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给出了离轴抽运固体激光器多模速率方程组在阈值附近的小信号求解方法, 用这种方法研究了模式随离轴量的变化以及厄米-高斯模的竞争行为. 抽运光斑较小时, 离轴量增加高阶模式依次出现; 抽运光斑较大时, 模式变化呈现复杂性. 用小信号近似得到的模式光子数比例与较高抽运功率下数值求解速率方程组的结果接近, 表明可以用该方法估算实际较高功率激光器的模式分布, 这可以方便这类激光器的研究. 对离轴抽运下的多厄米-高斯模激光器, 阈值附近的模竞争体现为, 随着抽运功率的增加, 第一个净增益由负变正的模式, 光子数随即开始增加, 增加趋势接近线性. 而第二个净增益由负变正的模式, 光子数并不立即开始增加, 而要等到抽运功率进一步增加后才开始增加, 其开始增加后第一个模式的增长趋势变缓. 从动态过程看, 各个模式经过交叉尖峰和交叉弛豫振荡竞争后, 逐渐达到稳态. 实验获得了HG00-HG50模光束, 实验所得到的模式分布与理论计算结果符合很好.To study the modes’ pattern and the modes’ competition behavior of an off-axis pumped solid-state laser, a small signal approximation method is derived, which simplifies the multiple-mode differential equations into liner algebraic equations. When the pump beam radius is small, the higher-order Hermite-Gaussian modes emerge successively with the off-axis displacement increasing, while the pattern evolution shows some complexity when the pump radius is larger. The percentage of the modes with a small pump power near the threshold, calculated with the small signal method, is close to that calculated at a higher pump power by directly solving the rate equations numerically. This indicates that we can estimate the modes’ pattern of an actual high power laser by using the small signal method. For a multiple Hermite-Gaussian modes off-axis pumped solid state laser, as the pump power increases, the photon number of the mode increases linearly as its net gain becomes positive, while that of the second mode with a smaller net gain does not increase immediately as it becomes positive successively. Larger pump power is required until the photon number begins to increase. The increasing slope of first mode decreases as the second mode begins to grow. The dynamics of the modes’ competition presents cross spiking and cross relaxation process before they become stable. Moreover, the outputs of the modes HG00-HG50 are experimentally demonstrated, and the spot evolution with the off-axis displacement agrees very well with the calculated result.
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Keywords:
- solid-state laser /
- off-axis pumped /
- modes competition /
- small signal approximation
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Google Scholar
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图 2 抽运功率0.25 W、抽运光半径0.075 mm时, 16个离轴量下的光斑
Fig. 2. Laser beam profiles with different mode distributions in the sixteen transverse displacements when the pump power is 0.25 W and the pump beam radius is 0.075 mm.
图 3 在阈值附近, HG00和HG10模光子数(a), 光子数比例(b); HG10和HG20模光子数(c), 光子数比例(d)随抽运功率的变化
Fig. 3. Photon numbers of the mode HG00 and HG10 (a), and their percentages (b); photon numbers of the modes HG10 and HG20 (c), and their percentages (d) near the threshold.
图 4 净增益随抽运功率的变化 (a) 0.08 mm; (b) 0.155 mm
Fig. 4. Dependence of the net gains on the pump power: (a) 0.08 mm; (b) 0.155 mm.
图 5 抽运功率0.5 W、抽运光半径0.15 mm时, 8个离轴量下的光斑
Fig. 5. Laser beam profiles with different mode distributions in the eight transverse displacements when the pump power is 0.5 W and the pump beam radius is 0.15 mm.
图 6 在阈值附近, HG00, HG10和HG20模光子数 (a), 光子数比例(b); HG10, HG20, HG30和HG40模光子数((c), (e)), 光子数比例((d), (f))随抽运功率的变化
Fig. 6. Photon numbers of the modes HG00 , HG10 and HG20 (a) and their percentages (b); photon numbers of the modes HG10, HG20 and HG30 ((c), (e)) and their percentages ((d), (f)) near the threshold.
图 7 净增益随抽运功率的变化 (a) 离轴量0.1 mm; (b) 离轴量0.2 mm
Fig. 7. Dependence of the net gains on the pump power: (a) 0.1 mm; (b) 0.2 mm.
图 8 光子数的动态变化过程 (a) ωp = 0.075 mm, Δx = 0.08 mm, Pa = 0.25 W; (b) ωp = 0.075 mm, Δx = 0.08 mm, Pa = 5 W; (c) ωp = 0.15 mm, Δx = 0.1 mm, Pa = 0.5 W; (d) ωp = 0.15 mm, Δx = 0.1 mm, Pa = 5 W; (e) ωp = 0.15 mm, Δx = 0.2 mm, Pa = 0.5 W; (f)ωp = 0.15 mm, Δx = 0.2 mm, Pa = 5 W
Fig. 8. Dynamics of the photon numbers: (a) ωp = 0.075 mm, Δx = 0.08 mm, Pa = 0.5 W; (b) ωp = 0.075 mm, Δx = 0.08 mm, Pa = 5 W; (c) ωp = 0.15 mm, Δx = 0.1 mm, Pa = 0.5 W; (d) ωp = 0.15 mm, Δx = 0.1 mm, Pa = 5 W; (e) ωp = 0.15 mm, Δx = 0.2 mm, Pa = 0.5 W; (f) ωp = 0.15 mm, Δx = 0.2 mm, Pa = 5 W.
图 11 光斑随激光功率保持不变
Fig. 11. The spot keeps unchanged with the variation of the laser power.
图 12 (a)激光功率随离轴量的变化; (b)模式能达到的最大功率
Fig. 12. (a) Dependence of the output power on the displacement; (b) the maximum powers of the modes.
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[1] Sayan Ö F, Gerçekcioğlu H, Baykal Y 2020 Opt. Commun. 458 124735
Google Scholar
[2] Beijersbergen M W, Allen L, van der Veen H E L O, Woerdman J P 1993 Opt. Commun. 96 123
Google Scholar
[3] Chu S C, Chen Y T, Tsai K F, Otsuka K 2012 Opt. Express 20 7128
Google Scholar
[4] 王亚东, 甘雪涛, 俱沛, 庞燕, 袁林光, 赵建林 2015 64 034204
Google Scholar
Wang Y D, Gan X T, Ju P, Pang Y, Yuan L G, Zhao J L 2015 Acta Phys. Sin. 64 034204
Google Scholar
[5] Yang Y J, Zhao Q, Liu L L, Liu Y D, Guzman C R, Qiu C W 2019 Phys. Rev. Appl. 12 064007
Google Scholar
[6] 付时尧, 高春清 2019 光学学报 39 0126014
Google Scholar
Fu S Y, Gao C Q 2019 Acta Opt. Sin. 39 0126014
Google Scholar
[7] Austin J, William J, Alan L, Linda M, Brandon C 2018 Opt. Express 26 2668
Google Scholar
[8] Willner A E, Zhao Z, Ren Y X, Li L, Xie G D, Song H Q, Liu C, Zhang R Z, Bao C J, Pang K 2018 Opt. Commun. 408 21
Google Scholar
[9] Forbes A 2017 Phil. Trans. R. Soc. A 375 20150436
Google Scholar
[10] Ngcobo S, Litvin I, Burger L, Forbes A 2013 Nat. Commun. 4 2289
Google Scholar
[11] Zhang M M, He H S, Dong J 2017 IEEE Photonics J. 9 1501214
Google Scholar
[12] Fang Z Q, Xia K G, Yao Y, Li J L 2015 IEEE J. Sel. Top. Quantum Electron. 21 1600406
Google Scholar
[13] Tuan P H, Liang H C, Huang K F, Chen Y F 2018 IEEE J. Sel. Top. Quantum Electron. 24 1600809
Google Scholar
[14] Shen Y J, Meng Y, Fu X, Gong M L 2018 Opt. Lett. 43 291
Google Scholar
[15] Kubodera K, Otsuka K 1979 J. Appl. Phys. 50 653
Google Scholar
[16] Chen Y F, Huang T M, Kao C F, Wang C L, Wang S C 1997 IEEE J. Quantum Electron. 33 1025
Google Scholar
[17] Shen Y J, Wang X J, Xie Z W, Min C J, Fu X, Liu Q, Gong M L, Yuan X C 2019 Light: Science & Applications 8 90
[18] Wang S, Zhang S L, Li P, Hao M H, Yang H M, Xie J, Feng G Y, Zhou S H 2018 Opt. Express 26 18164
Google Scholar
[19] 朱一帆, 耿滔 2020 69 014205
Google Scholar
Zhu Y F, Geng T 2020 Acta Phys. Sin. 69 014205
Google Scholar
[20] Liu Q Y, Zhao Y G, Ding M M, Yao W C, Fan X L, Shen D Y 2017 Opt. Express 25 23312
Google Scholar
[21] 付时尧, 高春清 2018 67 034201
Google Scholar
Fu S Y, Gao C Q 2018 Acta Phys. Sin. 67 034201
Google Scholar
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