光纤阵列准直器设计及其发散角测量

光纤阵列准直器设计及其发散角测量 Design of fiber array collimator and measurement of its divergence angle 1 3 first-author 刘若仙（1994-），女，博士研究生，2019年于大连理工大学获得硕士学位，主要从事光机系统设计与光电测量仪器方面的研究。E-mail： liuruoxian@mail.dlut.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554404&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554410&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554406&type= 22.01332855 32.00399780 刘若仙 （1994-），女，博士研究生，2019年于大连理工大学获得硕士学位，主要从事光机系统设计与光电测量仪器方面的研究。E-mail： liuruoxian@mail.dlut.edu.cn liuruoxian@mail.dlut.edu.cn 2 3 corresp E-mail： zhaoshiyuan@dlut.edu.cn E-mail： zhaoshiyuan@dlut.edu.cn 赵士元（1992-），男，博士，讲师，2015年、2021年于哈尔滨工业大学分别获得学士和博士学位，主要从事精密光学仪器与测量方面的研究。E-mail： zhaoshiyuan@dlut.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554418&type= 22.01332855 32.00399780 赵士元 （1992-），男，博士，讲师，2015年、2021年于哈尔滨工业大学分别获得学士和博士学位，主要从事精密光学仪器与测量方面的研究。E-mail： zhaoshiyuan@dlut.edu.cn zhaoshiyuan@dlut.edu.cn 2 3 2 2 3 赵明山（1960-），男，教授，博士生导师，1981年于曲阜师范大学获得学士学位，1986年于北京师范大学获得硕士学位，2003年于比利时根特大学获得博士学位，现为“辽宁省先进光电子技术重点实验室”主任，大连理工大学光子技术研究中心主任，主要从事信息光电子技术领域的研究与教学工作。E-mail： mszhao@dlut.edu.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554432&type= 22.01332855 32.00399780 赵明山 （1960-），男，教授，博士生导师，1981年于曲阜师范大学获得学士学位，1986年于北京师范大学获得硕士学位，2003年于比利时根特大学获得博士学位，现为“辽宁省先进光电子技术重点实验室”主任，大连理工大学光子技术研究中心主任，主要从事信息光电子技术领域的研究与教学工作。E-mail： mszhao@dlut.edu.cn 大连理工大学 机械工程学院，辽宁 大连 116024 大连理工大学 机械工程学院，辽宁 大连 116024 School of Mechanical Engineering， Dalian University of Technology， Dalian 116024， China School of Mechanical Engineering， Dalian University of Technology， Dalian 116024， China 大连理工大学 光电工程与仪器科学学院，辽宁 大连 116024 大连理工大学 光电工程与仪器科学学院，辽宁 大连 116024 School of Optoelectronic Engineering and Instrumentation Science， Dalian University of Technology， Dalian 116024， China School of Optoelectronic Engineering and Instrumentation Science， Dalian University of Technology， Dalian 116024， China 辽宁省先进光电子技术重点实验室，辽宁 大连 116024 辽宁省先进光电子技术重点实验室，辽宁 大连 116024 Key Laboratory of Advanced Optoelectronic Technology of Liaoning Province， Dalian 116024， China Key Laboratory of Advanced Optoelectronic Technology of Liaoning Province， Dalian 116024， China 传统基于单个固定折射率透镜和渐变式折射率透镜制备光纤阵列准直器的方法，存在阵元数扩展困难、封装工艺复杂且集成化困难等缺点。利用光学微透镜易阵列化且阵元特性一致性好等优点，提出了基于平凸微透镜阵列制备光纤阵列准直器的方法。根据高斯光学和矩阵光学理论，对光纤阵列准直器的准直特性进行了理论分析和仿真，确定了光纤阵列准直器的相关设计参数，据此加工制备了四阵元一维排布光纤阵列准直器，其阵元间距为250 μm。通过远场光斑法对光纤阵列准直器的主要性能参数远场发散角进行了测量，并采用蒙特卡洛法对测量不确定度进行了分析与评定。光纤阵列准直器各通道远场发散角的测量值分别为0.69°，0.67°，0.71°，0.68°，测量扩展不确定度为0.02°，该测量结果在设计容差（0.68±0.03）°之内。该光纤阵列准直器具有良好的准直特性，能够满足光纤通信系统中光纤阵列准直器小型化和集成化的需求。 The optical fiber array collimator is a major component in optical fiber communication systems， and its development is gradually moving toward array and integration. The traditional method of constructing a fiber array collimator is based on single-fixed and graded index lenses. It has disadvantages， such as difficulties in integrating and expanding the number of array elements and a complex packaging process. In this paper， a method of fabricating a fiber array collimator based on a plano-convex microlens array is proposed. The method utilizes the advantages of easy arraying of optical microlenses and good uniformity of array element characteristics. The theories of Gaussian and matrix optics were applied to analyze and simulate the collimation characteristics and determine the relevant design parameters of the fiber array collimator. A fiber array collimator with an array pitch of 250 μm is fabricated in this study. The far-field divergence angle is the main performance parameter measured using the far-field spot method. The measurement uncertainty is analyzed and evaluated using the Monte Carlo method. The measured values of far-field divergence angles of each channel in the fiber array collimator are 0.69°， 0.67°， 0.71°， and 0.68°， respectively， and the measurement expansion uncertainty is 0.02°， which is within the design tolerance limit of （0.68±0.03）°. The optical fiber collimator has good collimation characteristics and can meet the requirements of miniaturization and integration in the field of optical communication. 光纤通信 光纤阵列准直器 微透镜阵列 远场发散角 不确定度分析 optical fiber communication fiber array collimator microlens array far-field divergence angle uncertainty analysis 1 引 言 1 光纤阵列准直器是光纤通信系统的关键基础部件。随着新一代信息系统向阵列化、宽带化、小型化方向发展，基于微波光子技术的通信雷达等光电子系统对光纤阵列准直器的需求迅速增长，对其性能也提出了更高的要求。光纤阵列准直器急需向低成本、阵列化、集成化方向发展。现有工艺多采用固定折射率透镜（Conventional Lens， C-Lens）和渐变式折射率透镜（Gradient-index Lens， G-Lens）制备光纤阵列准直器，将多个单透镜依次与光纤阵列各通道光纤进行调试、匹配及封装，不断扩展成所需通道数的光纤阵列准直器 ［ 1 1 ］ 。武汉光迅通过将多个C-Lens与光纤阵列进行耦合封装，制备出一维多通道光纤阵列准直器 ［ 2 2 ］ 。该阵列准直器具有良好的准直特性和较大的工作距离，但阵元间距为3.9 mm，难以满足集成化更高的应用需求，且通道扩展过程中难以保证精确的阵元间距。Zabou等提出直接将透镜集成在光纤端面来制备准直器的方法，可以突破透镜尺寸及光学封装对结构紧凑性的限制 ［ 3 3 ］ 。该方法制备的光纤阵列准直器阵元间距在微米量级，但其工作距离有限，多应用在直接跟硅光芯片耦合的应用场合。此外，该方法设计及制造工艺复杂、价格昂贵，准直一致性差。向金山等提出利用衍射型菲涅尔透镜阵列制备光纤阵列准直器的方法 ［ 4 4 ］ ，通过微光学和微细加工技术制备菲涅尔透镜阵列，设计实现了高密度集成化的光纤阵列准直器，其阵元间距在微米量级。但菲涅尔透镜阵列浮雕结构依靠精确的光学设计，制备工艺复杂，难以保证通道间准直特性的一致性。此外，用以制备的光纤阵列准直器在不同工作波长条件下表现出不同的耦合损耗。近年来，人们采用单点金刚石车削与紫外固化压印技术 ［ 5 5 ］ 、光刻胶热熔法 ［ 6 6 ］ 等工艺制备微透镜阵列，制备工艺相对简单。所制备的微透镜阵列具有阵元填充率高、阵元位置精确、面形均匀度好等优点，且在一定的工作波长范围内耦合效率一致性好 ［ 7 7 ］ ，还可根据实际应用需求扩展成多维任意布局的光纤阵列准直器。采用微透镜阵列来制备光纤阵列准直器，阵元间距可以达到光纤自身纤芯直径这一极限值。本文基于该思路，将平凸微透镜阵列用于设计与制备光纤阵列准直器，通过对光纤阵列准直器的准直特性进行理论分析和仿真，确定光纤阵列准直器的相关设计参数，加工制备光纤阵列准直器。最后，通过远场光斑法对光纤阵列准直器的主要性能参数远场发散角进行测量，并采用蒙特卡洛法对测量不确定度进行分析与评定。 2 理论分析与参数设计 1 2.1　高斯光束变换理论 2 高斯光束在自由空间中的传播如 图1 图1 所示。 基模高斯光束在横截面内的场振幅分布按照高斯函数从中心向外平滑降落，由振幅降落到中心值 <math id="M1"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>e</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554453&type= 2.62466669 7.87400007 的点所定义的光斑半径 <math id="M2"><mi>ω</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> 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http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554595&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554590&type= 40.13199997 9.05933285 . 已知高斯光束在某位置的 <math id="M19"><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554852&type= 7.28133297 3.97933316 ，则可确定该位置处的 <math id="M20"><mi>ω</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554605&type= 7.78933382 3.55599999 ， <math id="M21"><mi>R</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554619&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554614&type= 8.12800026 3.55599999 ，即： <math display="block" id="M22"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mi mathvariant="normal">I</mi><mi mathvariant="normal">m</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554626&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554621&type= 40.47066879 23.79133224 ， 式中： <math id="M23"><mi mathvariant="normal">I</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554631&type= 4.23333359 2.79399991 表示取复数虚部， <math id="M24"><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554639&type= 4.31799984 2.79399991 表示取复数实部。 高斯光束复参数 <math id="M25"><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554852&type= 7.28133297 3.97933316 的传播及变换规律遵守ABCD定律 ［ 9 9 ］ ，即： <math display="block" id="M26"><mi>q</mi><mo>=</mo><mfrac><mrow><mi>A</mi><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>B</mi></mrow><mrow><mi>C</mi><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>D</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554655&type= 20.57399940 8.89000034 ， 式中： <math id="M27"><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mi>B</mi></mtd></mtr><mtr><mtd><mi>C</mi></mtd><mtd><mi>D</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554663&type= 12.53066730 8.55133343 是光学系统的光学变换矩阵 M 。 如果复参数 <math id="M28"><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554852&type= 7.28133297 3.97933316 的高斯光束顺次通过光学变换矩阵为 <math id="M29"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554679&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554675&type= 4.82600021 3.80999994 ， <math id="M30"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554687&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554682&type= 4.82600021 3.80999994 …… <math id="M31"><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>n</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554691&type= 4.82600021 3.80999994 的光学系统，ABCD定律仍成立，该光学系统的光学变换矩阵 M 为： <math display="block" id="M32"><mi mathvariant="bold-italic">M</mi><mo>=</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>…</mo><mo>…</mo><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554705&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554701&type= 30.81866837 3.80999994 . 2.2　光纤阵列准直输出原理 2 图2 图2 为光纤阵列准直器结构示意图，它由光纤阵列与平凸微透镜阵列组成，光纤阵列端面与平凸微透镜阵列之间的距离为阵列耦合参数 <math id="M33"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554890&type= 1.86266661 2.79399991 。通过在光纤阵列与平凸微透镜阵列之间注入光学胶水实现二者的耦合，即采用光路有胶的封装方式。 以光纤阵列准直器单个通道为例，平凸微透镜的准直光路示意图如 图3 图3 所示。光纤输出的光束近似为高斯光传输，束腰半径较小但发散角较大。利用平凸微透镜对高斯光束的变换作用，将光纤阵列端面出射的发散角较大的光束准直成发散角较小的近似平行光束。其中， <math id="M34"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554733&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554730&type= 3.72533321 3.72533321 为光纤阵列端面， <math id="M35"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554741&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554736&type= 1.86266661 2.79399991 为光纤阵列端面与平凸微透镜阵列之间的距离， <math id="M36"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554745&type= 2.45533323 2.79399991 ， <math id="M37"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554986&type= 2.79399991 2.79399991 分别为平凸微透镜阵列的厚度和曲率半径， <math id="M38"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554763&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554760&type= 3.72533321 3.72533321 为高斯光束出射后距离微透镜阵列曲面顶点 <math id="M39"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555067&type= 2.28600001 2.79399991 处的一个观察面。 当光纤阵列与微透镜阵列之间注入光学胶水时，平凸微透镜对光纤出射高斯光束的光学变换矩阵 <math id="M40"><mi mathvariant="bold-italic">T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554779&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554776&type= 2.87866688 2.79399991 为： <math display="block" id="M41"><mtable columnalign="left"><mtr><mtd><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mi>L</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac><mi>L</mi></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mo stretchy="false">(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi><mo stretchy="false">)</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>n</mi><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac><mi>L</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554784&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554782&type= 57.74266815 35.81399918 式中： <math id="M42"><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554789&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554786&type= 3.04800010 3.72533321 为空气折射率； <math id="M43"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554795&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554792&type= 1.86266661 2.79399991 为微透镜材料的折射率； <math id="M44"><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554804&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554801&type= 3.04800010 3.72533321 为光学胶水的折射率。定义 <math id="M45"><mi>S</mi><mo>=</mo><mo>-</mo><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub></mrow><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554811&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554808&type= 16.00200081 8.38199997 ，为平凸微透镜的物方截距 ［ 10 10 ］ 。 光纤阵列端面 <math id="M46"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554814&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554813&type= 3.72533321 3.72533321 出射的高斯光束经由平凸微透镜出射到距离透镜曲面顶点 <math id="M47"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555067&type= 2.28600001 2.79399991 处观察面 <math id="M48"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554818&type= 3.72533321 3.72533321 的整体光学变换矩阵 <math id="M49"><mi mathvariant="bold-italic">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554850&type= 3.97933316 2.96333337 为： <math display="block" id="M50"><mtable><mtr><mtd><mi mathvariant="bold-italic">M</mi><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mi>A</mi></mtd><mtd><mi>B</mi></mtd></mtr><mtr><mtd><mi>C</mi></mtd><mtd><mi>D</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mi>d</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mi mathvariant="bold-italic">T</mi><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced><mo>=</mo></mtd></mtr><mtr><mtd><mfenced separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac><mi>d</mi></mtd><mtd><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac><mi>d</mi><mo stretchy="false">)</mo><mi>b</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>⋅</mo><mi>L</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>⋅</mo><mi>d</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac><mi>d</mi><mo>⋅</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac><mi>L</mi></mtd></mtr><mtr><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac></mtd><mtd><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>R</mi></mrow></mfrac><mi>L</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac><mi>b</mi></mtd></mtr></mtable></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554823&type= 108.88134003 30.39533424 光纤阵列端面出射的高斯光束用 <math id="M51"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554846&type= 1.86266661 3.64066648 表示为： <math display="block" id="M52"><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi><mo>⋅</mo><msubsup><mrow><mi>ω</mi></mrow><mrow/><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mrow><mi>λ</mi></mrow></mfrac><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554832&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554830&type= 18.20333290 8.21266651 ， 式中： <math id="M53"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554991&type= 2.53999996 2.79399991 为光纤阵列各通道光纤的模场半径， <math id="M54"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554837&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554835&type= 1.77800000 2.79399991 为工作波长。令 <math id="M55"><mi>x</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi><mo>⋅</mo><msup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>λ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554838&type= 15.49400043 8.21266651 ，则距离微透镜阵列曲面顶点 <math id="M56"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555067&type= 2.28600001 2.79399991 处的高斯光束的 <math id="M57"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554847&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554846&type= 1.86266661 3.64066648 为： <math display="block" id="M58"><mi>q</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mi>A</mi><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>B</mi></mrow><mrow><mi>C</mi><msub><mrow><mi>q</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi>D</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mi>x</mi><mi>i</mi><mo>+</mo><mi>B</mi></mrow><mrow><mi>C</mi><mi>x</mi><mi>i</mi><mo>+</mo><mi>D</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>A</mi><mi>C</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>B</mi><mi>D</mi></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>A</mi><mi>D</mi><mi>x</mi><mo>-</mo><mi>B</mi><mi>C</mi></mrow><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554849&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554848&type= 97.79000092 9.05933285 ， 式中 A ， B ， C ， D 为光学变换矩阵 <math id="M59"><mi mathvariant="bold-italic">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554850&type= 3.97933316 2.96333337 的元素。根据高斯光束复参数 <math id="M60"><mi>q</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554853&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554852&type= 7.28133297 3.97933316 通过光学系统的变换特性，可知观察面 <math id="M61"><msub><mrow><mi>P</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554855&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554854&type= 3.72533321 3.72533321 处高斯光束的光斑半径 <math id="M62"><mi>ω</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554856&type= 11.85333347 3.55599999 与波面曲率半径 <math id="M63"><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554859&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554858&type= 12.19200039 3.55599999 为： <math display="block" id="M64"><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mi>ω</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mroot><mrow><mo>-</mo><mfrac><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mi mathvariant="normal">I</mi><mi mathvariant="normal">m</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mrow><mrow/></mroot></mrow></mfrac></mtd></mtr><mtr><mtd><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi mathvariant="normal">R</mi><mi mathvariant="normal">e</mi><mfenced open="{" close="}" separators="|"><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mfenced></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554860&type= 50.03799820 35.30599976 . 根据高斯光束的传输特性，经过平凸微透镜变换后的高斯光束束腰半径 <math id="M65"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 与远场发散角 <math id="M66"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 分别为： <math display="block" id="M67"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mroot><mrow><mfrac><mrow><mi>ω</mi><msup><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mo stretchy="false">[</mo><mfrac><mrow><mi mathvariant="normal">π</mi><mo>⋅</mo><mi>ω</mi><msup><mrow><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>λ</mi><mo>⋅</mo><mi>R</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></mfrac><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot><mo>=</mo><mroot><mrow><mfrac><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>A</mi><mi>D</mi><mi>x</mi><mo>-</mo><mi>B</mi><mi>C</mi><mi>x</mi></mrow><mrow><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot><mo>=</mo><mroot><mrow><mfrac><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac><mo>⋅</mo><mi>x</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><mi>n</mi><mi>R</mi></mrow></mfrac><mi>L</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac><mi>b</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>n</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mi>R</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554867&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554866&type= 155.44799805 20.57399940 ， <math display="block" id="M68"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi>λ</mi></mrow><mrow><mi mathvariant="normal">π</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554868&type= 12.95400047 8.38199997 . 由 式（12） 式（12） 可知，当微透镜阵列的相关参数确定后，变换后的高斯光束束腰半径 <math id="M69"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554870&type= 3.64066648 3.72533321 只与阵列耦合参数 <math id="M70"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554890&type= 1.86266661 2.79399991 有关。此外，高斯光束束腰半径 <math id="M71"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 越大，远场发散角 <math id="M72"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 越小，光纤阵列准直器的准直效果越好。 2.3　光纤阵列准直器参数设计 2 通过Matlab软件进行数值模拟仿真，分析阵列耦合参数 <math id="M73"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554890&type= 1.86266661 2.79399991 对光纤阵列准直器准直性能的影响，从而确定光纤阵列准直器的设计参数。仿真参数设置如 表1 表1 所示。 Parameter Value Wavelength/nm 1 550 Mode-field diameter/μm 10.5 Design radius of curvature/μm 315 Thickness/mm 1 <math id="M74"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554881&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554880&type= 1.60866666 2.37066650 1.444 <math id="M75"><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554884&type= 2.62466669 3.21733332 1.552 <math id="M76"><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554884&type= 2.62466669 3.21733332 1 高斯光束束腰半径 <math id="M77"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 与阵列耦合参数 <math id="M78"><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554890&type= 1.86266661 2.79399991 之间的关系如 图4 图4 所示。当阵列耦合参数 <math id="M79"><mi>b</mi><mo>=</mo><mn mathvariant="normal">0.026</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554892&type= 22.77533340 2.87866688 时，光纤阵列准直器出射的高斯光束的束腰半径最大， <math id="M80"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">83.06</mn><mtext> </mtext><mi mathvariant="normal">μ</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554895&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554894&type= 23.62199974 4.57200003 ，对应的远场发散角 <math id="M81"><mi>θ</mi><mo>=</mo><mn mathvariant="normal">0.68</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554897&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554896&type= 15.23999977 3.04800010 。 进一步分析可知，当高斯光束的束腰半径 <math id="M84"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 最大时，光纤阵列出射端面刚好位于平凸微透镜的物方焦点处，即当阵列耦合参数 b 等于平凸微透镜物方截距 <math id="M85"><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554908&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554907&type= 1.26999998 2.79399991 时，光纤阵列准直器的准直效果最好。 当 <math id="M86"><mi>b</mi><mo>=</mo><mi>s</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554910&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554909&type= 7.95866632 2.79399991 时， 式（12） 式（12） 可进一步简化成： <math display="block" id="M87"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>R</mi><mroot><mrow><mfrac><mrow><mi>λ</mi></mrow><mrow><mi mathvariant="normal">π</mi></mrow></mfrac><mo>⋅</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mo stretchy="false">(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msup><mrow><mo>-</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>x</mi></mrow></mfrac></mrow><mrow/></mroot><mo>=</mo><mfrac><mrow><mi>R</mi><mi>λ</mi><mo>⋅</mo><mroot><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot></mrow><mrow><mi mathvariant="normal">π</mi><mi>ω</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554912&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554911&type= 64.68533325 9.99066734 . 由 式（14） 式（14） 可知，光纤阵列准直器准直后的高斯光束束腰半径 <math id="M88"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 只与微透镜阵列的曲率半径 <math id="M89"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554986&type= 2.79399991 2.79399991 和光纤模场半径 <math id="M90"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554991&type= 2.53999996 2.79399991 有关，而与微透镜阵列的厚度 <math id="M91"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555032&type= 2.45533323 2.79399991 无关。 为制备一维四阵元光纤阵列准直器，通过六维精密调整装置，实时调整光纤阵列与微透镜阵列之间的相对位置，而后注入光学胶水实现二者的耦合封装。光纤阵列参数及微透镜阵列参数如 表2 表2 ~ 表3 表3 所示，阵列耦合参数 <math id="M92"><mi>b</mi><mo>=</mo><mn mathvariant="normal">0.026</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554922&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554921&type= 22.77533340 2.87866688 ，光学胶水固化后在1 550 nm波长条件下的折射率为1.552。其中，光纤阵列采用V型槽法制备 ［ 11 11 - 12 12 11-12 ］ ，微透镜阵列采用光刻胶热熔法制备。封装后的光纤阵列准直器实物图如 图5 图5 所示，其头部主体尺寸为2.5 mm×2.5 mm×10 mm。采用光学设计软件Zemax对上述设计参数的光纤阵列准直器进行建模仿真，其准直光路如 图6 图6 所示，准直效果良好。 Parameter Value Fiber type CORNING PM15-U25D Mode-field diameter/μm 10.5±0.5 Numerical aperture 0.09 Wavelength/nm 1 550 Number of fibers per array 1×4 Arrangement of the lenses Linear Array pitch/μm 250±0.5 Parameter Value Lens diameter/μm 240 Surface profil Plano-convex Design radius of curvature/μm 315 Wavelength range/nm 1 260~1 620 AR coating Single-sided， R < 0.5% Material Fused silica （ n =1.444@1 550 nm） Thickness/mm 1.0±0.025 Number of fibers per array 1×4 Arrangement of the lenses Linear Array pitch/μm 250±0.5 ROC uniformity （lens-lens in array） ±1% 2.4　理论设计误差 2 由 式（14） 式（14） 可知，若忽略光学胶水及微透镜阵列材料的折射率误差，光纤阵列准直器出射后的高斯光束束腰半径 <math id="M93"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 的理论设计误差 <math id="M94"><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554960&type= 5.50333309 3.72533321 仅与微透镜阵列曲率半径 <math id="M95"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554986&type= 2.79399991 2.79399991 的公差 <math id="M96"><mi mathvariant="normal">d</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554944&type= 4.57200003 2.79399991 及光纤模场直径 <math id="M97"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554950&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554948&type= 2.53999996 2.79399991 的公差 <math id="M98"><mi mathvariant="normal">d</mi><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554951&type= 4.40266657 2.79399991 有关，则束腰半径 <math id="M99"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 的理论设计偏差 <math id="M100"><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554962&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554960&type= 5.50333309 3.72533321 为： <math display="block" id="M101"><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>R</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi mathvariant="normal">d</mi><mi>ω</mi><mmultiscripts><mrow/><mprescripts/><mrow><mn mathvariant="normal">0</mn></mrow><mrow/></mmultiscripts></mrow><mrow><mi mathvariant="normal">d</mi><mi>ω</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554964&type= 41.06333542 11.17599964 ， 式中： <math display="block" id="M102"><mfrac><mrow><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">d</mi><mi>R</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>λ</mi><mo>⋅</mo><mroot><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot></mrow><mrow><mi mathvariant="normal">π</mi><mi>ω</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mi mathvariant="normal">d</mi><mi>R</mi><mo>=</mo><mn mathvariant="normal">0.259</mn><mi mathvariant="normal">d</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554970&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554968&type= 55.20266724 9.82133293 ， <math display="block" id="M103"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>ω</mi><mmultiscripts><mrow/><mprescripts/><mrow><mn mathvariant="normal">0</mn></mrow><mrow/></mmultiscripts></mrow><mrow><mi mathvariant="normal">d</mi><mi>ω</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>λ</mi><mi>R</mi><mo>⋅</mo><mroot><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow/></mroot></mrow><mrow><mi mathvariant="normal">π</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>n</mi><mo>-</mo><msub><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mi mathvariant="normal">d</mi><mi>ω</mi><mo>=</mo><mn mathvariant="normal">15.559</mn><mi mathvariant="normal">d</mi><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554974&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554972&type= 61.21400070 9.82133293 . 由此可见，光纤模场半径 <math id="M104"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554991&type= 2.53999996 2.79399991 的公差 <math id="M105"><mi mathvariant="normal">d</mi><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555001&type= 4.40266657 2.79399991 对于光纤阵列准直器的准直性能影响较大。由 表1 表1 ~ 表2 表2 可知，平凸微透镜阵列的曲率半径 <math id="M106"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554989&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554986&type= 2.79399991 2.79399991 、光纤模场半径 <math id="M107"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554993&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554991&type= 2.53999996 2.79399991 的公差 <math id="M108"><mi mathvariant="normal">d</mi><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554998&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39554996&type= 4.57200003 2.79399991 ， <math id="M109"><mi mathvariant="normal">d</mi><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555004&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555001&type= 4.40266657 2.79399991 分别为3.15 μm（±1%），0.25 μm，计算得到高斯光束束腰半径 <math id="M110"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 的理论设计误差 <math id="M111"><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">3.975</mn><mtext> </mtext><mi mathvariant="normal">μ</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555017&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555013&type= 25.48466682 4.57200003 ，根据 式（13） 式（13） 可得远场发散角的理论设计误差 <math id="M112"><mi mathvariant="normal">d</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">360</mn><mo>⋅</mo><mi>λ</mi></mrow><mrow><msup><mrow><mi mathvariant="normal">π</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mi mathvariant="normal">d</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0.03</mn><mo>°</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555024&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555021&type= 38.43866730 8.63599968 。那么，设计的光纤阵列准直器的远场发散角为（0.68±0.03）°。 3 实 验 1 3.1　远场发散角测量 2 3 . 1 . 1　测量原理 3 为了评价光纤准直器的准直特性，需要测量其远场发散角。远场光斑法通过测得远场区域不同位置处高斯光束的光斑直径计算得到高斯光束的远场发散角 ［ 13 13 - 14 14 13-14 ］ 。为了准确测量高斯光束的远场发散角 <math id="M113"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 ，定义 <math id="M114"><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555032&type= 2.45533323 2.79399991 为从高斯光束高斯束腰到高斯光束发散角 <math id="M115"><mi>θ</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0.99</mn><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555041&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555038&type= 21.67466736 3.97933316 处的距离，当 <math id="M116"><mfrac><mrow><mi>θ</mi><mfenced separators="|"><mrow><mi>z</mi></mrow></mfenced></mrow><mrow><mi>θ</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0.99</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555046&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555044&type= 20.65866661 8.29733276 时， <math id="M117"><mi>L</mi><mo>≈</mo><mn mathvariant="normal">7</mn><mo>×</mo><mfrac><mrow><mi mathvariant="normal">π</mi><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>λ</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">7</mn><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555049&type= 31.41133499 8.21266651 。因此，采用远场光斑法测量高斯光束远场发散角时，需要保证测量位置大于7倍瑞利距离 ［ 15 15 ］ 。 光纤阵列准直器单个通道出射的高斯光束传播示意图如 图7 图7 所示。取平凸微透镜曲面出射顶点为坐标原点 <math id="M118"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555195&type= 2.96333337 2.87866688 ，高斯光束传播方向为 <math id="M119"><mi>Z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555060&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555059&type= 2.53999996 2.79399991 轴正方向。其中， <math id="M120"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 为光纤阵列准直器出射高斯光束束腰半径； <math id="M121"><mi>d</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555067&type= 2.28600001 2.79399991 为高斯光束束腰位置距离坐标原点的轴向距离； <math id="M122"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 为高斯光束的远场发散角。 假设在远场区域 <math id="M123"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555091&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555089&type= 2.96333337 3.72533321 ， <math id="M124"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555319&type= 2.96333337 3.72533321 两处测得的光斑半径分别为 <math id="M125"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555354&type= 3.64066648 3.72533321 和 <math id="M126"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555405&type= 3.64066648 3.72533321 ，由 式（1） 式（1） 可得： <math display="block" id="M127"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><mi>λ</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">π</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555111&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555109&type= 43.26466751 10.15999985 ， <math display="block" id="M128"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mroot><mrow><mn mathvariant="normal">1</mn><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><mi>λ</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">π</mi><msubsup><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555117&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555114&type= 43.26466751 10.15999985 . 将 式（16） 式（16） 和 式（17） 式（17） 联立可得： <math display="block" id="M129"><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>λ</mi></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot><mo>-</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>λ</mi></mrow><mrow><mi mathvariant="normal">π</mi><msup><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mfenced open="|" close="|" separators="|"><mrow><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">π</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot><mo>-</mo><mroot><mrow><msup><mrow><mfenced separators="|"><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfrac><mrow><mi mathvariant="normal">π</mi><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mn mathvariant="normal">1</mn></mrow><mrow/></mroot></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555118&type= 141.90132141 14.30866718 . 令 <math id="M130"><mi mathvariant="normal">Δ</mi><mi>z</mi><mo>=</mo><mfenced open="|" close="|" separators="|"><mrow><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555125&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555123&type= 21.75933456 3.97933316 ，由 式（18） 式（18） 可知，通过测量相对距离 <math id="M131"><mi mathvariant="normal">Δ</mi><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555397&type= 4.14866638 2.79399991 的两处位置的光斑半径 <math id="M132"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555354&type= 3.64066648 3.72533321 ， <math id="M133"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555405&type= 3.64066648 3.72533321 ，可以求出光纤阵列准直器准直出射后的高斯光束束腰半径 <math id="M134"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 及远场发散角 <math id="M135"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 。 该光纤阵列准直器出射高斯光束的束腰半径 <math id="M136"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 理论值为83.06 μm，此时瑞利距离 <math id="M137"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi mathvariant="normal">π</mi><msup><mrow><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>λ</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">13.98</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555155&type= 36.57600021 8.21266651 。因此，取距离坐标原点 <math id="M138"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555161&type= 2.96333337 2.87866688 的轴向距离140 mm处为初始测量位置。考虑到实际操作空间，距离太远，测试光源的光输出功率难以满足光束分析仪的探测要求，选定相对测量距离 <math id="M139"><mi mathvariant="normal">Δ</mi><mi>z</mi><mo>=</mo><mn mathvariant="normal">60</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555168&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555166&type= 19.30400085 2.87866688 ，即 <math id="M140"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">200</mn><mtext> </mtext><mi mathvariant="normal">m</mi><mi mathvariant="normal">m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555172&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555171&type= 19.98133469 3.80999994 。 3 . 1 . 2　测量装置 3 基于远场光斑法的高斯光束远场发散角测量装置如 图8 图8 所示。该装置包括窄线宽激光器（Agilent 8164B）、多维度机械调节架、光纤准直器夹具、光学接杆组件、光学导轨和光束分析仪。待测光纤阵列准直器通过光纤准直器夹具固定在多维度机械调节架中，多维度机械调节架可以实现光纤阵列准直器六个自由度的调节。多维度机械调节架与光束分析仪分别通过高度可调节的光学接杆组件固定在光学导轨上。通过红外光显示卡辅助调整光束分析仪的高度，使光纤阵列准直器出射的高斯光束入射到光束分析仪的有效探测区域内。测量时，将多维度机械调节架固定在光学导轨中，通过光束分析仪在光学导轨中的左右滑动来改变与待测光纤阵列准直器之间的相对位置。测量过程中，所有元件都被安置在隔振光学平台上，且保证外界测量背景光不发生改变。 3 . 1 . 3　测量过程 3 激光器的工作波长为1 550 nm，输出光功率为4.25 mW，线宽典型值为100 kHz。光束分析仪为CINOGY公司CinCam系列光束分析仪，相机类型设置为CMOS-1203IR，响应波长为1 470~1 605 nm，可测光斑直径的为45 μm~4 mm。其配套软件在线监测界面如 图9 图9 所示，设置光束分析仪光斑尺寸测量方法为“Fit Gauss”中的13.53%，即 <math id="M141"><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555182&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555180&type= 3.38666677 8.12800026 处的光斑尺寸。通过光束分析仪配套软件对各测量位置处高斯光束各参数数据进行读取与保存，可以读取光斑位置、光斑尺寸及椭圆度等信息。采用远场光斑法测量高斯光束的远场发散角时，首先调整多维度机械调节架使得各通道测量光斑的椭圆度均≥99%，保证测试光源正入射到光束分析仪的探测面，以长轴光斑尺寸Width Major作为各测量位置处的光斑直径 <math id="M142"><mn mathvariant="normal">2</mn><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555183&type= 4.40266657 2.79399991 进行计算。 以光纤阵列准直器微透镜曲面出射顶点为坐标原点 <math id="M143"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555195&type= 2.96333337 2.87866688 ，高斯光束传播方向为 z 轴正方向。移动光束分析仪在光学导轨上的位置，取距离坐标原点 <math id="M144"><mi>O</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555196&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555195&type= 2.96333337 2.87866688 轴向距离140，200 mm处作为测量位置点 <math id="M145"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555198&type= 2.96333337 3.72533321 ， <math id="M146"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555319&type= 2.96333337 3.72533321 ，并将两处位置测得的光斑直径分别记为 <math id="M147"><mn mathvariant="normal">2</mn><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555207&type= 5.50333309 3.72533321 ， <math id="M148"><mn mathvariant="normal">2</mn><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555212&type= 5.50333309 3.72533321 ，各测量位置处光斑直径均为20次测量的平均值，由 式（18） 式（18） 计算得到高斯光束的束腰半径 <math id="M149"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 及远场发散角 <math id="M150"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 的最佳估计值。由于 式（18） 式（18） 为超越方程，故采用差分法由Matlab编程计算求解 <math id="M151"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 及 <math id="M152"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 。 3.2　测量结果 2 采用远场光斑法依次测量光纤阵列准直器4个通道出射高斯光束的束腰半径 <math id="M153"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555231&type= 3.64066648 3.72533321 及远场发散角 <math id="M154"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 ，测量结果如 表4 表4 所示。光纤阵列准直器各通道的远场发散角 <math id="M155"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 分别为0.69°，0.67°，0.71°，0.68°，与远场发散角理论设计值0.68°的差值均在0.03°以内，即制备的光纤阵列准直器满足设计要求。同时，各通道保持了较高的一致性。 Channel <math id="M156"><mn>2</mn><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555248&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555244&type= 4.74133301 3.21733332 /μm <math id="M157"><mn>2</mn><msub><mrow><mi>ω</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555254&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555251&type= 4.82600021 3.21733332 /μm <math id="M158"><msub><mrow><mi>ω</mi></mrow><mrow><mn>0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555261&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555258&type= 3.21733332 3.21733332 /μm <math id="M159"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555471&type= 1.60866666 2.37066650 /（°） 1 1 666 2 386 81.952 0.69 2 1 656 2 357 84.152 0.67 3 1 665 2 402 80.075 0.71 4 1 660 2 370 83.095 0.68 3.3　测量不确定度分析 2 由 式（18） 式（18） 可知，高斯光束远场发散角 <math id="M160"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 的测量误差源主要包括相对测量距离 <math id="M161"><mi mathvariant="normal">Δ</mi><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555397&type= 4.14866638 2.79399991 的测量误差以及高斯光束光斑半径 <math id="M162"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555354&type= 3.64066648 3.72533321 ， <math id="M163"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555405&type= 3.64066648 3.72533321 的测量误差，且测量过程中多种误差源同时存在。在进行测量不确定度评定时，GUM（Guide to the Uncertainty in Measurement）方法基于不确定度传播规律，主要适用于线性模型。采用远场光斑法测量远场发散角的测量模型为非线性模型，采用GUM方法所需的偏导计算比较困难，因而本文通过蒙特卡洛法（Monte Carlo Method， MCM）评定远场发散角的测量不确定度。MCM在建立数学模型、设定输入量概率密度函数的基础上，通过对概率分布随机采样进行分布传递，确定输出量的概率密度函数，从而得到输出量的估计值、标准不确定度以及在指定包含概率下的包含区间，进而实现对测量不确定度的评定 ［ 16 16 - 17 17 16-17 ］ 。 光学导轨的定位精度为0.5 mm，光束分析仪的光斑测量误差为10 μm，且两类误差均符合均匀分布。已知均匀分布函数 <math id="M164"><mi>R</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555300&type= 11.76866722 3.55599999 对应的概率密度函数（Probability Density Function， PDF）为 <math id="M165"><mi mathvariant="normal">g</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable><mtr><mtd><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfrac><mo>,</mo><mi>a</mi><mo>≤</mo><mi>ξ</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">其他</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555305&type= 40.63999939 14.39333248 。以通道1为例，光束分析仪在测量位置点 <math id="M166"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555315&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555313&type= 2.96333337 3.72533321 ， <math id="M167"><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555319&type= 2.96333337 3.72533321 处测得的光斑半径 <math id="M168"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555327&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555324&type= 3.64066648 3.72533321 ， <math id="M169"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555405&type= 3.64066648 3.72533321 分别为833，1 193 μm。采用Matlab MCM模型分析多误差源同时作用下的远场发散角测量不确定度的流程如下： （1）MCM输入：定义远场发散角 <math id="M170"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 为输出量 <math id="M171"><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555387&type= 3.13266683 2.79399991 ，相对测量距离 <math id="M172"><mi mathvariant="normal">Δ</mi><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555397&type= 4.14866638 2.79399991 及高斯光束光斑半径 <math id="M173"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555354&type= 3.64066648 3.72533321 ， <math id="M174"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555405&type= 3.64066648 3.72533321 分别为相关输入量 <math id="M175"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555370&type= 3.89466691 3.72533321 ， <math id="M176"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555375&type= 3.89466691 3.72533321 ， <math id="M177"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555385&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555382&type= 3.89466691 3.72533321 ；输出量 <math id="M178"><mi>Y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555389&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555387&type= 3.13266683 2.79399991 与输入量 <math id="M179"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555433&type= 22.01333237 3.97933316 之间的模型为所推导的理论测量 公式（18） 公式（18） 。相对测量距离 <math id="M180"><mi mathvariant="normal">Δ</mi><mi>z</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555397&type= 4.14866638 2.79399991 及高斯光束光斑半径 <math id="M181"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555401&type= 3.64066648 3.72533321 ， <math id="M182"><msub><mrow><mi>ω</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555405&type= 3.64066648 3.72533321 均满足均匀分布，对应分布函数分别为 <math id="M183"><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">59.5,60.5</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555411&type= 25.56933212 3.89466691 ， <math id="M184"><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">828,838</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555417&type= 21.67466736 3.89466691 ， <math id="M185"><msub><mrow><mi>R</mi></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></msub><mo stretchy="false">(</mo><mn mathvariant="normal">1188,1198</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555422&type= 25.23066711 3.89466691 。设置MCM试验样本量 M 为 <math id="M186"><msup><mrow><mn mathvariant="normal">10</mn></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555428&type= 4.82600021 3.04800010 。 （2）MCM传播：从输入量 <math id="M187"><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub><mfenced separators="|"><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555433&type= 22.01333237 3.97933316 的PDF中抽取 M 个样本值 <math id="M188"><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi><mo>,</mo><mi>r</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mn mathvariant="normal">3</mn><mo>,</mo><mi>r</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555441&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555439&type= 50.79999924 4.48733330 ，并对每个样本向量 <math id="M189"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn><mo>,</mo><mi>r</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn><mo>,</mo><mi>r</mi></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555447&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555443&type= 21.92866707 4.48733330 计算相应输出量 Y 的模型值。 （3）MCM输出及结果：将 M 个模型值按严格递增次序排序，并以200为子区间间隔绘制成直方图，得到输出量 Y 即通道1远场发散角 <math id="M190"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 的概率分布结果，如 图10 图10 所示。多误差源同时作用下，远场发散角 <math id="M191"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 的测量值满足正态分布，远场发散角 <math id="M192"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555458&type= 1.86266661 2.79399991 的估计值为0.69°，标准不确定度为0.01°，95%概率对称包含区间为 <math id="M193"><mfenced open="[" close="]" separators="|"><mrow><mo>-</mo><mn mathvariant="normal">0.02</mn><mo>°</mo><mo>,</mo><mn mathvariant="normal">0.02</mn><mo>°</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555464&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=39555462&type= 25.39999962 4.40266657 ，即扩展不确定度为0.02°，此时包含因子为2。 4 结 论 1 本文面向集成化、小型化阵列准直器的应用需求，提出了基于平凸微透镜阵列制备光纤阵列准直器的技术路线，详细阐述了其设计与性能评价的理论和方法。通过高斯光学和矩阵光学理论研究了平凸微透镜阵列对高斯光束的变换原理。基于光路有胶的耦合封装方式，推导了基于平凸微透镜阵列的光纤阵列准直器准直特性的解析表达式，仿真分析了阵列耦合参数对其准直效果的影响，为光纤阵列准直器的参数设计提供理论基础。最后，实验制备了四阵元光纤阵列准直器，并通过远场光斑法对其远场发散角进行测量。依据MCM方法对测量结果进行了测量不确定度的分析与评定。测量结果表明，光纤阵列准直器各通道远场发散角的测量值分别为0.69°，0.67°，0.71°，0.68°，扩展不确定度为0.02°，该测量结果均在设计容差（0.68±0.03）°之内。该光纤阵列准直器具有集成化高、性能指标一致性好、阵列易扩展、调试封装工艺简单等优势，为光通信领域中小型化、集成化的光纤阵列准直器提供了有效的技术手段。 参考文献 LEE S S . 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