive heater.In the second step, the laser crystals are placed in the same dewar system, and the liquid nitrogen evaporation rate is measured at different absorbed pump power levels during the lasing operation.Comparison of the measured evaporation rates enables accurate in-situ determination of thermal load.The method is employed for the well-known laser materials of Yb:YLF and Yb:YAG, and their fractional thermal load is determined as 1.7 and 1.5 times the quantum defect, respectively.The validity of the method is further confirmed by comparing the measured in situ temperatures of the crystals with temperature estimates based on detailed thermo-mechanical simulations.The presented method and the reported FTL values are anticipated to facilitate the laser community in predicting heat load-dependent dynamics and enhancing the performance of cryogenically cooled Yb-doped YLF and YAG-based solid-state lasers and amplifiers.

Appendix
For the temperature calculations, we employ the model described in one of our previous studies.For more details, we refer the interested reader to [6].Table 1 provides the related parameters used for simulating the thermal behavior of Yb:YLF and Yb:YAG crystals under continuous wave extraction.13414T − 1.3 W/(m.K) [49]

Fig. 2 .
Fig. 2. (a) A picture of the Yb:YLF crystal that is indium soldered to the copper heat sink.The crystal assembly is connected to the cold head of the dewar.(b) A picture of the resistive heating block mechanically mounted to the cold head of the dewar.The inset shows a picture of the 200 W cartridge resistive heating element used in the experiments.

Fig. 3 .
Fig.3.Measured evaporation rate of the dewar system as a function of the applied heat load via the resistive heater.

Fig. 4 .
Fig. 4. Measured CW laser performance of the cryogenically cooled (a) Yb:YLF and (b)Yb:YAG lasers using output couplers with a transmission of 40% and 25%, respectively.The insets display the near-field beam profiles of the laser beams at each data point.The slope efficiencies for the Yb:YLF and Yb:YAG lasers are calculated as 69% and 59%, respectively.

Fig. 5 .
Fig. 5. (a) Measured evaporation rates of cryogenic Yb:YLF and Yb:YAG systems during CW lasing operation as a function of absorbed pump power.(b) Measured relationship between absorbed pump power (on the left axis) and heat load (on the right axis) as a function of evaporation rate.(c) Estimated heat load as a function of absorbed power for the CW cryogenically cooled Yb:YLF and Yb:YAG lasers.