1 Introduction

Compared with multi-mode lasers, SLM lasers not only have the characteristics of strong stability and high beam quality but also feature narrow output spectral lines and good coherence. Because of these properties, they find wide-ranging applications in fields such as laser medicine, radar control, interferometric sensing, laser precision micromachining, and coherent communication [1,2,3,4,5]. Traditional ways to achieve SLM laser output include the short cavity method, F-P etalon method, unidirectional ring cavity method, pre-laser Q-switched mode selection method, torsion pendulum cavity method, and volume injection locking method [6,7,8,9,10,11]. According to the data, we have consulted, stable SLM laser output cannot be achieved by a single single-mode selection method under the conditions of high pump injection. Moreover, the conventional approach to achieve laser output in the visible band involves performing nonlinear frequency conversion on well-established gain media, which significantly reduces conversion efficiency.

Pr3+ stands out from other rare earth ions due to the significant number of radiative transitions it possesses within the visible light spectrum, including blue light (479 nm), green light (522 nm, 545 nm), orange light (604 nm, 607 nm), red light (639 nm, 670 nm) and deep red light (697 nm, 720 nm). In addition, 261 nm, 302 nm, 303 nm, 320 nm, 360 nm, and other ultraviolet band laser outputs can be realized by nonlinear frequency conversion. This has attracted the wide attention of many researchers [12,13,14,15]. Due to the serious non-radiative multi-phonon transition of the laser upper level of Pr3+, there are relatively few matrix materials suitable for doping Pr3+. At present, the reported crystals that can realize laser operation are Pr:YAP, Pr:LuAlO3, Pr, Mg:SrAl12O19, Pr3+:LiYF4, Pr:LiLuF4. In all materials, Pr3+ was doped in lithium yttrium fluoride crystal, that is Pr:YLF. Because of its long energy level life, good energy storage characteristics, mature production process and other advantages, it has been studied most widely [16]. For example, the continuous deep red SLM laser output of 720 nm has been realized using the combination of two F-P etalons [17]. Luo et al. from Xiamen University used a Brewster plate combined with two quarter-wave plates (QWPs) to form a torsion pendulum cavity method, achieving a continuous red SLM laser output of 640 nm [18]. Although Pr:YLF has made significant progress in the research of CW lasers, developing all-solid-state pulsed SLM lasers based on pre-laser technology is still slow. The question of how to obtain a Pr3+ doped SLM pulsed visible light laser with a high repetition rate, large pulse energy, and stable output has become a research hotspot in recent years. Among them, Jin et al. [19] investigated the high-repetition-rate Q-switched SLM Pr:YLF laser and optimized its performance [20]. In our follow-up research, we consider adding a nonlinear frequency conversion crystal, so that a SLM pulsed laser in the ultraviolet band can be obtained. The indicators proposed in the above literature are not yet achievable.

To address the above issues, the structure of the mode selection technology we used is composed of pre-laser Q-switched and three F-P etalon groups, although we add three F-P etalons in the resonant cavity, which increases some energy loss. However, it can further enhance the selection ability of the longitudinal mode under the condition of high energy injection through the micro-loss difference between adjacent longitudinal modes while reducing the spectral line width. Therefore, the SLM pulse laser with higher relative output energy can be effectively realized by using the coupling of pre-laser Q-switching mode selection technology and F-P etalon mode selection technology. We propose for the first time that the difference in transmission loss and the angle between adjacent longitudinal modes affects the SLM laser output when three F-P etalons are used for assisted mode selection. Our conjecture is verified based on the simulation results combined with laser experiments, resulting in single-pulse energy and linewidth of 11.1 μJ and 14 MHz, respectively.

2 Model construction and simulation

2.1 Establishment of the model

Based on the characteristics of Pr:YLF crystal, we introduce the boundary conditions of the acousto-optic Q-switch, the round-trip gain term, and the transmission loss difference between adjacent modes caused by three F-P etalons, and we combine these with the multimode competition theory. The resulting active Q-switched multimode rate equations are as follows:

$$ \frac{\partial N}{{\partial t}} = \frac{{P_{ab} }}{{hv\pi \omega^{2} l}} - \mathop \sum \nolimits_{i = 1}^{j} \gamma \sigma_{i} N\varphi_{i} - \frac{N}{{\tau_{f} }} $$
(1)
$$ \frac{{\partial \varphi_{i} }}{\partial t} = \varphi_{i} \left( {2\sigma_{i} cN\frac{{l_{0} }}{l} - \frac{{\delta_{z} }}{{t_{R} }}} \right) $$
(2)
$$ \sigma_{i} = \frac{{2\pi^{2} }}{3n\varepsilon h}|\mu |^{2} vg(v_{0} )\left( {1 + \frac{{4(v_{i} - v_{0} )}}{{(\vartriangle v_{H} )^{2} }}} \right)^{ - 1} $$
(3)
$$ T_{1} (v) = \frac{1}{{1 + \frac{{4R_{1} }}{{(1 - R_{1} )^{2} }}sin^{2} (2v\pi nh_{1} cos\theta_{1} /c)}} $$
(4)
$$ T_{2} (v) = \frac{1}{{1 + \frac{{4R_{1} }}{{(1 - R_{2} )^{2} }}sin^{2} (2v\pi nh_{2} cos\theta_{2} /c)}} $$
(5)
$$ T_{3} (v) = \frac{1}{{1 + \frac{{4R_{1} }}{{(1 - R_{3} )^{2} }}sin^{2} (2v\pi nh_{3} cos\theta_{3} /c)}} $$
(6)
$$ T_{z} = T_{1} (v_{i} )T_{2} (v_{i} )T_{3} (v_{i} ) $$
(7)
$$ L(v_{i} ) = 1 - [(1 - R_{1} )(1 - R_{2} )(1 - R_{3} )T_{{\varvec{z}}} ] $$
(8)
$$ L_{i} = 1 - L(v_{i} ) $$
(9)
$$ L_{{R^{\prime}}} = - (\ln (R_{1} ) + \ln (R_{2} ) + \ln (R_{3} )) $$
(10)
$$ L_{res} = - L_{{R^{\prime}}} - \ln (L_{i} ) - \ln (L_{s} ) - \ln (R) $$
(11)
$$ L_{0} (t) = \left\{ {\begin{array}{*{20}c} {\delta_{1} = \delta_{max} ,0 < t < t_{1} } \\ {\delta_{2} = - \ln (1 - \eta ),t_{1} < t < t_{2} } \\ {\delta_{3} = 0,t_{2} < t < T} \\ \end{array} } \right.\begin{array}{*{20}c} {} \\ {} \\ {} \\ \end{array} $$
(12)
$$ L_{tot} (t) = L_{res} + L_{0} (t) $$
(13)

Among them, N represents the number of population inversion, \({\varphi }_{i}\) is the number of photons of the ith mode, \({\sigma }_{i}\) is the stimulated emission cross-section of the ith mode, \({\tau }_{f}\) is the upper-level lifetime, and \(\gamma \) is the inversion factor. The loss term \({L}_{tot}\left(t\right)\) includes the reflection loss term \({L}_{{R}^{\mathrm{^{\prime}}}}\) of the F-P etalon group, the loss term \({L}_{i}\) of the i mode, the optical loss of the closed Q-switch \({L}_{s}\) and the step loss of the acousto-otic Q-switch \({L}_{0}\left(t\right)\). \(\delta =4\pi nvhcos\theta /c\) is the phase difference of two adjacent beams in the F-P etalon cavity, where h is the thickness of the metal plate and n is the refractive index of the F-P etalon. \({\delta }_{1}\) and \({\delta }_{2}\), respectively represent the loss of the beam in the acousto-optic Q-switch, and \(\eta \) is the loss term of the acousto-optic Q-switch itself. \(T\) is a period of the acousto-optic Q-switch, where \({t}_{1}\) and \({t}_{2}\) are the switching time in each period. We set the repetition frequency to 10 kHz, and use a uniformly broadened Pr:YLF solid-state laser to establish this model, with its related gain function being of the Lorentz type.

Formulas (7) and (8) demonstrate that the difference in transmittance loss between adjacent longitudinal modes significantly impacts the laser output characteristics of an SLM. The difference in transmittance between the central longitudinal mode and adjacent longitudinal modes increases as the insertion angle of the F-P etalon increases. Therefore, optimizing the insertion angle of the three F-P etalon groups can effectively realize loss modulation of the transmittance between different longitudinal modes.

2.2 The simulation and optimization of F-P etalon group auxiliary model selection performance

To verify the stable SLM laser output under the condition of high-power-energy injection, we selected a laser with a maximum output power of 13.5 W and three F-P etalons with thicknesses of 1 mm, 0.5 mm and 0.3 mm to assist mode selection. We use MATLAB software to simulate and analyze the influence of each F-P etalon angle on SLM laser output when the pump output power is maximized. The analysis revealed that the insertion angle of every F-P etalon can control the micro-loss difference between adjacent longitudinal modes, which improves the side mode suppression in the process of F-P etalon group auxiliary mode selection. The relevant parameters are shown in Table 1.

Table 1 Parameters of numerical simulation
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Under the same transmittance conditions, the thicker the F-P etalon, the stronger the interference filtering performance, indicating that the thickness of the F-P etalon has a significant impact on the compression of the laser gain spectral line. Therefore, compared to the 0.5 mm and 0.3 mm F-P etalons, the 1 mm F-P etalon is the preliminary compression of the longitudinal mode. We first simulated the relationship between the transmittance and the angle of the 1 mm F-P etalon, as shown in Fig. 1a. We aimed to ensure the maximum transmittance of the 1 mm F-P etalon and combined it with the 0.5 mm and 0.3 mm F-P etalons for fine mode selection. Point (a) represents the maximum transmittance point, at an angle of 1.07°.

Fig. 1
figure 1

The relationship between the micro-loss, transmittance, and different insertion angles between adjacent longitudinal modes

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Based on the angle of the 1 mm F-P standard at maximum transmittance, we simulated the relationship between adjacent longitudinal mode loss difference, transmittance, and angle of the 1 mm and 0.5 mm F-P standards, as shown in Fig. 1 (b). It can be seen that as the insertion angle of the 0.5 mm F-P etalon increases, the loss difference and transmittance of adjacent longitudinal modes gradually show periodic changes. This indicates that the insertion angle of the F-P etalon has a significant impact on the variation of its transmittance loss. So, when adjusting the angle, it is necessary to control the angle adjustment within a small range as much as possible, which can reduce insertion loss and obtain higher output power. In Fig. 1 (b), we can see that point (b) satisfies the condition of maximum adjacent longitudinal mode loss difference and maximum transmittance in the dual F-P etalon. We ultimately obtained a 0.5 mm F-P standard with an angle of 1.18°, corresponding to a coupling transmittance of 93.5%.

Similarly, after setting the insertion angles of 1 mm and 0.5 mm F-P etalons, we added 0.3 mm F-P etalon. The relationship between the loss difference, transmittance, and angle of adjacent longitudinal modes of the F-P etalon group was ultimately obtained, as shown in Fig. 1c. Among them, point (c) is the point that satisfies the maximum loss difference of adjacent longitudinal modes and the maximum transmittance; that is, the angle is 1.23°, and the transmittance is 89%.

The influence of each F-P etalon angle on SLM laser output is analyzed by simulation, and the optimal etalon angle parameters are finally determined. The angles of the 1 mm, 0.5 mm, and 0.3 mm F-P etalons are 1.07°, 1.18°, and 1.23°, respectively. The above parameters are used as indicators for subsequent experiments to achieve SLM laser output.

2.3 Results and analysis

Based on the above simulation results of the F-P etalon group angle and relevant parameters of the Pr3+:YLF laser, the Runge–Kutta method is used for the numerical solution.

From Fig. 2a, we can see that the establishment of the seed light in the pre-laser process is close to and more significant than the threshold state, which plays the role of the center frequency longitudinal mode to vibrate and suppress the adjacent longitudinal mode. Figure 2b shows that the photon number ratio of the central mode to the adjacent mode reaches 10:1, and all other longitudinal modes are completely suppressed. This indicates that the mode competition has been fully resolved, and the output laser is SLM state.

Fig. 2 a
figure 2

The trend of the number of photons and the number of population inversion in the cavity with time near the threshold; b the trend of the number of photons and the number of population inversion in the cavity with time after the Q switch is fully turned on

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To verify the correctness of the simulation, according to the initial and final conditions of the power ratio between the central longitudinal mode and the adjacent longitudinal mode and the number of longitudinal modes, we adopt the equation of the mode competition process and the mode competition intensity between the longitudinal modes under the compound mode selection operation mechanism. The specific expression is as follows [21]:

$$ \frac{{P_{i} }}{{P_{i + 1} }} = \left( {\frac{{G_{i} }}{{G_{i + 1} }}} \right)^{q} $$
(14)
$$ \frac{{G_{i} }}{{G_{i + 1} }} = \frac{{R_{i} RL_{C} L_{I} exp[2l\sigma_{i} N(t)]}}{{R_{i + 1} RL_{C} L_{I} exp[2l\sigma_{i + 1} N(t)]}} $$
(15)
$$ \frac{{P_{i} }}{{P_{i + 1} }} = [exp\left( {2l\sigma_{i} \left( {1 - \frac{{\sigma_{i + 1} }}{{\sigma_{i} }}} \right)\left( {N(t)} \right)} \right)^{{\frac{{t - t_{1} }}{{t_{i} }}}} \left( {\frac{{L_{i} }}{{L_{i + 1} }}} \right)^{{\frac{{t - t_{1} }}{{t_{i} }}}} $$
(16)

Among them, \({G}_{i}\) and \({G}_{i+1}\) are the annular gain coefficients of the adjacent modes, \({t}_{l}\) is the round-trip time, \(R\) and \({R}_{i}\) are the reflectivity of the resonant cavity endoscopy group, and \(N\left(t\right)\) is the number of upper-level population inversion. The above equations are combined into the active Q-switched multimode rate equations. When the power ratio between adjacent longitudinal modes is greater than or equal to 10, and the number of longitudinal modes is 1, we can consider that the state at this moment is SLM laser output. The number of upper-level population inversions changing with time is obtained by solving the differential equations, and the corresponding evolution results are finally obtained.

We can see that the number of longitudinal modes decreases with the increase in working time. Figure 3a shows the time-varying curve of the number of longitudinal modes and the power ratio between adjacent longitudinal modes obtained by using a double F-P etalon for auxiliary mode selection in the mode competition process. When the absorbed pump power is 7.4 W, the number of longitudinal modes is 30 when the working time reaches the spontaneous life, the power ratio between adjacent modes is 7.9, and the final output is MLM state. When we use three F-P etalon groups for auxiliary mode selection at the same pump absorption power, the final power ratio is 10, and the longitudinal modulus is 1. That is when the ring gain ratio in the mode competition reaches or exceeds 10, and the longitudinal mode number is 1, it can be considered as the SLM state, as shown in Fig. 3b.

Fig. 3
figure 3

In the process of mode competition, the time-varying curves of the number of longitudinal modes and the power ratio between adjacent longitudinal modes are obtained. a When the pump absorption power is 7.4 W, the state of the pre-laser Q-switched combined with the double F-P etalon; b When the pump absorption power is 7.4 W, the pre-laser Q-switching combines the state of three F-P etalon groups

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Through the above simulation results, we can prove the following points: 1. We first insert the parameters of the pre-laser Q-switched and double F-P etalon into the model. We found that when the absorbed pump power is set to the highest 7.4 W, the laser output state is MLM; that is, the mode selection effect of the mode selection element is weak. It is not ideal to realize SLM laser output under high power injection conditions. 2. Compared with the double F-P etalon, the triple F-P etalon group has higher mode selection performance and can maintain the SLM state under high energy injection. This shows that the micro-loss difference between the adjacent longitudinal modes of the F-P etalon directly affects the mode selection ability of the pre-laser Q-switch. This slight difference is continuously amplified after multiple oscillations. Therefore, the output state of the laser has changed. In the following experiments, we verify the correctness of the simulation again.

3 Experiment and discussion

The experiment setup includes an LD pump source, an input mirror (IM), a Pr:YLF crystal, an acoustic-optic Q-switching system (A-O), an F-P etalon group, a four-dimensional adjustment fixture, an output mirror (OC), as shown in Fig. 4. We used an InGaN laser diode with a 444 nm wavelength, a beam waist diameter of 200 μm, and a maximum absorbed pump power of 7.4 W as the pump source. The laser gain medium is a Pr:YLF crystal that is 3*3*5 mm3 in size and has a doping concentration of 0.5 at.%. The crystal is cut along the a-axis. In the experiment, the surface of Pr:YLF crystal is wrapped with indium foil and placed in a crystal clip of copper material, and the temperature is controlled at 20 °C to improve the stability of laser operation. In our experiments, the input mirror (IM), coated with a 639 nm high reflectivity coating (HR > 99.5%), is set as a flat mirror. An output mirror (OC) with a curvature radius of 100 mm and a transmittance of 3.5% was used in the experiment. The primary components of the pre-laser Q-switched system are the Q-switched crystal and digital signal generator. The stepped signal is achieved by varying the voltage loss and time of the digital signal generator. Additionally, the F-P etalon group comprises three glass plates with thicknesses of 1 mm, 0.5 mm, and 0.3 mm. These plates are further fine-tuned to narrow the linewidth, enabling SLM laser output. Notably, this fine-tuning process involves adjusting the etalon group using three four-dimensional adjustment fixtures. The angle adjustment range is ± 5°. The distance between each metal plate is approximately 5 mm. The illustration in Fig. 4 is a schematic of the insertion angle of each F-P etalon after optimization.

Fig. 4
figure 4

The schematic diagram of the overall experimental device of Pr:YLF SLM pulsed laser (inset, Optimized schematic diagram of F-P etalon group)

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We first conducted an experimental study of the 639 nm SLM laser output using a laser front Q-switch and a double F-P etalon. The thickness of the two F-P etalon are 1 mm and 0.3 mm, and their insertion angles are strictly controlled at 1.07° and 1.23°, respectively. As can be seen in Fig. 5a, with the increase of pump absorption power, the laser output state transitions from SLM to MLM. This indicates that the mode selection effect of the mode selection element gradually weakens.

Fig. 5
figure 5

The relationship between different etalon groups' output power and absorbed pump power. a Double F-P etalon; b Three F-P etalon groups

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Next, under the premise of ensuring that other conditions remain unchanged, we inserted a group of three F-P etalons in the resonator. The insertion angles of the three F-P etalons of 1 mm, 0.5 mm, and 0.3 mm are 1.07°, 1.18°, and 1.23°, respectively. Under the dual-mode selection of the pre-laser Q-switch and the three F-P etalon groups, the SLM laser output is achieved when the absorbed pump power reaches 2.2 W. Although the threshold increases with the increase of the number of mode selection elements in the resonant cavity. However, compared with Fig. 5a, b, we can clearly see that the SLM laser output energy becomes larger and the line width becomes smaller. Moreover, as the energy is injected from low to high, the laser output still maintains a relatively stable trend, showing an SLM state. When the absorption pump power is 6.32 W, it is the point to measure the maximum output power, as shown in point (a). We believe that the reason for the downward trend after point (a) is that when the pump power gradually increases to a certain extent, the temperature gradient distribution in the gain medium is distorted, resulting in a decrease in output power.

We utilized an F-P scanning interferometer (SA200-5B, Thorlabs) with a free spectral range of 1.5 GHz to record the SLM laser spectrum. To measure the linewidth of the Pr:YLF SLM laser, we employed the Lorentz spectrum waveform based on previous studies that offer an approximate solution for linewidth calculations. We use the software program to process the experimental data and calculate the SLM linewidth. Table 2 is based on the pre-laser Q-switch, and the double F-P etalon and the three F-P etalon groups are used to assist the mode selection to achieve the SLM laser output index. We can clearly see that under the condition that the repetition frequency is 10 kHz, each data has been improved. Among them, the pulse energy is increased by 56%. The spectral linewidth is compressed from 25 to 14 MHz, which is effectively improved by 44%. It is worth mentioning that the upper limit of pump absorption energy is changed from 4.7 W to 7.4 W, which is 57% higher than that of the original. This proves again that under the coupling effect of pre-laser Q-switched mode selection technology combined with three F-P etalon groups assisted mode selection, not only the SLM output energy of 639.7 nm is improved, but also the longitudinal mode selection ability under high energy injection conditions is effectively improved. It is undeniable that through the comparison and analysis of the experimental data. The above experiments verify our speculation, which is consistent with the simulation results.

Table 2 Comparison of experimental data in SLM state
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4 Conclusion

To achieve SLM laser output under high-power injection conditions, we established a theoretical model predicated on a multi-mode rate equation that combines pre-laser Q-switching with an F-P etalon group. We analyzed the influence of the angles of the three F-P etalons on the stable SLM laser output during the F-P etalon-assisted mode selection process through simulation. The experimental results aligned well with the theoretical predictions. Through simulation and comparative experiments, it can be seen that there is an important internal relationship between the stable SLM laser output and the angle parameters of the three F-P etalon groups, the insertion loss in the cavity, and the micro-loss difference between adjacent longitudinal modes. Ultimately, under high pump injection conditions, a 639.7 nm Pr:YLF SLM laser with a single pulse output energy of 11.1 μJ, a pulse width of 79.10 ns, and a linewidth of 14 MHz were obtained. Our theoretical model is potentially applicable to most solid-state lasers, offering a robust theoretical reference for the design and construction of experimental facilities. Moreover, this laser provides an ideal new light source for UV SLM laser research, carrying significant implications for the advancement of UV laser applications in various fields in the future.