混频对全视场外差动态干涉仪测量精度的影响

混频对全视场外差动态干涉仪测量精度的影响 Influence of frequency mixing on measurement accuracy of full-field heterodyne dynamic interferometer 1 2 first-author 杨宇琦（1997-），女，安徽人，硕士研究生，2019年于长春理工大学获得学士学位，主要从事光学精密检测技术的研究。E-mail： yangyuqi20@mails.ucas.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439225&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439232&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439227&type= 20.63299942 30.00000000 杨宇琦 （1997-），女，安徽人，硕士研究生，2019年于长春理工大学获得学士学位，主要从事光学精密检测技术的研究。E-mail： yangyuqi20@mails.ucas.ac.cn yangyuqi20@mails.ucas.ac.cn 1 2 corresp E-mail： liyang@aoe.ac.cn E-mail： liyang@aoe.ac.cn 李 杨（1986-），男，陕西人，博士，副研究员，硕士生导师，2008年于中国科学技术大学获得学士学位，2013年于中国科学院大学获得博士学位，主要从事光学精密检测、光学相干测量仪器研制等工作。E-mail： liyang@aoe.ac.cn http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439249&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439247&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439244&type= 21.42900085 30.00000000 李 杨 （1986-），男，陕西人，博士，副研究员，硕士生导师，2008年于中国科学技术大学获得学士学位，2013年于中国科学院大学获得博士学位，主要从事光学精密检测、光学相干测量仪器研制等工作。E-mail： liyang@aoe.ac.cn liyang@aoe.ac.cn 1 1 2 1 2 中国科学院 空天信息创新研究院，北京 100094 中国科学院 空天信息创新研究院，北京 100094 Aerospace Information Research Institute， Chinese Academy of Sciences， Beijing 100094， China Aerospace Information Research Institute， Chinese Academy of Sciences， Beijing 100094， China 中国科学院大学 光电学院，北京 100094 中国科学院大学 光电学院，北京 100094 School of Optoelectronics， University of Chinese Academy of Sciences， Beijing 100094， China School of Optoelectronics， University of Chinese Academy of Sciences， Beijing 100094， China 全视场外差动态干涉仪在面形测量方面具有测量精度高、抗干扰能力强等优点，适合用于长焦距面形的动态干涉测量以及大口径光学元件的测量。但是干涉仪系统内部光学元件性能不理想以及元件装配存在误差等，会在干涉光路中引入频率混叠，影响干涉仪的测量精度。为了分析混频对全视场外差动态干涉仪测量精度的影响，从混频产生的原因出发，建立了由混频引入的测量误差的理论模型，分析了混频程度对测量精度的影响，分析结果表明，测量误差与混频程度呈非线性正相关，混频会造成测量面形结果上叠加一个和干涉条纹相同频率的周期性误差。搭建了全视场外差动态干涉测量实验系统，验证了不同混频程度对面形测量精度的影响，当混频程度为0.029时，面形测量误差为0.053 λ ；当混频程度为0.120时，面形测量误差为0.110 λ ，与仿真分析结果吻合。本文的研究对研制高精度全视场外差动态干涉仪具有实际意义。 The full-field heterodyne dynamic interferometer has the advantages of a high measurement accuracy and strong anti-interference ability for surface shape measurement. It is suitable for large-focal length surface shape measurement and the measurement of large-aperture optical elements. However， the performance of the optical components in the interferometer system is not ideal， and there are errors in the assembly of components， which introduce frequency aliasing in the interferometer optical path and affect the measurement accuracy of the interferometer. To analyze the influence of frequency mixing on the measurement accuracy of the full-field heterodyne dynamic interferometer， a theoretical model of the measurement error introduced by frequency mixing is established according to the causes of frequency mixing， and the influence of the frequency mixing degree on the measurement accuracy is analyzed. The results indicate that the measurement error is nonlinearly and positively correlated with the degree of mixing and that the mixing causes a periodic error with the same frequency as the interference fringe on the measured surface shape. In this study， a full-field heterodyne dynamic interferometry experimental system is established to examine the effects of different mixing degrees on the surface measurement accuracy. When the mixing frequency is 0.029， the surface shape measurement error is 0.053λ. When the mixing frequency is 0.120， the surface shape measurement error is 0.110λ. The experimental results are consistent with the simulation results. The research presented in this paper has practical significance for the development of high-precision full-field heterodyne dynamic interferometers. 干涉测量 全视场外差 混频 测量误差 interferometry full-field heterodyne frequency mixing measurement error 1 引 言 1 随着航空航天、天文探测等技术的发展，对大口径光学镜面以及长焦光学元件的需求越来越大，对相应的光学检测精度要求也在不断提高。干涉仪作为一种光学检测仪器，具有非接触、精度高以及灵敏度高等优点 ［ 1 1 ］ 。但当干涉仪测量大口径以及长焦光学元件时，振动以及空气扰动等环境因素会影响测量结果。为了实现对不稳定环境状态下光学元件的面形测量，本课题组提出了结合全视场外差移相技术以及动态干涉测量技术全视场外差动态干涉仪，用于长焦距面形的动态干涉测量以及大口径光学元件的测量 ［ 2 2 ］ 。全视场外差移相通过两个低频差的声光移频器产生的拍频信号来实现移相，当外差干涉的频差调至合适频率时，通过高速探测器可以在单次测量过程中达到几十甚至上百次移相采样，实现动态干涉测量。目前，全视场外差干涉技术已成为研究热点 ［ 3 3 - 8 8 3-8 ］ 。 全视场外差动态干涉仪中采用泰曼-格林型光路 ［ 2 2 ］ ，干涉仪系统内部常因为光学元件性能不理想或元件装配问题，分光后的光路中会混入不同频率的光，影响相位测量精度，因此分析混频对外差干涉仪有着非常重要的意义。外差干涉仪常用于测距以及面形测量。对于测距式外差干涉仪，已有众多学者对干涉仪内部混频引入的误差，即非线性误差进行了分析 ［ 9 9 - 12 12 9-12 ］ 。2002年，埃因霍芬大学的Cosijins等针对激光源的偏振态、分光棱镜的性能以及检偏器的旋转角度等因素引起的非线性误差建立了误差理论模型，但因其分析的误差源不全面，所建立的理论模型并不完整 ［ 9 9 ］ 。2006年，哈尔滨工业大学的陈洪芳等发现偏振分光棱镜的分光性能不理想时，会引入亚纳米量级的非线性误差 ［ 10 10 ］ 。2008年，上海理工大学的侯文玫等发现非偏振分光棱镜对非线性误差的影响比偏振分光棱镜对非线性误差的影响更大 ［ 11 11 ］ 。但目前针对面形测量的外差干涉仪中由混频引入的面形测量误差的研究较少。本文从外差干涉测量原理入手，分析全视场外差动态干涉仪中产生混频的原因以及混频引入的测量误差对面形测量精度的影响，为高精度全视场外差动态干涉仪的研制奠定基础。 2 原 理 1 全视场外差动态干涉测量技术是一种利用具有固定频差的干涉光信号来实现全视场同步外差干涉探测的技术 ［ 2 2 ］ 。本文采用的干涉仪系统包括双频外差光源模块以及干涉测量模块，双频外差光源模块采用两个低差频的声光调制器（Acoustic Optical Modulator， AOM）提供拍频信号，干涉测量模块采用泰曼-格林型光路。 图1 图1 为全视场外差动态干涉仪原理。 激光经1/2波片（Half-wave Plate， HWP）调整分光比后，被偏振分光棱镜1（PBS1）分为偏振方向相互垂直的两束光，其中s偏振光被PBS1反射，p偏振光被PBS1透射。两束光经过AOM1与AOM2后，取其移频后的+1级（或 - 1级）光，再分别经过反射镜1（M1）与反射镜2（M2）反射，在PBS2处合束。合束光经过空间滤波后准直平行出射，进入干涉测量模块，再由PBS3分为偏振方向相互垂直的偏振光，其中频率为 <math id="M1"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439717&type= 2.96333337 4.57200003 的p偏振光进入测量光路，频率为 <math id="M2"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439366&type= 2.96333337 4.57200003 的s偏振光进入参考光路。两束光经过1/4波片（Quarter-Wave Plate， QWP）一个来回后，偏振方向都旋转90°，再经PBS3合束，成像镜组成像，与偏振片调节偏振方向后，参考光与测量光发生干涉，由面阵探测器接收干涉信息。 干涉光强为： <math display="block" id="M3"><mi>I</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439308&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439295&type= 100.24533844 9.22866726 ， 其中： <math id="M4"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439301&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439299&type= 12.27666664 4.91066647 为背景光强， <math id="M5"><mi>r</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439303&type= 11.26066685 3.97933316 为干涉条纹的调制度， <math id="M6"><mi mathvariant="normal">Δ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439325&type= 4.23333359 3.64066648 为两束光的频差， <math id="M7"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439329&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439328&type= 6.01133299 4.65666676 为测量光与参考光的初始相位差， <math id="M8"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439348&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439346&type= 6.01133299 4.65666676 为干涉光路中产生的干涉相位差， <math id="M9"><mi mathvariant="normal">Δ</mi><mi>L</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439350&type= 4.91066647 2.79399991 为被测光与参考光的光程差，由 <math id="M10"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439360&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439369&type= 19.04999924 7.95866632 即可解算出被测面的相位信息。 3 混频原理 1 泰曼-格林型全视场外差动态干涉仪对光的偏振方向要求很高，光路中偏振方向相互垂直的两束光的频率不同。在干涉仪的实际装配过程中，反射镜偏转光束、元件安装位置偏差以及装调后元件松动都会影响光束的偏振方向，干涉仪中的PBS，QWP，偏振片等偏振元件的性能不理想也会影响光束的偏振方向，这些因素会导致两种频率的光相互混叠，产生寄生条纹，影响面形测量精度。因此，本文主要针对干涉仪系统中对光束偏振方向有影响的PBS、QWP以及偏振片，分析干涉仪系统中混频产生的原因以及这些元件对混频的影响。 PBS作为偏振分光器件，可以将入射光分为偏振方向相互垂直的s偏振光与p偏振光。在AOM的作用下，p偏振光的频率为 <math id="M11"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439717&type= 2.96333337 4.57200003 ，s偏振光的频率为 <math id="M12"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439383&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439366&type= 2.96333337 4.57200003 。理想情况下，两束光都是单一频率的光，然而PBS的偏振分光性能并不完全理想，这会导致原本为单一频率的光中混入其他频率的光，如 图2 图2 （a）所示。图中， x 轴表示测量光路， y 轴表示参考光路， x 轴上的 <math id="M13"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439388&type= 8.80533314 4.91066647 中混入了部分 <math id="M14"><mi>s</mi><mi mathvariant="normal">'</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439402&type= 8.55133343 5.07999992 ， y 轴上的 <math id="M15"><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439407&type= 7.87400007 4.91066647 中混入了部分 <math id="M16"><mi>p</mi><mi mathvariant="normal">'</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439412&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439421&type= 9.56733322 5.07999992 ，产生混频现象，混频程度受PBS消光比的影响。干涉仪系统中含有多个PBS，这些PBS分别起到合束与分束的作用，但由于PBS安装位置不理想，经过PBS的光束偏振方向不正交，如 图2 图2 （b）所示。 x 轴上的 <math id="M17"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439444&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439427&type= 18.11866570 4.91066647 中混入了部分 <math id="M18"><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439446&type= 16.51000023 4.91066647 ， y 轴上的 <math id="M19"><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439440&type= 17.18733406 4.91066647 中混入了部分 <math id="M20"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439457&type= 17.52599907 4.91066647 ，从而产生混频，混频程度受角 <math id="M21"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439467&type= 2.96333337 3.72533321 与 <math id="M22"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439470&type= 2.96333337 3.72533321 的影响。同理，当PBS2与PBS3的透光轴未重合，如 图2 图2 （c）所示， x 轴上的 <math id="M23"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439491&type= 17.01799965 4.91066647 中混入了部分 <math id="M24"><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439509&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439506&type= 15.40933418 4.91066647 ， y 轴上的 <math id="M25"><mi>s</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439502&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439512&type= 16.08666801 4.91066647 中混入了部分 <math id="M26"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439519&type= 16.34066582 4.91066647 ，也会引入混频，混频程度与两个PBS透光轴之间的夹角 <math id="M27"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439525&type= 1.86266661 2.79399991 有关。在干涉仪系统的实际装调过程中，PBS2与PBS3的透光轴方向很难保持一致，需要严格控制PBS的安装角度，以防引入过大的混频。 1;2;3; QWP在干涉测量模块中起改变光的偏振方向的作用，光束经过QWP一个来回，其偏振方向旋转90°。当测量光与参考光中都有混频时，若QWP的方位角与相位延迟量有误差，光经过QWP一个来回后，混频程度会被改变。同理，偏振片在系统中可以调整光的偏振方向，让测量光与参考光发生干涉，当混频存在时，偏振片的方位角有误差不仅影响两束光的强度也影响混频程度。 4 混频对面形测量精度的影响 1 4.1　误差模型推导 2 将含有混频的测量光与参考光分别表示为： <math display="block" id="M28"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">i</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">i</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439577&type= 59.94400024 26.41600037 <math display="block" id="M29"><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub><mo>=</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>=</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">i</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mfenced open="[" close="]" separators="|"><mrow><mi mathvariant="normal">i</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>t</mi><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>+</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439613&type= 60.02866364 26.41600037 其中： <math id="M30"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439598&type= 4.48733330 4.65666676 与 <math id="M31"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439620&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439619&type= 4.48733330 4.65666676 分别为测量光与参考光的初始相位， <math id="M32"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439622&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439603&type= 4.48733330 4.65666676 与 <math id="M33"><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439627&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439639&type= 4.48733330 4.65666676 分别为在测量光路与参考光路中产生的相位， <math id="M34"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439629&type= 3.30200005 3.72533321 与 <math id="M35"><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439647&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439645&type= 3.30200005 3.72533321 分别为被测面与参考面的面形信息， <math id="M36"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439650&type= 2.96333337 3.72533321 与 <math id="M37"><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439669&type= 2.96333337 3.72533321 分别为不同频率的测量光与参考光的波长。振幅为 <math id="M38"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439673&type= 3.04800010 3.72533321 、频率为 <math id="M39"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439676&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439664&type= 2.96333337 4.57200003 的测量光中混入了部分振幅为 <math id="M40"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439667&type= 3.04800010 3.72533321 、频率为 <math id="M41"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439681&type= 2.96333337 4.57200003 的参考光；振幅为 <math id="M42"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439701&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439689&type= 3.04800010 3.72533321 ，频率为 <math id="M43"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439707&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439704&type= 2.96333337 4.57200003 的参考光中混入了部分振幅为 <math id="M44"><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439711&type= 3.04800010 3.72533321 、频率为 <math id="M45"><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439691&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439717&type= 2.96333337 4.57200003 的测量光。 测量光与参考光在干涉光路经过偏振片后产生干涉，在探测器上的光强为： <math display="block" id="M46"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced><mfenced separators="|"><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>+</mo><msub><mrow><mi>E</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow></mfenced></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo> </mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mi>t</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mi>t</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mi>t</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced open="[" close="]" separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mfenced separators="|"><mrow><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mi>t</mi><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></mrow></mfenced><mo>-</mo><mfenced separators="|"><mrow><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></mrow></mfenced><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439697&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439694&type= 151.72265625 54.52533722 令 <math id="M47"><mi mathvariant="normal">Δ</mi><mi>f</mi><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>f</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439731&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439721&type= 17.78000069 4.57200003 为两束光的频差， <math id="M48"><mi mathvariant="normal">Δ</mi><mi>L</mi><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">t</mi></mrow></msub><mo>-</mo><msub><mrow><mi>L</mi></mrow><mrow><mi mathvariant="normal">r</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439734&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439732&type= 20.99733353 3.72533321 为激光经过被测面与参考面一个来回产生的光程差，测量光与参考光的初始相位差为 <math id="M49"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">01</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">02</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439738&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439743&type= 24.55333328 4.65666676 ，干涉成像光路中产生的干涉相位差为 <math id="M50"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439748&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439740&type= 24.55333328 4.65666676 。因为两束光的频差很小，所以 <math id="M51"><mi>λ</mi><mo>≈</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≈</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439751&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439749&type= 17.27199936 3.72533321 ，干涉光强可表示为： <math display="block" id="M52"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced separators="|"><mrow><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mfenced separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439753&type= 153.24665833 20.23533440 提取只含 <math id="M53"><mi mathvariant="normal">Δ</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439757&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439768&type= 4.23333359 3.64066648 的拍频项为： <math display="block" id="M54"><mtable><mtr><mtd><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mi mathvariant="normal">'</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfenced><mo> </mo><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439772&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439774&type= 64.00800323 26.92399979 利用辅助角公式： <math display="block" id="M55"><mi>a</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>x</mi><mo>=</mo><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mi>x</mi><mo>-</mo><mi>φ</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439791&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439788&type= 61.63733673 5.41866684 ， 其中： <math display="block" id="M56"><mtable><mtr><mtd><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>φ</mi><mo>=</mo><mfrac><mrow><mi>a</mi></mrow><mrow><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>φ</mi><mo>=</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></mrow></mfrac><mo>,</mo></mtd></mtr><mtr><mtd><mi>φ</mi><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>b</mi></mrow><mrow><mi>a</mi></mrow></mfrac><mo>.</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439779&type= 29.80266571 30.64933395 可得干涉拍频项为： <math display="block" id="M57"><msub><mrow><mi>I</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mi mathvariant="normal">'</mi><mo>=</mo><mi>A</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>f</mi><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mo>-</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>ε</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439796&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439794&type= 73.32133484 9.22866726 ， 式中： <math display="block" id="M58"><mi>A</mi><mo>=</mo><mroot><mrow><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo> </mo><msup><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow/></mroot></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439802&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439799&type= 24.12999916 4.40266657 ， <math display="block" id="M59"><msub><mrow><mi>φ</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo> </mo><mo>=</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mi>D</mi></mrow><mrow><mi>C</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439805&type= 22.18266678 7.95866632 ， 其中： <math display="block" id="M60"><mtable><mtr><mtd><mi>C</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo> </mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439812&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439810&type= 61.38333893 20.23533440 <math display="block" id="M61"><mtable><mtr><mtd><mi>D</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow></mfenced><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></mrow></mfenced><mo>,</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439820&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439814&type= 61.21400070 20.23533440 式中： <math id="M62"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi><mi mathvariant="normal">Δ</mi><mi>L</mi></mrow><mrow><mi>λ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439833&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439822&type= 18.88066673 7.95866632 ，为被测相位差， <math id="M63"><msub><mrow><mi>φ</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439839&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439836&type= 3.30200005 4.65666676 为混频引入的相位误差。用混频光与测量光和参考光的光强比值表示混频程度， <math id="M64"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 为测量光的混频程度， <math id="M65"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 为参考光的混频程度，分别为： <math display="block" id="M66"><mi>α</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><msup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439850&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439830&type= 27.09333420 8.80533314 ， <math display="block" id="M67"><mi>β</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msup><mrow><msub><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439853&type= 27.34733391 8.80533314 . 所以混频引入的测量误差 <math id="M68"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 可表示为： <math display="block" id="M69"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>λ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mi>ε</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">π</mi></mrow></mfrac><mi mathvariant="normal">a</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mfrac><mrow><mo stretchy="false">(</mo><mroot><mrow><mi>α</mi></mrow><mrow/></mroot><mo>+</mo><mroot><mrow><mi>β</mi></mrow><mrow/></mroot><mo stretchy="false">)</mo><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfenced><mo>+</mo><mroot><mrow><mi>α</mi><mi>β</mi></mrow><mrow/></mroot><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfenced></mrow><mrow><mn mathvariant="normal">1</mn><mo>+</mo><mo stretchy="false">(</mo><mroot><mrow><mi>α</mi></mrow><mrow/></mroot><mo>+</mo><mroot><mrow><mi>β</mi></mrow><mrow/></mroot><mo stretchy="false">)</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfenced><mo>+</mo><mroot><mrow><mi>α</mi><mi>β</mi></mrow><mrow/></mroot><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mfenced separators="|"><mrow><mn mathvariant="normal">2</mn><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mn mathvariant="normal">2</mn><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></mrow></mfenced></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439868&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439860&type= 127.76200104 11.85333347 . 4.2　仿真分析 2 从 式（15） 式（15） 中可以看出，测量误差 <math id="M70"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 与混频程度 <math id="M71"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 ， <math id="M72"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 ，干涉相位差 <math id="M73"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440121&type= 6.01133299 4.65666676 以及被测相位差 <math id="M74"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439922&type= 4.65666676 3.72533321 有关。由此可以得到： （1）混频会在测量面形结果上叠加一个和干涉条纹相同频率的周期性误差。 （2）当 <math id="M75"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439890&type= 12.69999981 4.65666676 ，测量误差 <math id="M76"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 受混频程度 <math id="M77"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 与 <math id="M78"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 的影响， <math id="M79"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 与 <math id="M80"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 对 <math id="M81"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439934&type= 5.84200001 3.72533321 共同作用。 <math id="M82"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439922&type= 4.65666676 3.72533321 在两个周期内变化时，令 <math id="M83"><mi>α</mi><mo>=</mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440213&type= 8.80533314 3.64066648 ，其变化范围为0~0.4，仿真结果如 图3 图3 所示。从图中可以看出，在 <math id="M84"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 与 <math id="M85"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 的共同作用下， <math id="M86"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439948&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439960&type= 1.94733346 2.79399991 越大， <math id="M87"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 越大， <math id="M88"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 随 <math id="M89"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440075&type= 4.65666676 3.72533321 周期性变化。最大测量误差 <math id="M90"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 与 <math id="M91"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 呈非线性正相关，当 <math id="M92"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 在0到0.25内变化时， <math id="M93"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 的最大值为 <math id="M94"><mn mathvariant="normal">0.167</mn><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439996&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43439994&type= 11.17599964 2.79399991 ；当 <math id="M95"><mi>α</mi><mo>=</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0.096</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440106&type= 23.02933502 3.64066648 时， <math id="M96"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 的最大值超过 <math id="M97"><mn mathvariant="normal">0.1</mn><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440111&type= 7.45066643 2.79399991 。 1;2; （3）当 <math id="M98"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440029&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440023&type= 12.69999981 4.74133301 时，令 <math id="M99"><mi>α</mi><mo>=</mo><mi>β</mi><mo>=</mo><mn mathvariant="normal">0.1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440026&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440025&type= 19.30400085 3.64066648 ，取 <math id="M100"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440121&type= 6.01133299 4.65666676 在0~ <math id="M101"><mn mathvariant="normal">3</mn><mi mathvariant="normal">π</mi><mo>/</mo><mn mathvariant="normal">2</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440044&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440033&type= 7.45066643 3.30200005 ，可得 <math id="M102"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440121&type= 6.01133299 4.65666676 不同时 <math id="M103"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 与 <math id="M104"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440075&type= 4.65666676 3.72533321 的仿真曲线，如 图4 图4 所示。从 图4 图4 中可以看出， <math id="M105"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440121&type= 6.01133299 4.65666676 对 <math id="M106"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 与 <math id="M107"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440075&type= 4.65666676 3.72533321 的关系曲线造成一个左右的平移，会影响 <math id="M108"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 取到极值时 <math id="M109"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440076&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440075&type= 4.65666676 3.72533321 的值，即 <math id="M110"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440121&type= 6.01133299 4.65666676 只影响 <math id="M111"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 极值点的位置，而不会影响 <math id="M112"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 的值。 综上所述，混频会在测量面形结果上叠加一个和干涉条纹相同频率的周期性误差；测量误差 <math id="M113"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 主要受混频程度 <math id="M114"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 与 <math id="M115"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 的影响， <math id="M116"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440096&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440095&type= 1.94733346 2.79399991 ， <math id="M117"><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440087&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440098&type= 2.28600001 3.64066648 与最大测量误差 <math id="M118"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 呈非线性正相关，当 <math id="M119"><mi>α</mi><mo>=</mo><mi>β</mi><mo>></mo><mn mathvariant="normal">0.096</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440108&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440106&type= 23.02933502 3.64066648 时， <math id="M120"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 的最大值超过 <math id="M121"><mn mathvariant="normal">0.1</mn><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440112&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440111&type= 7.45066643 2.79399991 ；干涉相位差 <math id="M122"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>φ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440122&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440121&type= 6.01133299 4.65666676 会影响 <math id="M123"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 取到极值时的 <math id="M124"><msub><mrow><mi>Φ</mi></mrow><mrow><mi mathvariant="normal">m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440126&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440133&type= 4.65666676 3.72533321 ，不会影响 <math id="M125"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 。 5 实验与结果分析 1 为了验证不同程度混频对面形测量精度的影响，本文搭建了全视场外差动态干涉仪实验系统，如 图5 图5 所示。 干涉仪系统采用波长为633 nm的窄线宽激光器作为光源，激光通过偏振片校准偏振方向以及衰减器减弱光强后，由PBS1分束进入移频部分，两个声光移频器（AOM1与AOM2）的差频为5 Hz，HWP2与HWP3将通过AOM1与AOM2的光束偏振方向与PBS2的透光轴方向对准，两光束经过空间滤波器滤波及准直镜准直后通过内径为9 mm的软边光阑，软边光阑可以抑制光束透过光阑后，自由传播时产生的以轴上“亮点”为中心的菲涅尔衍射调制环，得到近乎平顶的超高斯分布 ［ 13 13 ］ 。在干涉测量模块中，正交双频光束分别进入测量光路与参考光路，之后分别携带面形精度为 λ /10的被测镜与参考镜的面形信息发生干涉，再通过帧频为20 frame/s、分辨率为1 200 pixel×1 200 pixel的CMOS相机采集5张干涉图，如 图6 图6 所示，掩模直径为1 000 pixel。采集数据后用四步相移法解算掩模区域的包裹相位，相位解包裹后，用Zernike多项式拟合去除平移、倾斜以及离焦项，得到被测镜的面形信息。 在PBS3与测量光路和参考光路中的QWP之间放置Thorlabs的PM100D型功率计，功率计探头为S130C型，测量精度为0.5%。先遮挡通过AOM2的光束，记录此时测量光路与参考光路的光强分别为 <math id="M126"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440158&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440145&type= 2.53999996 3.72533321 ， <math id="M127"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440163&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440160&type= 2.53999996 3.72533321 ，再遮挡通过AOM1的光束，记录此时测量光路与参考光路的光强分别为 <math id="M128"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440178&type= 2.53999996 3.72533321 ， <math id="M129"><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440187&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440166&type= 2.53999996 3.72533321 ，此时测量光与参考光的混频程度可分别表示为 <math id="M130"><mi>α</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mo>/</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440170&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440188&type= 13.54666710 4.14866638 ， <math id="M131"><mi>β</mi><mo>=</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>/</mo><msub><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440173&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440190&type= 13.80066681 4.82600021 。在软边光阑后插入一个HWP4，通过旋转HWP4改变入射光束的偏振方向与PBS3透光轴的夹角，从而改变混频量。在混频程度为0时，测量10次得到10组面形，计算平均面形作为基准面形。将不同混频下的平均面形减去基准面形得到误差面形，计算得到误差面形的峰谷值（Peak Value， PV），即实验所得测量误差 <math id="M132"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440197&type= 6.77333355 4.23333359 ，再将各个混频下的测量误差 <math id="M133"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440209&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440197&type= 6.77333355 4.23333359 与理论误差 <math id="M134"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440210&type= 5.84200001 3.72533321 相比较。 旋转半波片调节混频程度（ <math id="M135"><mi>α</mi><mo>=</mo><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440214&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440213&type= 8.80533314 3.64066648 ）为0，在实际测量过程中，无法完全消除混频，将混频程度为0.004近似为0时测量计算的平均面形作为基准面形，如 图7 图7 所示。然后，分别调整混频程度为0.029，0.060，0.097，0.120，0.157，0.180，0.207，测量得到不同混频程度下的平均面形与误差面形，如 图8 图8 所示。再将不同混频程度下的测量误差 <math id="M136"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440204&type= 6.77333355 4.23333359 和仿真的理论误差 <math id="M137"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440227&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440224&type= 5.84200001 3.72533321 相比较，结果如 表1 表1 所示。 1;2;3;4;5;6;7; Frequency mixing degree（ α = β ） <math id="M138"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440303&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440300&type= 4.99533367 3.21733332 /nm <math id="M139"><mi mathvariant="normal">Δ</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>ε</mi></mrow></msub><mi mathvariant="normal">'</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440281&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=43440305&type= 5.84200001 3.55599999 /nm 0.029 0.055 λ 0.053 λ 0.060 0.079 λ 0.076 λ 0.097 0.101 λ 0.096 λ 0.120 0.113 λ 0.110 λ 0.157 0.130 λ 0.134 λ 0.180 0.140 λ 0.141 λ 0.207 0.150 λ 0.152 λ 实验得到的测量误差拟合曲线与理论误差拟合曲线相比较，如 图9 图9 所示。从上述实验结果中可以看出，混频会在基础面形上叠加一个与干涉条纹相同频率的周期性误差，混频越大，面形测量误差越大，且不同混频程度的实验结果与理论仿真结果一致，证明混频程度与测量误差呈非线性正相关。 6 结 论 1 本文分析了全视场外差动态干涉仪中混频的产生原因，干涉仪中的PBS安装位置不理想时会引入混频，QWP的方位角与相位延迟量有误差以及偏振片的方位角有误差时会对干涉仪系统中的混频程度产生影响。然后，建立了由混频引起的测量误差理论模型，测量误差随被测相位差成周期性变化，且测量误差与混频程度呈非线性正相关。当混频程度超过0.096时，测量误差大于0.1λ。最后，搭建实验验证了不同混频程度对面形测量精度的影响，当混频为0.029时，测量误差为0.053λ，当混频为0.120时，测量误差为0.110λ，与理论仿真结果相符合。 对全视场外差动态干涉仪来说，混频越大，对其面形测量精度影响越大。结合混频的产生原因，为了降低混频的影响，在干涉仪实际装调过程中需严格控制PBS，QWP以及偏振片的安装位置，并选用性能好的光学元件以满足干涉仪的测量精度要求。本文的研究对全视场外差动态干涉仪的研制具有实际应用价值，对提高大口径光学镜面以及长焦光学元件的面形检测精度具有重要意义。 参考文献 MALACARA D . 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YANG Yuqi,LI Yang,LÜ Tong,et al.Influence of frequency mixing on measurement accuracy of full-field heterodyne dynamic interferometer[J].Optics and Precision Engineering,2023,31(09):1267-1276. 中国科学院战略性先导科技专项（No.XDC07040300） 2022-12-08 2023-01-05 2023-05-10 2023-05-15 37008322 光学精密工程 9 31