The recent developments in science and technology have resulted in an increasing demand for all-solid-state lasers. Several processes such as laser manufacturing, laser alignment, and laser ignition especially require all-solid-state lasers to operate with high efficiency, high output power, and high beam quality. Therefore, many researches in the field of laser technology have focused on improving the performance of all-solid-state lasers, and a couple of methods, such as mode matching and weak absorption pumping, have been developed.

Mode matching is one of the main techniques to obtain a laser with high beam quality. In this technique, the spatial distributions of the pump mode and the cavity mode will be highly overlapped in the laser crystal; thus, the fundamental-mode lasing light will be generated in the resonant cavity. In fact, the overlap efficiency that represents the quality of the mode matching also affects the overall efficiency of the laser system.^[1] By now, several authors have analysed the mode matching in all-solid-state lasers. Laporta et al., through the analysis of the space-dependent rate equation, reported the optimum ratio of the cavity mode size to the pump-mode size required for achieving a lower threshold and a higher output power.^[2] Chen et al. reported their calculation of the optimum design for the laser system, with regard to the pump beam quality.^[3] Their analysis indicates that the pump beam quality has a significant effect on the optimum size of the pump-mode and the corresponding location of the pump beam waist. Hajiesmaeilbaigi et al. conducted a numerical research on the design criteria for the laser system containing an active medium that has a small volume. Several factors including the optimum cavity mode size, pump beam quality, and minimum pump-mode size were analysed to achieve maximum output efficiency.^[4] Shayeganrad et al. analysed the effects of the active-medium length and pump beam quality, which determines the design of the laser resonator and coupling system.^[5] It should be noted that in all the reports mentioned above, the pump light was assumed to experience strong absorption within the gain medium; thus, merely the single-pass pumping was considered.

In general, a laser medium with a strong absorption of the pump light would generate heavy thermal load, which not only degrades the overall efficiency of the laser, but also threatens its safety. A widely applied solution to this problem is the adoption of weak-absorption pumping in the laser system, which ensures uniform heat distribution in the longitudinal direction of the active medium, thus reducing thermal effects such as end-effect and thermal-stress. Meanwhile, weak-absorption techniques such as in-band pumping could also improve the quantum efficiency of the laser system; thus providing the benefits of high efficiency and high output-power.^[6–29] However, a laser with weak-absorption pumping faces two problems. One problem is the under-utilization of the pump power. Due to the low absorption, the pump power might not be effectively absorbed, which results in a limitation of the lasing power. The other problem arises from the mode matching of the system for a low absorption of the pump light, which, combined with the required long absorption length of the laser medium, will strongly affect the mode matching of the system. Currently, both problems are solved by applying dual-pass pumping, in which the pump power passes through the active medium twice to ensure that more pump power is absorbed. However, no numerical analysis for the dual-pass pumping laser has been reported until now.

In this paper, a model for dual-pass pumping is established and a formula for the mode-matching function, with regard to dual-pass pumping, is provided. By assuming a high absorption efficiency of 90%, numerical calculations are performed for the mode matching of the all-solid-state laser, including both dual-pass pumping and weak absorption. The calculated results provide the optimum location of the pump-mode waist and the corresponding pump-mode size for both the forward pump beam and backward pump beam. These results also indicate that these optimum values would be affected by the pump beam quality as well as by the cavity mode size. By the numerical fitting of these results, simple formulas considering the beam quality of the pumping light, absorption coefficient, and beam waist of the cavity mode are obtained to express these optimum values. These formulas would be helpful when designing the laser system. To test the utility of the model, an Nd:YVO₄ laser was built and optimized by adopting the results of the calculation and there was a good agreement between the calculation and experiment results, which verifies the model and the numerical analysis.

The slope efficiency of a general all-solid-state laser can be given by^[1]

For our model, can be determined by assuming a fundamental mode, which is then represented as^[3]

Fig. 1.

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	Fig. 1. (color online) Typical model for the dual-pass pumping.

From Eqs. (2)–(13), the expression for the mode-matching function of the single-pass pumping should be^[4]

To achieve an efficient absorption of the pump power, the absorption efficiency is fixed at 90%, which is in agreement with . For the fixed α, l, , and , the optimum location of the pump-mode waist ( , and the optimum radius of the waist ( , can be obtained by solving the following equations:

Figure 2 shows the plot of the optimum pump-mode radius as a function of α and . The radius of the cavity mode is 0.5 mm. As shown by the plot, the optimum radii of the pump mode in the two directions can be regarded as identical, and they decrease with an increase in the value of α. Moreover, is an important factor that affects and . Being pumped by a source with a higher beam parameter product, the laser system tends to have a larger optimum pump beam size, which is consistent with the results of the previous study.^[3] Through the numerical analysis, the fitted formulas used to calculate the optimum radius of the pump beam are given by

Fig. 2.

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	Fig. 2. (color online) Dependence of the optimum radius of the pump-mode waist on α and .

Table 1.

Table 1.

Table 1. Factors for and . . /mm a1 a2 a3 b1 c1 c2 c3 d1 d2 d3 0.1 0.049 0.19 0.019 −4.56 0.15 0.14 −0.080 −0.19 −0.52 −0.11 0.2 0.0069 0.61 0.13 −4.59 0.37 0.10 −0.30 −0.32 −0.22 −0.023 0.3 5.39 0.0049 −5.23 −4.71 0.69 0.084 −0.64 0.023 0.47 −0.32 0.4 −0.27 −0.13 0.48 −4.63 0.17 0.26 −0.092 2.07 0.016 −2.42 0.5 −0.44 −0.12 0.66 −4.79 0.14 0.31 −0.067 0.31 0.083 −0.66 0.6 −0.37 −0.33 0.55 −4.96 0.15 0.31 −0.077 0.070 0.23 −0.41 0.7 −0.44 −0.40 0.59 −5.10 0.13 0.35 −0.049 0.10 0.18 −0.46 0.8 −0.55 −0.30 0.74 −5.33 0.11 0.39 −0.022 −0.89 −0.035 0.51 0.9 −0.74 −0.23 0.92 −5.56 0.11 0.38 −0.028 0.59 0.045 −0.98 1.0 −0.84 −0.24 1.01 −5.72 0.12 0.38 −0.035 0.082 0.20 −0.45

Table 1. Factors for and . .

Figure 3 shows the optimum location of the pump-mode waist in the forward direction as a function of α and , while was equal to 0.5 mm. It can be seen that in the range of , changes significantly as increases. However, as α increases, the effect of on reduces. Figure 4 shows the relationship between and α with a different value of . Compared to , is less dependent on . From the numerical analysis of both and , the following interesting but meaningful conclusion can be obtained:

Fig. 3.

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	Fig. 3. (color online) Optimum distance of the forward pump beam waist as a function of α and .

Fig. 4.

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	Fig. 4. (color online) Optimum distance of the backward pump beam waist as a function of α and .

To simplify the calculation, we fitted with both α and . The fitted formulas are

Table 2.

Table 2.

Table 2. Factors for . . /mm e1 e2 e3 f1 g1 g2 g3 h1 h2 h3 0.1 47.12 −0.42 3.77 −10.09 10.47 −0.39 −0.53 −0.50 −1.14 −0.55 0.2 78.91 −0.40 4.15 −10.34 24.42 −0.13 −10.39 −0.73 −0.20 −0.24 0.3 102.7 −0.34 0.30 −10.81 −21.98 0.088 36.77 1.18 0.070 −2.17 0.4 136.2 −0.45 9.83 −10.80 −119.9 0.019 136.4 −1.32 −0.10 0.21 0.5 154.7 −0.38 4.05 −11.11 −11.91 0.13 28.31 1.49 0.056 −2.58 0.6 187.6 −0.27 −22.79 −11.37 −9.04 0.16 25.94 0.96 0.081 −2.09 0.7 224.8 −0.21 −51.72 −11.57 −14.97 0.10 32.40 12.63 0.0071 −13.78 0.8 303.9 −0.14 −126.3 −11.92 −11.70 0.12 29.48 7.68 0.012 −8.86 0.9 599.5 −0.058 −421.5 −12.20 −5.68 0.20 23.41 0.60 0.11 −1.76 1.0 −509.6 0.052 684.9 −12.46 −4.09 0.24 21.87 0.39 0.15 −1.56

Table 2. Factors for . .

Figure 5 shows the plot of Fdual as a function of α and BPP obtained by inserting the values of , , , and in Eq. (15). For comparison, we calculated the optimum overlap function for the single-pass pumping, with a cavity mode radius of 0.5 mm, while the absorption efficiency was assumed to be 90% ( ). Obviously, the dual-pass pumping is more helpful in improving the overlap for shorter active mediums, which is especially obvious when the absorption efficient is significantly low.

Fig. 5.

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	Fig. 5. (color online) Optimum mode-matching function with respect to α and .

In order to verify the numerical analysis, an experiment was performed according to the calculations. The schematic setup for the experiment is shown in Fig. 6. The pumping source was a fibre-coupled laser diode operating at 914 nm. The coupling fibre had a diameter of and a numerical aperture of 0.22, which resulted in a of in the laser medium. The crystal had a dimension of 1.5× 1.5× 24 mm³ and an Nd-doping concentration of 1%, resulting in a dual-pass absorption efficiency of at the pump light. Both faces of the crystal were highly transmitting (HT) at 914 nm and 1064 nm. The optical collimation system consisted of three lenses whose focal lengths were 60 mm, 60 mm, and 160 mm, respectively. M1, the plane mirror, which was HT at 914 nm and highly reflecting (HR) at 1064 nm, served as one of the cavity mirror in the laser. M2 was a plane mirror, which was HT at 914 nm and HR at 1064 nm in a 45° direction. The OC was a plane mirror whose transmittance was 52.11% at 1064 nm. After transmitting through the laser medium, the remaining pump light could be reflected back to the Nd:YVO₄ crystal by the curved mirror M3, which had a radius of 100 mm and was HR coated at 914 nm. The laser cavity had a symmetric structure whose optical length was approximately 168 mm. This resulted in a fundamental-mode spot radius of mm for the 1064-nm oscillating beam in the laser medium. From the numerical analysis for the model, the optimization values for the pump light were obtained as , and . After these values were adopted, the relationship between the output power and the incident pump power was drawn, as shown in Fig. 7. It can be seen that the output power increases linearly with the increase in incident pump power, which resulted in a slope efficiency of ∼44%. This fits well with the numerical calculation of ∼45.2%, which was obtained by substituting the adopted and calculated values as well as into 1 13, and (14), and this verifies both the model and the numerical method. The three factors , , and were all regarded as ∼1,^[21,32] and the optimized overlap efficiency was obtained as 0.65.

Fig. 6.

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	Fig. 6. (color online) Experimental setup of the Nd:YVO4 laser with dual-pass pumping. LD: laser diode; OC: output coupler.

Fig. 7.

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	Fig. 7. (color online) Output power versus incident pump power.

It should be noted that in both the modelling and the experiment, the phase aberration, which is caused by the thermal lens and would result in degradation of the beam quality of the laser, was not taken into account. In general, to obtain a high-beam-quality laser, the ratio of the oscillating-beam spot radius to that of the pump light is always determined as some value.^[33] Our further research would analyse the phase aberration and study its influence on the dual-pass pumping laser as well as the corresponding optimization.

In conclusion, a model for a laser containing both dual-pass pumping and weak absorption was established. Numerical analysis was performed on the mode matching between the pump mode and the cavity mode. Through numerical calculations based on effective absorption, the optimum radius of the pump beam waist and the corresponding location was fitted to simple formulas. These, including the modelling and the numerical analysis could be helpful in the design of laser system that can achieve a high overlap. An Nd:YVO₄ laser was built and optimized by adopting the results of the calculation, and there was a good agreement between the results of the calculation and the experiment, which verified the model and the numerical method.