1 Introduction

Ytterbium-doped materials have been widely used over the last two decades for the construction of diode-pumped solid state lasers since they offer a long fluorescence lifetime in the millisecond range and a low quantum defect.

Furthermore, the availability of high-power laser diodes in the wavelength range between 920 and 980 nm, well suited for the absorption bands of Yb3+ in most host materials, enabled the planning, construction and successful operation of several high energy class facilities such as POLARIS [1], PEnELOPE [2], MERCURY [3] and LUCIA [4].

A major drawback of Yb3+-doped materials, however, especially when considering laser systems which are to be operated with pulsed laser diodes, is the thermally induced population of the lower laser level, due to the low separation of the intra-manifold energy levels. This induces quasi-three-level properties for the laser medium. As a consequence, a significant amount of pump energy needs to be transferred to the active medium before a population inversion can be built up. Since this required energy is not extractable during the subsequent amplification process, it significantly limits the overall efficiency of a laser system [5].

An approach to overcome this issue and to ultimately improve the efficiency of a laser system is to cool down the amplification medium to cryogenic temperatures, which enables four-level operation since the thermally induced population of the lower laser level is “frozen out.” Several amplifier systems operating in such conditions have been demonstrated to be highly efficient both in continuous wave [6, 7] and in pulsed laser regimes [8, 9]. In addition, a couple of large-scale facilities utilizing cryogenic cooling like DIPOLE [10], HiLASE [11], ELI [12] and GENBU [13] are under development. Further projects like HiPER [14] are in the planning phase.

However, cooling the active medium to cryogenic temperatures does not only affect the re-absorption of the emitted laser radiation, but also the general shape of the spectrum. Hence, the exact knowledge of the absorption and emission cross-sections as function of temperature and wavelength is crucial for the design and the simulation of any laser system.

The scope of this paper is to provide quantitative values of absorption and emission cross-sections for common Yb3+-doped laser materials as a function of temperature between 80 and 340 K.

The measurements presented in this paper have been carried out using 1 at.% Yb3+-doped Yttrium Aluminum Garnet (Yb:YAG) crystals of 5 and 15 mm thickness, a 15 at.% Yb3+-doped Lutetium Aluminum Garnet (Yb:LuAG) crystal of 1 mm thickness (both crystals were grown with Czochralski method by Crytur spol. s r.o., Czech Republic) and a 3 at.% Yb3+-doped Calcium Fluoride (Yb:CaF 2) ceramic of 6 mm thickness made by Incrom Ltd., Russia.

2 Experimental setup

The experimental setup used for the recording of the absorption and fluorescence spectra is shown in Fig. 1. It is equivalent to the setup used for the determination of spectral properties in the high temperature range as presented in [15].

Fig. 1
figure 1

Experimental setup for absorption and emission measurements. FLD fiber-coupled laser diode, WLS fiber-coupled white light source, ANDO optical spectrum analyzer, M 1,2,3 turning mirrors, SM 1,2 spherical mirrors with 200 mm radius of curvature, L 1,2 achromatic lenses

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The spectral data were recorded by an ANDO (AQ-6315 A) optical spectrum analyzer equipped with a 100-μm-diameter fiber cable, allowing for a maximum spectral resolution of 0.1 nm.

For the absorption measurements, a fiber-coupled white light source WLS100 by Bentham Instruments Ltd. was used, which emits light with a smooth spectral distribution between 800 and 1,150 nm. The fiber exit of the source was imaged into the sample using a spherical silver mirror with a radius of curvature of 200 mm and then re-imaged onto the fiber input of the spectrum analyzer using another spherical mirror of the same type.

The excitation source for the fluorescence measurements was a fiber-coupled 9 W laser diode manufactured by Lumics GmbH. To match the wavelength of the pump light to the absorption bands of the tested materials, the output wavelength of the diode could be tuned between 965 and 980 nm by adjusting the baseplate temperature. The exit of the 105 μm core diameter fiber was imaged onto the sample surface under an angle by a one-to-one telescope resulting in only a small spatial overlap between excitation and analysis beam path. Due to the minimized overlap volume, the measured fluorescence signals are only weakly affected by re-absorption. Our approach is equivalent to other approaches utilizing minimized measurement volumes to suppress radiation trapping in spectral measurements [16, 17]. Due to the re-absorption minimization, the setup is also suitable for the estimation of the radiative lifetime \(\tau_{\text{r}}\) by measuring the fluorescence decay with a fiber-coupled photodiode and an oscilloscope in the position of the optical spectrum analyzer. However, since other effects influencing the measured fluorescence lifetime such as non-radiative decay have been neglected, the determined value for \(\tau_{\text{r}}\) is considered a rough approximation only.

The sample was mounted in a modified ST300 cryostat (Janis Research Company, LLC) cooled with liquid nitrogen. The temperature of the sample was monitored with a temperature controller (Lake Shore Cryotronics, Inc.) equipped with a DT-670 diode. Furthermore, the sample temperature could be adjusted between 77 and 340 K by an electrical heating device. During the experiments, the sample was kept under vacuum conditions with maximum pressures of the order of 10−6 mbar to avoid condensation on the sample surfaces.

3 Data analysis

The absorption cross-sections \(\sigma_{\text{a}}\) were obtained using Lambert–Beer’s law [18] which is relating the spectral intensity of the white light source I(λ), which was transmitted through the sample to the spectral intensity I 0(λ) measured without the sample inserted into the beam path for each wavelength \(\lambda\):

$$\sigma_{a} (\lambda ) = {{\ln \left( {\frac{{I_{0} (\lambda )}}{I(\lambda )}} \right)} \mathord{\left/ {\vphantom {{\ln \left( {\frac{{I_{0} (\lambda )}}{I(\lambda )}} \right)} {N_{\text{dop}} l}}} \right. \kern-0pt} {N_{\text{dop}} l}}$$
(1)

Here, N dop is the density of dopant ions and l the thickness of the sample. Fresnel losses on the surface and other wavelength independent influences caused by the insertion of the samples into the setup were corrected by normalizing the measured transmission spectra at 810 nm and at 1,140 nm. In these spectral parts, absorption can be neglected. The remaining part of the measured spectrum was adapted to this normalization by applying a linear fit for the normalization factor between these two wavelengths.

To further increase the accuracy of the absorption measurements for Yb:YAG, two samples with a thickness 5 and 15 mm were measured. The final absorption spectrum was obtained by combining both absorption spectra, where the measurement with the thicker crystal was used for the low-absorbing spectral parts.

The emission cross-sections \(\sigma_{\text{e}}\) were calculated based on the absorption measurements using the McCumber (MC) relation also known as the reciprocity method [19, 20]:

$$\frac{{\sigma_{\text{a}} (\lambda )}}{{\sigma_{\text{e}} (\lambda )}} = \frac{{Z_{\text{l}} }}{{Z_{\text{u}} }} \cdot \exp \left( {E_{\text{ZL}} - \frac{hc}{\lambda }} \right)$$
(2)

Here, E ZL is the energy of the zero phonon line (ZPL), h is Planck’s constant and c the vacuum speed of light. Z l and Z u denote the partition functions for the lower and upper manifold, respectively, which are given by

$$Z_{m} = \mathop \sum \limits_{n} d_{n}^{m} \cdot \exp \left( { - \frac{{E_{n} }}{{k_{\text{B}} T}}} \right) .$$
(3)

Here, \(d_{n}^{m}\) is the degeneracy of the nth level in the mth manifold and E n the corresponding intra-manifold energy. T is the temperature and k B the Boltzmann factor.

Additionally, the Fuchtbauer–Ladenburg (FL) method [21] was used to determine the emission cross-sections directly from the spectral fluorescence intensity I f:

$$\sigma_{\text{e}} \left( \lambda \right) = \frac{{\lambda^{5} }}{{8\pi n^{2} \tau_{\text{r}} c}} \cdot \frac{{I_{\text{f}} (\lambda )}}{{\mathop \smallint \nolimits_{0}^{\infty } \lambda^{'} I_{\text{f}} \left( {\lambda^{'} } \right){\text{d}}\lambda^{'} }} .$$
(4)

Here \(\tau_{\text{r}}\) is the radiative lifetime and n the refractive index of the sample.

Both methods give independent results for the emission cross-sections. The MC-relation is valid for spectral parts where the absorption cross-sections can be measured with high accuracy. However, in the case of low temperatures, the MC-relation becomes sensitive to slight shifts of the ZPL as the Boltzmann distribution is getting steeper. To compensate for the temperature drift of the ZPL, we use the peak position of the corresponding spectral line in the absorption spectrum at each temperature for calculation.

In contrast, the FL-method leads to better results, when the underlying re-absorption is low. This yields higher accuracy for spectral parts with low absorption and at low temperatures. A valid result for the whole spectral range of interest can then be achieved by combining both methods. For this, the results from the MC-relation are used for the highly absorbing spectral parts only, while results from the FL-method are applied where absorption is negligible. Spectral parts with low but precisely measurable absorption are used to combine and cross-check the results from both approaches. In these spectral parts, the final spectrum is found by taking the average from the results of both methods. An example comparing results from the methods and the final emission cross-sections can be found in [15].

Input values used for the calculations are given in Table 1. Energy levels used for the calculation are assumed to be constant with temperature, since the small shifts will not significantly affect the partition functions. Changes with temperature of the refractive index n are also neglected as they can be considered too small (less than one percent over the whole temperature range [22, 23]) to cause a significant impact. The same is valid for the wavelength dependence.

Table 1 Values used for the calculation of the cross-sections
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Lifetimes were measured for the whole temperature range as described before. Within our measurements, no temperature dependence was found. As the absolute values for lifetime from our measurements might be influenced by parasitic effects, values used for our calculation were taken from the literature [15, 27–29]. In case of Yb:LuAG, the average of the values published in [28] and [29] was applied.

4 Results and discussion

All measurements were taken in temperature steps of 20 K starting at 80 K and ranging up to 340 K. For the sake of clarity, the displayed results are limited to selected temperatures.

4.1 Yb:YAG and Yb:LuAG

Yb:YAG and Yb:LuAG are very similar both in their atomic structure and spectral characteristic [28].

The derived cross-sections for Yb:YAG and Yb:LuAG are given in Figs. 2 and 3, respectively. In both cases, the absorption band ranging from about 915–940 nm decomposes into several peaks for low temperatures, while the peak height (exact values see Table 2) of the absorption cross-sections is increasing.

Fig. 2
figure 2

Absorption (upper plot) and emission cross-sections (lower plot) at different temperatures for Yb:YAG

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Fig. 3
figure 3

Absorption (upper plot) and emission cross-sections (lower plot) at different temperatures for Yb:LuAG

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Table 2 Peak absorption cross-sections for Yb:YAG and Yb:LuAG in the 940 nm band at different temperatures
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Considering the emission band at 1,030 nm, the peak emission cross-section at 80 K is about 5 times higher as compared to room temperature for both materials, as displayed in Fig. 4. However, the bandwidth is reduced from more than 10 nm at 340 K to about 1 nm full width at half maximum (FWHM) at 80 K. Furthermore, a shift to shorter wavelengths by 1 nm is observed for the peak position for the low-temperature case. Comparing both materials, Yb:LuAG offers a slightly higher peak cross-section for emission than Yb:YAG. This advantage is accompanied by a marginally lower FWHM bandwidth of the emission peak at room temperature, which was also observed in laser amplification experiments reported in [30]. In the low temperature range, Yb:YAG loses the bandwidth advantage as the 1,030-nm emission peak is split.

Fig. 4
figure 4

Comparative analysis of the emission cross-sections at 1,030 nm for Yb:YAG and Yb:LuAG. Upper plot displays the peak emission cross-section, the graph in the middle corresponds to the respective full width at half maximum of \(\sigma_{\text{e}}\) and the lowest figure the peak position. In the high temperature range data from [15] was included

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In the absorption band at 940 nm, Yb:YAG offers a higher cross-section than Yb:LuAG. But as displayed in Fig. 5, this is only applicable if a pump source with small bandwidth is available. In this plot, the product of doping concentration \(c_{\text{dop}}\) and thickness d, which is needed to absorb 90 % of the pump radiation, is given as function of the 1/e spectral width \({{\Updelta}}\lambda\) and center wavelength \(\lambda_{\text{c}}\) of the pump. Here, the spectral distribution of the pump source is assumed to be Gaussian, and Lambert–Beer’s law is used to calculate the absorption length. Due to the close to equally high absorption peaks at 936 and 940 nm, Yb:LuAG is suited better for pump sources with a broader bandwidth. This also holds true at temperatures of 80 K, where both materials have higher absorption cross-sections in combination with significant structuring of the spectrum. As a result, the higher peak values for \(\sigma_{\text{a}}\) are only usable with narrow bandwidth pump radiation.

Fig. 5
figure 5

Comparison of the absorption characteristics of Yb:YAG and Yb:LuAG at 300 and 80 K. The plots display the product of doping concentration \(c_{\text{dop}}\) and thickness d needed to absorb 90 % of the pump radiation as function of the 1/e spectral width \({{\Updelta}}\lambda\) and center wavelength \(\lambda_{\text{c}}\) of the pump

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4.2 Yb:CaF 2

In contrast to the previously presented materials, Yb:CaF 2 is typically used as a broadband amplification medium. The excited state lifetime of 1.9 ms is nearly the double of Yb:YAG. However, due to these features, the height of the absorption and emission cross-sections is relatively low.

As it can be seen in Fig. 6, the changes with temperature are much less pronounced than for the high gain materials discussed before. Emission and absorption cross-sections keep their broadband characteristic even at low temperatures, though the bands are more structured than at room temperature.

Fig. 6
figure 6

Absorption (upper plot) and emission cross-sections (lower plot) at different temperatures for Yb:CaF 2

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For low temperatures, a strong emission band at 990 nm emerges. In combination with pumping at the zero phonon line, this enables laser operation at an extremely low quantum defect [31].

The absorption band between 920 and 960 nm is very smooth and can be used without special requirements on the bandwidth of the pump source as shown in Fig. 7. For cryogenic temperatures, no significant structuring of this band is observed. The cross-sections are slightly higher than at room temperature with a dip at 940 nm. Additionally to this broadband, the zero phonon line at 980 can be used for pumping at room temperature as the available cross-section is much higher in this case. The high absorption cross-section for the zero phonon line at 80 K is only usable in combination with a narrow bandwidth pump source.

Fig. 7
figure 7

Comparison of the absorption characteristics of Yb:CaF 2 at 300 and 80 K. The plots display the product of doping concentration \(c_{\text{dop}}\) and thickness d needed to absorb 90 % of the pump radiation as function of the 1/e spectral width \({{\Updelta}}\lambda\) and center wavelength \(\lambda_{\text{c}}\) of the pump

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Though Yb:CaF 2 is mostly taken into consideration when the broad bandwidth is required to amplify femtosecond pulses, another feature makes this material attractive for pulsed high-power amplifiers.

A main contribution to the generation of heat within an active material is caused by the decay of the population inversion via spontaneous emission. The intensity distribution of the fluorescence light can be directly derived from the FL-method:

$$I_{\text{f}} (\lambda ) = {\text{const}} \cdot \frac{{\sigma_{\text{e}} (\lambda )}}{{\lambda^{5} }}$$
(5)

With this an effective emission wavelength \(\lambda_{\text{eff}}\) can be calculated:

$$\lambda_{\text{eff}} = \frac{{\mathop \smallint \nolimits \lambda \cdot I_{\text{f}} (\lambda ){\text{d}}\lambda }}{{\mathop \smallint \nolimits I_{\text{f}} (\lambda ){\text{d}}\lambda }}$$
(6)

Here, \(\lambda_{\text{eff}}\) is the wavelength of the average energy of the emitted photons. In Fig. 8, the effective wavelength of the fluorescence and the resulting fraction of absorbed pump light that is converted into heat is shown for Yb:CaF 2 as a function of temperature in comparison with Yb:YAG and Yb:LuAG. When decreasing the temperature, \(\lambda_{\text{eff}}\) is shifted toward longer wavelengths, which may be attributed to the increasing steepness of the Boltzmann distribution. This results in a higher heat load for cryogenic operation in general. In comparison with the high gain materials, Yb:CaF 2 benefits from a shorter \(\lambda_{\text{eff}}\) of the fluorescence. In combination with pumping at the zero phonon line, the heat load generated for the same pump power is two to three times smaller than for the other materials.

Fig. 8
figure 8

Upper plot effective wavelength of fluorescence \(\lambda_{\text{eff}}\) as a function of temperature in comparison for Yb:YAG, Yb:LuAG and Yb:CaF 2. Lower plot fraction of absorbed pump energy, which is converted to heat by the quantum defect of spontaneous emission for common pump wavelengths

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5 Conclusions

In conclusion, we have presented measured values of the absorption and emission cross-sections of Yb:YAG and Yb:LuAG as a function of temperature. For both materials, the emission cross-section at 80 K is about 5 times higher than that at room temperature, while the bandwidth is strongly reduced.

Additionally, we have presented cross-sections for Yb:CaF 2, which is of special interest for the generation and amplification of ultra-short pulses. Here, the cross-sections were much less affected by temperature, but the emission cross-section at 1,030 nm was still nearly doubled at 80 K as compared to room temperature. In contrast to the other tested materials, the spectrum showed only minor narrowing, which will allow amplifying ultra-short pulses at cryogenic temperatures with this gain medium.

Furthermore, we derived that the heat load caused by spontaneous emission in Yb:CaF 2 is significantly lower than in the high gain materials, which makes Yb:CaF 2 a very interesting material for the design and the realization of high-power laser systems.