1 Introduction

Diode pumped solid state single longitudinal mode (SLM) lasers, which have attracted increasing attention for requirements in many applications, including coherent communications, interferometric sensing, radar control and laser medicine [1,2,3,4]. However, current research on SLM laser technology focuses on the specific band of the mature gain medium. Therefore, it is inevitable to apply nonlinear frequency conversion to obtain values outside this range. The traditional method uses the nonlinear frequency conversion to obtain the SLM in the visible band. It is necessary to use two nonlinear frequency conversions to obtain the ultraviolet laser band. This operation significantly reduces the optical conversion efficiency. Therefore, the approach to achieve a high optical conversion efficiency of the visible or ultraviolet band of the laser method is crucial. This approach will provide the technical foundation for areas such as ultraviolet communication and high-speed laser photography.

The most efficient lasing in the visible region is achieved using Pr3+ doped fluoride materials. The reason is that Pr:YLF lasers can undergo several transitions in the green, orange, red, deep red, and dark-red spectral regions [5,6,7,8,9,10,11]. Because the laser output of the visible band can be directly obtained without nonlinear frequency conversion, it is the preferred crystal for investigations on visible SLM lasers. Some mode selection techniques, such as nonplanar ring oscillators, lasers inserted with two etalons, and twisted-mode lasers, have been used to produce SLM lasers, and thus, combine with nonlinear frequency conversion to achieve visible SLM lasers. However, it is challenging to achieve high single-frequency output power, owing to the introduction of many mode selection elements. The additional loss to restrain the generated redundant modes inevitably leads to the laser threshold. The pre-lase Q-switching technology is a potential mode selection method that opens in two stages to enable the slow pulse to build up with time. This technique is sometimes referred to as “self-injection-seeding”, as it can be considered a laser that is injection-seeded by its pre-lase signal [12,13,14]. In general, there are fewer inserted elements, and the total loss of the system is relatively low. The characteristics of the laser crystal and prelase Q-switching technology are combined to develop a new method with higher conversion efficiency than other mode selection techniques.

In this study, we established an effective model of the multimode rate equation to describe the operational procedure of the mode selection. In addition, we established an evaluation function to optimize the critical parameters of the pre-lase Q-switching Pr:YLF laser, which determines its mode selection action and output character optimization. The accuracy of the estimated parameters was verified, as a multi-mode rate equation analysis based on the estimated parameters replicated the experimental SLM laser performance. By performing simulations using the proposed multimode rate equation model and established estimation function, we designed and analyzed the pre-lase Q-switching SLM laser further. The experimental results showed that a pulse width of 88 ns, pulse energy of 4.41 μJ, and peak power of 50.1 W were obtained.

2 Theoretical modeling and discussion

For the SLM oscillation of pre-lase Q-switched Pr:YLF lasers, the output pulsation of each mode is governed by the gain coefficient distribution and mode competition for each mode and the intracavity loss. The SLM conversion efficiency and the single pulse energy of the output pulse trains were determined by setting the two-step signal of the Q-switch modulation time and other resonator parameters. The dynamics of the periodical modulation of pulse trains for the multilongitudinal-mode actively Q-switched Pr:YLF laser can be derived using the multimode laser rate equations by considering the double-step loss and duration time, as follows:

$$ \frac{{\partial N}}{{\partial t}} = \frac{{P_{{ab}} }}{{{\text{hv}}\pi \omega ^{2} l}}\frac{{\left( {N_{{tot}} - N} \right)}}{{N_{{tot}} }} - \sum\limits_{{{\text{i}} = 1}}^{j} {\gamma {\text{c}}\sigma _{{\text{i}}} \phi _{i} N - \frac{N}{{\tau _{f} }}} $$
(1)
$$ \frac{{\partial \phi _{{\text{i}}} }}{{\partial t}} = B\sigma _{{\text{i}}} \phi _{i} N - \frac{{ - \ln (R^{\prime}) - \ln (L_{i} ) - \ln (L_{c} ) + L_{{QS}} (t)}}{{t_{r} }} \cdot \phi _{i} $$
(2)
$$ T(\nu _{{\text{i}}} ) = \left( {1 + \frac{{4R}}{{(1 - R)^{2} }}\sin ^{2} \left( {\frac{{4\pi {\text{n}}\nu _{{\text{i}}} h\cos \theta }}{{2c}}} \right)} \right)^{{ - 1}} $$
(3)
$$ \sigma _{{\text{i}}} = \frac{{2\pi ^{2} }}{{3{\text{n}}\varepsilon ch}}\left| \mu \right|^{2} vg\left( {\nu _{0} } \right)\left( {1 + \frac{{4(\nu _{i} - \nu _{0} )^{2} }}{{(\Delta \nu _{H} )^{2} }}} \right)^{{ - 1}} $$
(4)
$$ L_{{\text{i}}} = (1 - R^{\prime})T(\nu _{i} ) $$
(5)
$$ L_{{{\text{tot}}}} = - l{\text{n}}\left( {R^{\prime}} \right) - l{\text{n}}\left( {L_{{\text{i}}} } \right) - l{\text{n}}\left( {L_{{\text{c}}} } \right) + L_{{QS}} \left( t \right) $$
(6)
$$ L_{{QS}} (t) = \left\{ {\begin{array}{*{20}c} {\delta _{1} = \delta _{{\max }} } & {} & {(0 < t < t_{1} )} \\ {\delta _{2} = - \ln (1 - \eta )} & {} & {(t_{1} < t < t_{2} )} \\ 0 & {} & {(t_{2} < t < T)} \\ \end{array} } \right\} $$
(7)

where \(N\) is the population inversion in the upper level, \(\phi _{{\text{i}}}\) is the laser photon number of \(i{\text{th}}\) mode, \(\sigma _{{\text{i}}}\) is the stimulated-emission cross section of the \(i{\text{th}}\) mode, B is a constant reflecting the basic parameters, \(L\left( {{\text{v}}_{i} } \right)\) is the loss per double pass of the \(i{\text{th}}\) mode, \(L_{{\text{i}}}\) is the loss term for the \(i{\text{th}}\) mode, \(\theta\) is the refraction angle of F–P etalon, \(\gamma\) is the inversion factor, \(L_{{\text{c}}}\) is the transmittance of the closed Q-switch, \(R\) is the reflectivity of the output mirror, \(R^{\prime}\) is the reflectivity of the etalon, \(- \ln \left( R \right)\) is the coupling loss, \(\nu _{{\text{i}}}\) is the frequency of the \(i{\text{th}}\) mode, \(\tau _{{\text{i}}}\) is the photon decay time, \(\eta\) is the diffraction efficiency of the acoustic–optic Q-switch, \(L_{{QS}} \left( {\text{t}} \right)\) is the loss function of the acoustic–optic Q-switch, \(\delta _{1}\) and \(\delta _{2}\) are the Q-switch losses in diffraction efficiency of the beam, T is the one period time of the Q-switch. In this study, the repeat frequency was set as 20 kHz. To understand the competition trend between the adjacent modes, we considered the equation for the loop gain as follows.

$$ \frac{{G_{{\text{i}}} }}{{G_{{i + 1}} }} = \frac{{R_{i} RL_{{\text{c}}} L_{i} exp{\text{[2}}l\sigma _{i} N(t){\text{]}}}}{{R_{{i + 1}} RL_{{\text{c}}} L_{i} exp{\text{[2}}l\sigma _{{i + 1}} N(t){\text{]}}}} $$
(8)

Although the loop gain ratio of the adjacent mode in the mode competition attained or exceeded 10, the main oscillation mode played a dominant role, and the state could be identified as an SLM operation. The loop gain ratio equation can be expressed as follows:

$$ 10 = \left[ {\exp \left( {2l\sigma _{i} \left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)(N(t))} \right)} \right]^{{\frac{{\left( {t - t_{1} } \right)}}{{t_{l} }}}} \left( {\frac{{L_{i} }}{{L_{{i + 1}} }}} \right)^{{\frac{{\left( {t - t_{1} } \right)}}{{t_{l} }}}} $$
(9)

where \(N(t)\) is the population inversion with respect to time, and \({\text{t}}_{l}\) is the loop transit time and a constant relative to the length of the resonator. We can derive the following expression.

$$ t = \frac{{t_{l} \cdot {\text{ln}}10}}{{\left[ {2l\sigma _{i} \left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)N(t) + \left( {\frac{{L_{i} }}{{L_{{i + 1}} }}} \right)} \right]}} + t_{1} $$
(10)

The optimal seed signal build-up time could be calculated through simulations. It was observed that under the SLM operation conditions, the optimal SLM of the mode competition time was extended and increased with decreasing cavity length. It is helpful to select the optimal parameter setting for the experiments. The laser system parameters used for the theoretical simulations were as follows:\(\tau _{{\text{f}}} = = 35.7\) μs, \(R = 0.98\), \(h = 0.3{\text{ mm}}\), \(R^{\prime} = 0.035\), \(\Delta {\text{v}}_{H} = 500{\text{ GHz}}\), \(\theta = 2^{ \circ } 3^{\prime}\), \(L = 95{\text{ mm}}\), \(l = 5{\text{ mm}}\), \(P_{{{\text{ab}}}} = 2.8{\text{ W}}\), and \(\omega = 200\) μm.

We derived the relationship between the total running time of the pre-lase Q-switched adjustment process and the seed signal build-up time. Moreover, we analyzed the cost time required to achieve SLM using different cavity lengths. In Fig. 1, P0/P1 is the approximate ratio of the powers of specific modes and the adjacent modes. It was observed that the cost time taken to achieve the SLM decreased is shorter with decreasing of cavity length. The mode competition time is expressed as \(t_{2} = t - t_{1}\). It should be noted that the total running time of the pre-lase Q-switched adjustment process needs to be limited at the microsecond level and is related to the spontaneous lifetime. If the total running time of the pre-lase Q-switched adjustment process exceeds the spontaneous lifetime, the spontaneous radiation plays a leading role and generates multimode oscillations. The numerical simulation results for the temporal evolution of the Q-switched pulse are presented in Fig. 2. Thus, we derived an evaluation function to optimize the seed signal build-up time, mode competition time, and other resonator parameters.

$$ \eta _{{\text{t}}} \left( {t_{1} ,t_{2} ,\theta ,L} \right) = \frac{t}{{\tau _{{\text{f}}} - t_{{\text{2}}} }} = \frac{1}{{\tau _{f} }}\left( {t_{1} + \frac{{t_{l} \ln 10}}{{2l\sigma _{i} \left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)N(t) + \ln \frac{{L_{i} }}{{L_{{i + 1}} }}}}} \right) $$
(11)
Fig. 1
figure 1

Curves for power ratio as function of running time at different cavity lengths

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Fig. 2
figure 2

Numerical simulation results for temporal evolution of Q-switched pulse

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By solving the evaluation function, we selected the optimal parameter under the SLM operation. The theoretical results is shown in Fig. 3, where ηt is the evaluation efficiency, and t1 is the seed signal build up time.

Fig. 3
figure 3

Relationship between evaluation efficiency and seed signal build-up time at different cavity lengths

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If \(\eta _{{\text{t}}} < 1\), the SLM operation is achieved, and the seed signal build-up time and mode competition time do not reach the optimal state.

If \(\eta _{{\text{t}}} = 1\), the SLM operation is achieved, and the seed signal build-up time and mode competition time attain reach the optimal state.

If \(\eta _{{\text{t}}} > 1\), we move back to the operation of multilongitudinal mode oscillations, and the end trend is an ordinary Q-switched pulse established in noise.

For the Q-switching laser, the \(N_{{\text{i}}}\) and \(N_{{\text{f}}}\) values were determined by solving the following system of equations.

$$ {\text{N}}_{i} - N_{f} = \Delta N_{t} \ln \frac{{N_{i} }}{{N_{f} }} $$
(12)
$$ \Delta {\text{N}}_{{\text{t}}} = \frac{{L_{{\text{Re} s}} }}{{2\sigma l}} $$
(13)

For the pre-lase Q-switching SLM laser, the accumulation of initial population inversion was determined by solving Eqs. (11, 12, 13), considering the evaluation function. The peak power based on the evaluation function is expressed as follows.

$$ P_{{M{\text{ax}}}} = \frac{{hvAl\ln \frac{1}{R}}}{{t_{r} }}\left[ {\left( {{{\frac{{{\text{t}}_{l} \ln 10}}{{\eta \tau _{f} - t_{1} }} + \ln \frac{{L_{i} }}{{L_{{i + 1}} }}} \mathord{\left/ {\vphantom {{\frac{{{\text{t}}_{l} \ln 10}}{{\eta \tau _{f} - t_{1} }} + \ln \frac{{L_{i} }}{{L_{{i + 1}} }}} {2\sigma _{i} l\left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)}}} \right. \kern-\nulldelimiterspace} {2\sigma _{i} l\left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)}}} \right) + \ln \frac{{L_{i} }}{{L_{{i + 1}} }} - N_{t} \left( {1 + \ln \frac{{N_{i} }}{{N_{t} }}} \right)} \right] $$
(14)

Using the equations above, we could also analyze the effect of the evaluation function on the peak power in the Q-switched Pr:YLF lasers. The optimal peak output power in the SLM operation was obtained when the efficiency reached \(\eta = 1\) (Fig. 4). In the laser experiments, the characteristics of the SLM Q-switched laser agreed reasonably well with the theoretical predictions. We found that the evaluation function significantly influenced the peak output power of the SLM Q-switched laser. The evaluation function was used for selecting the optimal parameters to achieve high power output based on the realization of the SLM operation.

Fig. 4
figure 4

Measurement results for pre-lase SLM laser at different evaluation efficiency values and spectra of laser under different conditions: a CW laser with multilongitudinal oscillation; (b) SLM oscillation

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The single pulse energy \(E = \int_{{{\text{t}}_{a} }}^{{{\text{t}}_{b} }} {P(t)dt}\) was obtained from the time integration of the output power. It should be noted that at \({\text{t}} = t_{a}\), \(N = N_{i}\), and at \({\text{t}} = t_{b}\), \(N = N_{{\text{f}}}\). Thus, we can derive (15):

$$ E = \frac{{{\text{hv}}V\ln R^{{ - 1}} }}{{\gamma L_{{\text{Re} s}} }}\left( {\left( {{{\frac{{{\text{t}}_{l} \ln 10}}{{\eta \tau _{f} - t_{1} }} + \ln \frac{{L_{i} }}{{L_{{i + 1}} }}} \mathord{\left/ {\vphantom {{\frac{{{\text{t}}_{l} \ln 10}}{{\eta \tau _{f} - t_{1} }} + \ln \frac{{L_{i} }}{{L_{{i + 1}} }}} {2\sigma _{i} l\left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)}}} \right. \kern-\nulldelimiterspace} {2\sigma _{i} l\left( {1 - \frac{{\sigma _{{i + 1}} }}{{\sigma _{i} }}} \right)}}} \right) - N_{f} } \right) $$
(15)

where \(N_{{\text{i}}} - N_{f}\) is the number of population inversion consumed during the pulse establishment. Hence, the energy in the cavity can be divided into two parts. A part is the energy storage \(E_{{\text{i}}}\) of the particle, which represents the initial energy stored in the laser crystal that can be converted into a laser. From the analysis above, it can be inferred that \(E_{{\text{i}}} = hvVN_{i}\). The other part is the remaining energy \(E_{{\text{f}}}\) in the laser crystal after the giant pulse is extinguished, and it can be expressed as \(E_{{\text{f}}} = hvVN_{f}\). We assumed that the energy utilization was \(\mu = 1 - {{E_{{\text{f}}} } \mathord{\left/ {\vphantom {{E_{{\text{f}}} } {E_{i} }}} \right. \kern-\nulldelimiterspace} {E_{i} }} = 1 - {{N_{f} } \mathord{\left/ {\vphantom {{N_{f} } {N_{i} }}} \right. \kern-\nulldelimiterspace} {N_{i} }}\); the energy utilization increased as \({{N_{{\text{f}}} } \mathord{\left/ {\vphantom {{N_{{\text{f}}} } {N_{{\text{i}}} }}} \right. \kern-\nulldelimiterspace} {N_{{\text{i}}} }}\) decreased. It was observed that the energy utilization depended on the density of the initial number of population inversion. Based on the evaluation function η, the optimal single-pulse energy could be determined when the efficiency reached \(\eta = 1\) (Fig. 5). From the numerical solutions, we found that the theoretical results were consistent with the experimental results obtained from the laser-diode-pumped actively pre-lase Q-switched Pr:YLF laser with the acoustic–optic modulator. The output performance was optimized.

Fig. 5
figure 5

Relationship between pulse energy and incident pump power, and oscilloscope trace of SLM laser pulse at maximum incident pump power

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3 Experimental setup

The experimental configuration of the diode-pumped Q-switched SLM 639-nm Pr:YLF laser is depicted in Fig. 6. The pump source was a 444 nm InGaN blue laser diode. The diameter of the fiber core was 200 μm, and the pump beam was focused on the Pr:YLF crystal using a 1:1 coupling system. The laser resonator was a plane-concave cavity. The input mirror was a flat mirror with a high-transmission coating at a pump wavelength of 444 nm and a high reflection of approximately 99.8% at 639 nm. The output coupler was a concave mirror with a radius of curvature of 100 mm, and it was coated with a partial transmission of about 2% at 639 nm. The laser gain medium was an uncoated a-cut 0.5at.% Pr3+:YLF crystal (3 × 3 × 5 mm). The crystal was placed in a water-cooled copper heat sink, and the temperature of the heat sink was maintained at 22 °C. An acoustic–optic Q-switched element was inserted into the cavity to generate Q-switched operation, and the signal was modulated using an acoustic–optic modulator. Fabry–Perot (F–P) etalons, each with a thickness of 0.3 mm, were inserted into the cavity, and the total effective cavity length of the laser was optimized to approximately 95 mm.

Fig. 6
figure 6

Schematic of pre-lase SLM Pr3+:YLF laser

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Figure 4 shows the mode spectra for the maximal output powers measured using a scanning F–P interferometer with a free spectral range of 10 GHz (SA210-5B, Thorlabs), the minimum finesse of 150, and a resolution of 67 MHz. For the SLM operations, the laser operated on only a longitudinal mode in the resonator. At the absorbed pump power of 2.8 W (an incident pump current of 1.63 A) and a repetition frequency of 20 kHz, maximum single-pulse energy of 4.41 \(\mu\) J was obtained in the SLM operation. Correspondingly, the pulse width and the pulse peak power were approximately 88 ns and 50.1 W, respectively.

4 Conclusions and discussion

We established an efficient optimization model of a continuous-wave end-pumped actively pre-lase Q-switched Pr3+:YLF SLM laser and analyzed the factors influencing the mode selection using pre-lase technology. We found that it is necessary to consider the effect of the evaluation function on the laser output characteristics during the SLM operation. In the laser experiments, the characteristics of the single-longitudinal Q-switched laser were in good agreement with the theoretical predictions. It is demonstrated that the output performance of the SLM laser is related to the optimization of the pre-lase Q-switching modulation time and other resonator parameters. This optimization method is applicable to all types of solid-state lasers. The proposed method helps select the insertion element parameters correctly and optimally modulate the pre-lase Q-switching two-step signal generator to achieve optimum laser output performance.