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Generation of high-quality circular Airy beams in laser resonator

Zhu Yi-Fan Geng Tao

Generation of high-quality circular Airy beams in laser resonator

Zhu Yi-Fan, Geng Tao
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  • A scheme for forming high-quality circular Airy beams inside the laser resonator is presented theoretically. The desired circular Airy beam can be generated when the common reflective mirror is replaced by a designed diffractive optical element. The mode generated in the proposed cavity can be stimulated by using the so-called eigenvector method. The calculated results show that the parameters of the beams can be controlled by changing the phase distribution of the diffractive optical element. The loss of the generated mode is very low, which is close to that of the fundamental Gaussian mode. The purity of the generated mode is very high, which is much better than that from the phase-only encoding method in Fourier space. The phase distribution of the diffractive optical element needs designing for a fixed resonator length. In practice, the real resonator length may not be equal to the designed resonator length. Thus, the influence of the alignment error of the resonator length is discussed in detail. The results show that the diffraction loss of the proposed system is still very small even when the error reaches up to 2 mm. Meanwhile, the purity of the generated mode decreases little. Then, the influence of etching depth errors and the decenter of the reflective mirrors are discussed in detail. Here we assume that the fluctuations are randomly distributed. The value of the maximum fluctuation is used to represent the etching depth error degree. The results show that the diffraction loss of the proposed system is more sensitive to production error, and the purity of the generated mode is more sensitive to alignment error. Thus, we estimate that the maximum etching depth error should be less than six percent of the wavelength, and the vertical distance between the centers of the two reflective mirrors should be less than 7 μm if one wants to obtain high-quality CAB with high efficiency. The requirements for precision are acceptable for existing microfabrication and operation technologies.
      Corresponding author: Geng Tao, Tao_Geng@hotmail.com
    [1]

    Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901

    [2]

    Minovich A E, Klein A E, Neshev D N, Pertsch T, Kivshar Y S, Christodoulides D N 2014 Laser. Photon. Rev. 8 221

    [3]

    Qian J, Liu B Y, Sun H X, Yuan S Q, Yu X Z 2017 Chin. Phys. B 26 114304

    [4]

    崔省伟, 陈子阳, 胡克磊, 蒲继雄 2013 物理学报 62 094205

    Cui S W, Chen Z Y, Hu K L, Pu J X 2013 Acta Phys. Sin. 62 094205

    [5]

    张泽, 刘京郊, 张鹏, 倪培根, Prakash J, 胡洋, 姜东升, Christodoulides D N, 陈志刚 2013 物理学报 62 034209

    Zhang Z, Liu J J, Zhang P, Ni P G, Prakash J, Hu Y, Jiang D S, Christodoulides D N, Chen Z G 2013 Acta Phys. Sin. 62 034209

    [6]

    Guo Y H, Huang Y J, Li X, Pu M B, Gao P, Jin J J, Ma X L, Luo X G 2019 Adv. Opt. Mater. 7 1900503

    [7]

    Li Z, Cheng H, Liu Z, Chen S, Tian J 2016 Adv. Opt. Mater. 4 1230

    [8]

    Fan Q, Zhu W, Liang Y, Huo P, Zhang C, Agrawal A, Huang K, Luo X, Lu Y, Qiu C, Lezec H J, Xu T 2019 Nano Lett. 19 1158

    [9]

    Efremidis N K, Christodoulides D N 2010 Opt. Lett. 35 4045

    [10]

    Papazoglou D G, Efremidis N K, Christodoulides D N, Tzortzakis S 2011 Opt. Lett. 36 1842

    [11]

    Liu K, Koulouklidis A D, Papazoglou D G, Tzortzakis S, Zhang X C 2016 Optica 3 605

    [12]

    Manousidaki M, Papazoglou D G, Farsari M, Tzortzakis S 2016 Optica 3 525

    [13]

    Manousidaki M, Fedorov V Y, Papazoglou D G, Farsari M, Tzortzakis S 2018 Opt. Lett. 4 3

    [14]

    Zhang P, Prakash J, Zhang Z, Mills M S, Efremidis N K, Christodoulides D N, Chen Z 2011 Opt. Lett. 36 2883

    [15]

    Davis J A, Cottrell D M, Zinn J M 2013 Appl. Opt. 52 1888

    [16]

    Davis J A, Cottrell D M, Sand D 2012 Opt. Express 20 13302

    [17]

    刘正楠, 耿滔, 邓攀 2019 中国激光 46 0209001

    Liu Z N, Geng T, Deng P 2019 Chin. J. Lasers 46 0209001

    [18]

    Fox A G, Li T 1961 Bell System Techical Journal 40 453

    [19]

    Chao J, Li B, Cheng Y, Wang Y 2007 Opt. Laser Technol. 39 490

    [20]

    Cheng Y Y, Wang Y Q, Hu J, Li J R 2004 Opt. Commun. 234 1

    [21]

    Bélanger P A, Paré C 1991 Opt. Lett. 16 1057

    [22]

    Leger J R, Chen D, Wang Z 1994 Opt. Lett. 19 108

    [23]

    Jiang Y, Zhu X, Yu W, Shao H, Zheng W, Lu X 2015 Opt. Express 23 29834

    [24]

    Li N, Jiang Y, Huang K, Lu X 2014 Opt. Express 22 22847

    [25]

    Zhou W, Bovik A C, Sheikh H R, Simoncelli E P 2004 IEEE Trans. Image Process 13 600

  • 图 1  (a) CAB初始面的光强分布; (b) CAB初始面的相位分布; (c) CAB的侧面光强分布

    Figure 1.  (a) Intensity distributions of the CAB at the initial plane; (b) phase distributions of the CAB at the initial plane; (c) intensity distributions of the CAB during propagation in the r-z plane.

    图 2  谐振腔示意图

    Figure 2.  Schematic of the laser resonator configuration for CAB generation.

    图 3  不同参数条件下, 使用Fox-Li方法计算获得的腔内光场模式分布 (a) r0 = 1 mm, w = 0.2 mm和a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm和a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm和a = 0.2

    Figure 3.  Calculation results of the intensity distributions of the modes by using Fox-Li method with different parameters: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

    图 4  理想CAB和使用不同方法产生的光束的径向光强分布 (a) r0 = 1 mm, w = 0.2 mm和a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm和a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm和a = 0.2

    Figure 4.  Radial intensity distributions of the ideal CAB and the beams produced by different methods: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

    图 5  理想CAB和使用不同方法产生的光束的光轴光强分布 (a) r0 = 1 mm, w = 0.2 mm和a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm和a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm和a = 0.2

    Figure 5.  On-axis intensity contrast of the ideal CAB and the beams produced by different methods: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

    图 6  光束参数为${r_0} = 1\;{\rm{mm}}$, $w = 0.2\;{\rm{mm}}$$a = 0.15$时, 系统对准误差对产生光束质量的影响 (a)基模的$\left| \gamma \right|$以及S与腔长误差${\delta _{\rm{l}}}$的关系; (b)基模的$\left| \gamma \right|$以及S与同轴度误差${\delta _{\rm{d}}}$的关系

    Figure 6.  The influence of the alignment errors on formation of the fundamental mode with ${r_0} = 1\;{\rm{mm}}$, $w = 0.2\;{\rm{mm}}$ and $a = 0.15$: (a) $\left| \gamma \right|$ and S of the fundamental mode as a function of $\delta_{\rm l}$; (b) $\left| \gamma \right|$ and S of the fundamental mode as a function of $\delta _{\rm d}$

    图 7  基模的$\left| \gamma \right|$以及S${\delta _{\rm{h}}}$的关系

    Figure 7.  $\left| \gamma \right|$ and S of the fundamental mode as a function of ${\delta _{\rm{h}}}$.

    表 1  不同参数条件下的衍射光学元件上的相位分布和计算获得的最大3个$\left| \gamma \right|$对应模式的光强分布

    Table 1.  The phase distributions of the diffractive optical elements, the three largest $\left| \gamma \right|$ and the calculated intensity distributions of corresponding modes with different parameters.

    CAB的参数衍射光学元件上的相位分布/rad$\left| \gamma \right|$光强分布
    r0 = 1 mm
    w = 0.2 mm
    a = 0.15
    0.9972
    0.9898
    0.9898
    r0 = 1.1 mm
    w = 0.22 mm
    a = 0.17
    0.9970
    0.9845
    0.9845
    r0 = 1.2 mm
    w = 0.25 mm
    a = 0.2
    0.9960
    0.9804
    0.9804
    DownLoad: CSV
  • [1]

    Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901

    [2]

    Minovich A E, Klein A E, Neshev D N, Pertsch T, Kivshar Y S, Christodoulides D N 2014 Laser. Photon. Rev. 8 221

    [3]

    Qian J, Liu B Y, Sun H X, Yuan S Q, Yu X Z 2017 Chin. Phys. B 26 114304

    [4]

    崔省伟, 陈子阳, 胡克磊, 蒲继雄 2013 物理学报 62 094205

    Cui S W, Chen Z Y, Hu K L, Pu J X 2013 Acta Phys. Sin. 62 094205

    [5]

    张泽, 刘京郊, 张鹏, 倪培根, Prakash J, 胡洋, 姜东升, Christodoulides D N, 陈志刚 2013 物理学报 62 034209

    Zhang Z, Liu J J, Zhang P, Ni P G, Prakash J, Hu Y, Jiang D S, Christodoulides D N, Chen Z G 2013 Acta Phys. Sin. 62 034209

    [6]

    Guo Y H, Huang Y J, Li X, Pu M B, Gao P, Jin J J, Ma X L, Luo X G 2019 Adv. Opt. Mater. 7 1900503

    [7]

    Li Z, Cheng H, Liu Z, Chen S, Tian J 2016 Adv. Opt. Mater. 4 1230

    [8]

    Fan Q, Zhu W, Liang Y, Huo P, Zhang C, Agrawal A, Huang K, Luo X, Lu Y, Qiu C, Lezec H J, Xu T 2019 Nano Lett. 19 1158

    [9]

    Efremidis N K, Christodoulides D N 2010 Opt. Lett. 35 4045

    [10]

    Papazoglou D G, Efremidis N K, Christodoulides D N, Tzortzakis S 2011 Opt. Lett. 36 1842

    [11]

    Liu K, Koulouklidis A D, Papazoglou D G, Tzortzakis S, Zhang X C 2016 Optica 3 605

    [12]

    Manousidaki M, Papazoglou D G, Farsari M, Tzortzakis S 2016 Optica 3 525

    [13]

    Manousidaki M, Fedorov V Y, Papazoglou D G, Farsari M, Tzortzakis S 2018 Opt. Lett. 4 3

    [14]

    Zhang P, Prakash J, Zhang Z, Mills M S, Efremidis N K, Christodoulides D N, Chen Z 2011 Opt. Lett. 36 2883

    [15]

    Davis J A, Cottrell D M, Zinn J M 2013 Appl. Opt. 52 1888

    [16]

    Davis J A, Cottrell D M, Sand D 2012 Opt. Express 20 13302

    [17]

    刘正楠, 耿滔, 邓攀 2019 中国激光 46 0209001

    Liu Z N, Geng T, Deng P 2019 Chin. J. Lasers 46 0209001

    [18]

    Fox A G, Li T 1961 Bell System Techical Journal 40 453

    [19]

    Chao J, Li B, Cheng Y, Wang Y 2007 Opt. Laser Technol. 39 490

    [20]

    Cheng Y Y, Wang Y Q, Hu J, Li J R 2004 Opt. Commun. 234 1

    [21]

    Bélanger P A, Paré C 1991 Opt. Lett. 16 1057

    [22]

    Leger J R, Chen D, Wang Z 1994 Opt. Lett. 19 108

    [23]

    Jiang Y, Zhu X, Yu W, Shao H, Zheng W, Lu X 2015 Opt. Express 23 29834

    [24]

    Li N, Jiang Y, Huang K, Lu X 2014 Opt. Express 22 22847

    [25]

    Zhou W, Bovik A C, Sheikh H R, Simoncelli E P 2004 IEEE Trans. Image Process 13 600

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    [2] LU XUAN-HUI, HUANG KAI-KAI. THE RESEARCH ON IMPROVING THE CHARACTERISTICS OF LASER OUTPUT WITH DIFFRACTIVE OPTICAL ELEMENTS. Acta Physica Sinica, 2001, 50(8): 1409-1414. doi: 10.7498/aps.50.1409
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  • Received Date:  16 July 2019
  • Accepted Date:  16 September 2019
  • Available Online:  05 December 2019
  • Published Online:  01 January 2020

Generation of high-quality circular Airy beams in laser resonator

    Corresponding author: Geng Tao, Tao_Geng@hotmail.com
  • Engineering Research Center of Optical Instruments and Systems, Ministry of Education, Shanghai Key Laboratory of Modern Optics and Systems, School of Optical-Electrical and ComputerEngineering, University of Shanghai for Science and Technology, Shanghai 200093, China

Abstract: A scheme for forming high-quality circular Airy beams inside the laser resonator is presented theoretically. The desired circular Airy beam can be generated when the common reflective mirror is replaced by a designed diffractive optical element. The mode generated in the proposed cavity can be stimulated by using the so-called eigenvector method. The calculated results show that the parameters of the beams can be controlled by changing the phase distribution of the diffractive optical element. The loss of the generated mode is very low, which is close to that of the fundamental Gaussian mode. The purity of the generated mode is very high, which is much better than that from the phase-only encoding method in Fourier space. The phase distribution of the diffractive optical element needs designing for a fixed resonator length. In practice, the real resonator length may not be equal to the designed resonator length. Thus, the influence of the alignment error of the resonator length is discussed in detail. The results show that the diffraction loss of the proposed system is still very small even when the error reaches up to 2 mm. Meanwhile, the purity of the generated mode decreases little. Then, the influence of etching depth errors and the decenter of the reflective mirrors are discussed in detail. Here we assume that the fluctuations are randomly distributed. The value of the maximum fluctuation is used to represent the etching depth error degree. The results show that the diffraction loss of the proposed system is more sensitive to production error, and the purity of the generated mode is more sensitive to alignment error. Thus, we estimate that the maximum etching depth error should be less than six percent of the wavelength, and the vertical distance between the centers of the two reflective mirrors should be less than 7 μm if one wants to obtain high-quality CAB with high efficiency. The requirements for precision are acceptable for existing microfabrication and operation technologies.

    • 由于具有无衍射、自加速和自修复等特性, 艾里光束自2007年被提出以来一直是光学领域的研究热点之一[1-5]. 为了产生艾里光束, 研究者通常使用傅里叶空间的立方相位调制方法[1], 近年来随着超表面技术的发展, 越来越多的研究者开始利用表面等离子激元产生艾里光束[6-8], 这种方法能在微纳尺度获得艾里光束, 在纳米微粒操作和光传感等领域有很大的应用前景. 将一维艾里光束做径向对称处理, 可以得到圆对称艾里光束(circular airy beam, CAB), CAB是一种自聚焦光束, 当其在自由空间传播时, 在焦点前光束能保持很低的光强分布, 而到达焦点时光强会突然提升数十甚至数百倍, 因此这种光束又被称为突然自聚焦光束(abruptly autofocusing wave)[9,10]. 由于CAB具有的这种突然自聚焦特性, 使得其在生物医学、激光加工、光学微操作和非线性光学等领域有着广阔的应用前景, 受到了越来越多的关注[11-14].

      CAB是一种特殊光束, 需要使用特定的方法来产生, 目前在可见光波段产生CAB的方法主要有两种. 一是使用具有径向3/2次方分布的相位板或空间光调制器(spatial light modulator, SLM)[15], 使用这种方法可方便地获得具有涡旋相位的CAB[16], 但缺点是无法针对CAB光环分布的疏密程度编码, 即无法得到所需径向比例系数的光束, 因此使用者较少. 另一种是在傅里叶空间使用经纯相位全息编码的相位板或SLM[10], 这种方法可以获得所需特定参数的光束, 因此是目前最常用的产生CAB的方法[11-14].

      但上述两种方法对光源的使用效率都很低, 例如第二种常用方法的理论衍射效率(即衍射成像的光通量与再现时照明光源的总光通量之比)低于3%, 经编码算法改进后虽有所提升, 但理论衍射效率仍低于7%[17], 严重阻碍了CAB的实际应用. 基于此, 本文提出了一种在谐振腔内产生可控高质量CAB光束的方法, 并对产生光束的质量和能量损耗问题进行了理论探讨. 研究结果表明, 该方法不仅光束参数可控, 能量损耗低, 且光束质量也明显优于纯相位编码方法. 本文最后还讨论了组装系统时产生的对准误差和衍射元件的加工误差对产生光束的影响, 结果显示本文提出的方法对误差有较好的容差性. 据我们所知, 目前在谐振腔内产生CAB的研究还未见报道.

    2.   理论方法
    • 在谐振腔中, 反射镜2的光场分布${U_2}$可由反射镜1的光场分布${U_1}$经衍射积分计算获得

      其中$\rho $为场点到源点的距离, $K\left( \theta \right)$为倾斜因子. 由于谐振腔腔长远远大于反射镜半径, 因此取$K\left( \theta \right) \approx 1$. 同理, 光波经反射镜2返回反射镜1上的光场${U_{1\_1}}$可由${U_2}$通过(1)式的衍射积分计算获得. 如此反复地迭代计算衍射积分, 当光波经过n次在腔内的往返传播以后, 到达镜面1的光场变为${U_{1\_n}}$, 当满足如下条件时可停止迭代.

      其中$\delta $为一趋于零的常数, 视计算精度要求取值, 本文取为10–3. 这种数值迭代方法称为Fox-Li方法[18].

      由于任意连续函数可作离散化处理, 那么(1)式中的${U_1}$${U_2}$可用列向量表示为

      p, q足够大, 即划分的单元区域足够的小, 由(1)式, 反射镜2上单元n处的光场可表示为

      其中${W_m}$为反射镜1上第m单元的面积. (4)式可改写为

      其中$A\left[ {n, m} \right] = - \dfrac{{\rm{i}}}{\lambda }\dfrac{{\exp ({\rm{i}}k\rho )}}{\rho }{W_m}$, 为${U_1}$${U_2}$的传输矩阵. 同理, 可计算得到${U_2}$${U_1}$的传输矩阵B. 那么, 光波从反射镜2出发, 经过反射镜1反射返回后, 在反射镜2上的光场可表示为

      其中$D = BA$为光场经过一个来回的传输矩阵, 它仅与谐振腔的几何尺寸有关.

      根据谐振腔的自再现理论, 光场的特征向量应满足:

      其中$\gamma $为方程特征值, 描述了光场经过一个来回光程的振幅衰减和相位变化. 比较(6)式和(7)式可知, $\gamma $即为传输矩阵D的特征值, 而D包含了腔内所有可能的光场模式分布. 那么, 求解腔内光场模式分布的问题转换为求解特征向量${U_1}$和相应的特征值$\gamma $的问题. 这种方法称为特性向量法[1921], 与Fox-Li方法相比, 特性向量法的优点在于可以一次算出一系列模式分布, 且计算不受光场初值的影响, 可适用于任何非规则形状的腔镜. 因此本文采用了特性向量法计算腔内的光场模式分布以及相应的模式能量损耗.

    • CAB在初始面上的光场分布可以表示为

      其中C为振幅常数, $Ai\left( \cdot \right)$为艾里函数, r为光场的径向坐标, 参数${r_0}$决定了主光环的半径, w为径向比例系数, 影响光环分布的疏密程度, $a$为指数衰减系数. 图1(a)图1(b)分别给出了CAB在初始面的光强和相位分布, 其中光强分布对最大值做了归一化处理, 计算参数为r0 = 1 mm, w = 0.2 mm, a = 0.15. 图1(c)给出了CAB的侧面光强分布, 从图1(c)可以看到在焦点前光强峰值沿抛物线轨迹传播, 并向焦点位置汇聚. 聚焦前, 光轴上的光强几乎为零, 在焦点处突然达到最大值, 显示出突然自聚焦特性.

      Figure 1.  (a) Intensity distributions of the CAB at the initial plane; (b) phase distributions of the CAB at the initial plane; (c) intensity distributions of the CAB during propagation in the r-z plane.

      产生CAB的谐振腔示意图如图2所示, 为了在腔内获得CAB, 我们使用了具有特定表面结构的衍射光学元件代替了其中的平面反射镜, 用于选取所需的激光模式, 凹面反射镜被设为光束输出镜.

      Figure 2.  Schematic of the laser resonator configuration for CAB generation.

      衍射光学元件的设计原理如下[12,22]: 设凹面反射镜上所需的输出光场分布为${U_1}\left( {{x_1}, {y_1}} \right)$, 那么由(1)式可得在衍射光学元件上的光场分布${U_2}\left( {{x_2}, {y_2}} \right)$. 如果衍射光学元件的反射率为

      其中$U_2^*\left( {{x_2}, {{{y}}_2}} \right)$${U_2}\left( {{x_2}, {y_2}} \right)$的共轭, ${\varphi _2}\left( {{x_2}, {y_2}} \right)$${U_2}\left( {{x_2}, {y_2}} \right)$的相位. 则经衍射光学元件再反射回凹面镜的光场为原光场函数的共轭, 即$U_1^*\left( {{x_1}, {y_1}} \right)$. 为了使得${U_1}\left( {{x_1}, {y_1}} \right)$能够自再现, 则需满足条件${U_1}\left( {{x_1}, {y_1}} \right) = U_1^*\left( {{x_1}, {y_1}} \right)$, 即

      其中${\varphi _1}\left( {{x_1}, {y_1}} \right)$${U_1}\left( {{x_1}, {y_1}} \right)$的相位. 从图1(b)可以看出, 由于CAB的相位呈$0 - {\text{π}}$环状分布, 能自动满足(10)式的要求. 由(9)式可知, 此时衍射光学元件为纯相位调制, 避免了使用振幅调制带来的能量损耗, 且可选择使用反射式的SLM来代替衍射光学元件.

      通过上述设计原理获得的CAB出现在凹面反射镜的凹面处, 因此如果输出镜为一般的平凹镜(即凹面反射镜外表面为平面), 当光波经凹面透射到图2中的虚线位置输出时, 光波在腔镜内部的传播会产生一个$\varDelta = n\left( {R - \sqrt {{R^2} - {r^2}} } \right)$的光程差, 其中n为反射镜的折射率, R为凹面的曲率半径, r为径向坐标. 在本文中凹面镜曲率半径远远大于反射镜截面半径, 满足$r \ll R$, 有$\varDelta \approx n{{{r^2}}}/{{2 R}}$, 因此以平凹镜输出时, 光波会附加一个相位畸变$\Delta \varphi \approx kn {{{r^2}}}/{{2 R}}$, 其中k为光束的真空波数. 所以, 在凹面镜的外侧我们使用了凸面结构对这一相位畸变进行了修正, 如图2所示. 在$r \ll R$$r \ll R'$的近似条件下, 可由简单的几何关系得

      其中$R'$为凸面的曲率半径.

    3.   结果与讨论
    • 本文在计算中设光束波长为$632.8\;{\rm{nm}}$(氦氖激光器), 凹面镜和衍射光学元件的截面半径为$4\;{\rm{mm}}$, 腔长$L = 1450\;{\rm{mm}}$, 凹面镜的曲率半径$R = 2000\;{\rm{mm}}$, 由(11)式可得外侧凸面的曲率半径$R' = 666.7\;{\rm{mm}}$. 衍射光学元件上的相位分布由(9)式可得$\theta \left( {{x_2}, {y_2}} \right)= $$ - 2{\varphi _{\rm{2}}}\left( {{x_2}, {y_2}} \right)$. 仿真计算使用了柱坐标系, 腔镜在径向和角向被分别划分为100和150个单元格. 由特征向量法可知, 光波在腔内经过一个来回后能量损耗比例为$1 - {\left| \gamma \right|^2}$. 换言之, 特征值的绝对值$\left| \gamma \right|$越大的模式, 能量损耗越小, 即越有可能在腔中被激发. 需要说明的是, 本文的讨论虽然以氦氖激光为例, 但设计方法同样适用于其他波段的激光器, 只是需根据不同的工作波长重新设计衍射元件的相位分布.

      表1列出了针对不同参数CAB设计的衍射光学元件上的相位分布, 以及利用这些相位分布计算出的最大3个$\left| \gamma \right|$所对应模式的光强分布. 从表1中可以看到, 不同参数计算所得的基模光束均为设计所需的CAB. 我们利用Fox-Li方法使用相同的参数进行了验证, 计算结果如图3所示, 所得模式与特征向量法的结果完全吻合. 腔内基模光场的主峰位置随着参数${r_0}$的增大逐渐外移, $\left| \gamma \right|$随之逐渐减小, 这表明为了获得较大${r_0}$的CAB光束, 应适当增加腔镜的半径. 如前所述, 腔内光场模式的单程损耗比例为$1 - {\left| \gamma \right|^2}$, 那么, 表1中3种情况的基模, 其单程损耗比例分别为0.56%, 0.60%和0.80%, 均小于1%, 接近产生高斯基模时光波的损耗(约0.14%), 说明这种方法产生的CAB能量损耗极小.

      CAB的参数衍射光学元件上的相位分布/rad$\left| \gamma \right|$光强分布
      r0 = 1 mm
      w = 0.2 mm
      a = 0.15
      0.9972
      0.9898
      0.9898
      r0 = 1.1 mm
      w = 0.22 mm
      a = 0.17
      0.9970
      0.9845
      0.9845
      r0 = 1.2 mm
      w = 0.25 mm
      a = 0.2
      0.9960
      0.9804
      0.9804

      Table 1.  The phase distributions of the diffractive optical elements, the three largest $\left| \gamma \right|$ and the calculated intensity distributions of corresponding modes with different parameters.

      Figure 3.  Calculation results of the intensity distributions of the modes by using Fox-Li method with different parameters: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

      为了直观地观察产生光束的质量, 图4给出了表1中3种情况基模输出光束的径向光强分布, 并与各自对应的理想CAB和使用纯相位编码法[10]产生的光束进行了比较. 从图4中可以看到, 纯相位编码方法产生的光束光强分布与理想的CAB基本能吻合, 其中次光环的峰值明显大于理想光束次光环的峰值, 且光环峰值不能按照从里到外的次序依次递减, 有跳变的现象存在. 本方法产生的光束, 其光强分布几乎与理想CAB完全重合, 重合度明显优于使用纯相位编码方法产生的光束. 这是由于本方法产生的CAB由腔内产生, 由于谐振腔的存在, 光波在腔内多次反射, 使得能量损失最小的基模得以保留, 而其他的损耗较大的高阶模式则加以抑制, 随着光波在腔内往返次数的增加, 混杂的高阶模式逐渐衰减, 最终获得了比腔外产生方法更纯净的CAB模式.

      Figure 4.  Radial intensity distributions of the ideal CAB and the beams produced by different methods: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

      为了考察产生光束的传输特性与理想CAB是否吻合, 图5给出了表1中3种情况基模输出光束在自由空间沿z轴传播的光轴强度分布, 并和各自对应的理想CAB和使用纯相位编码方法产生的光束进行了比较, 图中${I_0}$为各光束在初始面的主光环峰值. 从图5中可以看到, 纯相位编码方法产生的光束, 其焦点处的峰值明显大于理想光束, 这是因为相比于理想CAB, 这种光束在初始面的次光环和高阶次光环获得了更多的能量, 而已有的研究表明, CAB主光环对焦斑峰值几乎没有贡献, 可以使用压制主光环的调制方法提升光束的焦斑峰值[23,24]. 本方法产生的光束, 其光轴光强分布几乎与理想光束完全重合, 再一次证明腔内产生的光束质量明显优于目前常用的纯相位编码方法产生的光束.

      Figure 5.  On-axis intensity contrast of the ideal CAB and the beams produced by different methods: (a) r0 = 1 mm, w = 0.2 mm and a = 0.15; (b) r0 = 1.1 mm, w = 0.22 mm and a = 0.17; (c) r0 = 1.2 mm, w = 0.25 mm and a = 0.2.

      本文采用的方法需针对固定腔长设计衍射光学元件, 而在实际组装谐振腔时很难做到腔长与设计长度完全吻合, 因此需要考虑腔长误差对产生光束的影响. 另外组装系统时, 衍射元件中心和凹面镜的中心还不可避免的会产生同轴度误差. 为了评价这些误差对光束质量的影响, 引入结构相似性指数作为评判标准. 结构相似性理论是一种常用的图像质量测量的理论, 可以用来评估失真图像与参考图像的相似程度. 结构相似性指数可以表示为[25]

      其中A, B分别表示参考图像和失真图像, ${\sigma _{\rm{A}}}$${\sigma _{\rm{B}}}$分别表示参考图像和失真图像的标准差, ${\sigma _{{\rm{AB}}}}$为参考图像和失真图像的相关系数. $S({\rm{A}}, {\rm{B}})$的最大值为1, 其值越接近1说明失真图像越接近参考图像.

      图6(a)给出了光束参数为r0 = 1 mm, w = 0.2 mm和a = 0.15时, 基模特征值的绝对值$\left| \gamma \right|$以及结构相似性指数S与腔长误差${\delta _{\rm{l}}}$的关系, 这里腔长误差${\delta _{\rm{l}}}$定义为实际腔长与设计腔长的差值. 从图6(a)中可以看到, 即使腔长误差达到了$2\;{\rm{mm}}$, 对基模光束带来的不良影响都非常小, 基模的能量损耗仍然极小, 且光束质量保持在较高水准, 表明这种谐振腔系统对腔长误差具有极好的容差性.

      Figure 6.  The influence of the alignment errors on formation of the fundamental mode with ${r_0} = 1\;{\rm{mm}}$, $w = 0.2\;{\rm{mm}}$ and $a = 0.15$: (a) $\left| \gamma \right|$ and S of the fundamental mode as a function of $\delta_{\rm l}$; (b) $\left| \gamma \right|$ and S of the fundamental mode as a function of $\delta _{\rm d}$

      图6(b)给出了光束参数相同时, 基模的$\left| \gamma \right|$以及S与同轴度误差${\delta _{\rm{d}}}$的关系, 这里同轴度误差${\delta _{\rm{d}}}$定义为衍射元件中心和凹面镜中心之间的垂直距离. 从图6(b)中可以看出, 随着${\delta _{\rm{d}}}$的增加, $\left| \gamma \right|$下降速度较慢, S则下降速度较快. 这说明能量损耗受同轴度误差的影响较小, 而光束质量受同轴度误差影响较大. 计算结果表明, 当${\delta _{\rm{d}}} < 7\;\text{μ}{\rm{m}}$时, S值能保持在0.99以上, 即产生的光束质量较高. 因此建议实际组装系统时, 同轴度误差需小于$7\;\text{μ}{\rm{m}}$, 而这一精度要求以现有的机械调节技术并不难满足.

      除了系统组装时产生的对准误差以外, 衍射元件本身的加工误差也会对光束质量产生不利的影响. 如通常使用的离子束刻蚀技术会产生刻蚀误差, 即理想刻蚀深度与实际刻蚀深度之间的差值. 本文使用了较苛刻的误差分布模型-离散随机分布来分析刻蚀误差的影响, 即大小随机的刻蚀误差离散地分布在衍射元件表面, 并令其中的最大误差值为${\delta _{\rm{h}}}$, 用以表征加工精度. 图7给出了基模的$\left| \gamma \right|$以及S${\delta _{\rm{h}}}$的关系, 光束参数与图6相同. 从图7中可以看到, 随着${\delta _{\rm{h}}}$的增加, $\left| \gamma \right|$下降速度较快, S则下降速度较慢. 这说明能量损耗受刻蚀误差的影响较大, 而光束质量受刻蚀误差影响较小, 这一结论与同轴度误差的影响正好相反. 计算结果表明, 当${\delta _{\rm{h}}} < 0.06\lambda $(约$38\;{\rm{nm}}$)时, $\left| \gamma \right|$值能保持在0.97以上, 能量损耗仍保持较低水平. 因此在加工衍射元件时, 建议刻蚀误差小于$38\;{\rm{nm}}$, 以现有的微纳加工技术完全能满足这一精度要求.

      Figure 7.  $\left| \gamma \right|$ and S of the fundamental mode as a function of ${\delta _{\rm{h}}}$.

    4.   结 论
    • 本文提出了一种可以在谐振腔内产生CAB的方法, 且产生光束的参数可以通过改变衍射光学元件上的相位分布来进行调控. 研究发现, 该方法在能量损耗和光束质量两方面都明显优于目前常用的傅里叶空间的纯相位全息编码法, 且使用该方法组成的谐振腔系统对误差有很好的容差性. 但另一方面, 相位全息方法光路简单, 且使用了成熟的商业光学器件, 系统搭建要求低. 而腔内产生方法由于谐振腔腔长较长, 系统所占空间较大, 需额外定制加工较高精度的衍射光学元件, 且谐振腔的搭建对光路校准有较高的要求, 因此实现难度要高于相位全息法.

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