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
Fund Project:Project supported by the National Basic Research Program of China (Grant No. 2015CB352001), the Special Funds of the Major Scientific Instruments Equipment Development of China (Grant No. 2012YQ17000408), and the Natural Science Foundation of Shanghai, China (Grant No. 16ZR144600)
Received Date:16 July 2019
Accepted Date:16 September 2019
Available Online:05 December 2019
Published Online:05 January 2020
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. Keywords:circular Airy beam/ laser resonator/ eigenvector method/ diffractive optical element
$ {\rm{u}}\left( r \right) = C \cdot Ai\left( {\frac{{{r_0} - r}}{w}} \right)\exp \left( {a \frac{{{r_0} - r}}{w}} \right), $
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)和图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. 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 Figure1. (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. 图 2 谐振腔示意图 Figure2. Schematic of the laser resonator configuration for CAB generation.
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 Table1.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.
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 Figure3. 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. 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 Figure4. 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. 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 Figure5. 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.
其中A, B分别表示参考图像和失真图像, ${\sigma _{\rm{A}}}$和${\sigma _{\rm{B}}}$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. 图 6 光束参数为${r_0} = 1\;{\rm{mm}}$, $w = 0.2\;{\rm{mm}}$和$a = 0.15$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 Figure6. 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与${\delta _{\rm{h}}}$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. 图 7 基模的$\left| \gamma \right|$以及S与${\delta _{\rm{h}}}$的关系 Figure7.$\left| \gamma \right|$ and S of the fundamental mode as a function of ${\delta _{\rm{h}}}$.
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
图 2 谐振腔示意图
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
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
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
图 6 光束参数为
图 7 基模的
