Accurate values of the emission and absorption cross sections of Yb:YAG, Yb:LuAG, and as a function of temperature between room temperature and 200 °C are presented. For this purpose, absorption and fluorescence spectra were measured using a setup optimized to reduce the effect of radiation trapping. From these data, emission cross sections were retrieved by combining the Fuchtbauer–Ladenburg equation and the reciprocity method. Based on our measurements, simple estimations illustrate the effect of temperature shifts that are likely to occur in typical laser setups. Our results show that even minor temperature variations can have significant impact on the laser performance using Yb:YAG and Yb:LuAG as an active medium, while appears to be rather insensitive.
© 2012 Optical Society of America
For the simulation and the design of diode pumped solid-state lasers, reliable data on absorption and emission cross sections of the active laser medium are essential. Unfortunately, for many materials, especially for the less investigated Yb:LuAG, the available data is often not accurate enough, in particular if the temperature dependence needs to be considered. Even for well-known materials such as Yb:YAG, which has been used for more than a decade, the temperature dependence is not known. Moreover, data often vary significantly from one publication to another.
In the case of room temperature and above, the lack of data is even more severe. Spectral measurements are typically only carried out between room and liquid nitrogen temperatures [1,2]. At these low temperatures, one can take advantage of better thermo-mechanical properties of crystals to improve thermal conductivity , and in the case of quasi-three-level systems, to significantly reduce reabsorption. Nevertheless, in high power lasers, it is usually not possible to avoid heating of the laser material, even if very aggressive cooling schemes are applied . Hence, it is crucial to consider spectral alterations in order to optimize the laser performance. Furthermore, higher temperatures may lead to advantages such as broadening of the emission bandwidth.
In Section 2, we present measured absorption and emission spectra for three doped gain media, where the temperature is varied from room temperature up to 200 °C. In particular, we compare the two high gain media Yb:YAG and Yb:LuAG, which—due to their structural relation—are very similar in spectral behavior and which can therefore be used in laser systems likewise. Furthermore, measurements for are presented. This material has already attracted a lot of interest in the last few years because of its capability for high power broadband amplification  and the generation and amplification of pulses as short as 100 fs . In Section 3, basic calculations demonstrate the impact of temperature changes on the performance of a variety of laser systems.
2. SPECTRAL MEASUREMENTS
A. Measurement Setup
To perform highly accurate measurements with the temperature as a controllable parameter, we used a specially designed setup shown in Fig. 1, which allows us to measure absorption as well as emission spectra. All spectra have been obtained using an ANDO A-6315A/B optical spectrum analyzer offering a high dynamic range and a high spectral resolution. For the acquisition of absorption spectra, a fiber coupled white light source was used. This source is characterized by a sufficiently flat intensity distribution between 800 and 1150 nm. To guide the beam, silver coated mirrors were used to avoid any chromatic aberrations. As an excitation source for measuring the fluorescence spectra and the fluorescence lifetime, we used a fiber coupled 9 W laser diode. The center wavelength of the diode could be varied from 965 to 980 nm by changing its baseplate temperature.
The laser diode beam was focused onto the sample surface at a fixed angle of about 40° with respect to the detection beam path. Due to the small spatial overlap of the excited area in the material under investigation and the acceptance volume of the spectrum analyzer determined by the imaging system, a small measurement depth could be provided. By maximizing the fluorescence signal near the zero phonon line of each sample, we ensured that the measurement volume was at the surface of the sample. Thus the effect of reabsorption for the fluorescence spectra acquisition is minimized. Other approaches to minimize radiation trapping were published by Sumida and Fan  and Kuhn et al. .
B. Determination of the Cross Sections From the Measurements
From the measured data, the wavelength-dependent absorption cross sections were obtained by using Lambert–Beer’s law ,
To determine the emission cross sections , two commonly used approaches were employed. First, the emission cross sections were obtained by using their correlation to the absorption cross sections usually referred to as the McCumber relation or reciprocity method [10,11,12]:1.
This method has been widely used to calculate emission cross sections. However, since it directly relies on measured values for the absorption cross sections, it proves difficult to deduce reliable values for in spectral regions where the absorption is weak. This is particularly the case in spectral regions having significantly longer wavelengths than the zero phonon line, since small uncertainties in will result in huge errors of .
To overcome this, we additionally used the so-called Fuchtbauer–Ladenburg equation, which enabled us to directly calculate the emission cross sections from the fluorescence spectra :
Since both methods yield results with higher accuracy in different spectral regions, the determination of the emission cross sections using a combination of both approaches leads to a more reliable result. In Fig. 2, the results for at room temperature for both approaches and the combined emission cross sections are plotted. It is obvious that the values obtained from the calculation based on the reciprocity method are reasonably well defined close to the absorption bands, while there is an increasing uncertainty for longer wavelengths. In contrast, values deduced using the Fuchtbauer–Ladenburg (F–L) approach yield a good accuracy in these regions, while the values obtained close to the zero phonon line are too small due to radiation trapping. In an intermediate spectral region between about 990 and 1030 nm, both procedures lead to the same values, which is a proof for the validity of the calculations and the used parameters. For the combined final spectrum, an average of both methods is used within the area of agreement, while the higher energetic spectral parts only rely on the result from the reciprocity relation and the lower energetic spectral parts use only the results from the Fuchtbauer–Ladenburg equation. The spectra for other temperatures and the other two materials were derived in the same way.
C. Determination of Radiative Lifetimes
Lifetimes published in the literature are normally considered as fluorescence lifetimes. This quantity, however, includes effects such as radiation trapping or from parasitic transitions. In this case, the values are strongly dependent on the actual measurement setup and especially on the doping concentration and geometry of the characterized sample. As a result, the measured values differ significantly from the actual radiative lifetime, which does not include such effects. However, this is the relevant quantity, e.g., for determining the cross sections.
In doped crystals, the effect of parasitic transitions is typically small and can be neglected, especially at lower doping concentrations. The main effect causing a difference between radiative and fluorescence lifetimes is caused by reabsorption, which in general leads to higher values or amplified stimulated emission in case of high excitation. A common method to suppress these effects is to measure the lifetime within an area that is as small as possible. Reliable values for the radiative lifetimes of the investigated materials were achieved with different approaches in the past. Sumida and Fan  published results from a method where they placed a thin sample between two undoped and index-matched plates (sandwich). The measured values were significantly smaller than the values published earlier without these precautions for Yb:YAG  and Yb:LuAG . More recently, the pinhole method developed by Kuehn et al.  led to very similar results for Yb:YAG.
In addition to the values in the literature, we carried out measurements of the lifetime with our setup. The measurements were performed by reimaging the output of the fiber formerly connected to the optical spectrum analyzer onto a photodiode equipped with a 1000 nm high pass edge filter. The fluorescence signal was generated by exciting the sample with the fiber coupled laser diode in pulsed mode. The signal was recorded with a digital oscilloscope, and the decay time was determined by fitting the falling edge of the signal to an exponential function with the lifetime as the fitting parameter. This measurement was done at different temperatures for all materials. Within the accuracy of the measurement, no temperature dependence could be detected.
Because the radiation from the pump diode was coupled into the sample by imaging the fiber output to the sample surface resulting in a small focal spot, the fluorescence was effectively measured from a virtually thin sample. This reduced the influence of radiation trapping to a minimum, and hence the measured value of the lifetime is expected to be very close to the actual radiative lifetime.
Comparing our results with other measurements (see Table 2) from literature, we achieved good agreement in the case of Yb:YAG. For Yb:LuAG, we measured a fluorescence lifetime of () ms, a value that is about 10% smaller than those reported from other groups [16,4] (see Table 2). For samples of LuAG doped with more than 10 at. % , a reduction of the radiative lifetime was also reported in . The value of 0.83 ms given there for a 15 at. % sample matches our result within the error of our measurement. Since the effect is probably caused by quenching  and so far does not correspond to the real radiative lifetime, we used the higher values reported in the literature for the cross section calculations. In case of , we obtained a value of () ms, which is smaller than the 2.4 ms reported for such crystals . The reason for this difference is not quite clear. Since the values measured in our experiments fitted well for our cross section calculation to match the result from F–L-method and McCumber relation, we believe that our measurement is correct for the investigated crystal with a 3 at. % doping concentration.
D. Absorption Cross Sections
For Yb:YAG, the absorption measurements were taken with a 2 at. % doped crystal of 5 mm thickness. The absorption cross sections show a significant dependence on temperature (Fig. 3). Particularly, on the widely used pump bands at 940 nm, the absorption cross section is reduced by per 50 K and even about 20% per 50 K for the zero phonon line at 970 nm. Furthermore, since the absorption bands are getting broader in the high temperature case, adjacent spectral lines are joined and the spectrum is getting smoother, which also holds for the emission. In the wavelength range around 1030 nm, which is typically used for laser operation, the absorption increases with increasing temperature.
The absorption measurements for Yb:LuAG (Fig. 4), contrary to Yb:YAG, have been performed using a sample with a doping concentration specified by the vendor with 15 at. %. Unfortunately, it is so far not possible to get an accurate value for the doping concentration of our Yb:LuAG sample. To get comparable values, we estimated the doping value to be 18 at. % by adjusting the peak absorption cross section to be around 940 nm, which is the value given in other publications on this material [4,13,16]. In general, the thermally induced changes on the cross sections for Yb:LuAG are the same as for Yb:YAG, which is also valid for the general shape of the curves. Yb:LuAG features a slightly smaller absorption cross section around 940 nm, while it is higher at 970 nm as compared to Yb:YAG. The substructures in the main absorption band between 910 and 950 nm are more pronounced. In the high temperature case, Yb:LuAG shows a flat, almost plateaulike absorption distribution arising from the combination of two neighboring peaks.
In comparison to the other materials behaves completely differently, since it is a broadband amplification material. Our sample was doped with 3 at. % where spectral characteristics are known to correspond to the formation of -clusters [14,19]. The cross sections (Fig. 5) around the wavelength of 940 nm are lower than for the other materials investigated but broader and less structured. The general evolution of the cross sections for higher temperatures is similar as for Yb:YAG and Yb:LuAG, but the relative differences are smaller.
To achieve a direct comparison between the different materials, a normalized plot of the absorption cross sections (Fig. 6) at the prominent pump bands near the zero phonon line and around 940 nm highlights the similarities between Yb:YAG and Yb:LuAG. On the contrary, is less sensitive to temperature changes, which will be an advantage in applications where a temperature gradient cannot be avoided, e.g., in large aperture amplifiers of high power lasers.
E. Emission Cross Sections
The emission cross sections undergo similar changes with temperature as do the absorption cross sections. Again, the strong similarity for Yb:YAG and Yb:LuAG is easy to see (Figs. 7 and 8). The main emission line at 1030 nm is reduced by more than 20% per 50 K temperature increase in the case for Yb:YAG and Yb:LuAG, while the secondary emission line close to 1050 nm is vanishing in the edge of the 1030 nm peak. Furthermore, the spectral width of the main emission line is also broadened, which results in a larger amplification bandwidth. The most pronounced difference between Yb:YAG and Yb:LuAG from a spectral point of view is the peak emission cross section, which is about 30% higher for Yb:LuAG.
Again for , the variations with temperature are less pronounced than in the two other high gain materials (Fig. 9). The main emission band, which is slightly structured at room temperature, becomes smoother in the high temperature case, while the height of the band is reduced by a rate of slightly more than 10% per 100 K temperature rise.
Figure 10 shows the evolution of the main emission cross sections at 1030 nm. Again proves to be much less affected by temperature changes in comparison to the other materials. Between Yb:YAG and Yb:LuAG, no significant differences can be found, meaning, that these materials should perform comparably.
3. IMPACT ON LASER PERFORMANCE ILLUSTRATED BY SPECIFIC EXAMPLES
In this section, we present calculations to demonstrate some selected effects, such as temperature dependent gain for optical amplifiers and temperature dependent lasing thresholds for idealized laser systems. The investigated setups used for the calculations are virtual and simplified and are only intended to illustrate a general behavior.
A. Impact on Pulsed Laser Amplifiers
Often amplifiers are modeled under the assumption that within the operational scheme of the laser, the cross sections of the active medium are invariable. Actually, this is only valid if aggressive cooling schemes are applied. In unspecialized cooling schemes, significant temperature variations are likely to occur. Hence, the assumption of constant cross sections will not correctly describe the real laser performance.
To get an idea about the influence of thermally induced variations of the cross section, we have employed a model similar to the one used in . This model is based on solving the laser rate equation and modeling the radiation transport in a simplified manner.
For the pumping process, the inversion is numerically calculated by solving two coupled differential equations representing the temporal evolution of the inversion at a certain point in space [Eq. (6)] and the evolution of the pump Intensity for time [Eq. (7)]:
In a second step, the amplification of the laser pulse propagating through the material is calculated neglecting losses due to spontaneous emission and in a repetitive way by propagating the pulse in temporal slices through the crystal. This is done by solving
In this way, we have modeled a Yb:YAG amplifier pumped at a power density of and with rectangular pulses of 0.95 ms in duration at a wavelength of 940 nm. The crystal is assumed to be prepared with a 3 at. % doping concentration and a thickness of 8 mm. As the seed pulse, a top-hat profile in time with a duration of 2 ns and a fluence of at 1030 nm was assumed. This simplified layout of an amplifier applies to a Yb:YAG amplifier designed in such a way to achieve a gain of about 100 with an extraction fluence in the range of , which is close to saturation.
In Fig. 11, the effect of a temperature rise in the laser crystal is plotted as a function of the number of passes through the active medium. One can see that even a moderate temperature increase affects the amplifier output significantly. Operation at a higher temperature requires a much higher pump fluence for getting reasonable gain. This is due to the strong reduction of the emission cross section. Furthermore, it can be seen that the effect on the output energy is especially critical if saturation is not achieved within a fixed number of passes.
The reduction of the output power is one thing to keep in mind but can be compensated by increasing the launched pump power or increasing the number of passes. For large aperture amplifiers, another effect should be considered, caused by an inhomogeneous temperature distribution across the beam profile. In Fig. 12, the result for such a scheme is demonstrated. Here we assumed a pump distribution with a super-Gaussian shape in space. The seed pulse of the amplifier is modeled as a super-Gaussian with an order smaller than that of the pump distribution. The temperature distribution orthogonal to the optical axis is estimated to be Gaussian, with a maximum temperature rise of 50 K in the center. The -radius of the distributions is assumed to be 15 mm. All other parameters are taken from the model described before.
From our simulation, it can be seen that the original beam-shape will be distorted due to the lower small signal gain in the center of the beam, which leads to a dip in its profile. This dip is getting stronger during the nonsaturated amplification in the first five passes. In the subsequent passes, this dip is reduced; hence saturation occurs first in the wings and later in the center of the profile. Comparable results were obtained by performing the same calculations with the other materials characterized in this paper.
The outcome of these simulations is that even small changes in the temperature of the laser material cause differences in the emission cross sections that are not negligible in pulsed operated laser amplifiers.
B. Impact on cw-Operated Lasers
As already shown for amplifiers employing -doped gain media, variations in temperature have a significant influence on the population of the lower laser level and thus on the laser performance and the threshold behavior. Quasi-three-level lasers require the excitation of a significant fraction of the active ions; hence pump saturation effects must be taken into account. The reduced absorption of the pump light can be described by the absorption efficiency 
For quasi-three-level lasers, the population density , which must be reached for lasing, can be expressed as 21]
The overall lasing threshold is the sum of the transparency threshold and the cavity threshold [Eq. (12)]. Fig. 13 shows the transparency threshold for different crystal temperatures. There is a remarkable rise of the transparency threshold for the three crystals, from 20 to 200 °C. The most pronounced increase could be observed with Yb:LuAG and Yb:YAG. At 200 °C, the transparency threshold for both crystals is about 2.8 times higher as compared to the threshold at 20 °C. Another important parameter for laser design is the cavity threshold of a material. This quantity depends on the internal cavity losses and out-coupling factor but also on the absorption and emission cross section of crystal at the laser wavelength (Fig. 14). It can be observed that shows a different temperature dependence compared to Yb:YAG and Yb:LuAG. While the cavity threshold for the latter two materials nearly doubles from 20 to 200 °C, the threshold for increases only by a factor of 1.1 in the same temperature regime.
The temperature-dependent cross sections do not only affect lasing thresholds; significant temperature induced local absorption coefficient variations may be present in lasers, too. For further examinations, a simple analytical model is used. It is based on an axially cooled thin disk laser (Fig. 15). These lasers exhibit considerable temperature gradients between the cooled back side and the front side of the pumped crystal. Since is normally not used in thin disk lasers in the observed temperature regime, the simulation focuses on Yb:YAG and Yb:LuAG. The heat transfer inside the disk can be calculated analytically under the assumption of an approximately pure axially heat flow. This can be assumed if the pump mode has at least four times the diameter of the thickness of the disk . As the thermal conductivities of Yb:LuAG are of a similar order of magnitude as Yb:YAG, this approximation should be sufficient to demonstrate the influence of temperature variations on laser emission. More detailed simulations on the pumping process with temperature-dependent cross sections have been published by Toroghi et al.  and Najafi et al. .
The absolute heat deposition in the pumped volume of the crystal is given by 15) is then given by 
The absorbed pump power leads to an increase in the mean temperature of the crystal, which in turn affects the amount of the absorbed pump power. To model this effect, the absorbed pump power is determined. In a following step, the mean average temperature of the crystal is corrected accordingly. After that, the absorbed pump power is determined again. This cycle is repeated until the relative change in average temperature of the crystal is smaller than .
The thickness of the crystal and the doping concentration was chosen to lie in the range of current commercial thin disk lasers. In the following, we assumed that the measured absorption and emission spectra are still valid at the considered doping concentration. Parameters used for the simulation are given in Tables 3 and 4.
Figure 16 shows the mean temperature inside the pumped volume of the crystal as a function of pump intensity. Since the amount of absorbed pump light depends on the axial position in the crystal, the absorption cross section varies along the axial direction of the disk. Also, it should be mentioned that the absorption and emission cross sections will be a function of the radial position since a temperature gradient exists along this axis for real systems, too. Since a detailed investigation is very complex, we neglect this effect here.
Assuming a uniform heat deposition and an axial heat flow, the change in absorption cross section for the pump light can be calculated using Eq. (14). The results are shown in Fig. 17 for a pump intensity of . It can be seen that there is a significant change in the absorption cross section for the pump light of about 10% in maximum from the surface of the disk attached to the heat sink to the front face of the disk. This means that more heat is deposited in the area of higher absorption. As a consequence, this might lead to a slightly different axial temperature distribution, which is not yet included in our model.
Absorption and emission cross sections have been measured for the temperature range from 20 to 200 °C for Yb:YAG, Yb:LuAG, and . All materials showed a similar tendency for the dependence of cross sections on temperature. Nevertheless, this influence is much more pronounced for Yb:YAG and Yb:LuAG than for . The absorption cross sections in the spectral regions used for pumping are reduced, as well as the emission cross sections at laser wavelengths. In general, this leads to an overall worse performance for the laser operation at higher temperatures.Using the spectral data as a basis for laser simulations, we demonstrated that even small temperature changes in the range of 10 K will result in considerable differences in the performance of laser amplifiers. This expresses the need for sophisticated cooling designs and measures to reach a homogenous temperature distribution in high power solid-state lasers based on doped media. One might also consider knowingly using the dependence of the performance for, e.g., beam shaping, by applying a certain temperature profile to the gain medium.
In case of cw thin disk lasers, it could be shown that there is a great impact on threshold behavior in the observed temperature regime. In the case of Yb:YAG and Yb:LuAG, the transparency threshold increased by a factor of 2.8 in the temperature range from 20 to 200 °C, while experiences only a small increase of 1.1.
With the help of a simple model, it could be shown that the axial temperature profile causes a drop in pump light absorption in parts of the disk. For a pump intensity of , the change in absorption cross section for the pump light from one end to the other is about 10%.
This work was partly supported by the European Regional Development Funds (ERDF) through the Thuringian Ministry of Education, Science, and Culture (project numbers B514-09050 and B715-09012) and the European Social Fund (ESF) through the Thuringian Ministry of Economy, Employment, and Technology (project number 2011 FGR 0122).
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