The 3~5 µm band radiation is located at the atmospheric infrared transmission window and has ideal air transmission properties. As such a band contains absorption peaks of many molecules，the laser in the mid-infrared band can be widely used in military countermeasures，remote sensing technology，laser surgery，and atmospheric monitoring，etc^［1-7］. In some fields，the requirements are often needed on the size，outputted peak power，and repetition rate of a mid-infrared（MIR）laser. Among a variety of conventional solid-state lasers，lamp-pumped solid-state lasers usually have the low pump light-optical conversion efficiency and non-uniform absorption of the laser medium，which lead to the low output beam quality and electro-optical conversion efficiency^［6-9］. And at the same time，the output power of a lamp-pumped solid-state laser is not easy to be raised，which is prone to a large amount of energy waste. As most of the output bands of conventional lamp-pumped solid-state lasers are located in the near-infrared region，achieving the output of mid-infrared laser is basically based on the nonlinear optical method such as optical parametric oscillation（OPO）or optical parametric amplification（OPA）methods^［10-14］. Although the nonlinear optical procedures have already become mature，the instrumentation is somewhat complex，the number of system components is large，the price is expensive，and the malfunctions are also difficult to avoid. Quantum cascade lasers（QCLs）can also be used for the realization of the mid-infrared laser output. However，it is difficult to build the higher beam quality and high-power laser system at the same time for such semiconductor lasers^［15-21］. Being different from the lasers mentioned above，the holmium-doped barium yttrium fluoride laser crystal（Ho³⁺：BY₂F₈，Ho：BYF）directly-pumped by LDs can be applied to output the mid-infrared laser in which the optical system is simple and the cost of the whole equipment is relatively low^［22-28］. However，the quantum defect of the LD directly-pumped laser configuration is much larger than those of conventional near-infrared LD-pumped solid-state lasers. As a result，more heat would be generated during the laser operation. If the heat cannot be quickly dissipated，severe thermal damage may be caused inside the laser medium. In extreme cases，the MIR laser medium may even be broken or blasted^［29-33］. This is one of the primary issues that need to be urgently resolved for the development of mid-infrared LD directly-pumped solid-state lasers towards the practicality.

In order to alleviate the problems of thermal deposition of a Ho：BYF crystal and improve the power of a MIR laser，the specific surface area and the structure of the crystals have been theoretically examined in this paper. We refer to the crystal with the length near to the optimal value to obtain the laser output from the theoretical simulation，and propose that the configuration in which the laser crystal is shorter than the cooler might be benefit to enhance the heat dissipation ability of the pumped MIR crystal^［29］. And then，two experiments have been conducted to verify the simulation results. In the first experiment，the influence of LD pump sources on the outputted laser power has been investigated by using Crystal-I with the size of 3×3×10 mm. The experimental results show that under the conditions of the repetition rate of 15 Hz，pulse width of 90 µs，and current of 15 A，the single pulse energy of 2.81 mJ can merely be obtained by an energy meter，which corresponds to the average laser output power of 42.15 mW. However，under this pumping condition，the cracks can be observed inside the crystal，which indicates that 2.81 mJ should be near to the limitation that the crystal can critically withstand under such a cooling configuration in the experiment. In the second experiment，results of the simulation and experiment-1 have been well considered by using the optimal crystal length with the size of 3×3×8 mm（Crystal-II）. The experimental results show that the pulse energy of 7.28 mJ（corresponding to average laser output power of 109.2 mW）can be achieved under the repetition rate of 15 Hz，the pulse width of 90 µs，and the current of 14.5 A，respectively. As a result，a significant increase in laser outputted power has been successfully achieved. Note that there is no damage in the laser crystal under the improved the lasing structure. These results demonstrate that compared with the traditional method of embedding the laser crystal inside the cooling holder with the same size as the crystal，the structure of the cooler with the length little longer than that of the laser crystal can be really beneficial to improve the heat dissipation ability. The thermal stress of the laser crystal should also be partially relieved by employing this approach. The results reported in this paper is a further breakthrough of our team after we first reported the emission of a mid-infrared Ho：BYF laser directly-pumped by LDs in 2021^［29-33］. The study is thought to be helpful for the future practical applications of such MIR solid-state lasers.

In this paper，Ho：BYF is used to produce a MIR laser. The energy levels of Ho：BYF are illustrated in Fig. 1. The transition of ⁵I₈ → ⁵I₅ represents the process in which the electrons on the ground state energy level is pumped to the upper energy level by using LDs with the wavelength of 889 nm. Then，a 3.9 μm laser is directly generated through the stimulated emission process of ⁵I₅ → ⁵I₆. Note that both energy transfer up-conversion（ETU）and cross relaxation（CR）are important processes during the laser emission. The ETU and CR make the MIR lasing very complicated and both of them are all extremely essential for obtaining the MIR oscillation^［29-33］.

The first LD-directly-pumped solid-state MIR laser has been realized two years before by our laboratory using a 3×3×10 mm Ho：BYF crystal. As the outputted power was far from applications，such a MIR laser cannot be directly applied for any actual purposes. For the improvement of the output power of a Ho：BYF laser，heat accumulation inside the crystal is needed to study. At present，the main methods to alleviate heat accumulation of crystals are reducing the specific surface area of crystals or optimizing the cooling structure. Generally，the heat dissipation ability of laser crystals is related to the specific surface area（surface area/volume）presented by the following equation：

where K represents the specific surface area value of the crystal，a represents the crystal width，and L_c represents the crystal length，respectively. It is obvious that the shorter the crystal，the larger the specific surface area. Thus，shortening the length of the laser crystal should effectively enhance the heat dissipation ability of the laser crystal and laser output energy under the same pumping conditions.

In addition，the simulation of laser dynamics is also conducted for estimating the outputted pulse energy with different crystal lengths. According to our conclusions in Ref. 29，as shown in Fig. 2，the laser output energy changes with the crystal length. When the pump energy is set between 250 and 350 mJ，the optimal crystal length is around 8 mm.

Therefore，in order to enhance heat dissipation in crystals，the temperature and thermal stress inside the Ho：BYF medium with two kinds of lengths should be analyzed by using the same cooling configuration，as shown in Fig. 3. The corresponding thermal simulation of heat dissipation in crystals was carried out in detail for Crystal-I and Crystal-II by utilizing COMSOL. Figs. 4 and 5 show the temperature at Point A in Fig. 3 and thermal stress inside Crystal-I（3×3×10 mm）and Crystal-II（3×3×8 mm）under the same pumping conditions，respectively. The boundary conditions for COMSOL thermal simulation are listed in Table 1. From Fig. 4，although Crystal-II produces a little more temperature raising，which is also due to its smaller volume than that of Crystal-I，the minimum temperature of two crystals is almost the same before the next pump pulse reaches the crystal. Such phenomenon indicates that changing the pump length should be conducive to the heat dissipation of the crystal. As shown in Fig. 5，the thermal stress of Crystal-II is less than that of Crystal-I for the improved cooling structure. This means that it should be beneficial to alleviate the thermal stress of the crystal by adopting such a shorter crystal.

In order to verify the theoretical simulation results，two experiments have been carried out for the laser media with different lengths. Therefore，the influence of the pulse width，driving current，and repetition rate of the pump LDs on the outputted laser energy has been experimentally explored by using such two Ho：BYF crystals. The results will be inducted in Sections 3 and 4.

Fig. 6 shows the clamping diagram and section diagram of C-C' plane for Crystal-I. The Ho：BYF crystal was fixed in a rhombus-shaped holder combined by the upper and lower parts. The indium foil was intentionally paved between the brass seat and the crystal to fill the gaps as much as possible.

The schematic diagram and the photograph of our experimental setup are shown in Figs. 7 and 8，respectively. The light of pump LDs with the central wavelength of 889 nm was changed into a parallel beam by using the first plano-convex focus lens with the focal length of 75 mm. Then the paralleled LD radiation was focused by the second plano-convex focus lens（f = 75 mm）. The converge pump beam with the diameter of 650 μm enters into a plano-concave optical resonator consisting of a rear mirror（RM），an output mirror（OC），and Ho：BYF gain medium，Crystal-I. And then，a MIR laser with the wavelength of 3.9 μm can be directly obtained^［29］. In the experiments，a He-Ne laser was used to calibrate the optical path of the oscillator and a silicon wafer was used to filter out the components of the pump light mixed in the output beam. Actually，the measured output values are about a half of the real results because of the attenuation of such a silicon wafer.

In this experiment，the repetition rate of the pump laser was first set to 2 Hz，and the driving current of the pump LDs was set to 14，16，and 18 A，respectively. When the pulse width of the pump laser was changed from 60 to 130 µs，the laser output energy was observed by means of an energy meter.

Fig. 9 shows the experimental results. It can be seen that the output energy of the MIR laser increases with the pump pulse width roughly acting as multiple curves. Under the same pulse width，the laser output energy increases with the driving current of the pump. With the increase of the pump driving current，the laser output energy changes smoother. The laser output energy can reach up to 12 mJ when the pump driving current and pulse width are set to 18 A and 120 μs，respectively.

Then，as shown in Fig. 10，the relationship between the driving current of the pump LDs and the laser output energy under the repetition rate of the pump LDs of 2 Hz was investigated using the pump pulse width of 60，80，100，and 120 μs，respectively. It can be seen that the laser output energy increases with the driving current of the pump LDs. And the curves somewhat appear to be linear forms within a certain range，which indicates that the influence of the pump pulse width on the laser output energy is slightly stronger than that of the pump driving current. The variation of the laser output energy becomes more drastic for a wider pump pulse width.

Next，the laser output energy was examined for different pump repetition rates when the pulse width of the pump was set to 90 μs，and the driving current of the pump LDs were set to 14 and 15 A，respectively. As can be seen from Fig. 11，the laser output energy decreases with the pump repetition rate，and the corresponding energy of the pulse laser decreases，but the average output power of the MIR laser shows an increasing trend. The laser output energy increases with the pump energy under the same repetition rate. When the repetitive rate，the pulse width，and the drive current is 90 μs，15 Hz，and 15 A，respectively，the pulse energy can be obtained as about 2.81 mJ，which is converted to an average laser output power of 42.15 mW. After several ten seconds under such conditions，the crystal was suddenly destroyed and a deep crack was found at the input end of the crystal，which indicates that all the parameters here should correspond to the destroy threshold conditions that the laser crystal can withstand. Note that the pumping energy is approximately 221.8 mJ at that time.

Combined with the results of the above experiments，it has been analyzed that the pump pulse width and pump LD drive current have less influence on the laser output energy than the pump repetition rate. Therefore，in the following experiments，the values of pump pulse width and pump LD drive current were fixed，and the pump repetition rate was taken as a variable factor.

Fig. 12 shows the schematic clamping diagram and the section diagram of the C-C' plane of Crystal-II by employing the same cooling configuration as that for Crystal-I.

In order to obtain the high outputted laser energy without damaging the crystal，the pump pulse width at 90 μs，and the pump LD drive current at 14 and 14.5 A were set as safe values，and the repetition rate of the pump laser was changed to observe the change of the outputted laser energy.

Fig. 13 shows the energy and power diagrams of the laser output. From Fig. 13，we can see that the laser output energy still shows a decreasing trend with the pump repetition rate，while the overall output power of the laser still shows an increasing trend. In the case of the pump repetition rate of 15 Hz，the pulse width of 90 μs，the current of 14.5 A，the pulse energy can be measured by an energy meter as 3.64 mJ. Taking into account that such value is the result attenuated by the silicon wafer，and the actual laser output energy is exactly twice as much as the measured value，so the actual output energy of the MIR laser is 7.28 mJ（corresponding laser output average power of 109.2 mW）. Note that，there is no cracks in the crystal at that time.

It can be seen that，when using a new shorter laser crystal，the MIR laser output energy has been greatly improved. The structure that the length of the laser crystal is slightly shorter than the cooler has been proved to significantly improve the heat dissipation capacity of the laser crystal. Therefore，further optimization on the pump structure and the shape of a laser crystal should be the important way to achieve much higher outputted power of our MIR laser in the future.

In this paper，the length of a MIR medium，Ho：BYF，has been analyzed to earn the output energy as large as possible based on the theoretical simulations of laser dynamics. The theoretical computation of thermal dynamic reveals that the suitable length of a Ho：BYF crystal makes the output of a MIR laser be substantially enhanced and the heat dissipation ability significantly improve inside the laser crystal. The experimental results show that with the pump repetition rate of 15 Hz，the pulse width of 90 µs，and the drive current of 14.5 A，the pulse energy of the laser can achieve 7.28 mJ for a Ho：BYF crystal with the length near to the optimal value in theory. Such result is about four times as large as that in Ref. 29 for LD directly-pumped MIR lasers. Our work might lay the technical foundation for the realization of practical MIR solid-state lasers directly-pumped by LDs.