1. Introduction

Optically active materials can be cooled by optical excitation via anti-Stokes fluorescence, as proposed by Pringsheim in 1929 [1]. This technique is nowadays referred to as solid-state laser cooling, or optical refrigeration [2,3]. The first solid-state laser cooling was reported 66 years after Pringsheim’s initial work using an ytterbium (Yb)-doped fluorozirconate glass [4]. The advances in solid-state laser cooling, hereafter simply laser cooling, throughout the last two decades are remarkable. Cooling from room temperature down to below 100 K was realized [5,6]. In particular fluoride crystals exhibit a high quantum efficiency and thus an immense potential for laser cooling. Hence, Yb-doped LiYF₄ (YLF) crystals have been the work-horse material for this technology so far [3,7]. Thulium (Tm) and holmium (Ho) doped fluoride crystals also exhibit a high potential as laser cooling materials [8,9], but the availability of high-power pump sources in the 1 µm region suited for the excitation of Yb³⁺ ions makes Yb-doped fluoride crystals the best choice for practical applications [10]. The technology of solid-state laser cooling is highly demanded for all-solid-state optical cryocoolers [10], which are attractive alternatives to liquefied-gas-based cooling. Optical cryocooler devices benefit from their compactness, simplicity, as well as the absence of any mechanical vibrations, and thus warrant a high ruggedness and reliability. This technology also offers the possibility of focal cooling suited for medical treatments [11]. Moreover, solid-state laser cooling is the underlying principle of the concept of ‘athermal lasers’, also known as ‘radiation-balanced lasers’ [12,13].

The spectroscopic properties of many Yb-doped crystals are well-known since they are state-of-the-art materials for high power solid-state lasers [14–17]. Unfortunately, temperature-dependent spectroscopic data, including fluorescence lifetime as well as absorption and emission cross sections, are not readily available for most Yb-doped materials. However, such data are essential to reveal the potential of a material as a laser cooling medium. Moreover, they are also of high relevance for the application of Yb-doped gain media in cryogenically cooled laser oscillators and amplifiers [17–24].

The accurate determination of the spectroscopic parameters of Yb-doped materials is not trivial. They exhibit a strong spectral overlap between the absorption and the emission causing significant reabsorption of emitted photons. As a result, emission cross sections can be inaccurate when calculated from fluorescence spectra which undergo reabsorption. In addition, multiple reabsorption and reemission results in fluorescence decay times longer than the intrinsic fluorescence lifetime, an effect often referred to as radiation trapping [25]. Thus, spectroscopy of reabsorbing materials needs to be performed very thoroughly to avoid these effects.

Various reports on the spectroscopic properties of Yb:YLF exist [14,17,32,33,23,24,26–31]. However, these reports do not cover the full temperature range relevant for laser cooling applications. Moreover, high-temperature spectroscopic data are needed to evaluate the cooling efficiencies at elevated temperatures, e.g. for radiation-balanced lasers operating in this regime but such data have not been reported yet. Note that even lasers operated at ‘room temperature’ easily heat up by 100 K and their design might also benefit from these data. Further, many previous reports show a strong influence of reabsorption effects on the spectroscopic properties of Yb:YLF. In particular, the increase of the fluorescence lifetime with temperature is attributed to radiation trapping caused by a larger spectral overlap of absorption and emission [30,32], but on the other hand, a temperature-dependent radiative lifetime of Yb-doped materials has also been considered [34,35].

Here, we present a detailed spectroscopic characterization of Yb:YLF aiming to refine and extend the available spectroscopic data in particular with a focus on laser cooling applications. For this reason, we perform absorption and emission spectroscopy as well as fluorescence lifetime measurements in a wide range of temperatures from 17 K to 440 K. Reabsorption effects are carefully suppressed in the experiments. We determine the energetic positions of the Stark levels of Yb³⁺ ions in YLF and reveal an intrinsic temperature dependence of the radiative lifetime caused by the Boltzmann distribution in the upper Stark levels. With these refined spectroscopic data, we evaluate the cooling efficiency for different doping concentrations and purity levels, and find the minimum achievable temperatures to be below the liquid nitrogen temperature of 77 K for high purity Yb:YLF.

2. Determination of the energetic positions of Stark levels of Yb³⁺ in YLF

The accurate determination of the energetic positions of the Stark levels is essential to calculate their temperature dependent Boltzmann occupation and the cross sections by means of the reciprocity (or McCumber) relation [36]. We prepared two samples of different thickness from an Yb-doped YLF crystal grown at our institute by the Czochralski method [37]. YLF has a tetragonal crystal structure and possesses crystallographic a- and c-axes. Both samples were cut perpendicular to an a-axis. Inductively-coupled plasma optical emission spectroscopy (ICP-OES) revealed that 4.95 ± 0.05 at.% of Y ions in the YLF matrix are substituted by Yb ions, conventionally denoted as Yb(5 at.%):YLF. Spatially resolved X-ray fluorescence (XRF) (Bruker, TORNADO) measurements revealed homogeneously distributed Yb ions in these samples with a relative deviation of the concentration below ±0.5%. We identified the energetic positions of the Stark levels in the ²F_7/2 and ²F_5/2 multiplets of Yb³⁺ in YLF from the absorption and emission spectra recorded at 17 K using a high resolution monochromator (HORIBA, M1000) and a photomultiplier tube (Hamamatsu, R5108) as the detector. For both absorption and emission spectroscopy the spectral resolution was set to 0.08 nm to fully resolve the narrow zero-phonon line at 971.6 nm. The wavelength accuracy was evaluated to be better than 0.015 nm for the whole measurement range. The samples were cooled in a closed-cycle helium cryostat (Advanced Research Systems, DE204). In the absorption measurements, a broadband tungsten halogen lamp was used as the probe light, while a Ti:sapphire laser tunable from 700 to 1000 nm (M Squared, SolsTiS) was used as the excitation source in the emission spectroscopy. The wavelength dependent sensitivity of the entire measurement setup was characterized using a calibrated tungsten filament lamp (OSRAM, WI17G), and the recorded polarization-resolved fluorescence spectra were calibrated accordingly. Figure 1 shows the measured wavelength-dependent absorption coefficients (a) and fluorescence intensities (b) for σ (E ⊥ c) and π (E || c) polarization at 17 K. To measure the absorption coefficients for the entire spectral range with a good signal to noise ratio, a 180 µm thick sample was used to measure the strong absorption peaks above 955 nm, and another 4.5 mm thick sample was used to determine the weaker absorption features at wavelengths below 955 nm.

 

Fig. 1. Polarization-resolved (a) absorption coefficient and (b) fluorescence intensity spectra of the Yb(5 at.%):YLF samples recorded at 17 K. The absorption spectra left of the dashed line at 955 nm in (a) were taken with a sample of 4.5 mm thickness and are enhanced by a factor of 10 for clarity. The region above 955 nm was recorded with a 180 µm thick sample. In both figures the peaks assigned as the resonant transitions between the Stark levels are marked with black dots.

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For emission spectroscopy, again the 180 µm thick sample was used to minimize the influence of reabsorption at the zero-phonon line. The data obtained at 17 K enable us to determine the Stark-level positions of Yb³⁺ in YLF and compare it to existing data [27,28,33,38–40]. In both absorption and emission spectra we find the zero-phonon line transition at 971.6 nm. We assign the strong absorption peak at 960.2 nm to the transition E₁→E₆. Among the numerous vibronic absorption peaks in the region below 995 nm, the peak at 947.9 nm is assigned to the transition E₁→E₇. This is further confirmed by the growth of an emission peak for the transition E₇→E₁ with increasing temperature (cf. Figure 6). Three emission peaks of notable strength at 992.4, 995.1, and 1019.7 nm are assigned to the transitions E₅→E₂, E₅→E₃, and E₅→E₄, respectively. We consider the emission peak at 1008 nm, previously assigned to be the transition E₅→E₃ [28], to be a vibronic transition due to its relatively low intensity. This assignment is further supported by a comparison with the Raman spectra [27]. Figure 2(a) illustrates the resulting energy diagram of Yb³⁺ in YLF with the respective absorption and emission lines. Our assignment is consistent with the result in [27], and the difference in the determined Stark level positions is smaller than 5 cm⁻¹. Figure 2(b) illustrates the resulting calculated fractional occupancies of the three upper Stark levels.

 

Fig. 2. (a) Energy diagram of Yb³⁺ in the host material YLF with corresponding absorption and emission lines. (b) Fractional occupancy of the Stark levels in the ²F_5/2 multiplet based on the Boltzmann distribution as a function of temperature calculated by Eq. (6).

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3. Temperature-dependent fluorescence lifetime

The determination of the fluorescence lifetime of Yb-doped materials is not trivial due to radiation trapping. Therefore, the pinhole method as described in [25] was applied to get access to the intrinsic fluorescence lifetime at room temperature. The experimental setup is identical to the layout illustrated in the reference. We used a series of pinholes with different diameters between 0.6 and 2.5 mm. Note that we used a sample thicker than 2.5 mm; otherwise, for too small diameters the excited volume does not have a cubic-relation with the pinhole diameter and the dependence of the lifetime on the pinhole diameter is not linear [25]. As an excitation source we used an optical parametric oscillator (GWU-Lasertechnik, versaScan) set to a wavelength of 960 nm and delivering 5-ns pulses at 10 Hz repetition rate. The detection setup was same as described in the previous section, and time-resolved fluorescence signals were recorded by an oscilloscope. We performed two measurements for each temperature with the detection set to 995 nm and 1020 nm, and confirmed unchanged measured fluorescence lifetimes by the detection wavelength. We observed a very good linear relation of the measured lifetime with the pinhole diameter as shown in Fig. 3(a), where the longest measured lifetime of 2.75 ms was 25% longer than the extrapolated fluorescence lifetime, unaffected by radiation trapping, determined to be 2.20 ± 0.03 ms (95% confidence interval). This clearly underlines the need to avoid radiation trapping during the determination of the fluorescence lifetime of Yb-doped materials to obtain reliable results. Multiphonon relaxation is strongly suppressed in Yb³⁺-doped materials, because the ground state ²F_7/2 and the excited state ²F_5/2 are typically separated by ≈10⁴ cm⁻¹, and thus an order of magnitude larger than the highest phonon energies in typical laser crystals. This allows for external quantum efficiencies well in excess of 99% [41]. For this reason, we interpret the extrapolated fluorescence lifetime of 2.20 ms as the radiative lifetime at room temperature.

 

Fig. 3. (a) Fluorescence lifetime measured with pinholes of different diameter. The intrinsic fluorescence lifetime is extrapolated to be 2.20 ± 0.03 ms. (b) Time-dependent fluorescence signal at 17 K and 440 K from a Yb(5 at.%):YLF sample of a 180 µm thickness suppressing radiation trapping. (c) Fluorescence lifetime versus temperature measured with a Yb(5 at.%):YLF sample of a 180 µm. Circular symbols represent the measured lifetime. Diamond symbols correspond to the fluorescence lifetime corrected for remaining radiation trapping by Eq. (5). Star symbol indicates the lifetime at room temperature determined by the pinhole method used as τ₀ in Eq. (5). Dashed line is the result of fitting with Eq. (7) with the radiative lifetimes of each individual Stark level τ_i denoted in the figure.

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The temperature dependence of the fluorescence lifetime of Yb:YLF was examined with the 180 µm thick Yb:YLF sample. The measured room temperature lifetime of this sample was found to be 2.3 ms. This is only 4.5% longer than the lifetime obtained by the pinhole method, which shows the limited influence of radiation trapping for this sample geometry. A closed-cycle helium cryostat and a ceramic heater were employed to vary the temperature from 17 K to 300 K and 300 K to 440 K, respectively. Figure 3(b) shows the decay curves at 17 K and 440 K in which we clearly identify a temperature dependence of the fluorescence lifetime. Bensalah et al. reported the same lifetime of 1.94 ms at 12 K in Yb(0.5 at.%):YLF [27]. The fluorescence decay is single exponential for all the temperatures. The fast decay reported in Yb(25 at.%):YLF [32] is not observed in our Yb(5 at.%):YLF sample. The circular symbols in Fig. 3(c) show the measured fluorescence lifetime at different temperatures. We found a drop of the lifetime with decreasing temperature. This trend is consistent with previous reports on the fluorescence lifetime of Yb:YLF, where it was attributed to radiation trapping caused by a larger spectral overlap between the absorption and emission spectra at higher temperatures [30]. However, the room temperature lifetime of 2.3 ms measured with this very thin sample compared to the pinhole-lifetime of 2.20 ± 0.03 ms excludes radiation trapping as a single explanation for the observed much stronger temperature dependency. In order to evaluate the remaining radiation trapping at different temperatures T, we define the reabsorption factor for uniaxial crystals as

In the equation, λ is the wavelength, and σ_em and σ_abs are emission and absorption cross sections for π- and σ-polarization, respectively. This factor becomes unity when the absorption and emission spectra are identical, and would be zero if there is no overlap. The absorption cross section σ_abs can be expressed using the emission cross section σ_em by the reciprocity relation [36] given as

Here, h is Planck’s constant, k_B is the Boltzmann constant, E_ZPL is the energy of the zero-phonon line transition at 10292 cm⁻¹, Z_k with i = 1, 2, 3, 4 and 5, 6, 7 is the partition function of the lower and upper multiplet Z_l and Z_u, respectively, and g_i is the degeneracy of the respective Stark level. The emission cross section can be rewritten using the fluorescence intensity by the Füchtbauer-Ladenburg equation for uniaxial crystals [42] given as

Note that the resulting reabsorption factor is independent from the radiative lifetime and the actual cross sections, but only utilizes the shape of the fluorescence spectra I_ζ(λ,T) and the energetic positions of the Stark levels. Instead of using the fluorescence spectra, the reabsorption factor can be also rewritten using the absorption spectra α_ζ(λ,T) in a very similar way.

Figure 4 shows the calculated reabsorption factor with respect to temperature using the fluorescence spectra (Eq. (4)) and the absorption spectra. The separately derived reabsorption factors are in a good agreement which indicates a limited influence of remaining reabsorption in the fluorescence spectra on f_reabs even at elevated temperatures. The reabsorption factor cannot drop to zero due to the remaining spectral overlap at the zero-phonon line. We corrected the remaining radiation trapping in the measured temperature-dependent fluorescence lifetime using the model presented in [43,44] but modified to consider temperature dependency. The measured fluorescence lifetime τ_meas can thus be written as

Here, τ₀ is the intrinsic fluorescence lifetime, and β is a temperature independent parameter accounting for the geometry of the sample and setup. Assuming the pinhole lifetime to be equal to the intrinsic fluorescence lifetime τ₀, the parameter β is determined to be 0.11 for the 180 µm thick Yb:YLF sample from the ratio of the lifetimes measured with and without pinhole method. With this value, the intrinsic fluorescence lifetime τ₀ can be calculated for all temperatures using the respective reabsorption factor. The temperature-dependent fluorescence lifetimes after the correction for the remaining radiation trapping are shown in Fig. 3(c). As clearly seen, the observed temperature dependency in fluorescence lifetime is still not solely explained by the radiation trapping. We thus conclude that the fluorescence lifetime of Yb:YLF, assumed to be identical with the radiative lifetime, is indeed temperature dependent.

 

Fig. 4. Reabsorption factor with respect to temperature calculated by Eq. (4) using the fluorescence spectra and using the absorption spectra.

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A temperature-dependent radiative lifetime of Yb-doped materials has already been considered [34,35] but corresponding experimental findings were previously explained by radiation trapping [30,32,45,46]. We attribute the observed temperature dependency to the distribution of population in the Stark levels of the upper ²F_5/2 multiplet. A transition from one of the upper Stark levels E_i to one of the lower Stark levels E_j possesses a sub-level transition cross section σ_ij [35]. Accordingly, each upper Stark level has an individual radiative lifetime. The fractional occupancy of each upper Stark level f_i (i = 5, 6, and 7) based on the Boltzmann distribution, shown in Fig. 2(b), is given by

We emphasize that this model can be applied for any Yb-doped material, and thus their radiative lifetime may also be a temperature-dependent parameter. This result motivates further investigations on the temperature-dependent lifetime of Yb³⁺ ions in other host materials.

4. Temperature-dependent spectra of Yb:YLF

We thoroughly characterized absorption and emission spectra of Yb:YLF at temperatures between 17 K and 440 K. The fluorescence measurements were carried out with the setup described in Section 2. To accurately evaluate the emission cross sections of Yb:YLF, we used the sample of 180 µm thickness, which had been proven to inhibit reabsorption. The spectral resolution of the measurements was optimized for each temperature to fully resolve the narrowest features, which in most cases were found at the zero-phonon line. Figure 5 shows the polarization-resolved absorption and emission cross section spectra at room temperature. The emission cross sections σ_em were derived by two different approaches: the reciprocity relation (Eq. (2)) and the Füchtbauer-Ladenburg equation (Eq. (3)). Since the temperature variation of refractive index dn/dT of Yb:YLF is only in the order of 10⁻⁶ K⁻¹ [47], it is not taken into account in Eq. (3). Figure 5 also shows the emission cross sections calculated by the reciprocity relation using the absorption cross sections σ_abs which were obtained from absorption coefficients α calculated from transmission measurements and the Yb concentration c_Yb by a relation α = σ_abs·c_Yb. The identical shape of the emission cross section spectra independently obtained by both methods confirms that the reabsorption effect in the 180 µm thick sample is widely suppressed, and the use of the Füchtbauer-Ladenburg equation is reliable here. It is also a proof for the accurate intensity calibration of the measurement setup. The emission cross sections obtained by the Füchtbauer-Ladenburg equation are slightly lower than those from the reciprocity relation. The difference of ≈10% cannot be attributed to the much lower inaccuracy in the Yb concentration by ICP-OES. In addition, the absence of the ultraviolet absorption originating from Yb²⁺[42], confirmed by transmission measurements, ensures the determined Yb atomic concentration can be considered to be the Yb³⁺ concentration. In previous reports, the value stated for the room temperature emission cross section peak at 995 nm for π-polarization varies between 0.7·10⁻²⁰ and 1.27·10⁻²⁰ cm², mainly because different radiative lifetimes are used in the Füchtbauer-Ladenburg equation [14,26,27,29–31,33]. We evaluate the accuracy of our room temperature radiative lifetime of 2.2 ms to be better than ±5%. Therefore, non-reciprocity caused by vibronic transitions is suspected to play a role in the origin of this discrepancy [48].

 

Fig. 5. Polarization-resolved absorption and emission cross sections of Yb:YLF at 300 K. The emission cross sections drawn with solid lines are calculated by the Füchtbauer-Ladenburg (F-L) equation (Eq. (3)) using the radiative lifetime of 2.2 ms determined by the pinhole method. The emission cross sections shown with dashed lines are obtained using the reciprocity relation (Eq. (2)) from the absorption cross sections.

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We also recorded the fluorescence spectra of Yb:YLF for a wide range of temperatures from 17 K up to 440 K. Figure 6 shows the polarization-resolved emission cross sections calculated by the Füchtbauer-Ladenburg equation at selected temperatures in a logarithmic scale. The temperature-dependent effective radiative lifetimes given by Eq. (7), corresponding to the dashed line in Fig. 3(c), are used in Eq. (3). One clearly recognizes the correlation between the temperature-dependent emission cross sections and the population of the upper Stark levels shown in Fig. 2(b). For example, the intensity of the peaks at 960.2 nm (6→1 in Fig. 6) and at 947.9 nm (7→1 in Fig. 6) reflects the population of the corresponding Stark levels E₆ and E₇, respectively. The spectrum at 17 K confirms that the level E₅ is the only populated Stark level at this temperature.

 

Fig. 6. Emission cross sections of Yb:YLF at selected temperatures between 17 K and 440 K. The cross section values were calculated from the fluorescence spectra by the Füchtbauer-Ladenburg equation using the temperature-dependent radiative lifetime model shown as dashed line in Fig. 3(c). The observed resonant transition peaks are labeled.

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Figure 7(a) shows the temperature dependency of the peak absorption cross section at the main laser pump wavelength of 960 nm for π-polarization. The data were derived from the emission cross sections in Fig. 6 using the reciprocity relation (Eq. (2)). This practice underestimates the absorption by ≈10%, which we confirmed by comparison with measured absorption data at room temperature (cf. Figure 5). In Fig. 7(b) and 7(c) the emission cross sections at the main laser wavelengths of 995 and 1020 nm for π-polarization are shown with respect to temperature. All data could be fitted with a sigmoidal Boltzmann function denoted in Figs. 7(a)-(c) with the respective fit parameters. Please note that the physical meaning of the fit parameters is limited because the change of the cross sections with temperature is only partially caused by a changing Boltzmann population and other relevant effects such as line-broadening with temperature are not considered. Demirbas et al. proposed polynomial fit equations for emission cross sections at representative wavelengths for the temperature range of 78–300 K [31]. Those at 995.1 and 1019.7 nm are shown with dashed lines in Fig. 7(b) and 7(c), respectively. Their emission cross section values are ≈10% larger around 300 K but in good agreement at 80 K as compared with our results, since they used a radiative lifetime of 2.0 ms for all temperatures. Our peak emission cross section values at 80 K are also consistent with another recent report [30].

 

Fig. 7. (a) Absorption cross section values at 960.2 nm and (b), (c) emission cross section values at 995.1, and 1019.7 nm for π-polarization with respect to temperature. Each data set is fitted by a sigmoidal Boltzmann function. The best fit parameters are shown in the respective figures. Dashed lines in (b) and (c) are the proposed fit equations in [31].

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5. Yb:YLF for laser cooling

A simple but useful model for laser cooling was first established by Hoyt et al. [49]. The model provides a temperature-dependent cooling efficiency η_c based on the external quantum efficiency η_ext and a background absorption coefficient α_b by

One of the essential parameters for laser cooling based on anti-Stokes fluorescence is the mean fluorescence wavelength λ_f(T). This value is usually measured in the actual laser cooling setup because the detected fluorescence usually undergoes reabsorption and its influence depends on the pump and sample geometry. Thus, the calculated mean fluorescence wavelength λ_f for uniaxial crystals, defined as

Figure 8 shows the temperature dependency of the mean fluorescence wavelength calculated from the fluorescence data with a 180 µm thick Yb:YLF sample. To intentionally show the influence of reabsorption, the mean fluorescence wavelength derived from data recorded with a 4.5 mm thick Yb:YLF sample is also presented in Fig. 8. For comparison, the linear fit equation presented in [7] is shown with the dashed line in the figure.

 

Fig. 8. Mean fluorescence wavelength λ_f from 17 to 440 K calculated from fluorescence spectra recorded with Yb:YLF samples of 180 µm and 4.5 mm thickness. A fitting with a sigmoidal Boltzmann function is applied to the values for the 180 µm sample above 50 K. The equation and the best fit parameters are also shown in the figure. Realignment and corresponding change of the reabsorption conditions with different temperature controllers above and below room temperature explains the minor discontinuity for the 4.5 mm sample. The dashed line is the linear fit equation presented in [7].

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As in Section 3 we confirmed that reabsorption effects in the fluorescence spectra are negligibly small for the 180 µm sample, its mean fluorescence wavelength can thus be considered to be the intrinsic mean fluorescence wavelength of Yb:YLF. To our knowledge this is the first report on the mean fluorescence wavelength not influenced by reabsorption. The data for the 180 µm sample are of particular relevance for laser cooling of micrometer-sized crystals [50], where the effect of reabsorption should be very limited. It is also reliable for thin samples with similar dimensions like the sample used here. Usual samples for laser cooling experiments have larger volume, so the amount of red shift in the mean fluorescence wavelength can be even larger than for the 4.5 mm sample. However, at temperatures below 50 K, the calculated mean fluorescence wavelengths for our two samples are very close, confirming that the influence of reabsorption on the mean fluorescence wavelength is very limited at these temperatures as seen in Fig. 4. This shows that the geometry of the sample becomes less relevant with decreasing temperature. It is also worth noting that the temperature dependency of the mean fluorescence wavelength can be considered to be linear when the fluorescence undergoes strong reabsorption [7].

The temperature-dependent resonant absorption coefficient α_r(λ,T) is required for calculating the cooling efficiency according to Eq. (8). The corresponding values at 1020 nm, a typical pump wavelength for cooling Yb:YLF, easily become as low as 10⁻⁴ cm⁻¹ at low temperatures. Such low values are not accessible by conventional absorption spectroscopy. Thus, the absorption spectra were again calculated using the reciprocity relation from the emission cross sections acquired with the 180 µm sample. The corresponding values for 20 K intervals in a temperature range between 50 K and 290 K for both polarizations are shown in Fig. 9.

 

Fig. 9. Temperature-dependent absorption cross sections of Yb:YLF calculated from the emission cross sections by the reciprocity relation (Eq. (2)). The temperature interval between each line is 20 K.

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With these data, we calculated the cooling efficiencies for Yb:YLF crystals of three different doping levels, 5 at.%, 10 at.%, and 20 at.%, using Eq. (8). Maximum cooling efficiencies η_c,max and optimum pump wavelengths λ_opt, defined as a wavelength to maximize cooling efficiency, with respect to temperature are plotted in Fig. 10. We calculated for various background absorptions α_b, to account for different purity levels as well as a possible temperature dependence of the background absorption as reported in [6], and a fixed external quantum efficiency η_ext of 99.5%. Note that we used the mean fluorescence wavelength obtained with the 180 µm sample. Figure 10 shows the cooling efficiencies in Yb:YLF also at elevated temperatures for the first time. The improved cooling efficiency here originates from the increasing population of the Stark levels E₆ and E₇ with temperature as shown in Fig. 2(b). This effect was recently experimentally confirmed with Yb:YAG [51]. Note that the longest pump wavelength in the calculation was 1070 nm as the absorption cross sections derived by the reciprocity relation above this wavelength are noisy since even the emission cross sections are fairly low above 1070 nm. Thus, for the data points with λ_opt = 1070 nm, even higher cooling efficiencies are obtainable at longer pump wavelengths. The calculations show that an Yb(20 at.%):YLF crystal with a reasonable background absorption of 4·10⁻⁴ cm⁻¹ exhibits more than 6% cooling efficiency at 440 K. This corresponds to an enhancement of ≈30% in cooling efficiency compared to room temperature and may open a way to realize more efficient radiation-balanced lasers [12,13] operated at elevated temperatures.

 

Fig. 10. Optimum pump wavelengths λ_opt (top) and maximum cooling efficiencies (bottom) for (a) 5 at.%, (b) 10 at.%, and (c) 20 at.% Yb:YLF for π-polarization calculated for different background absorption coefficients. The external quantum efficiency was assumed to be 99.5% for all cases. Due to the lack of data for longer wavelengths, the optimum pump wavelength was restricted to a maximum value of 1070 nm. This underestimates the maximum cooling efficiency for the data points with λ_opt = 1070 nm.

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The zero-crossing temperature of each maximum cooling efficiency in Fig. 10 corresponds to the MAT_g. The MAT_g is always achieved at a pump wavelength of 1020 nm regardless of the doping concentration and the background absorption. Since the absorption at this wavelength corresponds to the resonant transition E₄→E₅, the population in the level E₄ determines the absorption efficiency at low temperatures and accordingly the MAT_g. The calculated MAT_g for three external quantum efficiencies with respect to background absorption for five different Yb doping levels is shown in Fig. 11. It illustrates the significant decrease of the MAT with reduced background absorption.

 

Fig. 11. Global minimum achievable temperature (MAT_g) of 1 at.%, 2.5 at.%, 5 at.%, 10 at.%, and 20 at.% Yb:YLF for an external quantum efficiency η_ext of (a) 98.5%, (b) 99.0%, and (c) 99.5% with respect to background absorption coefficient α_b. The excitation is at 1020 nm for π-polarization to achieve the MAT_g for all cases.

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As can be seen in Fig. 11(a) an external quantum efficiency of 98.5% is insufficient to cool down to the liquid nitrogen (LN) temperature of 77 K. The calculated results show the prospect to cool a highly pure Yb(20 at.%):YLF crystal down to ≈70 K assuming a background absorption in the order of 10⁻⁵ cm⁻¹ and an external quantum efficiency higher than 99%. In a cooling setup, the mean fluorescence wavelength of bulk samples is longer than the presented intrinsic mean fluorescence wavelength due to reabsorption, but a MAT_g of ≈70 K would still be reasonable since at such low temperatures the impact of reabsorption becomes very small (cf. Figure 8). The investigations in [6] reveal a strong drop of the background absorption by an order of magnitude between 300 K and 110 K in an YLF crystal doped with Yb and co-doped with a small amount of Tm. In view of this result, it seems realistic to achieve cooling down to 70 K using crystals with room temperature background absorption in the order of 10⁻⁴ cm⁻¹. Increasing the Yb concentration in YLF crystals without degrading the crystal quality and purity is thus considered to be the main challenge of crystal growth technology for the practical realization of cooling to such low temperatures.

6. Conclusion

We presented a detailed spectroscopic investigation of Yb:YLF for a wide range of temperatures between 17 K and 440 K. We were able to widely suppress radiation trapping by utilizing a very thin sample, and further corrected the measured fluorescence lifetimes by the temperature-dependent radiation trapping model with a reabsorption factor. The corrected data still exhibit a temperature dependence of the radiative lifetime originating from the Boltzmann distribution in the Stark levels of the ²F_7/2 multiplet. To our knowledge, this is the first experimental proof directly showing such a temperature dependence of the radiative lifetime in Yb:YLF. This finding is also highly relevant for other host materials doped with Yb³⁺ since the precise radiative lifetime is important as the fluorescence lifetime at low temperatures was often erroneously taken as the radiative lifetime for all temperatures.

Further, we re-evaluated the emission cross sections in a wide temperature range by fully resolving all spectral features and strongly suppressing reabsorption effects. The derived sigmoidal Boltzmann fit equations for the temperature-dependent peak absorption and emission cross sections are also useful for the design of cryogenically operated laser oscillators and amplifiers based on Yb:YLF. These values also enabled to calculate the intrinsic mean fluorescence wavelength of Yb:YLF and to evaluate the influence of reabsorption on this parameter. Calculations of the MAT_g reveal that an external quantum efficiency higher than 99% and a background absorption in the order of 10⁻⁵ cm⁻¹ are required to reach the LN temperature of 77 K or even lower values. Based on recent findings [6] we estimate that sufficiently low background absorption can be realized in crystals with a realistic value of 10⁻⁴ cm⁻¹ at room temperature. Such crystals would furthermore provide significantly enhanced cooling efficiencies, thus increasing the efficiency limit of radiation-balanced lasers in particular when operated at elevated temperatures.