1 Introduction

High-power Holmium (Ho) solid-state lasers at ~ 2.1 μm are attractive for extensive applications such as wind lidar [1], countermeasures [2], medical surgeries [3], and material processing [4], owing to their lasing wavelengths falling within the atmospheric window and water absorption band. Additionally, they can also serve as the ideal pump sources for generating mid-infrared lasers toward 3–12 μm via nonlinear frequency conversion [5, 6].

Thulium (Tm)-sensitized Ho materials, compatible with commercial laser diodes (LDs) around 785 nm pumping [7], stand as a good candidate for advancing the development of high power miniaturized Ho lasers. Over the past decades, Tm–Ho co-doped lasers employing different hosts, including YAG, YLF, YAP, and YVO4, have been successfully demonstrated in the generation of lasers around 2 μm [8,9,10,11]. Among these materials, Tm,Ho:YAG stands out for its high thermal conductivity and robust mechanical properties, making it an ideal choice for high-power room-temperature (RT) laser operations. Its mature preparation process and cost-effectiveness further expand its applicability across a wide range of uses. Several LD end-pumped rod Tm,Ho:YAG lasers have been studied, producing output powers ranging from several tens of milliwatts to a maximum of 1.2 W [8, 12, 13]. However, power scaling in this geometry is hindered by the presence of strong thermally induced lensing and stress. Therefore, the adoption of a side-pumping geometry, characterized by the high pump power density and the reduced thermal gradient, is more favorable for high output energy or power. For example, a maximum output power of ~ 37 W was achieved by LD side-pumped rod Tm,Ho:YAG laser at a temperature of 6 ℃ [14]. A significant challenge in the pursuit of higher output power with Tm,Ho:YAG lasers at RT is the substantial heat generation as a result of multiple up-conversion processes. Given the quasi-three-level nature of the co-doped system, one can expect higher power output from these lasers by lowering the operating temperature, such as employing cryogenic cooling [15]. Nevertheless, this benefit is offset by the added system complexity, which can be costly. Most recently, our group has demonstrated a high brightness Tm,Ho:YAG laser at near RT using a laser rod with low dopant concentrations of Tm3+ and Ho3+ but maintaining a high dopant concentration ratio [16]. This approach has the potential to reduce the up-conversion losses, thereby mitigating the depletion of the population inversion in the upper laser level, which favors both high-power and high-energy operations [17, 18]. Therefore, it is worthwhile to assess the feasibility of such dopant method using rate equations and to explore the output limitations of Tm,Ho:YAG laser.

In this paper, we studied an LD side-pumped rod Tm,Ho:YAG laser with a high dopant concentration ratio of 30 (3.0 at.% Tm3+ and 0.1 at.% Ho3+). Through careful design of the system elements, we achieved a maximum continuous-wave (CW) output power of 87.7 W at 2.09 μm with an optical-to-optical (o–o) efficiency of 13.3% and a slope efficiency of 21.2%. Additionally, the laser output characteristic was analyzed using a simplified quasi-three-level model for the side-pumped geometry. The simulation results agreed well with experimental data, indicating that a modified co-operative up-conversion rate was well suited for modeling the side-pumped Tm,Ho:YAG laser with a high dopant concentration ratio. These findings held great importance for guiding the subsequent development of the Tm–Ho co-doped lasers.

2 Numerical model

The Tm–Ho co-doped laser system exhibits a quasi-three-level characteristic around RT. The rapid energy transfer between Tm3+ and Ho3+ ions allows their upper level to be treated as a combined upper level (CUL) [19]. While multiple up-conversion energy transfer mechanisms exist, for simplicity purposes, we primarily focus on the Tm–Ho co-operative up-conversion, which is predominantly responsible for depleting the population of the upper laser level within this system [20]. Therefore, the rate equation for this CUL in a steady state can be described as [19]:

$$ \frac{{{\text{d}}N_{u} }}{{{\text{d}}t}} = R_{p} (r,z) - \frac{{N_{u} }}{\tau } - \frac{{f_{Tm} f_{Ho} }}{2}uN_{u}^{2} - \frac{\sigma c}{n}\Delta N\Phi (r,\theta ,z) = 0, $$
(1)

where \(N_{u}\) represents the populations of both \(^{{3}} {\text{F}}_{{4}}\) and \({}^{{5}}{\text{I}}_{{7}}\). \(R_{p}\) is the normalized spatial pump distribution. For a side-pumped rod geometry, we assume a uniform top-hat distribution for the pump intensity within the rod. Consequently, \(R_{p}\) can be expressed as [21]:

$$ R_{p} \left( {r,z} \right) = \left\{ {\begin{array}{*{20}c} {\frac{R}{{\pi \omega_{p}^{2} l}},} & {r \le \omega_{p} ,} \\ {0,} & {r > \omega_{p} ,} \\ \end{array} } \right. $$
(2)

where \(\omega_{p}\) is the pump beam radius. The total pump rate \(R\) is defined as [19]:

$$ R = \frac{{\eta_{a} \eta_{p} }}{{h\nu_{p} }}P_{in} , $$
(3)

where \(\eta_{a}\) and \(\eta_{p}\) denote the absorption efficiency and pump quantum efficiency, respectively. \(h\nu_{p}\) is the energy of each pump photon, and \(P_{in}\) is the total incident pump power. The term \(N_{u} /\tau\) in (1) represents the spontaneous emission from CUL and its lifetime \(\tau\) is defined as follows [19]:

$$ \frac{1}{\tau } = \frac{{f_{Tm} }}{{\tau_{Tm} }} + \frac{{f_{Ho} }}{{\tau_{Ho} }}, $$
(4)

where \(\tau_{Tm}\) and \(\tau_{Ho}\) are the lifetime of \(^{{3}} {\text{F}}_{{4}}\) and \({}^{{5}}{\text{I}}_{{7}}\) level of Tm3+ and Ho3+, respectively. \(f_{Tm}\) and \(f_{Ho}\) represent their fractional populations in the CUL, following a Boltzmann distribution that is directly influenced by the temperature T. The final two terms on the right-hand side of Eq. (1) represent the co-operative up-conversion loss and simulated emission. Here, \(u\) corresponds to the up-conversion rate, \(\sigma\) is the stimulated emission cross-section of Ho3+, \(c\) is the speed of light, \(n\) is the refractive index of YAG, \(\Delta N\) is the population inversion, and \(\Phi\) is the spatial photon density distribution.

The total photon number in the cavity in steady state can be described by the following equation [22]:

$$ \frac{{{\text{d}}\Omega }}{{{\text{d}}t}} = \frac{\sigma c}{n}\iiint_{crystal} {\Delta N\Phi \left( {r,\theta ,z} \right)}{\text{d}}V - \frac{\Omega }{{\tau_{c} }}, $$
(5)

where the population inversion ΔN is

$$ \Delta N = ff_{Ho} N_{u} - f_{l} N_{Ho} , $$
(6)

with \(f = f_{u} + f_{l}\), \(f_{u}\) and \(f_{l}\) are the fractional occupation of upper and lower stark level of Ho3+, which is directly influenced by the temperature \(T\) [19]. \(N_{Ho}\) is the total Ho3+ doping concentration. The photon lifetime \(\tau_{c}\) in the cavity is expressed as [22]:

$$ \tau_{c} = \frac{{2l_{o} }}{{c\left[ {\delta - \ln (1 - t)} \right]}}, $$
(7)

where \(\delta\) is the round-trip cavity loss, and \(t\) is the transmission of the output coupler. The spatial photon density distribution is given by \(\Phi (r,\theta ,z) = \Omega \phi_{{{\text{mn}}}} (r,\theta ,z)\), where \(\Omega\) is the total photon number in the cavity and \(\phi_{{{\text{mn}}}} (r,\theta ,z)\) denotes the normalized spatial photon density of \({\text{TEM}}_{{{\text{mn}}}}\) mode. Therefore, for a side-pumped laser with circular plane-parallel cavity, \(\phi_{{{\text{mn}}}} (r,\theta ,z)\) can be described as [23]:

$$ \phi_{{{\text{mn}}}} (r,\theta ,z) = A\left( {\frac{{2r^{2} }}{{\omega_{l}^{2} }}} \right)^{{\text{n}}} \left[ {L_{{\text{m}}}^{{\text{n}}} \left( {\frac{{2r^{2} }}{{\omega_{l}^{2} }}} \right)} \right]^{2} \cos^{2} \left( {{\text{n}}\theta } \right)\exp \left( { - \frac{{2r^{2} }}{{\omega_{l}^{2} }}} \right) $$
(8)

and

$$ \iiint_{cavity} {\phi_{{{\text{mn}}}} (r,\theta ,z)}{\text{d}}V = 1, $$
(9)

where \(L_{{\text{m}}}^{{\text{n}}} \left( X \right)\) is the generalized Laguerre polynomials, \(A\) is the normalization constant, \(\omega_{{{\text{mn}}}}\) is the \({\text{TEM}}_{{{\text{mn}}}}\) mode radius. For \({\text{TEM}}_{00}\) mode, \(\phi_{00} (r,\theta ,z)\) can be written as

$$ \phi_{00} (r,\theta ,z) = \frac{2}{{\pi \omega_{00}^{2} l_{o} }}\exp \left( { - \frac{{2r^{2} }}{{\omega_{l}^{2} }}} \right), $$
(10)

where \(l_{o} = l_{c} + (n - 1)l_{r}\) is the one-way optical path, \(l_{c}\) is the total cavity length, and \(l_{r}\) is the full length of the laser rod. The distribution of higher-order transverse modes can also be acquired using Eq. (8). It is noteworthy that the integration (4) is taken over the crystal volume since the population inversion only occurs inside the crystal and outside the crystal is zero [22]. By solving Eqs. (1)–(10), an implicit expression for \(\Omega\) can be derived. Subsequently, the output power can be calculated using the following equation:

$$ P_{out} = \frac{{ch\nu_{l} t\Omega }}{{2l_{o} }}, $$
(11)

where \(h\nu_{l}\) is the laser photon energy.

3 Experimental setup

We designed a compact side-pumped laser head and its cross-section view is shown in Fig. 1. The laser head consisted of a Tm,Ho:YAG rod, a glass flow tube, a diffusive optical reflector (DOR), and three LD arrays (LDAs). The Tm,Ho:YAG rod (3.0 at.% Tm3+ and 0.1 at.% Ho3+) had dimensions of 3 mm in diameter and 74 mm in length, with a co-doped region of 36 mm in the middle of the rod. To alleviate thermal lensing, two 19 mm undoped YAG caps were diffusion-bonded at both ends of the rod. This technique could effectively spread the thermal loading over a greater length of the rod, thereby reducing the thermal gradient, particularly under high intensity pumping [24]. The side surface of the rod was polished to avoid optical loss, while both end facets of the rod were cut flat and anti-reflectance (AR) coated at 2090 nm. The temperature of the rod was regulated through cooling water at near RT of 12 ℃ and the flow rate was set to 8 L/min. In order to prevent water condensation on the rod end facets, our laboratory maintained controlled conditions with a temperature of 22 ℃ and a relative humidity of 30 RH.

Fig. 1
figure 1

Cross-sectional view of the LDAs side-pumped laser head

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Three CW LDAs were placed around the diffusive reflector in a threefold symmetric geometry, each LDA could deliver a maximum output power of 220 W. The emission central wavelength of LDA was located at 782 nm and the LDA working temperature was controlled by deionized water through bonded copper heat sinks with an accuracy of \(\pm\) 0.1℃. To match the LDA emission wavelength with the absorption band of Tm3+, we investigated the variations concerning different cooling temperatures, drive currents, and flow rates. Our findings indicate a relationship of 0.28 nm/℃, 0.01 nm/A, and 0.6 nm/(L/min), respectively. Consequently, we adopted three LDAs with closely matched wavelengths operating at 11 ℃ with a flow rate of 1.5 L/min. At maximum drive current, the LDAs had a central wavelength of 780.2 nm with a full width at half maximum (FWHM) of ~ 1.8 nm. This wavelength fell within the broader absorption band of Tm3+, thereby avoiding uneven changes in the pump distribution due to wavelength shifts [25].

The pump beam entered the reflector directly through a narrow slit (36 mm long and 3 mm wide) in the DOR. Afterward, it underwent refraction by the glass tube and water before being absorbed by the rod. The inner surface of the DOR had a reflectivity of around 97%. To achieve uniform pump distribution within the rod, a simulation was conducted using the Ray-Tracing method to determine the appropriate dimensions of the reflector. The absorption coefficient used in the simulation was 3 cm−1. Only the central cross-section pump distribution of the rod was considered because light propagates equally in \(\pm\) z direction [26]. The simulated absorbed pump distribution for different reflector inner diameters \(D_{r}\) is shown in Fig. 2. It is evident from Fig. 2 that a smaller DOR size with a diameter of 10 mm resulted in a higher absorbed power density. Nevertheless, a DOR with this size had a non-uniform pump distribution, resembling a Gaussian-like profile, which led to significant thermally induced birefringence due to an enlarged thermal gradient [27]. The utilization of DORs with diameters of 13 mm and 17 mm both contributed to the attainment of a uniform top-hat pump distribution. Upon a comparison of the absorbed power density, a DOR with an inner diameter of 13 mm was eventually chosen. As a result, the calculated absorption efficiency was approximately 70%.

Fig. 2
figure 2

Simulated absorbed pump distribution at the central section of the Tm,Ho:YAG rod for different reflector inner diameters in both the x and y directions

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To achieve high power output from the Tm,Ho:YAG laser, a short plane-parallel symmetric resonator with two flat mirrors was employed, as illustrated in Fig. 3. The rear flat mirror M1 was high-reflectance (HR) coated (R > 99.8%) at 2090 nm, while M2 was an output coupler with a partial transmission coating at 2090 nm. The total length of the resonator was only 90 mm. The entire system was compact, rugged, and convenient for adjusting.

Fig. 3
figure 3

Experimental setup for the high-power LDAs side-pumped Tm,Ho:YAG laser oscillator

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4 Results and discussion

First, the fluorescence distribution at an end facet of the Tm,Ho:YAG rod was measured by an infrared camera (Spiricon Pyrocam IIIHR) at the pump power of 400 W, as depicted in Fig. 4. The observed pump profile displayed a roughly top-hat shape in both horizontal and vertical directions, indicating a uniform pump distribution within the rod. This distribution agreed well with our above-mentioned simulation results. The high absorption efficiency and uniform pump distribution in the rod were of great benefit for high output power.

Fig. 4
figure 4

Fluorescence image at an end facet of the Tm,Ho:YAG rod detected by an infrared camera at a pump power of 400 W

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To investigate the cavity stability, we performed measurements of the thermal focal length of Tm,Ho:YAG at different pump powers using a unstable-cavity method [28], as shown in Fig. 5. The thermal focal length exhibited a decreasing trend as the pump power increased. For instance, at a pump power of 297 W, 462 W, and 650 W, the measured thermal focal length was approximately 160.3 mm, 100.3 mm, and 70.3 mm, respectively. Additionally, the dioptric power against pump power was also plotted in Fig. 6, yielding a linear relationship of 0.022 m/W. Figure 6 displays the relationship between the calculated fundamental mode radius and thermal focal length based on the ABCD propagation matrix for a Gaussian beam. It is evident that under given cavity parameters, the laser operated in a stable zone across the entire pump range.

Fig. 5
figure 5

Measured thermal focal length and dioptric power of Tm,Ho:YAG rod versus total LDA pump power

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Fig. 6
figure 6

Simulated fundamental mode radius at the center of the Tm,Ho:YAG rod as a function of thermal focal length

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Before examining the output characteristics of Tm,Ho:YAG laser, we applied the Findlay–Clay method to determine both the small signal \(g_{0}\) and the round-trip loss \(\delta_{r}\) of the cavity [29]. Using three output couplers with transmissions of 7.8%, 10%, and 12.5% and fitting the measured thresholds, we derived \(g_{0}\) to be 0.046 \({\text{cm}}^{ - 1}\) and \(\delta_{r}\) to be 0.05. Subsequently, the optimal transmission value of the output coupler could be calculated using the following equation [23]:

$$ - \ln \left( {1 - T_{opt} } \right) = \left( {\sqrt {2g_{0} l/\delta_{r} } - 1} \right)\delta_{r} , $$
(12)

where the \(l\) was the doped length of the rod. Then, the optimal transmission was calculated to be approximately 7.6%. Thus, an output coupler with 7.8% transmission was chosen for subsequent experiments, as it was the closest to the optimal value.

The output power of the Tm,Ho:YAG laser was measured using a power meter (Gentenc-EO, MAESTRO). The evolution of the CW output power as a function of the incident pump power is shown in Fig. 7. It was found from Fig. 7 that the pump threshold was 219 W. As the pump power was increased, the output power of the laser progressed monotonically. At a pump power of 660 W, the laser delivered a maximum output power of 87.7 W with an o–o efficiency of 13.3% and a slope efficiency of 21.2%. It is noteworthy that no thermal rollover effect was observed in the output power, which aligned with our previous calculations. By further increasing the pump power, we anticipated the laser could produce an output power beyond 100 W. Meanwhile, the beam quality factor M2 of this laser at full output power was measured using an infrared camera (Spiricon Pyrocam IIIHR). From the beam caustic measurement, the values were determined to be \(M_{x}^{2} = 31.94\) and \(M_{y}^{2} = 34.74\) in the x and y axes, respectively. A near-field two-dimensional (2D) beam image was captured as displayed in the inset in Fig. 7. The image revealed a multimode Laguerre–Gaussian beam pattern with a clear \({\text{TEM}}_{41}\) mode profile that exhibited peak intensity in the central region. The multimode output should be attributed to the short cavity geometry mainly.

Fig. 7
figure 7

Tm,Ho:YAG laser output power versus total incident pump power for an output coupler with 7.8% transmission. Red solid curve represents the calculated output power based on \({\text{TEM}}_{11} + {\text{TEM}}_{41}\) modes. Inset: near-field 2D beam intensity profile

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To fully understand the output characteristics of Tm,Ho:YAG laser, we performed the numerical calculation by solving Eqs. (1) to (11) by taking into account the multimode operation. The \({\text{TEM}}_{{{00}}}\) mode radius used in the simulation was determined to be 220 μm according to the previous calculations. Therefore, the radii of higher-order modes such as \({\text{TEM}}_{{{11}}}\) and \({\text{TEM}}_{{{41}}}\) can be derived using this value [30]. Regarding the choice of the co-operative up-conversion rate \(u\) for Tm,Ho:YAG, reported values fell within the ranges of 2 × 10−17 to 1.4 × 10−16 \({\text{cm}}^{3} {\text{/s}}\) [22]. It is worth noting that these experimental data were obtained for a dopant concentration of 6.0 at.% Tm3+ and 0.5 at.% Ho3+, which differs from the concentration used in our experiment. Nonetheless, it has been reported that the up-conversion rate of Tm,Ho:YAG exhibited a specific relationship with the product of Tm3+ and Ho3+ dopant concentrations [22]. Therefore, it was reasonable to assume that u was approximately to be 1 × 10−18 \({\text{cm}}^{3} {\text{/s}}\) in our simulation. All the values of the related parameters are listed in Table 1.

Table 1 Related parameters used in the simulation of multimode laser operation
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The calculated result verify that evolution of simulated output power based on \({\text{TEM}}_{11}\) and \({\text{TEM}}_{41}\) modes (red curve) is in close agreement with the experimental data as shown in Fig. 7. As the pump power increased, the higher-order modes reached the lasing threshold and competed with the lower-order ones, resulting in a multimode operation that exhibited increased slope efficiency at high pump levels. It is important to emphasize that the actual multimode output was more complex than the simulations suggested, as it was not simply the superposition of each oscillating mode’s contributions. Overall, the consistency between the simulation and experiment results suggested that the adjusted up-conversion rate was well-suited for Tm,Ho:YAG laser operation with a high dopant concentration ratio.

The output power stability of Tm,Ho:YAG laser at full output power was monitored and the results are displayed in Fig. 8. For an interval of 15 min, the RMS power fluctuation was estimated to be 0.46% (Std. Dev.), indicating an excellent stability of this laser operation. Furthermore, the output spectrum of the laser was recorded using a spectrometer (Ocean Optics, NIRQuest256, 850 nm–2500 nm, resolution: ~ 9.5 nm) as shown in the inset of Fig. 8. It can be observed that the central wavelength of the laser was located at 2090 nm with an FWHM linewidth of 9.6 nm.

Fig. 8
figure 8

Evaluation of power stability at full output power over 15 min. Inset: measured wavelength and linewidth of Tm,Ho:YAG laser

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The investigation of the cooling temperature-dependent Tm,Ho:YAG laser performance at full output is presented in Fig. 9. Altering the cooling temperature from 8 ℃ to 22 ℃ resulted in a nearly linear decrease in the maximum laser output power, dropping from 89.5W to 81.6W at a rate of 0.56 W/℃. These results suggest that operating at lower temperatures can enhance output power, but it comes with inherent risks, including the potential for water condensation on coatings and crystal fractures.

Fig. 9
figure 9

Maximum output power of Tm,Ho:YAG laser as a function of the cooling temperature of the laser rod

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5 Conclusions

In summary, we have successfully demonstrated a high power CW Tm,Ho:YAG laser operating at near RT. Each element of the side-pumped laser head and the output coupler were meticulously optimized. By employing a Tm,Ho:YAG crystal rod with 3.0 at.% Tm3+ and 0.1 at.% Ho3+, an output power of 87.7 W at 2.09 μm was obtained, with an optical-to-optical efficiency of 13.3% and a slope efficiency of 21.2%. The laser characteristics, including the thermal lens effect, beam quality, and power stability were investigated. Furthermore, we utilized a combined quasi-three-level rate equation with a modified up-conversion rate to numerically analyze the multimode laser operation, and the simulation agreed well with our experimental data. This consistency highlights the benefits of employing a high dopant concentration ratio in Tm,Ho:YAG, facilitating the attainment of high-power laser output. These results not only advance the understanding of Tm–Ho co-doped lasers but also pave the way for further development of the high-power and high-energy 2 μm lasers.