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Design and analysis of 90° image rotating four-mirror non-planar ring resonator based on mid-infrared optical parametric oscillator beam quality optimization

Liu Jing-Liang Chen Xin-Yu Wang Rui-Ming Wu Chun-Ting Jin Guang-Yong

Design and analysis of 90° image rotating four-mirror non-planar ring resonator based on mid-infrared optical parametric oscillator beam quality optimization

Liu Jing-Liang, Chen Xin-Yu, Wang Rui-Ming, Wu Chun-Ting, Jin Guang-Yong
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  • Mid-infrared optical parametric oscillator (OPO) operating in the mid-infrared transmission window (3—5 μm wavelength range) is one of hot issues in the field of laser system. It has many applications in environmental detection, remote sensing, and medicine. Besides, this laser system is used as a key component of infrared countermeasures. The optical damage limit of nonlinear crystal is a great challenge to the mid-infrared OPO which is pumped by a nanosecond laser source. Therefore, the pump beam diameter should be appropriately increased to avoid damaging the crystal when scaling a nanosecond OPO to high pulse energy. The result of this design is that the Fresnel number in the cavity is increased and the beam quality is deteriorated. In order to improve the beam quality of mid-infrared OPO laser, a 90° image-rotating four-mirror non-planar ring resonator structure is designed. The advantages of this design include the general ring resonators, such as greatly reduced feedback into the pump laser and the avoidance of optical damage caused by standing wave cavity structure. Most importantly, the image rotating cavity can uniform the beam in the cavity and improve the beam quality. In this paper, the equivalent sphere representation of a four-mirror nonplanar ring resonator is established, and the image rotation angle of this special cavity structure is calculated. Based on this method, the parameters related to the 90° image rotating resonator structure suitable for mid-infrared OPO operation are designed. The self-reproduction of the transverse mode in the axially-asymmetric resonator is further established. It is found that the transverse mode in the resonator is gradually uniformed as the rotation angle of the image changes from 0° to 90°. When the rotation angle is 90°, the fundamental mode and the high-order mode both exhibit very good central symmetry. Finally, the mid-infrared ZnGeP2 OPO laser with the 90° image rotating resonator structure is used to verify the improvement of beam quality. The beam quality of $M_X^2=1.81 $ and $M_Y^2=1.61$ are achieved. It can be proved that the 90° rotating four-mirror non-planar ring resonator has a significant effect on the optimization of the output beam quality of the mid-infrared OPO laser system.
      Corresponding author: Jin Guang-Yong, jgycust@163.com
    [1]

    Mürtz M, Hering P 2008 Mid-Infrared Coherent Sources and Applications: Online Monitoring of Exhaled Breath Using Mid-Infrared Laser Spectroscopy (Vol. 1) (Germany: Springer) p535

    [2]

    Geiser P, Willer U, Walter D, Schade W 2006 Appl. Phys. B 83 175

    [3]

    Waynant R W, Ilev I K, Gannot I 2001 Philos. Trans. R. Soc. B 359 635

    [4]

    Stoeppler G, Schellhorn M, Eichhorn M 2012 Laser Phys. 22 1095

    [5]

    任钢, 钟鸣, 李彤, 牛瑞华, 曾饮勇, 龚赤冲, 何衡湘, 于淑范, 王滨 2006 红外与激光工程 3 5

    Ren G, Zhong M, Li T, Niu R H, Zeng Q Y, Gong C C, He H X, Yu S F, Wang B 2006 Infrared Laser Eng. 3 5

    [6]

    于永吉, 陈薪羽, 成丽波, 王超, 吴春婷, 董渊, 李述涛, 金光勇 2015 物理学报 22 234

    Yu Y J, Chen X Y, Cheng L B, Wang C, Wu C T, Dong Y, Li S T, Jin G Y 2015 Acta Phys. Sin. 22 234

    [7]

    王礼, 杨经纬, 蔡旭武, 王金涛, 吴海信, 吴先友, 江海河 2014 中国激光 41 37

    Wang L, Yang J W, Cai X W, Wang J T, Wu H X, Wu X Y, Jiang H H 2014 Chinese J. Lasers 41 37

    [8]

    Kadwani P, Gebhardt M, Gaida C, Shah L, Richardson M 2013 CLEO: Applications and Technology JW2A 29

    [9]

    姚宝权, 王月珠, 柳强, 王骐 2001 中国激光 28 693

    Yao B Q, Wang Y Z, Liu Q, Wang Q 2001 Chinese J. Lasers 28 693

    [10]

    Rustad G, Øystein Farsund, Arisholm G 2010 SPIE Solid State Lasers and Amplifiers IV, and High-Power Lasers Brussels, Belgium April 12−16, 7721 77210J

    [11]

    Lippert E, Fonnum H, Arisholm G, Stenersen K 2010 Opt. Express 18 26475

    [12]

    Haakestad M W, Fonnum H, Lippert E 2014 Opt. Express 22 8556

    [13]

    Shen Y J, Yao B Q, Cui Z, Duan X M, Ju Y L, Wang Y Z 2014 Appl. Phys. B 117 127

    [14]

    Qian C P, Shen Y J, Dai T Y, Duan X M, Yao B Q 2016 SPIE High-Power Lasers and Applications VIII Beijing, China October 12−14, 10016 100160G

    [15]

    安然, 范小贞, 卢建新, 文侨 2018 物理学报 67 074201

    An R, Fan X Z, Lu J X, Wen Q 2018 Acta Phys. Sin. 67 074201

    [16]

    蔡小天, 李霄, 赵国民 2017 光学学报 37 1219001

    Cai X T, Li X, Zhao G M 2017 Acta Opt. Sin. 37 1219001

    [17]

    方洪烈 1981 光学谐振腔理论 第23页

    Fang H L 1981 The Principle of the Optical Resonator (Vol. 1) (Beijing: Science Press) p23 (in Chinese)

    [18]

    张楚宾 1959 球面三角学 (北京: 高等教育出版社) 第14页

    Zhang C B 1959 Spherical Trigonometry (Vol. 1) (Beijing: Higher Education Press) p14 (in Chinese)

    [19]

    吕百达 2003激光光学 光束描述、传输变换与光腔技术物理(北京: 高等教育出版社) 第13页

    Lu B D 2003 Laser Optics: Beam Characterization, Propagation and Transformation, Resonator Technology and Physics (Vol. 3) (Beijing: Higher Education Press) p13 (in Chinese)

    [20]

    汪之国, 肖光宗, 丁志超, 卢广峰, 杨开勇 2015 中国激光 42 s102009

    Wang Z G, Xiao G Z, Ding Z C, Lu G F, Yang K Y 2015 Chinese J. Lasers 42 s102009

  • 图 1  中红外OPO四镜非平面环形腔示意图

    Figure 1.  Schematic diagram of a four-mirror non-planar ring resonator mid-infrared OPO laser.

    图 2  非平行入射平面的参考系和像旋转示意图

    Figure 2.  Diagram of reference frame and image rotation for nonparallel planes of incidence.

    图 3  简单四镜非平面环形腔示意图

    Figure 3.  Example of a four-mirror nonplanar ring resonator.

    图 4  四镜非平面环形腔的两种等效球体表示方式 (a) 透明等效单位球体; (b) 非透明等效单位球体

    Figure 4.  Two equivalent sphere representations of a four-mirror nonplanar ring resonator: (a) Transparent equivalent unit sphere; (b) non-transparent equivalent unit sphere

    图 5  90°像旋转四镜非平面环形腔三视图及单位球表示

    Figure 5.  Diagram and unit sphere representation of a 90° image rotating four-mirror non-planar ring resonator.

    图 6  不同旋转角下四镜非平面环形腔内横模光强分布 (a) 0°旋转角光强分布; (b) 5°旋转角光强分布; (c) 45°旋转角光强分布; (d) 90°旋转角光强分布

    Figure 6.  The intensity distribution of transverse mode in a four-mirror non-planar ring resonator at different rotation angles: (a) The intensity distribution at 0° rotation angle; (b) the intensity distribution at 5° rotation angle; (c) the intensity distribution at 45° rotation angle; (d) the intensity distribution at 90° rotation angle.

    图 7  90°像旋转四镜非平面环形腔内横模光强分布 (a) TEM00模; (b) TEM01模; (c) TEM10

    Figure 7.  The intensity distribution of transverse mode in a 90° image rotating four-mirror non-planar ring resonator: (a) TEM00 mode; (b) TEM01 mode; (c) TEM10 mode

    图 8  不同腔型下中红外ZnGeP2 OPO输出光束质量 (a) 平平腔; (b) 90°像旋转四镜非平面环形腔

    Figure 8.  The beam quality based on ZnGeP2 OPO in different resonators: (a) Plano-plano resonator; (b) 90° image rotating four-mirror non-planar ring resonator.

  • [1]

    Mürtz M, Hering P 2008 Mid-Infrared Coherent Sources and Applications: Online Monitoring of Exhaled Breath Using Mid-Infrared Laser Spectroscopy (Vol. 1) (Germany: Springer) p535

    [2]

    Geiser P, Willer U, Walter D, Schade W 2006 Appl. Phys. B 83 175

    [3]

    Waynant R W, Ilev I K, Gannot I 2001 Philos. Trans. R. Soc. B 359 635

    [4]

    Stoeppler G, Schellhorn M, Eichhorn M 2012 Laser Phys. 22 1095

    [5]

    任钢, 钟鸣, 李彤, 牛瑞华, 曾饮勇, 龚赤冲, 何衡湘, 于淑范, 王滨 2006 红外与激光工程 3 5

    Ren G, Zhong M, Li T, Niu R H, Zeng Q Y, Gong C C, He H X, Yu S F, Wang B 2006 Infrared Laser Eng. 3 5

    [6]

    于永吉, 陈薪羽, 成丽波, 王超, 吴春婷, 董渊, 李述涛, 金光勇 2015 物理学报 22 234

    Yu Y J, Chen X Y, Cheng L B, Wang C, Wu C T, Dong Y, Li S T, Jin G Y 2015 Acta Phys. Sin. 22 234

    [7]

    王礼, 杨经纬, 蔡旭武, 王金涛, 吴海信, 吴先友, 江海河 2014 中国激光 41 37

    Wang L, Yang J W, Cai X W, Wang J T, Wu H X, Wu X Y, Jiang H H 2014 Chinese J. Lasers 41 37

    [8]

    Kadwani P, Gebhardt M, Gaida C, Shah L, Richardson M 2013 CLEO: Applications and Technology JW2A 29

    [9]

    姚宝权, 王月珠, 柳强, 王骐 2001 中国激光 28 693

    Yao B Q, Wang Y Z, Liu Q, Wang Q 2001 Chinese J. Lasers 28 693

    [10]

    Rustad G, Øystein Farsund, Arisholm G 2010 SPIE Solid State Lasers and Amplifiers IV, and High-Power Lasers Brussels, Belgium April 12−16, 7721 77210J

    [11]

    Lippert E, Fonnum H, Arisholm G, Stenersen K 2010 Opt. Express 18 26475

    [12]

    Haakestad M W, Fonnum H, Lippert E 2014 Opt. Express 22 8556

    [13]

    Shen Y J, Yao B Q, Cui Z, Duan X M, Ju Y L, Wang Y Z 2014 Appl. Phys. B 117 127

    [14]

    Qian C P, Shen Y J, Dai T Y, Duan X M, Yao B Q 2016 SPIE High-Power Lasers and Applications VIII Beijing, China October 12−14, 10016 100160G

    [15]

    安然, 范小贞, 卢建新, 文侨 2018 物理学报 67 074201

    An R, Fan X Z, Lu J X, Wen Q 2018 Acta Phys. Sin. 67 074201

    [16]

    蔡小天, 李霄, 赵国民 2017 光学学报 37 1219001

    Cai X T, Li X, Zhao G M 2017 Acta Opt. Sin. 37 1219001

    [17]

    方洪烈 1981 光学谐振腔理论 第23页

    Fang H L 1981 The Principle of the Optical Resonator (Vol. 1) (Beijing: Science Press) p23 (in Chinese)

    [18]

    张楚宾 1959 球面三角学 (北京: 高等教育出版社) 第14页

    Zhang C B 1959 Spherical Trigonometry (Vol. 1) (Beijing: Higher Education Press) p14 (in Chinese)

    [19]

    吕百达 2003激光光学 光束描述、传输变换与光腔技术物理(北京: 高等教育出版社) 第13页

    Lu B D 2003 Laser Optics: Beam Characterization, Propagation and Transformation, Resonator Technology and Physics (Vol. 3) (Beijing: Higher Education Press) p13 (in Chinese)

    [20]

    汪之国, 肖光宗, 丁志超, 卢广峰, 杨开勇 2015 中国激光 42 s102009

    Wang Z G, Xiao G Z, Ding Z C, Lu G F, Yang K Y 2015 Chinese J. Lasers 42 s102009

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  • Received Date:  11 November 2018
  • Accepted Date:  09 June 2019
  • Available Online:  26 November 2019
  • Published Online:  01 September 2019

Design and analysis of 90° image rotating four-mirror non-planar ring resonator based on mid-infrared optical parametric oscillator beam quality optimization

    Corresponding author: Jin Guang-Yong, jgycust@163.com
  • Jilin Key Laboratory of Solid Laser Technology and Application, College of Science,Changchun University of Science and Technology, Changchun 130022, China

Abstract: Mid-infrared optical parametric oscillator (OPO) operating in the mid-infrared transmission window (3—5 μm wavelength range) is one of hot issues in the field of laser system. It has many applications in environmental detection, remote sensing, and medicine. Besides, this laser system is used as a key component of infrared countermeasures. The optical damage limit of nonlinear crystal is a great challenge to the mid-infrared OPO which is pumped by a nanosecond laser source. Therefore, the pump beam diameter should be appropriately increased to avoid damaging the crystal when scaling a nanosecond OPO to high pulse energy. The result of this design is that the Fresnel number in the cavity is increased and the beam quality is deteriorated. In order to improve the beam quality of mid-infrared OPO laser, a 90° image-rotating four-mirror non-planar ring resonator structure is designed. The advantages of this design include the general ring resonators, such as greatly reduced feedback into the pump laser and the avoidance of optical damage caused by standing wave cavity structure. Most importantly, the image rotating cavity can uniform the beam in the cavity and improve the beam quality. In this paper, the equivalent sphere representation of a four-mirror nonplanar ring resonator is established, and the image rotation angle of this special cavity structure is calculated. Based on this method, the parameters related to the 90° image rotating resonator structure suitable for mid-infrared OPO operation are designed. The self-reproduction of the transverse mode in the axially-asymmetric resonator is further established. It is found that the transverse mode in the resonator is gradually uniformed as the rotation angle of the image changes from 0° to 90°. When the rotation angle is 90°, the fundamental mode and the high-order mode both exhibit very good central symmetry. Finally, the mid-infrared ZnGeP2 OPO laser with the 90° image rotating resonator structure is used to verify the improvement of beam quality. The beam quality of $M_X^2=1.81 $ and $M_Y^2=1.61$ are achieved. It can be proved that the 90° rotating four-mirror non-planar ring resonator has a significant effect on the optimization of the output beam quality of the mid-infrared OPO laser system.

    • 中红外传输窗口3—5 μm波段内的光参量振荡器(OPO)是当下激光系统的研究热点之一, 其在环境检测、遥感、医疗等领域有非常重要的作用[1-4]. 除此之外, 其还是红外对抗系统的核心部件[5,6]. 因此此类激光器的高脉冲能量和高平均功率的发展是必然趋势. 由于其多数采用纳秒级激光系统进行抽运[7,8], 因此对于损伤阈值较低的中红外非线性晶体来说是一项非常大的挑战. 为了降低晶体损伤的风险, 通常会采用增大抽运光斑半径的方式进行优化. 这种方式的引入会导致谐振腔内菲涅耳数提高, 模式辨别能力降低, 激光输出光束质量恶化. 因此在高能中红外OPO激光系统中光束质量的优化是亟待解决的问题之一. 近年来, 关于光束质量优化方面也有许多研究报道[9-16], Rustad等[10] 采用腔内两块非线性晶体正交走离放置的方式实现光束质量${M^2} = 2 \times 2$输出, 但是两块晶体的放置需要精确调节. Lippert等[11]和Haakestad等[12]采用主振荡器功率放大器 (MOPA)结构抽运三镜环形腔实现光束质量为${M^2} = 3$的中红外激光输出, 由于其要求腔内光束两次通过晶体, 因此晶体的放置位置同样需要精确调节, 并且MOPA结构整体系统相对复杂, 对于小型化集成化的要求存在一定限制. Shen等[13]和Qian等[14]采用平面环形腔结构实现了光束质量为${M^2} \approx 3$的中红外激光和${M^2} \approx 1.6$的远红外激光, 但是平面环形腔会导致系统体积变大, 同样不适于小型化集成化的要求.

      本文将基于非平面环形腔结构, 对中红外OPO光学谐振腔进行优化设计, 并对所设计的腔型结构进行深入分析, 详细讨论非平面环形腔图像旋转对腔内模式的调控. 最终经过试验验证, 完成中红外OPO激光系统的光束质量优化, 对小型化集成化中红外激光系统的设计有一定的指导意义.

    2.   非平面环形腔型设计
    • 本文所采用的中红外OPO四镜非平面环形腔结构如图1所示, 抽运光由镜M1入射进入中红外非线性晶体, 经过相位匹配非线性频率变换产生中红外参量输出, 未发生频率变换的抽运光从镜M2输出, 避免再次返回前级抽运源而引起激光系统的损坏. 镜M2为参量光输出耦合镜, 其他三个腔镜镀有参量光高反射膜. 中红外参量光在四镜非平面腔镜内以${{\rm{M}}_2} \to {{\rm{M}}_3} \to {{\rm{M}}_4} \to {{\rm{M}}_1} \to {{\rm{M}}_2} \cdot \cdot \cdot $单向多次反射产生振荡. 由于在相邻两镜之间光束反射形成的平面存在角度, 即为像旋转角. 由此参量光在经过反射之后会产生图像以及偏振方向的旋转, 经过对腔镜之间的像旋转角以及总像旋转角的设计, 并在腔内插入半波片控制参量光的偏振方向, 最终可以使反射之后再次经过非线性晶体的参量光保持偏振方向不变, 形成参量光增益. 与此同时, 在特定像旋转角度下, 参量光经过多次像旋转形成高度中心对称的激光模式输出, 实现中红外OPO高光束质量激光输出.

      Figure 1.  Schematic diagram of a four-mirror non-planar ring resonator mid-infrared OPO laser.

      对此我们对中红外OPO四镜非平面环形腔的像旋转角进行计算与设计, 并从光场模式以及实验测量验证所设计的腔型结构对中红外OPO激光输出光束质量的改善效果.

    • 对于非平面镜面反射来说, 光束在传输过程中会发生图像旋转, 如图2所示. 假设光入射镜M1上的初始参考系为$[X,Y]$, 此处所定义的直角坐标为$Y$轴位于光在镜M1的入射平面(光传播矢量和法线所组成的平面)内, $Z$轴位于传播方向上. 光从镜M1反射之后, $Z'$参考系将沿新的传播方向进行对齐, 此时参考系变为$[X',Y']$. 在新的参考系下, 图像的$y'$值相对于初始参考系中的$y$值出现反转, 而$x$值不发生变化. 由此参考系$[X,Y]$的像坐标$(x,y)$在新的参考系$[X',Y']$中将变为$(x',y')$. 在镜M2上的入射平面与在M1上的平面并不为同一个面, 所以在从镜M1反射的光入射到镜面M2时参考系需要进行旋转, 此处定义旋转之后的参考系为$[X'',Y'']$, 从镜M2反射之后的参考系为$[X''',Y''']$, 由于两镜之间参考系的转换需要围绕传播方向旋转角度$\gamma $, 即镜子M1和M2的入射平面之间的角度. 采用传输矩阵的方法可得, 从初始参考系到最终参考系的总转换为

      Figure 2.  Diagram of reference frame and image rotation for nonparallel planes of incidence.

      其中M为镜面反射的像变换矩阵, R为两镜之间坐标旋转的变换矩阵[17], 有:

      由此应用于形成谐振腔的一组环形腔镜上, 在这种情况下, 初始和最终参考系可以经过旋转又变为相同的参考系. 以镜M1之后坐标变换之前作为光束传播起点, 此处分别采用三镜和四镜环形腔进行变换:

      其中$({x_{\rm{0}}},{y_{\rm{0}}})$是初始参考系中一个点的坐标, $({x_{\rm{1}}},{y_{\rm{1}}})$是光束在谐振腔中环程一次之后的坐标. 可以将(4)式和(5)式改写为:

      式中${{R}}_{ij}^{ - 1} = {{M}}{{{R}}_{ij}}{{M}}$Rij的逆矩阵.

      由此可以得出结论, 在腔镜为奇数时与旋转矩阵Rij有关, 而腔镜为偶数时与旋转矩阵${ R}_{ij}^{-1} $有关. 如果谐振腔组成为偶数镜, 则(7)式所示的变换矩阵的分组清楚地表明腔中的像发生了真正的旋转. 然而, 如果谐振腔组成为奇数镜, 则会额外存一个镜面反射变换矩阵M, 由此腔中的像经过旋转后又叠加了一次镜面反转, 导致$Y$轴坐标出现镜面反转, 使腔内模式单向反演形成镜像而不能实现重合叠加的可能, 除非在腔中加入相关色散补偿元件, 否则这种“假旋转”将不会达到预期光束旋转叠加改善光束质量的效果. 由此我们将主要考虑四镜环形谐振腔的设计.

      四镜环形腔内光往返一周的像旋转角为连续镜面的入射平面之间的角度之和, 其中偶数支路和奇数支路采用相反的z方向. 图3所示光束沿腔镜循环反射${{\rm M}_1} \to {{\rm M}_2} \to {{\rm M}_3} \to {{\rm M}_4}$. 为了方便计算, 可以将每个镜面上的入射平面表示在一个单位球面上, 将四个单位矢量k1, –k2, k3和–k4绘制在一个单位球中, 如图4(a)所示. 其中 M1的入射平面为–k4k1所在的平面, M1上的入射角为–k4k1所形成的弧角的一半, 镜M1到镜M2的像旋转角为两镜的入射平面之间的夹角的负值, 在图中用${\gamma _{\rm{1}}}$表示, 其他镜面同理. 所以往返一周的旋转角度可以由下式表示:

      Figure 3.  Example of a four-mirror nonplanar ring resonator.

      Figure 4.  Two equivalent sphere representations of a four-mirror nonplanar ring resonator: (a) Transparent equivalent unit sphere; (b) non-transparent equivalent unit sphere

      图4(a)可以进一步将像旋转角的表示绘制在一个非透明球体上, 如图4(b)所示. 以$Z$轴与球面的交点$O$作为辅助点连接每个矢量点, 可得:

      由此可知, 像旋转角可以简单地由连接传播单位矢量在单位球面上的弧所组成的球面四边形的面积进行表示.

    • 在OPO激光系统中, 对于腔内传播光束必须要考虑其偏振变换特性. 基于此我们对上述四镜非平面环形腔进行进一步设计, 采用具有90°像旋转的四镜环形腔. 图5(a)为该设计的正视图、俯视图和左视图, 从俯视图所在x-y平面的矩形开始进行扭转, 支路1和支路3相对于y-z平面倾斜角为$\alpha $, 而支路2和支路4相对于该平面倾斜角为$\beta $. 图5(b)为对应单位球体表示, 从该腔型的对称性来看, 构成四边形的四个三角形是相同的. 设定图像旋转角度为90°, 则由(9)式可得:

      Figure 5.  Diagram and unit sphere representation of a 90° image rotating four-mirror non-planar ring resonator.

      如果进一步将角度A限制为45°, 则角度B = 67.5°, 这种角度设定可以使支路1和支路4所在的平面与支路1和支路2所在的平面垂直. 由此光线通过支路4在镜M1处反射所得的s偏振光再经过支路1在镜M2处反射会变为p偏振光. 同理, 在镜M3和M4之间也会发生s和p偏振的转换. 但是, 因为B≠45°, 镜M2和M3之间或镜M4和M1之间偏振不会发生反转. 这种设计在OPO腔型结构设计中是非常有用的.

      这种腔型结构在实际的激光系统中应用如图1所示. 在长支路${L_1}$中加入单个非线性激光晶体, 确定非线性晶体的切割参数及摆放位置之后, 可以使其e和o偏振光位于反射镜M1和M2的正交入射平面中. 由于s和p偏振在反射镜M3和M4之间反转, M3处反射时s和p之间的任何反射相移都被反射镜M4处的相移抵消. 因此从晶体的信号光沿水平极化方向传播, 在通过支路${L_{\rm{2}}}$时采用半波片将传播极化改变为垂直, 因此镜M2和M3之间的相移抵消, 随后光束通过镜M3, M4和M1时光的偏振以像旋转的相同方式进行旋转, 最终光束经过镜M1之后偏振方向又变回水平方向. 由此光束在围绕谐振腔完成一次环程之后, 水平极化波最终又会变回水平极化波, 因此谐振腔的本征极化就是晶体e和o极化, 实现OPO环程振荡条件.

      对于以上腔型参数的确定, 可以采用球面三角形余弦定理[18]进行求解:

      由此可得$a = {\rm{40}}{\rm{.0}}{{\rm{6}}}$°, 同理可得$b = {\rm{5}}{7.235}$°, $c = {\rm{6}}{5.53}$°. 由于$a = {\text{π}}/2 - \beta $, 可以得到$\beta = {49.94}$°. 同理得到$\alpha = {\rm{32}}{\rm{.76}}{{\rm{5}}}$°, 而由弧角$c$的值可以得出光线在四个谐振腔镜的入射角为$c/2 = {32.765}$°.

      图5可知该结构下的四个支路的长度存在关系: ${L_1} = {L_3}$, ${L_2} = {L_4}$, 相邻两支路的长度比由下式给出:

      最终可得相邻两支路的长度比为

    3.   非平面环形腔内光场模式分布分析
    • 在非对称轴环形腔中, 衍射面与观察面之间不是自由空间, 而是由ABCD变换矩阵表征的复杂光学系统, 则根据Collins公式可以得出, 用4阶变换矩阵的空间域中的广义Huygens-Fresnel衍射积分公式[19]

      式中A, B, C, D均为2 × 2矩阵. ${E_1}({x_1},{y_1},0)$${E_2}({x_2},{y_2},z)$分别为波源点和传播点处光场的复振幅; $\lambda $为光波波长, $k = 2{\text{π}}/\lambda $. 上述所设计的OPO腔结构中加入了半波片, 由于半波片只起到相位延迟的作用, 只改变光场的偏振方向, 不改变光场分布, 因此光在腔中垂直透射过半波片, 并不影响腔内模式的变换, 此处可以忽略不计.

      根据激光谐振腔中的模式自再现条件, 在环形腔中光束环程一周后再次到达波源点的场分布与初始场分布完全相同, 除了其振幅相差一个复常数因子$\xi $, 因此有$\xi {E_1}(x{}_1,{y_1}) = {E_2}(x{}_2,{y_2})$.

      对于(15)式我们采用有限元传输矩阵算法(FEM)进行求解[20], 将源场积分区域按照一定的顺序划分成足够多个单元, 这样原有的连续函数${E_1}({x_1},{y_1})$离散为向量$U = {[{E_1}[1],{E_1}[2],\cdots,{E_1}[g]]^{\rm{T}}}.$ 当源场划分足够精细时, 每个点的复振幅波动将会很小, 可以近似认为均匀分布, 因此每个点与积分变量$x$,$y$无关, 由此(15)式可以写为

      式中$n=1,2,\cdots,g $, 由此可以得到如下关系:

      式中

      $ \xi$为方程的本征值, 表示光在谐振腔中环程一周之后的振幅衰减和相位变化.

      通过(17)式可以看出传输矩阵${{V}}$特征值的求解过程即是本征值$\xi $的求解过程, 而不同阶的模式分布即对应不同特征值下的特征向量. 由此可见传输矩阵${{V}}$包含了对腔内所有模式和光束特性的描述.

      对于文中所研究的四镜非平面环形谐振腔结构, 可得光在腔内环程一周的ABCD矩阵为:

      式中${{M}}(M)$,${{M}}(L)$${{M}}(\gamma )$分别表示反射腔镜、自由空间传输及坐标旋转的4阶变换矩阵. 分别表示为:

      式中$R$为反射镜曲率$\theta $为入射角度, 由于本文中设计的谐振腔镜主要是平面镜, 因此该矩阵将直接变为单位矩阵.

      式中$L$为两腔镜之间的距离.

      式中$\gamma $为上文所说的两镜之间的像旋转角.

      由此基于之前所设计的谐振腔结构, 所选总腔长为L = 150 mm, 腔内初始光束直径为d = 700 μm. 根据(10)式—(14)式可以计算得出在不同旋转角下的各支路腔长参数. 由此仿真给出当总旋转角为0°, 5°, 45°, 90°时腔内TEM00, TEM10和TEM01横模光强分布图, 如图6所示.

      Figure 6.  The intensity distribution of transverse mode in a four-mirror non-planar ring resonator at different rotation angles: (a) The intensity distribution at 0° rotation angle; (b) the intensity distribution at 5° rotation angle; (c) the intensity distribution at 45° rotation angle; (d) the intensity distribution at 90° rotation angle.

      图6可以看出, 在旋转角为0°时, 即平面环形腔中, 腔内光强分布类似于典型的平平两镜谐振腔中的光强分布, 高阶模的光场分布不为中心对称, 而是在某个方向产生分离. 随着旋转角的增加, 腔内基模在横向和纵向出现轻微椭圆变化, 而高阶模出现逐渐融合的现象, 当旋转角为90°时, 基模以及高阶模都具有非常好的中心对称性.

      图7为所设计90°像旋转四镜非平面环形腔腔内横模光强分布三维视图. 可以看出, 其光场均匀化并不是变为一般意义上的高斯光束, 而是变为环形光斑. 考虑到输出激光主要是以基模激光为主, 再由此叠加像旋转均匀化处理之后的高阶模式激光, 可以预见该腔型结构对输出激光光束质量的优化有着非常优良的效果.

      Figure 7.  The intensity distribution of transverse mode in a 90° image rotating four-mirror non-planar ring resonator: (a) TEM00 mode; (b) TEM01 mode; (c) TEM10 mode

    4.   中红外非平面环形腔OPO实验测量
    • 基于以上所设计的90°像旋转非平面环形腔, 采用中红外ZnGeP2 非线性晶体进行OPO实验测量. 实验采用重复频率6 kHz, 脉冲宽度21 ns, 光束质量${M^{\rm{2}}} \approx 1.{\rm{5}}$、中心波长2090.7 nm的Ho:YAG激光作为抽运源, ZnGeP2晶体的尺寸为5 mm × 5 mm × 15 mm, 切割角为$\theta = {\rm{5}}{{\rm{5}}^ \circ }$, OPO腔长为150 mm, 其中长支路为44 mm, 短支路为31 mm. 在抽运功率为21.5 W时, 最终获得5.97 W的3—5 μm激光输出. 同时, 实验对比测量了腔长同为150 mm的平平腔和90°像旋转非平面环形腔的输出光束质量, 如图8所示. 从图中可以看出, 平平腔的光束质量为$M_X^2 = {\rm{3}}{\rm{.19}}$, $M_Y^2 = {\rm{3}}{\rm{.49}}$, 非平面环形腔的光束质量为$M_X^2 = 1.81$$M_Y^2 = 1.61$的激光输出, 由此可见90°像旋转非平面环形腔有助于中红外 OPO输出光束质量的改善.

      Figure 8.  The beam quality based on ZnGeP2 OPO in different resonators: (a) Plano-plano resonator; (b) 90° image rotating four-mirror non-planar ring resonator.

    5.   结 论
    • 本文设计了一种改善中红外OPO激光输出光束质量的90°像旋转四镜非平面环形腔结构. 通过采用单位球等效的方式计算非平面环形腔的像旋转角, 并由此确定中红外OPO 90°像旋转谐振腔结构相关参数. 建立在非对称轴环形腔中光场模式自再现模型, 分析不同像旋转角对腔内模式的调控作用. 采用中红外ZnGeP2 OPO, 对所设计的腔型参数进行实验测量, 实现了光束质量$M_X^2 = 1.81$$M_Y^2 = 1.61$. 最终证明90°像旋转四镜非平面环形腔对于中红外OPO激光系统输出光束质量的优化具有良好的效果, 对实际激光系统的设计有一定的指导意义.

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