Thin disk laser experiments with Yb:LuAG (Yb:Lu3Al5O12) were performed leading to 5 kW of output power and an optical-to-optical efficiency exceeding 60%. Comparative analyses of the laser relevant parameters of Yb:LuAG and Yb:YAG were carried out. While the spectroscopic properties were found to be nearly identical, investigations of the thermal conductivities revealed a 20% higher value for Yb:LuAG at Yb3+-doping concentrations of about 10%. Due to the superior thermal conductivity with respect to Yb:YAG, Yb:LuAG offers thus the potential of improved performance in high power thin disk laser applications.
©2010 Optical Society of America
In the last two decades the thin disk laser  has been proven to be a suitable laser design for multi-kW output power with high beam quality . To exploit the advantages of the thin disk laser, the thickness of the gain medium needs to be as low as possible for efficient heat removal. Therefore, high doping concentrations are required to allow for sufficient absorption.In addition, a high thermal conductivity is necessary to improve the heat transport to the heat sink. Up to now Yb3+-doped Y3Al5O12 (Yb:YAG) is the gain material of choice for most commercial applications: the Yb3+-ion exhibits a small quantum defect and a simple two-manifold energy level structure, avoiding detrimental loss processes like upconversion, cross relaxation, and excited state absorption, thus allowing high doping concentrations. In combination with the relatively strong coupling of the electronic 4f-4f-transitions to the phonons of the host lattice , which leads to broad absorption lines, this laser ion is highly suitable for efficient high power diode laser pumping. Even though there is a rapid progress in the development of new, more efficient host materials for the Yb3+-ion, e. g. the sesquioxides , YAG is still preferred in commercial applications due to the possibility to grow large crystals in reproducible and very good optical quality by the well established Czochralski-technique. Furthermore, most of the newly developed Yb3+-doped gain materials for thin disk laser applications [5–7] are pumped at the zero phonon line around 975 nm, whereas Yb:YAG is pumped at a much broader absorption band around 940 nm. Using e.g. the sesquioxides in commercial applications will thus require the implementation of new pump diodes and new coating structures, optimized for the different pump wavelength, not to mention the challenges of the crystal growth .
In contrast, Yb3+-doped Lu3Al5O12 (Yb:LuAG), which is examined in this work, is an alternative to Yb:YAG that would not require substantial changes in commercial applications as it has nearly identical optical properties compared to Yb:YAG and can be pumped at a similar wavelength with the same pump source. Like Yb:YAG, Yb:LuAG can also be simply fabricated by the Czochralski-technique. But compared to Yb:YAG, where the Yb3+-ion of the atomic mass 173 g/mol substitutes an Y3+-ion of the atomic mass 89 g/mol, LuAG provides Lu3+-sites (mass 175 g/mol) for the dopant ion. This very low mass difference of the dopant and the substituted ion leads to a low phonon scattering rate . Thus, at high doping concentrations which are necessary for efficient thin disk laser operation, the thermal conductivity of Yb:LuAG is expected to be higher than that of Yb:YAG, even though the latter provides a better thermal conductivity for the undoped host [10–12]. A similar approach led to the development of many other very suitable host materials for the Yb3+-ion containing lutetium instead of yttrium, e.g. LuVO4 , KLu(WO4)2 , LiLuF4 , and also Lu2O3 .
In this work a comparative study of LuAG and YAG crystals doped with Yb3+-ions regarding their spectroscopic, thermal, and laser properties is presented. Particular attention was paid to the investigation of the Yb3+-concentration dependency of the thermal conductivity. These investigations predict a better suitability of Yb:LuAG for high power thin disk laser applications compared to Yb:YAG. Furthermore, for the first time Yb:LuAG is applied as an active medium in the thin disk laser setup. In initial experiments with up to 25 W of output power, 72% slope efficiency and an optical-to-optical-efficiency of 62% were obtained. In high power experiments performed in cooperation with Trumpf Laser GmbH & Co. KG the output power could then be scaled to 5 kW from one disk and a maximum optical-to-optical efficiency of more than 60% could be reached. This performance is comparable to that of Yb:YAG despite no optimization of the pump wavelength was accomplished for Yb:LuAG.
2. Fabrication, spectroscopic and thermal investigations
2.1 Crystal growth
The crystals investigated in this work were grown by the Czochralski method using rhenium-crucibles. In previous examinations it has been shown that rhenium-crucibles provide several advantages compared to the more common iridium-crucibles. In particular, rhenium-crucibles are available in a much higher purity, leading to a lower contamination of the melt by impurities . Furthermore, the melting point of LuAG of 2060°C  is higher than for YAG (1940°C ) (see Tab. 1) and quite close to the applicability of iridium-crucibles with their melting point of 2447°C . In contrast, with 3186°C the melting point of rhenium is one of the highest of all elements .
For the determination of the Yb3+-concentration dependency of the thermal conductivity Yb:LuAG crystals of different Yb3+-doping concentrations were grown. In addition, two high quality 10% and 15% (with respect to the dodecahedral site) Yb3+-doped LuAG crystals were grown for laser experiments. By scanning electron microscopy with energy dispersive X-ray spectrometry (SEM/EDS) the Yb3+-distribution coefficient of both crystals was determined to be 1.0. This indicates a more homogeneous doping concentration in contrast to YAG, which exhibits a distribution coefficient of 1.1 to 1.2.
2.2 Spectroscopic investigations
Fluorescence lifetime measurements applying the pinhole method [27,28] resulted in 965 µs and 985 µs for the 10% and 15% doped Yb:LuAG, respectively. The value for the 10% doped crystal is lower than the value of 1317 µs for the same material reported by Brenier et al. . It is assumed that in  reabsorption effects led to increased values for the measured lifetime, which are eliminated by the pinhole method. For the 15% Yb3+-doped crystal the corresponding value of 830 µs reported by Brenier et al. is much lower than the presented data above. In this case the explanation can probably be found in the lower purity of the crystals in , which led to quenching of the fluorescence lifetime . The lifetimes in this work obtained by the pinhole method are very similar to the 951 µs reported for Yb(16.5%):YAG investigated with the same method .
The absorption cross sections were determined from transmission spectra of Yb(10%):LuAG and Yb(7%):YAG  measured with a Varian CARY 5000 spectrophotometer with a spectral resolution of 0.6 nm. For the determination of the emission cross sections, fluorescence spectra were recorded with a Fourier-transform spectrometer (Equinox 55, Bruker Optik GmbH, Germany) at a resolution of 0.5 cm−1. As the results of the Füchtbauer-Ladenburg method  are influenced by reabsorption in the short wavelength range and the reciprocity method  results in a weak signal to noise ratio in the longer wavelength range, a combination of the resulting spectra was used to avoid both effects. In this way, reliable values for the emission cross sections σem over the whole emission bandwidth of the Yb3+-ion were obtained.
As the ratio of the partition functions Zu and Zl is needed, the reciprocity relation requires the exact knowledge of the energetic positions of the three Stark energy levels of the upper 2F5/2 and the four Stark levels of the lower 2F7/2 multiplet of the Yb3+-ion:17,32–35]. To obtain comparable spectra, the Stark level energies for Yb3+ in YAG and LuAG given in  were used, as these were determined under the same conditions for both materials. The resulting room temperature absorption and emission cross section spectra for Yb:YAG and Yb:LuAG are shown in Fig. 1 , where the vertical line marks the changeover between the results of the reciprocity and the Füchtbauer-Ladenburg method. As expected, due to the similarity of both cubic host materials (see Tab. 1) the spectra of the materials are nearly identical.
The absorption maximum of Yb:YAG at 941 nm with a cross section of 8.20 × 10−21 cm2 is about 14% higher than the absorption maximum of Yb:LuAG at 940 nm (7.22 × 10−21 cm2). However, this difference becomes less important when considering the emission bandwidth of high power laser diodes. A convolution of the absorption spectra of Yb:YAG and Yb:LuAG with a Gaussian curve simulating the emission spectrum of a high power laser diode revealed that for a realistic bandwidth of 5 nm the effective maximum absorption only differs by about 5%. It has to be mentioned, that in this case the highest absorption in Yb:LuAG is obtained for a center wavelength of 938 nm, whereas for Yb:YAG it remains at 941 nm. However, both pump wavelengths can be covered with standard Yb:YAG laser diodes by temperature tuning of about 10 K .
In contrast, the emission cross section at the laser wavelength around 1030 nm is about 20% higher for Yb:LuAG (σem = 2.59 × 10−20 cm2) than for Yb:YAG (σem = 2.14 × 10−20 cm2). This results in a slightly lower threshold for Yb:LuAG which contributes to a good pump light absorption, because the bleaching at the pump wavelength is less pronounced than in case of Yb:YAG. Also, the extractable gain for the laser wavelength in Yb:LuAG is higher, allowing for a higher transmission of the output coupling mirror which makes the resonator more tolerant with respect to intracavity losses.
2.3 Thermal conductivity
The thermal conductivity κ of a solid state material is given by the relation12,26]. One can intuitively understand that the disorder due to Yb3+-doping is larger in the YAG lattice than in the LuAG lattice due to the very large mass difference between Y3+-ions and Yb3+-ions (see Tab. 1). Thus, despite the fact that for the undoped YAG the thermal diffusivity (and also the thermal conductivity) is higher than for LuAG, this is not necessarily the case for higher Yb3+-concentrations.
To evaluate the Yb3+-concentration dependency of the thermal conductivity in Yb:LuAG and Yb:YAG the thermal diffusivities at room temperature were measured for crystals with different doping concentrations using an ai-Phase Mobile 1 (ai-Phase CO, Ltd., Japan) which is based on the temperature wave analysis . According to the Neumann-Kopp rule , the specific heat capacities for the examined doping concentrations were linearly interpolated from literature values for undoped YAG, LuAG and YbAG (see Tab. 1). The shift of the density due to a linear change of the lattice constant with increasing Yb3+-concentration [11,39] and the mass difference of Yb3+ and the substituted ion were also taken into account. The resulting thermal conductivities obtained from Eq. (2) are shown in Fig. 2 .
If the thermal conductivities for the undoped (κ0) and 100% doped (κ100) material are known, the doping concentration dependency of the thermal conductivity κ can be calculated by the following equation :Fig. 2).
The measurements presented in Fig. 2 show, that indeed the room temperature thermal conductivity of undoped YAG (9.5 W/(m·K)) is by about 20% higher than for undoped LuAG (7.7 W/(m·K)). However, according to the fit, κ becomes already equal for both materials at an Yb3+-concentration of about 3% due to a much stronger decrease with Yb3+-concentration in Yb:YAG. For an Yb3+-concentration of 10%, which is a typical doping concentration used in thin disk lasers, both the fit and the experimental data result in a 20% higher thermal conductivity for Yb:LuAG (7.4 W/(m·K)) than for Yb:YAG (6.1 W/(m·K)). Fan et al. have defined the thermo-optic figure of merit FOM :
It has to be mentioned, that at a measurement error of just 5% for the thermal diffusivity, the data shown in Fig. 2 are about 10% lower than previously reported thermal conductivities for Yb:YAG with comparable doping concentrations [12,26,41]. Although an accurate determination of the thermal conductivity is difficult (cf. Tab. 1), e. g. due to the uncertainty of cp, it cannot be excluded that this is a general issue of the temperature wave method compared to the more common laser-flash method . However, the ratios between the Yb:LuAG and Yb:YAG values should not be affected.
3. Laser experiments
3.1 Theoretical considerations
The analytical zero-dimensional model  is a simple approach to determine suitable operation parameters, e. g. the thickness of the active medium in the thin disk laser setup. In this model, all laser parameters are assumed to be constant within the pumped volume. Due to the low thickness and the multiple passes of the pump light through the disk shaped gain medium this approximation is reasonably good to obtain useful information and compare two different gain media. The influence of thermal effects is neglected in the first instance and treated in a separate simulation. Therefore, the resulting theoretical efficiencies are higher than observed in the experiments shown in the next section.
For comparison of 10% Yb3+-doped LuAG and YAG, internal losses of 0.1% and pump intensities of 6 kW/cm2 at a pump wavelength of the 940-nm-peak are assumed, which are typical values for such simulations . Figure 3a shows the optical-to-optical efficiency for an output coupling transmittance of 3%. The rise of the optical-to-optical efficiency with increasing disk thickness is based on the improvement of the absorption efficiency. Due to the effect of reabsorption at the laser wavelength, the laser threshold is also increasing with the disk thickness and becomes dominant at thicker disks decreasing the optical-to-optical efficiency. This results in an optimum thickness of 185 µm for Yb(10%):YAG. Because of the slightly lower absorption efficiency at this pump wavelength, the optimum thickness of 200 µm for Yb(10%):LuAG is a bit larger . But as shown in Fig. 3a, in the range from about 160 µm to 230 µm the efficiency is just varying by about 0.5% or less. In this region Yb(10%):LuAG and Yb(10%):YAG can be considered having practically the same optical-to-optical efficiency. For both optimum disks the fraction of absorbed pump power after 24 pump light passes is calculated to be higher than 98.8%. As the quantum defect and Yb3+-densities can be considered identical in Yb:LuAG and Yb:YAG (see Tab. 1), the thermal load is almost the same for both materials.
To evaluate the thermal effects, the heat equation was solved for the respective disks assuming a 1-dimensional heat flux perpendicular to the surface of the crystal and using the thermal conductivities determined in the previous section as well as a total heat resistance of 12 mm∙K/W for the HR-coating and the fluid interface . The results predict that due to its higher thermal conductivity, for a specified average disk temperature a 5% higher pump intensity can be applied to the 200 µm thick Yb(10%):LuAG disk compared to the 185 µm thick Yb(10%):YAG disk (see Fig. 3b). Assuming comparable laser performances, Yb:LuAG disks would thus allow for a 5% higher maximum output power if the disk temperature is considered as the limiting factor. This difference even increases for more sophisticated cooling approaches with a lower total heat resistance, e.g. when gluing the disks onto diamond heat sinks. It has to be mentioned that in this simulation the dependency of the thermal conductivity on the temperature  was not considered due to the lack of reliable data.
3.2 Experimental results
The disks, prepared with different thicknesses around the optimum values, had an anti-reflective coating for the pump and the laser wavelength on their front side. On their backside, being mounted onto a water-cooled copper heat sink a highly reflective coating for the pump and the laser wavelength was applied.
For the demonstration of the first thin disk laser operation with Yb:LuAG, a commercial thin disk laser pump head (TGSW Stuttgart, Germany) aligned for 24 pump passes through the active medium was used. A 600 µm fiber-coupled laser diode (JENOPTIK Laserdiode GmbH, Germany) which was imaged onto a 1.2 mm pump spot on the disk served as pump source. In these experiments the diode emission wavelength was tuned by adapting the cooling temperature to match the absorption of Yb:LuAG at 938 nm. The multimode I-cavity was formed simply by the HR-backside of the disk and one curved output mirror.
In Fig. 4a the laser characteristics of Yb:LuAG and Yb:YAG disks with nearly identical doping concentrations and thicknesses, pumped with the same laser diode at their respective optimum pump wavelengths are presented. The materials exhibit a very similar laser performance with slope efficiencies of more than 70% and an identical optical-to-optical efficiency of 62% for each disk.Although the thickness of the Yb:LuAG disk is lower than the optimum value of 200 µm, no remarkable difference in efficiency is expected as can be seen in Fig. 3a. This comparison demonstrates that Yb:LuAG is very well suited as alternative gain material to Yb:YAG in the thin disk laser setup. For higher power experiments, where heat removal is crucial, Yb:LuAG is expected to benefit more from its better thermal conductivity.
Figure 4b shows the dependency of the slope efficiency on the transmission TOC of the output coupling mirror at the laser wavelength for a Yb(10%):LuAG disk of 220 µm thickness. It can be seen that the best results in this power region were obtained for TOC between 1.5% and 3.5%. This crystal also delivered the highest slope efficiency of all examined Yb:LuAG disks with 72% at TOC = 2.2%.
To prove the suitability of Yb:LuAG for output powers in the kW-regime, first high power experiments were performed at Trumpf Laser GmbH & Co. KG. The 11% Yb3+-doped crystals were grown, prepared, and mounted under the same conditions (and by the same suppliers) as the standard Yb:YAG crystals usually used in commercially available Trumpf-Lasers. In Fig. 5 the laser characteristics of the Yb:LuAG and the Yb:YAG disks are shown, obtained using a thin disk laser setup, which was optimized for Yb:YAG. Both materials delivered maximum optical-to-optical efficiencies of more than 60%, comparable to the results in the low power experiments and output powers of 5.0 kW and 5.4 kW for Yb:LuAG and Yb:YAG, respectively.
As mentioned in the spectroscopic investigations section the optimum pump wavelength for Yb:LuAG would be 938 nm (see Fig. 1). However, in these initial experiments the cooling temperature of the pump diode laser was adjusted to perfectly fit the absorption of Yb:YAG at 941 nm at the maximum pump power of 9.2 kW. As a consequence, the emission wavelength of the diode laser shifted from shorter wavelengths to this value while increasing the pump power. Obviously, the optimum pump wavelength for Yb:LuAG is reached at a pump power around 4 kW (see Fig. 5), where the highest optical-to-optical efficiency for this material was obtained. The drop in efficiency for higher incident pump powers is thus attributed to a wavelength drift to even longer wavelengths and is not caused by a thermal rollover. Further experiments with a pump wavelength optimized for Yb:LuAG are in progress.
Nevertheless, it can be seen that the maximum optical-to-optical efficiency of Yb:LuAG of 61.9% is slightly higher than the maximum for Yb:YAG of 60.7%. From this fact it can be deduced that with an optimized pump wavelength, Yb:LuAG has the potential to deliver higher efficiencies and output powers.
This paper reports on the first thin disk laser operation with Yb:LuAG as gain material and presents a detailed analysis of the relevant laser parameters in comparison to Yb:YAG. While the spectroscopic properties of both materials are shown to be very similar, investigations of the thermal properties reveal a 20% higher thermal conductivity of Yb:LuAG at the typical doping concentrations used in thin disk lasers. Theoretical calculations suggest, that this should allow for higher pump power densities and thus higher laser output powers in kW-thin disk lasers, where heat removal is crucial.
Compared to other high efficiency thin disk laser materials as for example the Yb-doped sesquioxides  which are pumped at their zero phonon line around 975 nm, a major advantage of Yb:LuAG is its pump wavelength of 938 nm. This pump wavelength allows for the use of the same pump diodes and coatings like for the established Yb:YAG which is typically pumped at 940 nm.
In first thin disk laser experiments, with an output power of 25 W at 71% slope efficiency Yb(10%):LuAG delivered very similar laser properties compared to Yb(10%):YAG. Subsequent high power thin disk laser experiments performed at Trumpf Laser GmbH & Co KG did not yet demonstrate the advantage of Yb:LuAG, since the pump wavelength was adjusted for Yb:YAG in these experiments. However, the maximum optical-to-optical efficiency was even slightly higher for Yb:LuAG, promising a superior performance compared to Yb:YAG when operated the adequate pump wavelength. The maximum output power of 5 kW obtained in these experiments represents the highest that has ever been obtained with this material.
We thank the BMBF (contract no. 13N 8382) and the Joachim Herz Stiftung for their financial support. Furthermore, we would like to thank Trumpf Laser GmbH & Co. KG for performing the high power laser experiments and the accreditation to publish these data. We also thank Dr. H.-J. Bernhardt (Institute of Geology, Mineralogy and Geophysics, Bochum) for performing the EDS-measurements and Dr. D. Klimm (Institut für Kristallzüchtung, Berlin) for providing experimental and theoretical data for our thermal conductivity measurements.
References and links
1. A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable Concept for Diode-Pumped High-Power Solid-State Lasers,” Appl. Phys. B 58, 365–372 (1994).
2. A. Giesen and J. Speiser, “Fifteen Years of Work on Thin-Disk Lasers: Results and Scaling Laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007). [CrossRef]
3. A. Ellens, H. Andres, M. L. H. ter Heerdt, R. T. Wegh, A. Meijerink, and G. Blasse, “Spectral-line-broadening study of the trivalent lanthanide-ion series. II. The variation of the electron-phonon coupling strength through the series,” Phys. Rev. B 55(1), 180–186 (1997). [CrossRef]
5. C. Kränkel, R. Peters, K. Petermann, P. Loiseau, G. Aka, and G. Huber, “Efficient continuous-wave thin disk laser operation of Yb:Ca4YO(BO3)3 in EIIZ and EIIX orientations with 26 W output power,” J. Opt. Soc. Am. B 26(7), 1310–1314 (2009). [CrossRef]
6. C. Kränkel, J. Johannsen, R. Peters, K. Petermann, and G. Huber, “Continuous-wave high power laser operation and tunability of Yb:LaSc3(BO3)4 in thin disk configuration,” Appl. Phys. B 87(2), 217–220 (2007). [CrossRef]
7. C. R. E. Baer, C. Kränkel, O. H. Heckl, M. Golling, T. Südmeyer, R. Peters, K. Petermann, G. Huber, and U. Keller, “227-fs pulses from a mode-locked Yb:LuScO3 thin disk laser,” Opt. Express 17(13), 10725–10730 (2009). [CrossRef] [PubMed]
8. R. Peters, C. Kränkel, K. Petermann, and G. Huber, “Crystal growth by the heat exchanger method, spectroscopic characterization and laser operation of high-purity Yb:Lu2O3,” J. Cryst. Growth 310(7-9), 1934–1938 (2008). [CrossRef]
9. P. G. Klemens, “Thermal Resistance due to Point Defects at High Temperatures,” Phys. Rev. 119(2), 507–509 (1960). [CrossRef]
10. G. A. Slack and D. W. Oliver, “Thermal Conductivity of Garnets and Phonon Scattering by Rare-Earth Ions,” Phys. Rev. B 4(2), 592–609 (1971). [CrossRef]
11. Y. Kuwano, K. Suda, N. Ishizawa, and T. Yamada, “Crystal growth and properties of (Lu,Y)3Al5O12,” J. Cryst. Growth 260(1-2), 159–165 (2004). [CrossRef]
12. R. L. Aggarwal, D. J. Ripin, J. R. Ochoa, and T. Y. Fan, “Measurement of thermo-optic properties of Y3Al5O12, Lu3Al5O12, YAlO3, LiYF4, LiLuF4, BaY2F8, KGd(WO4)2, and KY(WO4)2 laser crystals in the 80-300 K temperature range,” J. Appl. Phys. 98, 103514 (2005). [CrossRef]
14. G. Palmer, M. Schultze, M. Siegel, M. Emons, U. Bünting, and U. Morgner, “Passively mode-locked Yb:KLu(WO4)2 thin-disk oscillator operated in the positive and negative dispersion regime,” Opt. Lett. 33(14), 1608–1610 (2008). [CrossRef] [PubMed]
15. A. Bensalah, Y. Guyot, A. Brenier, H. Sato, T. Fukuda, and G. Boulon, “Spectroscopic properties of Yb3+:LuLiF4 crystal grown by the Czochralski method for laser applications and evaluation of quenching processes: a comparison with Yb3+:YLiF4,” J. Alloy. Comp. 380(1-2), 15–26 (2004). [CrossRef]
16. D. Fagundes-Peters, N. Martynyuk, K. Lünstedt, V. Peters, K. Petermann, G. Huber, S. Basun, V. Laguta, and A. Hofstaetter, “High quantum efficiency YbAG-crystals,” J. Lumin. 125(1-2), 238–247 (2007). [CrossRef]
17. A. Brenier, Y. Guyot, H. Canibano, G. Boulon, A. Ródenas, D. Jaque, A. Eganyan, and A. G. Petrosyan, “Growth, spectroscopic, and laser properties of Yb3+-doped Lu3Al5O12 garnet crystal,” J. Opt. Soc. Am. B 23(4), 676–683 (2006). [CrossRef]
18. J. L. Caslavsky and D. J. Viechnicki, “Melting behaviour and matastability of yttrium aluminium garnet (YAG) and YAlO3 determined by optical differential thermal analysis,” J. Mater. Sci. 15(7), 1709–1718 (1980). [CrossRef]
19. H. Okamoto, “The Ir-Re (Iridium-Rhenium) System,” J. Phase Equilibria 13(6), 649–650 (1992). [CrossRef]
20. F. Euler and J. A. Bruce, “Oxygen coordinates of compounds with garnet structure,” Acta Crystallogr. 19(6), 971–978 (1965). [CrossRef]
21. A. A. Kaminskii, Laser Crystals: Their Physics and Properties (Springer Verlag, Heidelberg, 1990).
22. T. B. Coplen, “Atomic Weights of the Elements 1999 Technical Report,” Pure Appl. Chem. 73(4), 667–683 (2001). [CrossRef]
23. W. Koechner, Solid-State Laser Engineering (Springer-Verlag, Heidelberg, Germany, 1999).
24. D. Klimm, data from FactSage 5.3 (personal communication, 2009).
25. N. P. Padture and P. G. Klemens, “Low Thermal Conductivity in Garnets,” J. Am. Ceram. Soc. 80(4), 1018–1020 (1997). [CrossRef]
26. F. D. Patel, E. C. Honea, J. Speth, S. A. Payne, R. Hutcheson, and R. Equall, “Laser Demonstration of Yb3Al5O12 (YbAG) and Materials Properties of Highly Doped Yb:YAG,” IEEE J. Quantum Electron. 37(1), 135–144 (2001). [CrossRef]
27. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett. 32(13), 1908–1910 (2007). [CrossRef] [PubMed]
28. C. Kränkel, D. Fagundes-Peters, S. T. Fredrich, J. Johannsen, M. Mond, G. Huber, M. Bernhagen, and R. Uecker, “Continuous wave laser operation of Yb3+:YVO4,” Appl. Phys. B 79, 543–546 (2004). [CrossRef]
29. K. Petermann, G. Huber, L. Fornasiero, S. Kuch, E. Mix, V. Peters, and S. A. Basun, “Rare-earth-doped sesquioxides,” J. Lumin. 87–89, 973–975 (2000). [CrossRef]
30. P. F. Moulton, “Spectroscopic and laser characteristics of Ti:Al2O3,” J. Opt. Soc. Am. B 3(1), 125–132 (1986). [CrossRef]
31. D. E. McCumber, “Einstein Relations Connecting Broadband Emission and Absorption Spectra,” Phys. Rev. 136, A954–957 (1964). [CrossRef]
32. R. A. Buchanan, K. A. Wickersheim, J. J. Pearson, and G. F. Herrmann, “Energy Levels of Yb3+ in Gallium and Aluminum Garnets. I. Spectra,” Phys. Rev. 159(2), 245–251 (1967). [CrossRef]
33. D. L. Wood, “Energy Levels of Yb3+ in Garnets,” J. Chem. Phys. 39(7), 1671–1673 (1963). [CrossRef]
34. J. A. Koningstein, “Energy Levels and Crystal-field Calculations of Trivalent Ytterbium in Yttrium Aluminum Garnet and Yttrium Gallium Garnet,” Theor. Chim. Acta 3(3), 271–277 (1965). [CrossRef]
35. G. A. Bogomolova, L. A. Bumagina, A. A. Kaminskii, and B. Z. Malkin, “Crystal field in laser garnets with TR3+ ions in the exchange charge model,” Sov. Phys. Solid State 19 (1977).
36. J. Kawanaka, H. Nishioka, N. Inoue, and K. Ueda, “Tunable continuous-wave Yb:YLF laser operation with a diode-pumped chirped-pulse amplification system,” Appl. Opt. 40(21), 3542–3546 (2001). [CrossRef]
37. J. Morikawa, C. Leong, T. Hashimoto, T. Ogawa, Y. Urata, S. Wada, M. Higuchi, and J. Takahashi, “Thermal conductivity/diffusivity of Nd3+ doped GdVO4, YVO4, LuVO4, and Y3Al5O12 by temperature wave analysis,” J. Appl. Phys. 103, 063522 (2008). [CrossRef]
38. E. S. R. Gopal, Specific Heats at Low Temperatures (Plenum Press, New York, 1966).
39. X. Xu, Z. Zhao, P. Song, G. Zhou, J. Xu, and P. Deng, “Structural, thermal, and luminescent properties of Yb-doped Y3Al5O12 crystals,” J. Opt. Soc. Am. B 21(3), 543–547 (2004). [CrossRef]
40. T. Y. Fan, D. J. Ripin, R. L. Aggarwal, J. R. Ochoa, B. Chann, M. Tilleman, and J. Spitzberg, “Cryogenic Yb3+-Doped Solid-State Lasers,” IEEE J. Sel. Top. Quantum Electron. 13(3), 448–459 (2007). [CrossRef]
41. R. Gaumé, B. Viana, D. Vivien, J.-P. Roger, and D. Fournier, “A simple model for the prediction of thermal conductivity in pure and doped insulating crystals,” Appl. Phys. Lett. 83(7), 1355–1357 (2003). [CrossRef]
42. W. J. Parker, R. J. Jenkins, C. P. Butler, and G. L. Abbott, “Flash Method of Determining Thermal Diffusivity, Heat Capacity, and Thermal Conductivity,” J. Appl. Phys. 32(9), 1679–1684 (1961). [CrossRef]
43. K. Contag, M. Karszewski, C. Stewen, A. Giesen, and H. Hugel, “Theoretical modelling and experimental investigations of the diode-pumped thin-disk Yb:YAG laser,” Quantum Electron. 29(8), 697–703 (1999). [CrossRef]
44. K. Contag, Modellierung und numerische Auslegung des Yb:YAG Scheibenlasers (PhD-thesis, IFSW, University of Stuttgart, 2002).