Objective Since Er ³⁺,Yb ³⁺∶glass is used in laser diodes (LD) pumped lasers, several studies on thermal properties of Er ³⁺,Yb ³⁺∶glass have been reported. However, there are still few related reports on the effect of pumping repetition frequency and the material thermo-optic coefficient on output energy. The pumping repetition frequency of LD and thermo-optic coefficient of Er ³⁺,Yb ³⁺∶glass affect the thermal focal length and ultimately change the output energy of the Er ³⁺,Yb ³⁺∶glass laser. Thus, in this study, we analyze the effect of pumping repetition frequency and the thermo-optic coefficient on output energy to guide material selection and output energy analysis for the end-pumped high output energy Er ³⁺,Yb ³⁺∶glass laser.

Methods In this paper, we design a passively Q-switched laser with LD end-pumped to measure output energy using a plano-concave cavity structure and Co ²⁺∶MgAl₂O₄ as the Q-switching crystal (Fig. 1). The measuring thermal focal length experiment is based on the critical cavity method and uses a flat-flat cavity to measure the thermal focal length. In the measurement of slope efficiency, the variation of the output energy with pumping power is measured without Co ²⁺∶MgAl₂O₄ in Fig. 1 under the same pumping conditions. Additionally, the thermal focal length formula, the matrix calculation method of the stable cavity, and the rate equation are used to analyze the phenomena in the above experiments.

Results and Discussions In a passively Q-switched LD end-pumped laser, we obtained that the output pulse energy of EY1, EY2, and EY3 decreases from 417, 401, and 378 μJ to 356, 337, and 329 μJ as the pumping repetition frequency increases from 10 Hz to 28 Hz (Fig. 2). First, we verified the phenomenon that the thermal focal length decreases with increasing pumping repetition frequency theoretically and experimentally to analyze the above phenomenon. Second, by numerical simulation of the resonator, it is found that in the plane-concave cavity, the beam radius of the front surface of the resonator decreases with increasing thermal focal length when the thermal focal length is greater than f_Tm(the thermal focal length corresponding to the minimum beam radius of the front surface of the resonator) (Fig. 5). Finally, the rate equation is solved using the Runge-Kutta algorithm to obtain the trend of the output pulse energy gradually decreasing with increasing pumping repetition frequency (Fig. 6). This is because the output pulse energy is positively correlated with the beam radius of the front surface of the resonator. After performing the free-running operation, the slope efficiencies of the three Er ³⁺,Yb ³⁺∶glass samples, EY1, EY2, and EY3 were 22.16%, 22.82%, and 23.14%, respectively (Fig. 7). Their thermo-optic coefficients are 34×10 ^-7, 29×10 ^-7, and 24×10 ^-7 K ^-1, respectively. The Q-switched output pulse energy after adding Co ²⁺∶MgAl₂O₄ to the above non-saturable cavity are 323, 363, and 391 μJ (Table 3). When the thermal focal length increases with a decrease in the thermo-optic coefficient (Fig. 3), the cross-sectional area of the laser beam in Er ³⁺,Yb ³⁺∶glass increases accordingly. The inversion population participating in stimulated radiation in Er ³⁺,Yb ³⁺∶glass also increases, which increases the slope efficiency. Moreover, since the Q-switched output energy is positively correlated with the beam radius of the front surface of the resonator, the Q-switched output energy also shows an increasing trend.

Conclusions In this study, we reveal the relationship among thermo-optic coefficient, pumping repetition frequency, slope efficiency, and Q-switched output pulse energy of an eye-safe solid-state laser with Er ³⁺,Yb ³⁺∶glass as the gain medium experimentally and theoretically, when the thermal focal length is greater than the value at the minimum point. The thermal focal length in Er ³⁺,Yb ³⁺∶glass decreases with an increase in the pumping repetition frequency, which decreases the cross-sectional area in the gain medium, and finally reduces the Q-switched output pulse energy. Thus, the Q-switched output pulse energy is negatively related to the LD pumping repetition frequency. Since the same pumping conditions, Er ³⁺,Yb ³⁺∶glass with lower thermo-optic coefficient has a larger thermal focal length, larger cross-sectional area of the stimulated radiation beam in Er ³⁺,Yb ³⁺∶glass, higher slope efficiency in a non-saturable cavity, and higher Q-switched output pulse energy. Thus, the Q-switched output pulse energy is negatively correlated with thermo-optic coefficient and pumping repetition frequency. Therefore, we can reduce the thermal lens effect by reducing the LD pumping repetition frequency and selecting low thermo-optical coefficient Er ³⁺,Yb ³⁺∶glass to achieve high output pulse energy, which will help human eye-safe solid-state lasers use this material further to develop the ranging, lidar, and medical fields.