1 Introduction

2 µm laser, located in the atmospheric window and eye-safe band, has lots of applications, such as medical treatment, atmospheric remote sensing, material processing, nonlinear optical frequency conversion, and so on [1]. Laser diode pumped thulium ion (Tm3+) doped crystals are the effective way to generate 2 µm laser [2], especially Tm3+-doped yttrium aluminum oxide (YAP) crystal, which belongs to the space group D162h, owns large effective emission cross-section, excellent thermal and mechanical properties and strong natural birefringence [3,4,5]. The establishment and solution of rate equation model is an effective way to obtain the output performance of lasers, which can also lay a foundation for analyzing experimental phenomena. However, the existing rate equation model of Tm3+ doped lasers either ignore various effects [6,7,8,9,10], or regard those effects as a fixed loss [5, 7], which lead to significant deviation between experimental results and theoretical simulation results, thus losing the guiding significance of rate equation theory for conducting experiments.

The reabsorption in Tm3+-doped laser was found and considered for the first time by Risk W P in 1988 [11], which was computer-only with no experiments. Then the upconversion loss and ground state loss of both Tm3+ and Ho3+ doped lasers were calculated by Rustad G et al. in 1996 [6]. In 1997, Taira T et al. made a great progress in quasi-three-level modeling. [7]. In 2007, a more intuitive model was established by O.A. Burry et al. [5], for comparative analysis and optimization of Tm: YAG and Tm: YAP lasers, in which reabsorption is considered as a fixed loss and included in the loss coefficient γ. However, introducing reabsorption only as a loss into the rate equation, there may be significant deviations between the experimental results and the theoretical simulation results of Tm3+-doped lasers. Although the reabsorption effect reduces the output power of the laser, the population inversion of the laser material increases to a certain extent, which means that a more accurate rate equation model is needed to describe the reabsorption process and improve the simulation accuracy of laser output performance.

In this paper, a new rate equation model of continuous wave (CW) Tm: YAP laser with fully consideration of reabsorption is developed. The models that ignore the reabsorption of thulium ion at all (Model 1) or only takes it as a fixed loss (Model 2) are also used to verify the advantage of the new model (Model 3). Laser performance of LD end-pumped CW Tm: YAP crystal is achieved experimentally to indicate the correctness of Model 3.

2 Energy level structure and transition of thulium ion

The quasi-three-level generation scheme of thulium laser is shown in Fig. 1.

Fig. 1
figure 1

Scheme of quasi-three-level thulium laser

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The quasi-three-level generation scheme of thulium laser is shown in Fig. 1. The energy manifolds indicated 3H4, 3H5, 3F4 and 3H6, recorded as by 4, 3, 2, 1. And because the lower laser level is one of the Stark sublevels of ground level, the generation scheme is a quasi-three-level. Tm3+ ions located at multiplet 1 will be extracted to manifold 4 after the absorption of pump photons, which can be expressed by pump rate Rp as formula (1).

$$R_{p} = \frac{{P_{a} }}{{Al_{a} hv_{p} }}$$
(1)

where h is Plank constant, vp is the frequency of pumping laser. Pa is the absorbed pumping power, connected with pumping power Pi that injected into the laser material, as Pa = Pi(1 − exp(− σaNTmla)), in whichσa is the absorption cross-section at the pumping wavelength, NTm is the thulium concentration, la is the length of the crystal. Thus, 1 − exp(− σaNTmla) can be understood as an absorption ratio of pump power. This idea of the ratio can also be used to describe the reabsorbed power of 1.9 µm laser. The pump laser can be seen as an ideal cylindrical light, whose mode cross-section A describes the space pumping laser passes, expressed by A = πrp2, in which rp is the radius of the pumping beam.

Then Tm3+ ions on manifold 3H4 will be downwards in three ways: spontaneous emission, thermal relaxation, and cross relaxation. The thermal relaxation assumed as instantaneous process can be ignored [7, 9]. An ion on manifold 3H4 will interact with a surrounding ion on 3H6, one of them loses energy and another absorbs, both transition to 3F4, which can be shown as (3H4, 3H6) to (3F4, 3F4), and this process is the cross relaxation of Tm3+-doped laser. Cross relaxation and the spontaneous emission determine the average lifetime of Tm3+ on manifold 3H4 (t4), which can be written as t4 = 0.541 ms [12, 13]. To better illustrate the conjunction between t4 and cross relaxation, cross relaxation rate γcr is introduced, expressed as formula (2).

$$\gamma_{{{\text{cr}}}} = \frac{1}{{t_{4} }}\left( {\frac{{N_{{{\text{tm}}}} }}{{N_{0} }}} \right)^{2}$$
(2)

where N0 is the thulium concentration at that the cross-relaxation rate is equal to the rate of the thulium ions transitions from multiplet 4 to the lower multiplets. γcr can be accurately calculated when taking N0 = 0.5 at. % or other values around 0.5 at.% to 1.6 at.% [5, 12, 13]. N0 is of great difficult to obtain [12, 13], however, γcr is mainly used for calculating quantum yield ηQY, and changing t4 or N0 would only make ηQY a variation from 1.88 to 2. Therefore, N0 of 0.5 at. % and t4 = 0.541 ms can be considered as an accurate hypothesis [5], which caused ηQY = 1.972 in the formula. Also, cross-relaxation of (3H5, 3H6) to (3F4, 3F4) is ignored due to the lifetime of 8 μs for multiplet 3H5. Also, β43 = 0.55, β42 = 0.05 here, and β41 is not involved in the rate equation [6].

Upconversion means the ions on multiplet 3F4 may interact with each other, one of them loses energy and another absorbs, then transits to different multiplets, which degrades the laser performance. The value of the upconversion rate is denoted by μ3 for (3F4, 3F4) to (3H5, 3H6) and μ4 for (3F4, 3F4) to (3H4, 3H6). As the total upconversion rate is determined mainly by μ3, the value can be μ = 1.885 × 10–18 cm3/s [5, 6].

As shown in Fig. 1, 1.9 μm radiative transition is generated between 3H6 and 3F4 Stark sub-level. Due to the laser photon energy is equal to the energy difference between 3H6 and 3F4, the laser photon produced has a chance to be reabsorbed when counteracting with an ion on 3H6 in Tm: YAP crystal, named reabsorption, which can be expressed as a rate. This can cause an increase of particle density on 3F4 but a descend on 3H6, also the decrease of laser photons, which should be added in the model to describe the energy transfer process more precisely. Therefore, the reabsorption rate Rre can be expressed as formula (3).

$$\left\{ {\begin{array}{*{20}c} {R_{{{\text{re}}}} = \frac{{P_{{{\text{io}}}} \left( {1 - \exp \left( { - 2\sigma_{a} N_{{{\text{tm}}}} l_{a} } \right)} \right)}}{{Al_{a} hv}}} \\ {\gamma_{i} = 2\sigma_{a} N_{{{\text{tm}}}} l_{a} } \\ \end{array} } \right.$$
(3)

The Pio is the power of 1.9 μm laser oscillating in the resonator, which is expressed as:

$$P_{io} = \frac{hv}{{t_{r} }}q$$
(4)

where the value tr = 2L/c is the time of photons double passing along the resonator, q is number of photons. Pao = Pio(1 – exp(− 2σaNTmla)) is reabsorption ratio of laser photons, sharing the same structure in pump rate R, and the denominator of Rre is also the same to R. Considering the high reflectivity, and the laser keep generating along with pump light in the crystal, index ‘2’ in Pa and tr is introduced here to describe laser double passing the resonator [5]. σa is absorption cross section on laser wavelength, which is 0.324 × 10–21 cm2 considering the experiments in this paper [5, 6, 14].

In formula (3), γi is reabsorption loss, often ignored (Model 1) or be taken as a fixed loss (Model 2) in previous works.

3 Novel theoretical model and simulations

A model considering the reabsorption as a rate can be expressed as formula (5).

$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{{\text{d}}N_{4} }}{{{\text{d}}t}} = R + \mu_{4} N_{2}^{2} - \frac{{N_{4} }}{{t_{4} }} - \gamma_{cr} N_{4} } \\ {\frac{{{\text{d}}N_{3} }}{{{\text{d}}t}} = \mu_{3} N_{2}^{2} + \beta_{43} \frac{{N_{4} }}{{t_{4} }} - \frac{{N_{3} }}{{t_{3} }}} \\ \end{array} } \\ {\frac{{{\text{d}}N_{2} }}{{{\text{d}}t}} = 2\gamma_{cr} - 2\left( {\mu_{3} + \mu_{4} } \right)N_{2}^{2} + \frac{{N_{3} }}{{t_{3} }} + \beta_{42} \frac{{N_{4} }}{{t_{4} }} - \frac{{N_{2} }}{{t_{2} }} - \frac{{c\left( {\sigma_{e} N_{2} - \sigma_{a} N_{1} } \right)}}{{Al_{a} }}q{ + }\boxed{{\text{R}}_{{{\text{re}}}} }} \\ \end{array} } \\ {N_{tm} = N_{1} + N_{2} + N_{3} + N_{4} } \\ \end{array} } \\ {\frac{{{\text{d}}q}}{{{\text{d}}t}} = c\left( {\sigma_{e} N_{2} - \sigma_{a} N_{1} } \right)q - \frac{q}{{\boxed{t_{c} }}}} \\ \end{array} } \right.$$
(5)

where Rre is reabsorption rate. β4i, in which i = 1…3 are the coefficients of luminescence branching, which indicates the destination of spontaneous emission by index “i”. q is a quantity of laser photons in cavity.

In formula (5), ignore Rre and change the expression of tc, Model 1 can be obtained. It can delete Rre and only take the reabsorption as the loss to get Model 2. Reabsorption components, as reabsorption rate Rre and photon lifetime tc, have been considered and circled in red, can be seen as Model 3, which actually express the essence of reabsorption, that ascend of population inversion and descend of photons in resonator. Reabsorption gain rate only appears in the dynamic equation describing N2, and the loss of reabsorption on the quantity of photons is already included in the photon lifetime tc, which is expressed as tc = 2L/cγ in the model. γ is double-passing loss in the cavity, γ = γi + (γ1 + γ2) + γL. And γi = 2σaNTmla is the double-passing reabsorption loss. γ1, γ2 are losses of the mirrors, γ1 = − ln (R1), γ2 = − ln (R2). γL is the cavity loss and can be taken as γL = 0.015 [7, 9].

Considering the concentrations of thulium ions on N3 and N4 are small enough, NTm = N1+N2. The density of inversion particles for quasi-three-level model is expressed as N = N2-N1σa/σe [5], therefore, formula (6) is obtained (based on CW output).

$$\left\{ {\begin{array}{*{20}c} {\eta_{QY} R - \frac{{c\sigma_{e} }}{{Al_{a} }}Nq - \frac{{N_{2} }}{{t_{2} }} - \mu N_{2}^{2} + R_{{{\text{re}}}} = 0} \\ {\left( {c\sigma_{e} N - \frac{1}{{t_{c} }}} \right)q = 0} \\ \end{array} } \right.$$
(6)

Assuming \(\eta_{QY} = \frac{{\beta_{42} + \beta_{43} + 2\gamma_{cr} t_{4} }}{{1 + \gamma_{cr} t_{4} }}\) as a quantum yield, ηQY = 1.972 after calculation. Thus, μ = μ3 + μ4(2 − ηQY) = 1.885 × 10–18. It has been proved that quantum yield is close to 2 due to the existence of cross relaxation [5,6,7, 9].

The critical density of inversion particles Nc, the output laser power Pre (by solving formula (6) to obtain q) and the threshold pump power Pthreshre (by setting output power is 0 W) can be obtained after solving the Model 3, the results can be expressed as formula (7). It is different from the results of Model 2, as shown in formula (8), [5].

$$\left\{ {\begin{array}{*{20}c} {N = \frac{1}{{c\sigma_{e} t_{c} }}} \\ {{\text{Pthresh}}_{{{\text{re}}}} = \frac{{Al_{a} hv_{p} }}{{\eta_{{{\text{QY}}}} \left( {1 - \exp \left( { - 2\sigma_{p} N_{{{\text{tm}}}} l_{a} } \right)} \right)}}\left( {\frac{{N_{2} }}{{t_{2} }} + \mu N_{2}^{2} } \right)} \\ {P_{{{\text{re}}}} = \frac{hv}{{t_{r} }}\gamma_{2} \frac{{Al_{a} t_{c} t_{r} }}{{t_{r} - t_{c} \left( {1 - \exp \left( { - 2\sigma_{a} N_{{{\text{tm}}}} l_{a} } \right)} \right)}}\left( {\eta_{{{\text{QY}}}} \frac{{P_{i} \left( {1 - \exp \left( { - 2\sigma_{p} N_{{{\text{tm}}}} l_{a} } \right)} \right)}}{{Al_{a} hv_{p} }} - \frac{{N_{2} }}{{t_{2} }} - \mu N_{2}^{2} } \right)} \\ \end{array} } \right.$$
(7)
$$\left\{ {\begin{array}{*{20}c} {N = \frac{1}{{c\sigma_{e} t_{c} }}} \\ {{\text{Pthresh}} = \frac{{Al_{a} hv_{p} }}{{\eta_{{{\text{QY}}}} \left( {1 - \exp \left( { - 2\sigma_{p} N_{{{\text{tm}}}} l_{a} } \right)} \right)}}\left( {\frac{{N_{2} }}{{t_{2} }} + \mu N_{2}^{2} } \right)} \\ {P = \frac{{hv\gamma_{2} Al_{a} t_{c} }}{{t_{r} }}\left( {\eta_{{{\text{QY}}}} \frac{{P_{i} \left( {1 - \exp \left( { - 2\sigma_{p} N_{{{\text{tm}}}} l_{a} } \right)} \right)}}{{Al_{a} hv_{p} }} - \frac{{N_{2} }}{{t_{2} }} - \mu N_{2}^{2} } \right)} \\ \end{array} } \right.$$
(8)

And N2 can be expressed as formula (9), works for both Model 1, 2 and 3, but tc must be the respective expressions, same as other expressions including tc.

$$N_{2} = \frac{{\sigma_{a} N_{tm} + \frac{1}{{ct_{c} }}}}{{\sigma_{e} + \sigma_{a} }}$$
(9)

All parameters used are shown in Table 1.

Table 1 The parameters used in the model [5, 6, 8, 9, 12, 14]
Full size table

Simulation on output power versus input pump power for Model 1, 2, and 3 is shown in Fig. 2.

Fig. 2
figure 2

Model simulation and comparison

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The threshold is 2.75 W, 3.17 W and 3.18 W respectively, and the slope efficiency are 70.5%, 15.38% and 30.73% for Model 1, 2, and 3, respectively. The simulation result of Model 1 has a significant deviation from previous experimental reports [5, 6]. While, completely treating reabsorption as a loss can result in low slope efficiency of Tm: YAP laser, as Model 2. Therefore, it is necessary to comprehensively consider the impact of reabsorption on the output performance of Tm: YAP lasers in theory.

4 Experimental results and analyses

To further demonstrate the accuracy of Model 3, experiment on laser characteristics of Tm: YAP crystal is carried out. The schematic of LD end-pumped Tm: YAP laser is shown in Fig. 3.

Fig. 3
figure 3

Schematic of the LD end-pumped Tm: YAP laser

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The pump source is a fiber coupled laser diode with central wavelength of 795 nm and maximum output power of 70 W. The radius and numerical aperture of the fiber are 200 µm and 0.22, respectively. The focus coupling ratio of the lens group is 1:1, and the focal lengths coated with high transmissions at 795 nm (T > 99.5%) are f1 = 35 mm and f2 = 75 mm, respectively. The mode matching between pump mode and laser mode is optimized by changing the pump beam waist radius and its location. M1 is a plane mirror, coated with anti-reflection at 795 nm (R < 0.5%) and high reflection (R > 99.5%) at 1940 nm. M2 is a concave output mirror with curvature radius of 200 mm and transmittance of 10% at 1940 nm. The Tm3+ doping concentration and the dimension of b-cut Tm: YAP crystal is 3.5 at. % and Φ3 × 14 mm3, respectively. Both ends of the crystal are coated with anti-reflection at 795 nm (R < 0.5%) and 1940 nm (R < 0.5%). The crystal is wrapped in indium foil with the thickness of 0.05 mm and placed in a copper heat sink, which is cooled by water which is kept at 18 °C. The cavity length is 78 mm.

The output power versus input pump power is achieved, as shown in Fig. 4, by using a power meter F150A (OPHIR, Jerusalem, Israel). Threshold pump power of 5.19 W and maximum output power of 7.1 W under pump power of 27.13 W is obtained. The slope efficiency is 31.3%. However, saturation occurs at higher pump power. Higher pump power means severely thermal accumulation in Tm: YAP crystal, leading to an enhancement of the upconversion of Tm: YAP, and a deterioration of the mode matching between pump mode and laser mode. The stability of the resonant cavity has changed, too.

Fig. 4
figure 4

Output power versus input power of Tm: YAP laser

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The central wavelength of Tm: YAP laser is measured to be 1936.89 nm by a spectrometer (AQ6370 of Yokogawa, Musashino, Tokyo, Japan), as shown in Fig. 5.

Fig. 5
figure 5

Central wavelength of Tm: YAP laser

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The simulation results under Model 1, Model 2, Model 3, and experimental results are shown in Fig. 6.

Fig. 6
figure 6

Simulation results under three models and experimental results

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The experimental threshold is 5.19 W, which is higher than that of 2.75 W, 3.17 W and 3.18 W for the simulation results of Model 1, 2, and 3. At pump power of 6.35 W, output power of 2.6 W, 0.37 W, 0.97 W and 0.46 W is obtained for simulation results of Model 1, 2, 3 and experimental result, which means the simulation results of Model 2 are closer to the actual situation at lower pump power. At pump power of 27.13 W, output power of 17.24 W, 2.76 W, 7.36 W is obtained for Model 1, 2, 3. While, the experimental result is 7.1 W, which is closer to the simulation result of Model 3. The deviation of the slope efficiency between the simulation of Model 3 (30.73%) and experimental result (31.3%) is 0.57%. At low pump power, the photon density in the resonator is small, leads to lower probability of counteract between photons and thulium ions. As mentioned in formula (3), more q lead to higher Rre, which means Model 3 owns higher accuracy at high pump power.

The experimental results under resonator length of 51 mm and 105 mm are achieved, to further illustrate the accuracy of the Model 3.

As shown in Fig. 7, when the resonator length is 51 mm, the slope efficiency of experiment and Model 1, 2, 3 are 31.2%, 70.4%, 11.53%, 30.72%, respectively. And the slope efficiency under resonator length of 105 mm is 30.68%, 70.44%, 11.53%, 30.70%. As simulation results indicate, the three models exhibit similar slope efficiencies with different resonant cavity lengths. At the laser oscillation threshold, the output power in experiment is like the simulation result of Model 2, which means that when the pump power is low, the reabsorption loss has a more significant impact on the output power of the laser. While, beyond threshold, the output power in experiment is closer to the simulation results of Model 3, which means that considering reabsorption effect as a fixed loss is unreasonable. Deviation between experimental results and theoretical simulations mainly come from two aspects. The change in cavity length will lead to not only the changes in mode matching between the pump laser and the oscillating laser, but also the changes in the cavity loss. However, the slope efficiency of the experimental results is consistent with the simulation results of Model 3, which proves the accuracy of the model proposed in this paper.

Fig. 7
figure 7

Simulation results and experimental results with different resonator length

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5 Conclusion

In summary, a new theoretical model based on rate equations considering reabsorption is established to simulate the output characteristics of CW Tm: YAP laser more precisely for the first time. Compared to the model that completely ignore the reabsorption of thulium ion or only takes it as a fixed loss, the simulation results of the new model can better predict the output performance of Tm: YAP laser. Output power of 7.1 W at input pump power of 27.13 W with resonator length of 78 mm is achieved in experiment, which is close to the simulation result of 7.36 W with the new model. The deviation of the slope efficiency between the simulation (30.73%) and experimental result (31.3%) is 0.57%. Experimental results with resonator length of 51 mm and 105 mm are obtained, which prove the accuracy of the model proposed in the paper. Further research can focus on refining each parameter, considering the field distribution of particles, and introducing Gaussian beams, to improve Model 3 and predict output transverse mode and longitudinal mode in simulation.