1 Introduction

Tm:YAG lasers operating at 2 μm wavelengths have attracted a lot of attention owing to their strong absorption in water, as well as the eye-safe wavelength bands, which have been utilized extensively in soft-material processing, medical diagnostics, direct optical communications, and so on^[1^–3^]. One of the main challenges is related to the induced laser performance degeneration that arises from waste heat generated inside the crystal, which is detrimental to the cited applications. Increasing the heat dissipation capability and analyzing thermal problems in Tm:YAG lasers are thus critical for laser developers.

Efficient heat removal is realized by increasing the surface-to-volume ratio of the gain medium, as in fiber, slab, and disk lasers^[4^–7^]. Among them, disk architectures have several advantages, including efficient cooling capacity, weak nonlinear effect, and power scaling without compromising beam quality^[8^]. The highest published output power of a Tm:YAG disk laser is 24 W with M2 value of 11.7 and slope efficiency of 31% when pumped at 785 nm, in which the presence of cross-relaxation (CR) can increase the pump quantum efficiency^[9^,10^]. In addition, the use of a 1 μm laser as the pump source has also proved successful in generating 2 μm radiation^[11^]. This is a readily available and cost-effective pump method, because the 1 μm laser (e.g., lamp-pumped Nd:YAG laser,Yb³⁺ doped laser) is commercially mature with high power, high reliability, and mass production.Very recently, an intra-cavity pumped Tm:YAG disk laser in multi-transverse mode operation was reported with an average output power of 760 mW in quasi-continuous wave mode and 17 mW in continuous wave mode^[12^]. The relatively low absorption coefficient of 1 μm laser for a disk-shaped gain medium is compensated by the intra-cavity pumping method^[11^].Nevertheless, its poor pumping stability and strong nonlinear coupling can degrade the output laser performances significantly^[13^,14^]. It is then difficult to study the operation characteristics of 1 μm pumped Tm:YAG disk laser in detail.

On the other hand, because a high population inversion is necessary to compensate for the limitation of the disk’s thickness, the intensities of transitions in competition, up conversion (UC) and excited state absorption (ESA), cannot be ignored, as is done in rod and fiber-shaped lasers^[4^]. The enhancement of output power is then limited and more heat is generated in Tm:YAG disk lasers. Therefore, verifying the proportions of all transitions and evaluating their effects on operation characteristics and heat generation are extremely important forTm:YAG disk lasers.

In the present paper, a dual parabolic mirror 20-pass pumping scheme is designed for the1 μm pumped Tm:YAG disk laser. The multi-pass pumping method ensures high pumping stability and efficient thermal management of the gain medium^[15^]. This scheme can also enable the measurement of the unabsorbed pump power in real time. The absorption properties of 1 μm pump light are then studied efficiently. In addition, a numerical model considering stimulated emission (SE), fluorescence, non-radiative decay, UC, and ESA is developed to evaluate the proportions and fractional thermal loads of all transitions quantitatively in view of the absorption, output, and temperature properties in this laser.Experimental and theoretical results show that the UC, ESA, and SE compete with each other to consume the1 μm pump photons, and the proportions of UC and ESA are more dominant than that of SE in a1 μm pumped Tm:YAG disk laser, leading the absorbed pump power to decrease abnormally after lasing. The strong UC and ESA effects are the main part of total heat generation for two thicknesses of gain medium. In addition, the small thickness of gain medium with better heat dissipation capacity has relatively weak UC and ESA and low total fractional thermal load, which can improve the output laser performances. Consequently, a 0.5 mm Tm:YAG disk laser delivered an average power of 1.05 W with slope efficiency of 13.2%, and the measurement of it yielded M_x² = 2.02 and M_y² = 2.03.

2 Theoretical analysis

The energy diagram of a Tm:YAG crystal is shown in Figure 1, where ${\lambda}_{{P}}$ and ${{\lambda}}_{{L}}$ are the wavelengths of the pump and output laser, respectively.

In Figure 1, the processes of heat generation in a Tm:YAG crystal could be divided into the following four parts: quantum defect heating, non-radiative decay heating from the ³F₄ state, fluorescence heating from the ³F₄ state, and UC- and ESA-induced heating. The heating energy density due to the quantum defect caused by SE can be described as

$$\begin{align}{Q}_L=\Delta n{\eta}_L\left(h{\upsilon}_P-h{\upsilon}_L\right),\end{align}$$ (3)

where $h$ is the Planck constant, ${\upsilon}_P$ is the frequency of the pump photon, ${\upsilon}_L$ is the frequency of the laser photon, and $\Delta n$ is the density of pump photons in laser medium. Here ${\eta}_L$ is the laser extraction efficiency, which is defined as the fraction of absorbed pump photons consumed by SE.

The heat energy density generated by non-radiative decay in the ³F₄ state can be described as

$$\begin{align}{Q}_{NR}^{3F4}=\Delta n\left(1-{\eta}_L\right)\left(1-{\eta}_R^{3F4}\right)\left(1-{\eta}_{UC\&ESA}\right)h{\upsilon}_P,\end{align}$$ (4)

where ${\eta}_R^{3F4}$ is the radiative quantum efficiency of the ³F₄ state. Meanwhile, the heating energy density caused by fluorescence decay in the ³F₄ state can be described as

$$\begin{align}\!\!\!{Q}_F^{3F4}=\Delta n\left(1-{\eta}_L\right){\eta}_R^{3F4}\left(1-{\eta}_{UC\&ESA}\right)\left(h{\upsilon}_P-h{\upsilon}_F^{3F4}\right),\end{align}$$ (5)

where ${\upsilon}_F^{3F4}$ is the frequency of average fluorescence wavelength from the ³F₄ state to the ³H₆ state and ${\eta}_{UC\& ESA}$ is the quantum efficiency of competing transitions (UC and ESA), representing the percentage of upper-level laser particles for UC and ESA. The additional absorption efficiency is then denoted by $\left(1-{\eta}_L\right){\eta}_{UC\& ESA}$, which represents the fraction of absorbed pump photons consumed by UC and ESA transitions. Here $\left(1-{\eta}_L\right)\left(1-{\eta}_{UC\& ESA}\right)\left(1-{\eta}_R^{3F4}\right)$ and $\left(1-{\eta}_L\right)\left(1-{\eta}_{UC\& ESA}\right){\eta}_R^{3F4}$ represent the fraction of absorbed pump photons consumed by non-radiative decay and fluorescence decay in the ³F₄ state, respectively.

The UC- and ESA-induced heating includes non-radiative decay heating from the ¹G₄ state ${Q}_{NR}^{1G4}$ and fluorescence heating from the ¹G₄ state ${Q}_F^{1G4}$, which can be described as

$$\begin{align}{Q}_{NR}^{1G4} =\Delta n\left(1-{\eta}_L\right)\left(1-{\eta}_R^{1G4}\right){\eta}_{UC\&ESA}h{\upsilon}_P,\end{align}$$ (6)

$$\begin{align}{Q}_F^{1G4}& =\Delta n\left(1-{\eta}_L\right){\eta}_{UC\&ESA}{\eta}_R^{1G4}\left(h{\upsilon}_P-h{\upsilon}_F^{1G4}/3\right),\end{align}$$ (7)

where ${\eta}_R^{1G4}$ is the radiative quantum efficiency for the ¹G₄ manifold and${\upsilon}_F^{1G4}$ is the frequency of the equivalent fluorescence wavelength from the¹G₄ state.

The total fractional thermal load of a Tm:YAG crystal is given by

$$\begin{align}{\eta}_h&=\frac{\left({Q}_L+{Q}_{NR}^{3F4}+{Q}_F^{3F4}+{Q}_{NR}^{1G4}+{Q}_F^{1G4}\right)}{\Delta{n}_Rh{\upsilon}_P} =\underset{\mathrm{laser}\kern0.33em \mathrm{heating}}{\underbrace{\eta_L\left(1-\frac{\lambda_P}{\lambda_L}\right)}}\notag\\[2pt]&\quad+\underset{\begin{array}{c}\mathrm{fluorescence}\kern0.33em\mathrm{and}\kern0.33em \mathrm{non}\text{-}\mathrm{radiative}\\[-3pt] \kern0.66em\mathrm{decay}\kern0.33em \mathrm{heating}\kern0.33em \mathrm{in}\kern0.33em {}^{3}\mathrm{F}_4\kern0.33em\mathrm{state}\end{array}}{\underbrace{\left(1-{\eta}_L\right)\left(1-{\eta}_{UC\&ESA}\right)\left(1-{\eta}_R^{3F4}\frac{\lambda_P}{\lambda_F^{3F4}}\right)}}\notag\\&\quad+\underset{\mathrm{UC}\text{-}\kern0.33em \mathrm{and}\kern0.33em\mathrm{ESA}\text{-}\mathrm{induced}\kern0.33em\mathrm{heating}}{\underbrace{\left(1-{\eta}_L\right){\eta}_{UC\&ESA}\left(1-{\eta}_R^{1G4}\frac{\lambda_P}{3{\lambda}_F^{1G4}}\right)}}.\end{align}$$ (8)

It is straightforward to show that the first term of Eq. (8) represents laser heating, the second term represents fluorescence heating and non-radiative decay in the ³F₄ state, and the third term represents UC- and ESA-induced heating via fluorescence and non-radiative decay in the¹G₄ state. In addition, the laser extraction efficiency is given by

$$\begin{align}{\eta}_L=\frac{\frac{I_L}{h{\upsilon}_L}{g}_L{M}_R}{\frac{I_P}{h{\upsilon}_P}{\alpha}_P},\end{align}$$ (9)

where ${I}_P$ is the average pump intensity in the crystal, ${I}_L$ is the power intensity of the circulating laser radiation inside the resonator,${M}_R$ is the number of the laser passing through the crystal per resonator round trip,${\alpha}_P$ is the absorption coefficient for pump radiation, and ${g}_L$ is the gain coefficient for laser. When the pump intensity is higher than laser threshold, the laser starts lasing and gives ${g}_L{M}_R=\delta /d$, where $d$ is the thickness of the gain medium. Here $\delta$ is the round trip loss and can be described as

$$\begin{align}\delta =-\ln \left(1-{T}_{oc}\right)-\ln \left(1-{L}_{loss}\right),\end{align}$$ (10)

where ${T}_{oc}$ is the output coupler transmission and ${L}_{loss}$ accounts for residual losses, owing to the scattering during one round trip in the resonator. Equation (9) is simplified as

$$\begin{align}{\eta}_L=\frac{\frac{I_L}{h{\upsilon}_L}\frac{\delta}{d}}{\frac{I_P}{h{\upsilon}_P}{\alpha}_P}=\frac{\lambda_L}{\lambda_P}\frac{I_L\delta}{I_{abs}}=\frac{\lambda_L}{\lambda_P}\frac{I_{out}\delta }{I_{abs}{T}_{oc}},\end{align}$$ (11)

where ${I}_{out}$ is the output power intensity and ${I}_{abs}$ is the absorbed pump power intensity. When the pump intensity is lower than laser threshold, the laser extraction efficiency is zero. The total fractional thermal load is simplified as

$$\begin{align}{\eta}_h&=\left(1-{\eta}_{UC\&ESA}\right)\left(1-{\eta}_R^{3F4}\frac{\lambda_P}{\lambda_F^{3F4}}\right)\notag\\&\quad +{\eta}_{UC\& ESA}\left(1-{\eta}_R^{1G4}\frac{\lambda_P}{3{\lambda}_F^{1G4}}\right).\end{align}$$ (12)

Based on the absorbed pump power ${P}_{abs}$ and the total fractional thermal load ${\eta}_h$, the central temperature of Tm:YAG disk is calculated by using the finite element method.The measured temperature data are then fitted with the calculated data by adjusting ${\eta}_{UC\& ESA}$ and ${\eta}_R^{1G4}$. Consequently, the proportions and fractional thermal load of all transitions are derived. Parameters used in the simulation are listed in Table 2.Table 2

Basic parameters used for the model.

Parameters	Value
Loss factor for laser in resonator(${L}_{loss}$)	2%
Cu–W plate thickness (${d}_{CuW}$)	2 mm
Thermal conductivity of Cu–W plate(${k}_{CuW}$)	385 W/(m·K)
Thermal conductivity of Tm:YAG crystal(${k}_{Tm: YAG}$)	7 W/(m·K)[19]
Convective heat transfer coefficient(${K}_{heat}$)	10 W/(m2·K)[20]
Radiative quantum efficiency of3F4 state (${\eta}_R^{3F4}$)	0.98[21]
Temperature of coolant fluid (${T}_{cool}$)	292 K

3 Experimental setup

The Tm:YAG disk crystal was mounted on a water-cooled copper heatsink at 18°C with a doping concentration of 3.5%, and its thickness and pump spot size were 0.5 and 1.7 mm, respectively. In addition, the front surface of the Tm:YAG disk had an antireflection coating at both 1 and2 μm wavelengths, and the back surface had a highly reflective coating in the same wavelength band. The absorption coefficient of Tm:YAG crystal at 1 μm was around 1.37× 10^–3 mm^–1. Moreover, the range of dioptric power of the 0.5 mm disk was measured as 0.1–0.93 m^–1[22^]. A V-shaped cavity with large stable area was designed to match the dioptric power. M1 was a concave mirror with curvature radius of 1 m, and it had a high reflectivity at 2 μm. OC1 was a plane output coupler at 2 μm with a transmittance of 1%, which was the optimum transmissivity by measuring the highest output power. The distances between components in the cavity were L₁ = 0.455 m and L₂ = 0.25 m. Meanwhile, a 1 mm thick Tm:YAG disk laser was also analyzed as a comparison, which shared the same coating parameters and doping concentration as the 0.5 mm disk. To realize mode matching in terms of dioptric power of the thick disk, the cavity lengths of two arms were adjusted toL₁ = 0.55 m and L₂ = 0.25 m, respectively. The number of pumping passes and pumped spot size were the same as in the 0.5 mm thickness case. In addition, to study the properties of absorbed pump power and output power, P1 and P2 were used to measure the unabsorbed pump power ${P}_{unabs}$ and output laser power ${P}_{out}$, respectively.

4 Results and discussion

During the experiments, a Yb:YAG thin disk laser was used as the pump source, and the maximum input pump power of the Tm:YAG disk laser was ${P}_{in}$ = 65.8 W. To prevent disk damage, the pump power was limited such that the disk temperature never exceeded 125°C.

4.1 Operation characteristics of a 1 μm pumped Tm:YAG disk laser

In practice, the reflection of 1 μm laser in the multi-pass pumping scheme is incomplete^[15^]. Considering the loss of the multi-pass pumping scheme, the absorbed pump power ${P}_{abs}$ is given by

$$\begin{align}{P}_{abs}={P}_{in}-{P}_{unabs}-{P}_{in}\left(1-{R}_p\right),\end{align}$$ (13)

where ${R}_p$ is the total reflectivity of the Tm:YAG multi-pass pumping scheme with the measured value of 77%.

On the other hand, the absorbed pump power of the Tm:YAG disk is significantly different under lasing and non-lasing conditions, as shown in Figure5. Under lasing condition, the absorbed pump power of Tm:YAG decreases

obviously for both 0.5 and 1 mm thicknesses of gain medium. This abnormal phenomenon is very interesting, and shows us the dynamic absorption properties of 1 μm pump light for a Tm:YAG disk before and after lasing, which is equivalent to the consumption of population inversion on the upper laser level. In general, the depletion of upper laser level induces an increased absorption of pump light for a laser medium with simple energy level, e.g., Yb:YAG orNd:YAG^[23^]. However, the Tm:YAG crystal shows the abnormal changes of absorbed pump power when pumped by the 1 μm light. This is attributed to the significant UC and ESA effects. To show the operation characteristics of1 μm pumped Tm:YAG disk laser clearly, an operating principle diagram under non-lasing and lasing conditions is presented in Figure 7. The absorbed pump power of Tm:YAG mainly consists of two parts: direct absorption via SA and CR, and additional absorption via UC and ESA. Above the laser threshold, the SE, UC, and ESA compete with each other to consume the population inversion on the ³F₄ state, and this shows the dynamic absorption properties of the pump light. Under the non-lasing condition, no SE occurs, and the significant UC and ESA effects allow high absorption of the 1 μm laser. At the input pump power of 57.8 W, the absorbed pump power of the 0.5 mm disk was 22.5 W. Under the lasing condition, the generation of SE makes the direct absorbed pump power increase due to the depletion of the ³F₄ state. However, the consumption of population inversion on the ³F₄ state can suppress the intensities ofUC and ESA. The additional absorbed pump power, which is used for UC and ESA, decreases accordingly. As the intensities of UC and ESA were more dominant than that of SE, the absorbed pump power of the gain medium was decreased to 18.7 W. On the other hand, the increased temperature of Tm:YAG is mainly caused by the UC and ESA when pumped by 1 μm light, and the temperature of Tm:YAG thus drops from 121.1 to91.1°C after lasing at 2 μm, as shown in Figure7.Figure7

Operating principle diagram of the 0.5 mm Tm:YAG disk laser under (a) non-lasing and (b) lasing conditions. The thicknesses of the arrows are proportional to the intensities of the transitions.

Figure7

Operating principle diagram of the 0.5 mm Tm:YAG disk laser under (a) non-lasing and (b) lasing conditions. The thicknesses of the arrows are proportional to the intensities of the transitions.

4.2 Evaluation intensities of UC and ESA by calculating ${\eta}_{UC\& ESA}$and${\eta}_R^{1G4}$

As mentioned previously, the Tm:YAG disk laser has significant UC and ESA, which has been described qualitatively. In this section, the intensities of UC and ESA are evaluated quantitatively by calculating${\eta}_{UC\& ESA}$ and ${\eta}_R^{1G4}$. The finite element method is used to calculate the central temperature of the gain medium on account of ${P}_{abs}$ and ${\eta}_h$. The additional absorption efficiency ${\eta}_{UC\& ESA}$ and radiative quantum efficiency for the ¹G₄ manifold${\eta}_R^{1G4}$ are obtained from a least-squares fit to the measured temperature data.

The dependences of central temperature for a 0.5 mm Tm:YAG disk laser on absorbed pump power are shown in Figure 8(a). When the absorbed pump power is below 0.1 W, the power density is <240 W/cm³, which is less than the ESA threshold^[11^,12^]. The ESA effect can then be neglected, i.e., ${\eta}_{UC\& ESA}$= 0. When the absorbed pump power is above the ESA threshold, the best fit to data gives ${\eta}_{UC\& ESA}$ = 99.6% and ${\eta}_R^{1G4}$ = 0.3% for the crystal by using the finite element method. The calculated central temperature can be obtained based on the thermal load model. It can be seen that the temperatures of Tm:YAG disk are directly proportional to the absorbed pump power, and are significantly different under lasing and non-lasing conditions. Under lasing condition, the intensities ofUC and ESA, denoted by $\left(1-{\eta}_L\right){\eta}_{UC\& ESA}$, are decreased due to the increased laser intensity ${\eta}_L$ via SE. As the increased temperature of Tm:YAG is mainly caused by the UC and ESA when pumped by 1 μm light^[12], the maximum temperature of Tm:YAG is thus decreased from122.3°C to 108.5°C at the absorbed pump power density of 2.0 kW/cm³. Moreover, the central temperature of a 1 μm pumped Tm:YAG disk crystal was comparable with the 785 nm pumped case, where the disk temperature of 150°C was reached at an absorbed pump power density of 3.1 kW/cm^3[10^]. This is due to the fact that the UC and ESA effects are significant for both 785 nm and1 μm pumped Tm:YAG disk lasers.Figure8

Central temperature of (a) 0.5 mm Tm:YAG disk and (b) 1 mm Tm:YAG thick disk versus absorbed pump power under lasing and non-lasing conditions.

Figure8

Central temperature of (a) 0.5 mm Tm:YAG disk and (b) 1 mm Tm:YAG thick disk versus absorbed pump power under lasing and non-lasing conditions.

On the other hand, when the absorbed pump power is above the ESA threshold(0.2 W of absorbed pump power), the 1 mm thick disk exhibits approximately ${\eta}_{UC\& ESA}~=~99.7\%$ and ${\eta}_R^{1G4} = 0.2\%$ by using the same method. As the heat dissipation of the 1 mm thick disk is lower than the 0.5 mm disk, the temperature of the 1 mm thick disk is much higher than in the0.5 mm case at the same absorbed pump power, as shown in Figure 8(b). At the input pump power of 38.1 W, the central temperature of the gain medium has increased by 50% from 70°C to 105°C. In addition, the intensity of UC and ESA for the 1 mm thick disk is higher than in the 0.5 mm case at the same absorbed pump power, and this is attributed to the weaker intensity of SE for the 1 mm thick disk, as shown in Figure 6.

Fig. 9. Proportions of all the transitions for (a) 0.5 mm Tm:YAG disk laser and (b) 1 mm Tm:YAG thick disk laser under the lasing condition.

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On the other hand, the tendencies of the 1 mm Tm:YAG thick disk are similar to the 0.5 mm case over the range of absorbed pump power. At the absorbed pump power of 16.5 W, the laser proportion of the 1 mm thick disk is 19%, which is lower than that of the 0.5 mm disk, in which a laser proportion of 24% is delivered. Conversely, compared with the 0.5 mm disk, the proportion of UC andESA for the 1 mm Tm:YAG disk increases from 75% to 82%, indicating that the 1 mm Tm:YAG disk has higher UC- and ESA-induced heating, thereby a higher total fractional thermal load. The higher heat generation, as well as poorer heat dissipation capacity, makes the temperature of the 1 mm thick gain medium much higher than that of the 0.5 mm case at the same absorbed pump power, as shown inFigure 8. Therefore, we expect that the Tm:YAG disk with lower thickness and better heat dissipation capacity can achieve higher power and efficiency via this scheme.

5 Conclusion

In summary, we have reported a 2 μm Tm:YAG disk laser pumped by a 1 μm laser via a 20-pass pumping scheme. This scheme enables the evaluation of absorption, output, and temperature properties. The competitive transitions of UC and ESA particularly challenge the scaling of output powers for 2 μm disk lasers. To analyze the effects of UC and ESA, a numerical model accounting forSE, fluorescence, non-radiative decay, UC, and ESA has been developed. More broadly, the proportions and fractional thermal loads of all transitions have been derived quantitatively based on the model.Experimental and theoretical results show that the UC and ESA effects are dominant in 1 μm pumped Tm:YAG disk lasers, leading the absorbed pump power to decrease abnormally after lasing. The strongUC and ESA are the main sources of total heat generation of the gain medium. In addition, although theTm:YAG disk with small thickness has a lower absorbed pump power, its higher heat dissipation capacity enables the relatively weak UC and ESA and low total fractional thermal load, which are conducive to the promotion of laser performance. As a result, the 0.5 mm Tm:YAG disk has a maximum output power of1.05 W, limited only by the available pump power, and the measurement of it yielded M_x² = 2.02, M_y² = 2.03. Finally, this model can be used for any Tm³⁺-doped disk laser, which lays the groundwork for analyzing the operation characteristics and evaluating the dynamic processes of all transitions for disk-shaped Tm³⁺-doped lasers. Further reducing the thickness of the gain medium, together with increasing pump number and spot size, will be propitious to scaling the Tm³⁺ disk lasers’ output power without compromising beam quality and mode stability in the near future.