CN102621072A

CN102621072A - Polarization and birefringence measurement system

Info

Publication number: CN102621072A
Application number: CN2012100881880A
Authority: CN
Inventors: 范真节; 林妩媚; 邢廷文; 刘学峰
Original assignee: Institute of Optics and Electronics of CAS
Current assignee: Institute of Optics and Electronics of CAS
Priority date: 2012-03-29
Filing date: 2012-03-29
Publication date: 2012-08-01
Anticipated expiration: 2032-03-29
Also published as: CN102621072B

Abstract

The invention provides a polarization and birefringence measurement system, which improves the measurement accuracy of polarization and stress birefringence by adopting a multi-wave plate combination, wherein a light source module: the device comprises a light source, a collimation and beam expansion system, a polarizer and a sample to be detected. The optical signal modulation module: the device comprises a multi-stage phase difference coaxial wave plate, an analyzer, a stepping motor driving card and a stepping motor. The data acquisition and processing module: comprises an image acquisition card, a light intensity detector and a computer. The invention well avoids the influence caused by environment and measurement condition changes such as air flow, temperature change and some uncertain vibration modes. The system uses more than one independent phase modulation element to suppress noise, averaging the major error sources. The system adopts three wave plates to measure polarization and birefringence, well avoids nonlinear errors caused by nonlinear influence of the system and greatly improves the measurement precision.

Description

A polarization and birefringence measurement system

技术领域 technical field

本发明属于现代光学测试技术领域，具体涉及一种偏振和双折射测量系统，其对偏振和双折射分布进行精确测量。The invention belongs to the technical field of modern optical testing, in particular to a polarization and birefringence measurement system, which can accurately measure the distribution of polarization and birefringence.

背景技术 Background technique

当光刻投影物镜NA＞0.8时，光束的偏振特性会严重影响成像质量，特别是p偏振会降低成像对比度，影响系统分辨率。另外，偏振光的点扩散函数不对称，光强分布沿着偏振方向扩散。对于线条图形，当偏振方向与线条不平行时，光强透过率较高，有利于成像。因此，根据掩模图形，采用相应的偏振光束配合离轴照明，能进一步提高系统的分辨率和成像质量。When the NA of the lithography projection objective lens is greater than 0.8, the polarization characteristics of the beam will seriously affect the imaging quality, especially the p-polarization will reduce the imaging contrast and affect the system resolution. In addition, the point spread function of polarized light is asymmetric, and the light intensity distribution spreads along the polarization direction. For line graphics, when the polarization direction is not parallel to the line, the light intensity transmittance is higher, which is conducive to imaging. Therefore, according to the mask pattern, the resolution and imaging quality of the system can be further improved by using the corresponding polarized light beam with off-axis illumination.

同时，增大数值孔径将能够捕获更多的衍射级数和更大角度的光线，这无疑加剧了偏振效应的负面影响。由于大角度光线的互相影响以及干涉效应，导致空间图像对比度被大为削弱。因此光刻设备的光学部件必须使用低双折射的石英材料，因为光学镜头对双折射现象非常敏感而双折射现象又与偏振效应紧密联系。At the same time, increasing the numerical aperture will be able to capture more diffraction orders and larger angles of light, which undoubtedly exacerbates the negative impact of polarization effects. Due to the mutual influence and interference effect of light at a large angle, the contrast of the spatial image is greatly weakened. Therefore, the optical components of lithography equipment must use low-birefringence quartz materials, because optical lenses are very sensitive to birefringence and birefringence is closely related to polarization effects.

浸没式光刻技术导致数值孔径激增从而引发偏振效应，在193nm ArF浸没式光刻机中不得不使用低双折射的石英材料，因此必须对石英材料中的应力双折射分布进行测量，获取数据代入仿真以保证光刻机的顺利研制。同时193nm ArF的浸没式光刻机中相当部分熔石英透镜尺寸直径上超过300mm，材料的双折射测量采用采样点的测量方式进行，要保证测量精度需要尽可能多的采样点。经过仿真分析，由于材料的双折射效应可能引起4～6nm波像差变化，将严重影响193nm曝光光学系统的性能，所以必须对材料的偏振效应和双折射分布进行精确地测量，然而材料的偏振和双折射模式和分布的快速精确测量在空间和时间上是十分困难的，但对于193nm ArF浸没式光刻机的研制却是必须的。Immersion lithography technology leads to a sharp increase in numerical aperture, which causes polarization effects. In the 193nm ArF immersion lithography machine, low birefringence quartz materials have to be used. Therefore, it is necessary to measure the stress birefringence distribution in the quartz material and obtain data into Simulation to ensure the smooth development of the lithography machine. At the same time, in the 193nm ArF immersion lithography machine, a considerable part of the fused silica lens has a diameter exceeding 300mm. The birefringence measurement of the material is carried out by sampling point measurement. To ensure the measurement accuracy, as many sampling points as possible are required. After simulation analysis, because the birefringence effect of the material may cause 4-6nm wave aberration changes, which will seriously affect the performance of the 193nm exposure optical system, it is necessary to accurately measure the polarization effect and birefringence distribution of the material. However, the polarization effect of the material Fast and accurate measurement of birefringence modes and distributions is very difficult in space and time, but it is necessary for the development of 193nm ArF immersion lithography machine.

由于波像差质量的可靠性与精密光学成像系统如线宽90纳米的光刻曝光系统息息相关，对于线宽为22纳米甚至16纳米的曝光系统，光刻投影物镜的数值孔径大于1，为了满足高分辨率的要求，波长非常短，例如现在是193纳米，随着波长的缩短，散射的程度相当严重，测量系统的稳定性和测量结果的可重复性是我们面临的一个巨大挑战。所以，发展一种具有更高精度、更稳定、具有更高可重复性的偏振双折射测量系统才能满足工程研究领域的需求。Since the reliability of wave aberration quality is closely related to precision optical imaging systems such as lithography exposure systems with a line width of 90 nm, for exposure systems with a line width of 22 nm or even 16 nm, the numerical aperture of the lithography projection objective lens is greater than 1, in order to meet High resolution requires very short wavelengths, such as 193 nanometers now. As the wavelength shortens, the degree of scattering is quite serious. The stability of the measurement system and the repeatability of the measurement results are a huge challenge we face. Therefore, the development of a polarization birefringence measurement system with higher precision, stability and repeatability can meet the needs of engineering research.

综上所述，必须探索一种新的偏振和双折射测定方法，建立一套能对材料的偏振效应和双折射分布进行精确测量的系统，即偏振和双折射多级相差共轴旋转精密测量系统，对光刻曝光系统中材料的偏振效应和双折射分布进行准确测量，以提高系统的分辨率和成像质量。To sum up, it is necessary to explore a new method of polarization and birefringence measurement, and establish a system that can accurately measure the polarization effect and birefringence distribution of materials, that is, the polarization and birefringence multi-level phase difference coaxial rotation precision measurement The system can accurately measure the polarization effect and birefringence distribution of materials in the lithography exposure system to improve the resolution and imaging quality of the system.

另外，该测量系统还可以用于宝石、晶体鉴定以及医药方面等等，总之，随着光刻技术的向前发展，需要曝出的线条宽度越来越细，对材料的偏振效应和双折射效应要求越来越高，建立一套能精确测定材料偏振效应和双折射分布的光学系统变得十分迫切。In addition, the measurement system can also be used in gemstones, crystal identification, and medicine, etc. In short, with the development of lithography technology, the width of the lines that need to be exposed is getting thinner and thinner, and the polarization effect and birefringence of materials The effect requirements are getting higher and higher, and it is very urgent to establish an optical system that can accurately measure the polarization effect and birefringence distribution of materials.

发明内容 Contents of the invention

本发明针对偏振和双折射效应进行精确测量，并对测量系统中光学元件进行物理模型建立。提供一种偏振和双折射测量系统，其核心部件为多级相差共轴波片，通过旋转多级相差共轴波片对光进行模式化，得到旋转角度和测量系统光强间对应关系，通过数学运算精确解析出被测面积上经过模式化的偏振和应力双折射分布。通过采用多波片组合以提高偏振和应力双折射的测量精度。The invention carries out precise measurement for polarization and birefringence effects, and establishes physical models for optical elements in the measurement system. A polarization and birefringence measurement system is provided, the core component of which is a multi-stage phase-contrast coaxial wave plate. By rotating the multi-stage phase-contrast coaxial wave plate to model the light, the corresponding relationship between the rotation angle and the light intensity of the measurement system is obtained. Through Mathematical operations precisely resolve the patterned polarization and stress birefringence distributions across the measured area. The measurement accuracy of polarization and stress birefringence is improved by using multi-wave plate combination.

本发明的技术方案为：一种偏振和双折射测量系统，主要包括光源模块、光信号调制模块和数据采集处理模块，其中光源模块包括光源、准直扩束系统，起偏器和待测样品；光信号调制模块包括多级相差共轴波片、检偏器和步进电机；数据采集处理模块包括光强探测器和计算机；波长为193.368nm的ArF激光器光源发出的光经过准直扩束系统后会聚光束变为远心光束，并入射到起偏器，远心光束经起偏器后变为偏振光，偏振光入射到待测样品上，光束通过待测样品后进入光信号调制模块，光束经多级相差共轴波片和检偏器后入射到光强探测器上，检偏器与起偏器指标性能匹配；多级相差共轴波片是由尺寸不等的多个波片构成的圆盘，多级相差共轴波片由步进电机驱动，通过驱动其旋转实现对光强的调制；光强探测器获取的光强信息经数据采集卡将数据存入计算机，并通过计算机的运算模块进行偏振和双折射数据参数的运算并储存，同时计算机通过D/A板给步进电机发送驱动信号以控制步进电机驱动的起止时间和旋转角度，步进电机带动多级相差共轴波片旋转，旋转角度值可通过计算机进行控制，光强探测器获取不同方位角时的强度信息，利用计算机对获取的强度信息进行处理和数学解析，得到光学材料的偏振和双折射模式和分布。The technical solution of the present invention is: a polarization and birefringence measurement system, which mainly includes a light source module, an optical signal modulation module and a data acquisition and processing module, wherein the light source module includes a light source, a collimator beam expander system, a polarizer and a sample to be tested ; The optical signal modulation module includes a multi-stage phase difference coaxial wave plate, a polarizer and a stepping motor; the data acquisition and processing module includes a light intensity detector and a computer; the light emitted by the ArF laser light source with a wavelength of 193.368nm is collimated and expanded After the system, the converging beam becomes a telecentric beam and enters the polarizer. The telecentric beam passes through the polarizer and becomes polarized light. The polarized light is incident on the sample to be tested, and the beam enters the optical signal modulation module after passing through the sample to be tested. , the light beam is incident on the light intensity detector after passing through the multi-stage phase difference coaxial wave plate and the analyzer, and the performance of the polarizer and the polarizer are matched; the multi-stage phase difference coaxial wave plate is composed of multiple wave plates of different sizes The multi-level phase difference coaxial wave plate is driven by a stepping motor, and the light intensity is modulated by driving its rotation; the light intensity information obtained by the light intensity detector is stored in the computer through the data acquisition card, and Calculate and store polarization and birefringence data parameters through the computing module of the computer. At the same time, the computer sends a driving signal to the stepping motor through the D/A board to control the start and stop time and rotation angle of the stepping motor. The stepping motor drives the multi-stage The phase difference coaxial wave plate rotates, and the rotation angle value can be controlled by a computer. The light intensity detector obtains the intensity information at different azimuth angles, and the computer processes and mathematically analyzes the obtained intensity information to obtain the polarization and birefringence of the optical material. mode and distribution.

本发明与现有技术相比具有以下优点：Compared with the prior art, the present invention has the following advantages:

1、该系统采用共光路原理，很好的避免了环境和测量条件变化如空气流动，温度变化以及某些不确定的振动方式带来的影响。1. The system adopts the principle of common optical path, which well avoids the influence of changes in the environment and measurement conditions such as air flow, temperature changes and certain uncertain vibration modes.

2、该系统使用超过一个独立的位相调制元件来抑制噪声，平均主要误差源。2. The system uses more than one independent phase modulation element to suppress noise and average out major error sources.

3、该系统采用三个波片来对偏振和双折射进行测量，很好地避免了由于系统的非线性影响引起的非线性误差，大大提高了测量精度。3. The system uses three wave plates to measure polarization and birefringence, which avoids the nonlinear error caused by the nonlinear influence of the system and greatly improves the measurement accuracy.

附图说明 Description of drawings

图1为一种偏振和双折射测量系统测量原理示意图；Fig. 1 is a schematic diagram of the measurement principle of a polarization and birefringence measurement system;

图2为多级相差共轴波片示意图；Figure 2 is a schematic diagram of a multi-stage phase difference coaxial wave plate;

图3为偏振和双折射测量系统数学解析示意图；Fig. 3 is the mathematical analysis schematic diagram of polarization and birefringence measurement system;

图中标号说明：1-光源、2-准直扩束系统、3-起偏器、4-待测样品、5-多级相差共轴波片、6-检偏器、7-光强探测器、8-计算机、9-步进电机、10-光源模块、11-光信号调制模块、12-数据采集处理模块。Explanation of symbols in the figure: 1-light source, 2-collimator beam expander system, 3-polarizer, 4-sample to be tested, 5-multi-stage phase difference coaxial wave plate, 6-analyzer, 7-light intensity detection Device, 8-computer, 9-stepper motor, 10-light source module, 11-optical signal modulation module, 12-data acquisition and processing module.

具体实施方式 Detailed ways

为了更好地说明本发明的目的和优点，下面结合附图对本发明作进一步说明。In order to better illustrate the purpose and advantages of the present invention, the present invention will be further described below in conjunction with the accompanying drawings.

图1为一种偏振和双折射测量系统测量原理示意图，包括光源模块10、光信号调制模块11和数据采集处理模块12，所述光源模块10包括光源1、准直扩束系统2、起偏器3、待测样品4；光信号调制模块11包括多级相差共轴波片5、检偏器6；数据采集处理模块12包括光强探测器7、计算机8和步进电机9。如图1所示，光源1为波长为193.368nm的ArF激光器，ArF激光器发出的光经过准直扩束系统2后会聚光束变为远心光束，并入射到起偏器3，光束经起偏器3后变为偏振光，偏振光入射到待测样品4上，光束透过待测样品4后进入光信号调制模块11，光束经多级相差共轴波片5和检偏器6入射到光强探测器7上，检偏器6与起偏器3指标性能匹配。多级相差共轴波片5由尺寸不等的多个波片构成的圆盘，多级相差共轴波片5由高精度步进电机9驱动，通过驱动其旋转实现对光强的调制。光强探测器7获取的光强信息经数据采集卡将数据存入计算机8，并通过计算机8的运算模块进行偏振和双折射数据参数的运算并储存，并可以同时在显示器上显示光斑图像，同时计算机8通过D/A板给步进电机9发送驱动信号以控制步进电机9的驱动的起止时间和旋转角度，步进电机9带动多级相差共轴波片5旋转，旋转角度值可通过计算机进行控制，光强探测器7获取不同方位角时的强度信息，利用计算机8对获取的强度信息进行处理和数学解析，得到光学材料的偏振和双折射模式和分布。Fig. 1 is a schematic diagram of the measurement principle of a polarization and birefringence measurement system, including a light source module 10, an optical signal modulation module 11, and a data acquisition and processing module 12. The light source module 10 includes a light source 1, a collimating beam expander system 2, a polarizer The optical signal modulation module 11 includes a multi-stage phase difference coaxial wave plate 5 and a polarizer 6 ; the data acquisition and processing module 12 includes a light intensity detector 7 , a computer 8 and a stepping motor 9 . As shown in Figure 1, the light source 1 is an ArF laser with a wavelength of 193.368nm. The light emitted by the ArF laser passes through the collimated beam expander system 2, and the convergent beam becomes a telecentric beam, and enters the polarizer 3. The beam is polarized The polarized light is transformed into polarized light after the detector 3, and the polarized light is incident on the sample 4 to be tested, and the beam enters the optical signal modulation module 11 after passing through the sample 4 to be tested, and the beam is incident on the On the light intensity detector 7, the index performance of the analyzer 6 and the polarizer 3 is matched. The multi-stage phase-difference coaxial wave plate 5 is a disk composed of multiple wave plates of different sizes. The multi-stage phase-difference coaxial wave plate 5 is driven by a high-precision stepping motor 9, and the light intensity can be modulated by driving its rotation. The light intensity information obtained by the light intensity detector 7 is stored in the computer 8 through the data acquisition card, and the calculation module of the computer 8 is used to calculate and store the polarization and birefringence data parameters, and the spot image can be displayed on the display at the same time. Simultaneously computer 8 sends drive signal to stepper motor 9 by D/A board to control the start-stop time and the angle of rotation of the drive of stepper motor 9, stepper motor 9 drives multi-stage phase difference coaxial wave plate 5 to rotate, and the value of angle of rotation can be Controlled by a computer, the light intensity detector 7 acquires intensity information at different azimuth angles, and the acquired intensity information is processed and mathematically analyzed by a computer 8 to obtain polarization and birefringence modes and distributions of optical materials.

多级相差共轴波片5设计技巧：(1)如图2所示，在一个圆盘内，三种波片必须具有高度的平行性，当光束通过旋转的波片时，它们的光轴必须保持相同。否则，就必须引入空间方位角，从而使计算和数据处理变得复杂化，甚至带来不可控制的错误；(2)关于三种波片所在一个圆周中所占角度，现给出一种最优解，1/4波片1/2圈，1/2波片1/4圈，3/4波片1/4圈；(3)每种波片的连接必须准确无误，每个连接处必须紧密成一线，否则，在测量时，当步进电机转至连接处时，可能没有光通过。Multi-stage phase difference coaxial wave plate 5 Design skills: (1) As shown in Figure 2, in a disc, the three wave plates must have a high degree of parallelism. When the beam passes through the rotating wave plate, their optical axes must remain the same. Otherwise, it is necessary to introduce the spatial azimuth angle, which complicates the calculation and data processing, and even brings uncontrollable errors; (2) Regarding the angles occupied by the three wave plates in a circle, an optimal Excellent solution, 1/4 wave plate 1/2 turn, 1/2 wave plate 1/4 turn, 3/4 wave plate 1/4 turn; (3) The connection of each wave plate must be accurate, each connection It must be in a tight line, otherwise, when measuring, no light may pass through when the stepper motor is turned to the connection.

三种波片在圆周中所占角度分配方法：对于入射的线偏振光，经过1/4波片，将1/4波片旋转一周，位相改变4π；经过1/2波片，将1/2波片旋转一周，位相改变8π；经过3/4波片，将3/4波片旋转一周，位相改变12π。因此，1/4波片旋转半圈，位相改变一个周期2π；1/2波片旋转1/4圈，位相改变一个周期2π；3/4波片旋转1/4圈，位相改变1.5个周期3π。依此类推，还可以选用别的波片组合：如可以选用1/4波片1/2圈，1/3波片3/8圈，5/12波片1/8圈；The distribution method of the angle occupied by the three wave plates in the circumference: for the incident linearly polarized light, after passing through the 1/4 wave plate, the 1/4 wave plate is rotated once, and the phase changes by 4π; after passing through the 1/2 wave plate, the 1/4 When the 2-wave plate rotates once, the phase changes by 8π; when the 3/4 wave plate rotates once, the phase changes by 12π. Therefore, if the 1/4 wave plate rotates half a circle, the phase changes by 2π for one period; for the 1/2 wave plate rotates for 1/4 circle, the phase changes for one period for 2π; for the 3/4 wave plate rotates for 1/4 circle, the phase changes for 1.5 periods. 3π. And so on, you can also choose other wave plate combinations: for example, you can choose 1/4 wave plate 1/2 circle, 1/3 wave plate 3/8 circle, 5/12 wave plate 1/8 circle;

图3为偏振和双折射测量系统数学解析示意图，主要包括以下几个模块：光源模块10、光信号调制模块11和数据采集处理模块12。FIG. 3 is a schematic diagram of the mathematical analysis of the polarization and birefringence measurement system, which mainly includes the following modules: a light source module 10 , an optical signal modulation module 11 and a data acquisition and processing module 12 .

光源模块10：包括光源1和准直扩束系统2，起偏器3，待测样品4。Light source module 10 : including a light source 1 , a collimator beam expander system 2 , a polarizer 3 , and a sample 4 to be tested.

光信号调制模块11：包括多级相差共轴波片5、检偏器6。该模块的核心部分是多级相差共轴波片5，它是把1/4波片、1/2波片、3/4波片嵌于同一圆盘上，必须精确地控制步进角度而旋转。在波片前面，一个标准的准直扩束光学系统2使会聚光束变为远心光束，通过起偏器3和待测样品4后照射在多级相差共轴波片5上；在多级相差共轴波片5后面，放置一个与起偏器3指标性能一样的检偏器6。Optical signal modulation module 11 : including a multi-stage phase difference coaxial wave plate 5 and a polarizer 6 . The core part of this module is the multi-stage phase difference coaxial wave plate 5, which is to embed the 1/4 wave plate, 1/2 wave plate, and 3/4 wave plate on the same disc, and the step angle must be precisely controlled. rotate. In front of the wave plate, a standard collimating beam expander optical system 2 makes the converging beam into a telecentric beam, which passes through the polarizer 3 and the sample to be measured 4 and then irradiates on the multi-stage phase difference coaxial wave plate 5; Behind the phase difference coaxial wave plate 5, an analyzer 6 with the same index performance as the polarizer 3 is placed.

为了精确地测量偏振/双折射，光源模块10成为了待测系统，输出的位相和偏振态尽可能精确地测量。由于一些不确定因素会影响到测量的精度：特定方向上的偏振度，起偏角，两偏振矢量之间的滞后角等，所以采用多个波片来进行测量，以减小误差从而大大提高了测量精度。In order to accurately measure polarization/birefringence, the light source module 10 becomes the system under test, and the phase and polarization state of the output are measured as precisely as possible. Since some uncertain factors will affect the accuracy of the measurement: the degree of polarization in a specific direction, the angle of polarization, the lag angle between the two polarization vectors, etc., so multiple wave plates are used for measurement to reduce errors and greatly improve measurement accuracy.

数据采集处理模块12：包括光强探测器7、计算机8和步进电机9。Data collection and processing module 12 : includes a light intensity detector 7 , a computer 8 and a stepper motor 9 .

由激光光源发出的单色性好的光经过准直扩束系统2后，再经过起偏器3，变为偏振光，经过待测样品4，再经过多级相差共轴波片5，当用步进电机9控制多级相差共轴波片5旋转任意角度时，经过多级相差共轴波片5调制后的偏振光经过检偏器6到达光强探测器7，从光强探测器7上可以读出此时的透射光强，以相同的角速度转动波片，从而得到多组光强值，光强信息经图像采集卡将数据存入计算机8并储存，主要通过D/A板给步进电机9发送驱动信号，从而带动多级相差共轴波片5旋转，旋转角度由计算机8控制，光强探测器7摄取不同方位角时的强度信息，然后通过计算机8内的运算模块对图像进行数据处理，便求得了待测样品4的斯托克斯矢量和偏振度，通过计算机8可以直接读取的二者的测试结果。The light with good monochromaticity emitted by the laser light source passes through the collimator beam expander system 2, then passes through the polarizer 3, becomes polarized light, passes through the sample to be tested 4, and then passes through the multi-stage phase difference coaxial wave plate 5, when When the stepping motor 9 is used to control the rotation of the multi-stage phase-difference coaxial wave plate 5 at any angle, the polarized light modulated by the multi-stage phase-difference coaxial wave plate 5 reaches the light intensity detector 7 through the analyzer 6, and then passes through the light intensity detector 7. 7 can read the transmitted light intensity at this time, and rotate the wave plate at the same angular velocity to obtain multiple sets of light intensity values. The light intensity information is stored in the computer 8 through the image acquisition card and stored, mainly through the D/A board. Send a drive signal to the stepper motor 9, thereby driving the multi-stage phase difference coaxial wave plate 5 to rotate, the rotation angle is controlled by the computer 8, the light intensity detector 7 picks up the intensity information at different azimuth angles, and then passes through the calculation module in the computer 8 Data processing is performed on the image to obtain the Stokes vector and the degree of polarization of the sample 4 to be tested, and the test results of the two can be directly read by the computer 8 .

计算机内的描述光强-多级相差共轴波片的数学物理模型的运算模块如下：The calculation module of the mathematical-physical model describing the light intensity-multilevel phase difference coaxial wave plate in the computer is as follows:

1.斯托克斯模型描述：1. Stokes model description:

S′＝M×SS'=M×S

$(\begin{matrix} {s the s}_{00}^{' '} \\ {s the s}_{11}^{' '} \\ {s the s}_{22}^{' '} \\ {s the s}_{33}^{' '} \end{matrix}) = = (\begin{matrix} {m m}_{0000} & {m m}_{0101} & {m m}_{0202} & {m m}_{0303} \\ {m m}_{1010} & {m m}_{1111} & {m m}_{1212} & {m m}_{1313} \\ {m m}_{2020} & {m m}_{21 twenty one} & {m m}_{22 twenty two} & {m m}_{23 twenty three} \\ {m m}_{3030} & {m m}_{3131} & {m m}_{3232} & {m m}_{3333} \end{matrix}) (\begin{matrix} {s the s}_{00} \\ {s the s}_{11} \\ {s the s}_{22} \\ {s the s}_{33} \end{matrix})$

2.斯托克斯模型下的穆勒矩阵描述：2. The description of the Mueller matrix under the Stokes model:

旋转矩阵：Rotation matrix:

$R R ((θ θ)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & cos cos 22 θ θ & sin sin 22 θ θ & 00 \\ 00 & - - sin sin 22 θ θ & cos cos 22 θ θ & 00 \\ 00 & 00 & 00 & 11 \end{matrix})$

偏振旋转矩阵描述：Polarization rotation matrix description:

对于偏振方向为水平方向：For the polarization direction to be horizontal:

P(θ)＝R(-θ)P(0)R(θ)P(θ)=R(-θ)P(0)R(θ)

$P P ((00)) = = (\begin{matrix} 11 / / 22 & 11 / / 22 & 00 & 00 \\ 11 / / 22 & 11 / / 22 & 00 & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

当旋转开始位置为水平位置时，偏振旋转矩阵表示如下：When the rotation start position is the horizontal position, the polarization rotation matrix is expressed as follows:

$P P ((θ θ,, 00)) = = (\begin{matrix} 11 & cos cos 22 θ θ & sin sin 22 θ θ & 00 \\ cos cos 22 θ θ & {cos cos}^{22} 22 θ θ & cos cos 22 θ θ sin sin 22 θ θ & 00 \\ sin sin 22 θ θ & cos cos 22 θ θ sin sin 22 θ θ & {sin sin}^{22} 22 θ θ & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

当旋转开始位置为垂直位置，偏振旋转矩阵表示如下：When the rotation start position is the vertical position, the polarization rotation matrix is expressed as follows:

$P P ((θ θ,, 00)) = = (\begin{matrix} 11 & - - cos cos 22 θ θ & - - sin sin 22 θ θ & 00 \\ - - cos cos 22 θ θ & {cos cos}^{22} 22 θ θ & cos cos 22 θ θ sin sin 22 θ θ & 00 \\ - - sin sin 22 θ θ & cos cos 22 θ θ sin sin 22 θ θ & {sin sin}^{22} 22 θ θ & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

在垂直位置：In vertical position:

$P P ((9090)) = = \frac{11}{22} (\begin{matrix} 11 & - - 11 & 00 & 00 \\ - - 11 & 11 & 00 & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

3.多级相差共轴波片描述：3. Description of multi-level phase difference coaxial wave plate:

a.波片滞后角为

快轴与X轴所成角度为θ，则穆勒矩阵如下：a. The lag angle of the wave plate is

The angle formed by the fast axis and the X axis is θ, then the Mueller matrix is as follows:

${M m}_{c c} ((φ φ,, 22 θ θ)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 θ θ + + cos cos φ φ {sin sin}^{22} 22 θ θ & ((11 - - cos cos φ φ)) sin sin 22 θ θ cos cos 22 θ θ & - - sin sin φ φ sin sin 22 θ θ \\ 00 & ((11 - - cos cos φ φ)) sin sin 22 θ θ cos cos 22 θ θ & {sin sin}^{22} 22 θ θ + + cos cos φ φ {cos cos}^{22} 22 θ θ & sin sin φ φ cos cos 22 θ θ \\ 00 & sin sin φ φ sin sin 22 θ θ & - - sin sin φ φ cos cos 22 θ θ & cos cos φ φ \end{matrix})$

b.1/4波片快轴竖直方向：b. Vertical direction of fast axis of 1/4 wave plate:

$Q Q ((00)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & 11 & 00 & 00 \\ 00 & 00 & 00 & - - 11 \\ 00 & 00 & 11 & 00 \end{matrix})$

c.1/4波片快轴水平方向：c. 1/4 wave plate fast axis horizontal direction:

$Q Q ((9090)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & 11 & 00 & 00 \\ 00 & 00 & 00 & 11 \\ 00 & 00 & - - 11 & 00 \end{matrix})$

d.1/4波片，快轴与X轴成一角度θ：d. 1/4 wave plate, the fast axis forms an angle θ with the X axis:

$W W ((θ θ,, 9090)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & sin sin 22 θ θ \\ 00 & sin sin 22 θ θ cos cos 22 θ θ & {sin sin}^{22} 22 θ θ & - - cos cos 22 θ θ \\ 00 & - - sin sin 22 θ θ & cos cos 22 θ θ & 00 \end{matrix})$

e.1/2波片，快轴与X轴成一角度 $θW (θ, 180) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos^{2} 2 θ - \sin^{2} 2 θ & 2 \sin 2 θ \cos 2 θ & 0 \\ 0 & 2 \sin 2 θ \cos 2 θ & \sin^{2} 2 θ - \cos^{2} 2 θ & 0 \\ 0 & 0 & 0 & - 1 \end{matrix})$ e.1/2 wave plate, the fast axis is at an angle to the X axis $θW (θ, 180) = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos^{2} 2 θ - \sin^{2} 2 θ & 2 \sin 2 θ \cos 2 θ & 0 \\ 0 & 2 \sin 2 θ \cos 2 θ & \sin^{2} 2 θ - \cos^{2} 2 θ & 0 \\ 0 & 0 & 0 & - 1 \end{matrix})$

f.3/4波片，快轴与X轴成一角度θf.3/4 wave plate, the fast axis forms an angle θ with the X axis

$W W ((θ θ,, 270270)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & - - sin sin 22 θ θ \\ 00 & sin sin 22 θ θ cos cos 22 θ θ & {sin sin}^{22} 22 θ θ & cos cos 22 θ θ \\ 00 & sin sin 22 θ θ & - - cos cos 22 θ θ & 00 \end{matrix})$

4.如图3所示偏振和双折射测量系统数学解析图：4. The mathematical analysis diagram of the polarization and birefringence measurement system as shown in Figure 3:

a.光源组件10是指激光光源1出射光束经过准直扩束系统2后，再经过一水平起偏器3，并通过待测样品4。如果用Ms表示待测样品4的穆勒矩阵，其相位延迟角为δ，方位角为φ，则a. The light source assembly 10 means that the output beam of the laser light source 1 passes through the collimator beam expander system 2 , then passes through a horizontal polarizer 3 , and passes through the sample 4 to be tested. If Ms is used to represent the Mueller matrix of sample 4 to be tested, its phase delay angle is δ, and its azimuth angle is φ, then

$Ms Ms. ((δ δ,, φ φ)) = = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ & ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ & - - sin sin δ δ sin sin 22 φ φ \\ 00 & ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ & {sin sin}^{22} 22 φ φ + + cos cos δ δ {cos cos}^{22} 22 φ φ & sin sin δ δ cos cos 22 φ φ \\ 00 & sin sin δ δ sin sin 22 θ θ & - - sin sin δ δ cos cos 22 φ φ & 00 \end{matrix})$

水平起偏器3的斯托克斯矩阵表示如下：The Stokes matrix of the horizontal polarizer 3 is expressed as follows:

${S S}_{00} = = (\begin{matrix} {s the s}_{00} \\ {s the s}_{11} \\ {s the s}_{22} \\ {s the s}_{33} \end{matrix}) = = (\begin{matrix} 11 \\ 11 \\ 00 \\ 00 \end{matrix})$

则光源组件10的斯托克斯矢量：Then the Stokes vector of the light source component 10:

${S S}_{s the s} = = {M m}_{s the s} {S S}_{00}$

$= = (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ & ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ & - - sin sin δ δ sin sin 22 φ φ \\ 00 & ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ & {sin sin}^{22} 22 φ φ + + cos cos δ δ {cos cos}^{22} 22 φ φ & sin sin δ δ cos cos 22 φ φ \\ 00 & sin sin δ δ sin sin 22 φ φ & - - sin sin δ δ cos cos 22 φ φ & 00 \end{matrix}) (\begin{matrix} 11 \\ 11 \\ 00 \\ 00 \end{matrix})$

$= = (\begin{matrix} 11 \\ {cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ \\ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ \\ sin sin δ δ sin sin 22 φ φ \end{matrix})$

b.调制组件11包括多级相差共轴波片5和检偏器6，通过多级相差共轴波片5的旋转而产生位相角，四分之一波片调制器的穆勒矩阵是：B. Modulation assembly 11 comprises multistage phase difference coaxial wave plate 5 and polarizer 6, produces phase angle by the rotation of multistage phase difference coaxial wave plate 5, and the Muller matrix of quarter wave plate modulator is:

${M m}_{q q} = = P P ((9090)) W W ((9090))$

$= = \frac{11}{22} (\begin{matrix} 11 & - - 11 & 00 & 00 \\ - - 11 & 11 & 00 & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & sin sin 22 θ θ \\ 00 & sin sin 22 θ θ cos cos 22 θ θ & {sin sin}^{22} 22 θ θ & - - cos cos 22 θ θ \\ 00 & - - sin sin 22 θ θ & cos cos 22 θ θ & 00 \end{matrix})$

$= = \frac{11}{22} (\begin{matrix} 11 & - - {cos cos}^{22} 22 θ θ & - - sin sin 22 θ θ cos cos 22 θ θ & - - sin sin 22 θ θ \\ - - 11 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & sin sin 22 θ θ \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

二分之一波片调制器的穆勒矩阵是：The Mueller matrix for a half-wave plate modulator is:

${M m}_{h h} = = P P ((9090)) W W ((180180))$

$= = \frac{11}{22} (\begin{matrix} 11 & - - 11 & 00 & 00 \\ - - 11 & 11 & 00 & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 θ θ - - {sin sin}^{22} 22 θ θ & 22 sin sin 22 θ θ cos cos 22 θ θ & 00 \\ 00 & 22 sin sin 22 θ θ cos cos 22 θ θ & {sin sin}^{22} 22 θ θ - - {cos cos}^{22} 22 θ θ & 00 \\ 00 & 00 & 00 & - - 11 \end{matrix})$

$= = \frac{11}{22} (\begin{matrix} 11 & - - {cos cos}^{22} 22 θ θ + + {sin sin}^{22} 22 θ θ & - - 22 sin sin 22 θ θ cos cos 22 θ θ & 00 \\ - - 11 & {cos cos}^{22} θ θ - - {sin sin}^{22} 22 θ θ & 22 sin sin 22 θ θ cos cos 22 θ θ & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

四分之三波片的穆勒矩阵为：The Mueller matrix for a three-quarter wave plate is:

${M m}_{t t} = = P P ((9090)) W W ((270270))$

$= = \frac{11}{22} (\begin{matrix} 11 & - - 11 & 00 & 00 \\ - - 11 & 11 & 00 & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} 11 & 00 & 00 & 00 \\ 00 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & - - sin sin 22 θ θ \\ 00 & sin sin 22 θ θ cos cos 22 θ θ & {sin sin}^{22} 22 θ θ & cos cos 22 θ θ \\ 00 & sin sin 22 θ θ & - - cos cos 22 θ θ & 00 \end{matrix})$

$= = \frac{11}{22} (\begin{matrix} 11 & - - {cos cos}^{22} 22 θ θ & - - sin sin 22 θ θ cos cos 22 θ θ & sin sin 22 θ θ \\ - - 11 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & - - sin sin 22 θ θ \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix})$

c.探测单元12是指用来对偏振和双折射进行信号接收的二维阵列相机，并通过计算机控制和软件分析对所获得的信号进行处理。c. The detection unit 12 refers to a two-dimensional array camera used to receive signals for polarization and birefringence, and processes the obtained signals through computer control and software analysis.

对于λ/4波片，将光源组件10与调制组件11进行作用，得到四分之一波片作用后的斯托克斯矩阵：For the λ/4 wave plate, the light source component 10 and the modulating component 11 are used to obtain the Stokes matrix after the action of the quarter wave plate:

${S S}_{q q} = = {M m}_{q q} {S S}_{s the s}$

$= = \frac{11}{22} (\begin{matrix} 11 & - - {cos cos}^{22} 22 θ θ & - - sin sin 22 θ θ cos cos 22 θ θ & - - sin sin 22 θ θ \\ - - 11 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & sin sin 22 θ θ \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} 11 \\ {cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ \\ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ \\ sin sin δ δ sin sin 22 φ φ \end{matrix})$

$= = \frac{11}{22} (\begin{matrix} 11 - - {cos cos}^{22} 22 θ θ (({cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ)) - - sin sin 22 θ θ cos cos 22 θ θ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ - - sin sin 22 θ θ sin sin δ δ sin sin 22 φ φ \\ - - 11 + + {cos cos}^{22} 22 θ θ (({cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ)) + + sin sin 22 θ θ cos cos 22 θ θ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ + + sin sin 22 φ φ sin sin δ δ sin sin 22 φ φ \\ 00 \\ 00 \end{matrix})$

上式斯托克斯矩阵中第一行表示强度I(θ)，通过对第一行进行化简，最后得到如下等式：The first row in the Stokes matrix above represents the intensity I(θ). By simplifying the first row, the following equation is finally obtained:

$22 \times \times I I ((θ θ)) = = ((11 - - \frac{{cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ}{22})) + + ((- - sin sin φ φ sin sin 22 φ φ)) sin sin 22 θ θ$

$+ + \frac{((cos cos δ δ - - 11)) sin sin 22 φ φ cos cos 22 φ φ}{22} sin sin 44 θ θ + + ((- - \frac{{cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ}{22})) cos cos 44 θ θ$

令θ＝ωt(ω表示波片旋转的角速度，t表示时间)， $a_{0} = 1 - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ a₂＝-sinδsin2φ， $a_{4} = \frac{(\cos δ - 1) \sin 2 φ \cos 2 φ}{2},$ $b_{4} = - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ 则上式可以化简为：2×I(ω)＝a₀+a₂sin(2ωt)+a₄sin(4ωt)+b₄cos(4ωt)Let θ=ωt (ω represents the angular velocity of wave plate rotation, t represents time), $a_{0} = 1 - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ a ₂ =-sinδsin2φ, $a_{4} = \frac{(\cos δ - 1) \sin 2 φ \cos 2 φ}{2},$ $b_{4} = - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ Then the above formula can be simplified as: 2×I(ω)＝a ₀ +a ₂ sin(2ωt)+a ₄ sin(4ωt)+b ₄ cos(4ωt)

对于λ/2波片，将光源组件10与调制器件11作用，最后得到二分之一波片作用后的斯托克斯矩阵：For the λ/2 wave plate, the light source assembly 10 and the modulation device 11 are used to act, and finally the Stokes matrix obtained after the half wave plate is acted on:

${S S}_{h h} = = {M m}_{h h} {S S}_{s the s}$

$= = \frac{11}{22} (\begin{matrix} 11 & - - {cos cos}^{22} 22 θ θ + + {sin sin}^{22} 22 θ θ & - - 22 sin sin 22 θ θ cos cos 22 θ θ & 00 \\ - - 11 & {cos cos}^{22} 22 θ θ - - {sin sin}^{22} 22 θ θ & 22 sin sin 22 θ θ cos cos 22 θ θ & 00 \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} 11 \\ {cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ \\ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ \\ sin sin δ δ sin sin 22 φ φ \end{matrix})$

$= = \frac{11}{22} (\begin{matrix} 11 + + ((- - {cos cos}^{22} 22 θ θ + + {sin sin}^{22} 22 θ θ)) (({cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ)) + + ((- - 22 sin sin 22 θ θ cos cos 22 θ θ)) ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ \\ - - 11 + + (({cos cos}^{22} 22 θ θ - - {sin sin}^{22} 22 θ θ)) (({cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ)) + + 22 sin sin 22 θ θ cos cos 22 θ θ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ \\ 00 \\ 00 \end{matrix})$

以上矩阵中第一行表示光强度I(θ)，对其进行化简，最后得到以下等式：The first row in the above matrix represents the light intensity I(θ), which is simplified, and finally the following equation is obtained:

2×I(θ)＝1+(cosδ-1)sin2φcos2φsin4θ-(cos²2φ+cosδsin²2φ)cos4θ2×I(θ)＝1+(cosδ-1)sin2φcos2φsin4θ-(cos ² 2φ+cosδsin ² 2φ)cos4θ

令θ＝ωt(ω表示波片旋转的角速度，t表示时间)，a₀＝1，a₂＝0，a₄＝(cosδ-1)sin2φcos2φ，b₄＝-(cos²2φ+cosδsin²2φ)，则上式可以化简为：2×I(ω)＝a₀+a₂sin(2ωt)+a₄sin(4ωt)+b₄cos(4ωt)Let θ=ωt (ω represents the angular velocity of wave plate rotation, t represents time), a ₀ =1, a ₂ =0, a ₄ =(cosδ-1) sin2φcos2φ, b ₄ =-(cos ² 2φ+cosδsin ² 2φ ), then the above formula can be simplified as: 2×I(ω)＝a ₀ +a ₂ sin(2ωt)+a ₄ sin(4ωt)+b ₄ cos(4ωt)

所以上式可以化简为：So the above formula can be simplified as:

2×I(ω)＝a₀+a₂sin(2ωt)+a₄sin(4ωt)+b₄cos(4ωt)2×I(ω)＝a ₀ +a ₂ sin(2ωt)+a ₄ sin(4ωt)+b ₄ cos(4ωt)

对于3λ/4波片，将光源组件10与调制器件11作用，最后得到四分之三波片作用后的斯托克斯矩阵：For the 3λ/4 wave plate, the light source assembly 10 and the modulation device 11 are used to act, and finally the Stokes matrix after the three-quarter wave plate is obtained:

${S S}_{t t} = = {M m}_{t t} {S S}_{s the s}$

$= = \frac{11}{22} (\begin{matrix} 11 & - - {cos cos}^{22} 22 θ θ & - - sin sin 22 θ θ cos cos 22 θ θ & sin sin 22 θ θ \\ - - 11 & {cos cos}^{22} 22 θ θ & sin sin 22 θ θ cos cos 22 θ θ & - - sin sin 22 θ θ \\ 00 & 00 & 00 & 00 \\ 00 & 00 & 00 & 00 \end{matrix}) (\begin{matrix} 11 \\ {cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ \\ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ \\ sin sin δ δ sin sin 22 φ φ \end{matrix})$

$= = \frac{11}{22} (\begin{matrix} 11 - - {cos cos}^{22} 22 θ θ (({cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ)) - - sin sin 22 θ θ cos cos 22 θ θ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ + + sin sin 22 θ θ sin sin δ δ sin sin 22 φ φ \\ - - 11 + + {cos cos}^{22} 22 θ θ (({cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ)) + + sin sin 22 θ θ cos cos 22 θ θ ((11 - - cos cos δ δ)) sin sin 22 φ φ cos cos 22 φ φ - - sin sin 22 θ θ sin sin δ δ sin sin 22 φ φ \\ 00 \\ 00 \end{matrix})$

$22 \times \times I I ((θ θ)) = = ((11 - - \frac{{cos cos}^{22} 22 φ φ + + cos cos δ δ {sin sin}^{22} 22 φ φ}{22})) + + sin sin δ δ sin sin 22 φ φ sin sin 22 θ θ$

令θ＝ωt(ω表示波片旋转的角速度，t表示时间)， $a_{0} = 1 - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ a₂＝sinδsin2φ， $a_{4} = \frac{(\cos δ - 1) \sin 2 φ \cos 2 φ}{2},$ $b_{4} = - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ 则上式可以化简为：2×I(ω)＝a₀+a₂sin(2ωt)+a₄sin(4ωt)+b₄cos(4ωt)Let θ=ωt (ω represents the angular velocity of wave plate rotation, t represents time), $a_{0} = 1 - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ a ₂ =sinδsin2φ, $a_{4} = \frac{(\cos δ - 1) \sin 2 φ \cos 2 φ}{2},$ $b_{4} = - \frac{\cos^{2} 2 φ + \cos δ \sin^{2} 2 φ}{2},$ Then the above formula can be simplified as: 2×I(ω)＝a ₀ +a ₂ sin(2ωt)+a ₄ sin(4ωt)+b ₄ cos(4ωt)

在上式中，I(θ)可以通过光强探测器直接获得，待测元件厚度L，入射光波长为λ，Δn表示非常光与寻常光由于传播速度不同引起的折射率的差异，即双折射Δn。由公式δ＝2π(n2-n1)L/λ，可知：

In the above formula, I(θ) can be directly obtained by the light intensity detector, the thickness of the component to be measured is L, and the wavelength of the incident light is λ. Refraction Δn. From the formula δ=2π(n2-n1)L/λ, we can know:

从而可以求出待测样品的双折射分布。对于多级相差共轴波片，上面对运用每种波片进行双折射测量进行了公式推导，通过平均化处理，最后可以得到待测样品的双折射分布。Thus, the birefringence distribution of the sample to be tested can be obtained. For the multi-stage phase-contrast coaxial wave plate, the above formula is derived for the birefringence measurement using each wave plate. After averaging, the birefringence distribution of the sample to be tested can finally be obtained.

对于偏振测量，从获得的斯托克斯矢量可以获得任意光的偏振态信息：For polarization measurements, the polarization state information of arbitrary light can be obtained from the obtained Stokes vector:

$tg tg 22 θ θ = = \frac{{S S}_{11}}{{S S}_{22}},,$ $ξ ξ = = \frac{11}{22} arcsin arcsin \frac{{S S}_{33}}{{(({S S}_{11}^{22} + + {S S}_{22}^{22} + + {S S}_{33}^{22}))}^{\frac{11}{22}}},,$ $e e = = tgξ tgξ = = \frac{b b}{a a},,$ $ρ ρ = = {(({S S}_{11}^{22} + + {S S}_{22}^{22} + + {S S}_{33}^{22}))}^{\frac{11}{22}} / / {S S}_{00},,$

其中θ为椭圆的方位角，表示椭圆的取向；ξ为椭圆的椭圆角，ξ正负表征偏振光是右旋偏振光和左旋偏振光；e表示椭圆的椭率；ρ描述偏振度，其值从非偏振光情况下的0到全偏振光情况下的1之间变化。Among them, θ is the azimuth angle of the ellipse, indicating the orientation of the ellipse; ξ is the ellipse angle of the ellipse, and the positive and negative of ξ indicate whether the polarized light is right-handed or left-handed; e is the ellipticity of the ellipse; ρ describes the degree of polarization, and its value Varies from 0 for unpolarized light to 1 for fully polarized light.

当然，也可以多次重复上述测量，对采集到的多个测量结果求平均值，这样可以进一步提高测量精度。Of course, the above measurement can also be repeated many times, and the average value of the collected measurement results can be calculated, so that the measurement accuracy can be further improved.

本技术领域中的普通技术人员应当认识到，以上的实施例仅是用来说明本发明，而并非用作为对本发明的限定，只要在本发明的实质精神范围内，对以上所述实施例变化，变型都将落在本发明权利要求书的范围内。Those of ordinary skill in the art should recognize that the above embodiments are only used to illustrate the present invention, rather than as a limitation to the present invention, as long as within the scope of the spirit of the present invention, changes to the above embodiments , modifications will fall within the scope of the claims of the present invention.

Claims

1. polarization and birefringence measurement system, it is characterized in that: this measuring system comprises three modules: light source module (10), light signal modulation module (11) and digital sampling and processing (12); Wherein: light source module (10) comprises light source (1), collimating and beam expanding system (2), the polarizer (3) and testing sample (4); Light signal modulation module (11) comprises multistage coaxial wave plate (5) and the analyzer (6) of differing; Digital sampling and processing (12) comprises light intensity detector (7), computing machine (8) and stepper motor (9); The light that light source (1) sends becomes telecentric beam through collimating and beam expanding system (2) post-concentration light beam; And incide the polarizer (3); Light beam becomes polarized light behind the polarizer (3), polarized light incides on the testing sample (4), and light beam sees through that testing sample (4) is laggard goes into light signal modulation module (11); Light beam differs coaxial wave plate (5) and analyzer (6) incides on the light intensity detector (7) through multistage, analyzer (6) and the polarizer (3) index Performance Match; The multistage coaxial wave plate (5) that differs is the disk that is made up of a plurality of wave plates that size does not wait, and the multistage coaxial wave plate (5) that differs is driven by stepper motor (9), realizes the modulation to light intensity through driving its rotation; The intensity signal that light intensity detector (7) obtains deposits data in computing machine (8) through data collecting card; And the computing module through computing machine (8) carries out the computing of polarization and birefringence data parameters and stores; Simultaneous computer (8) sends beginning and ending time and the anglec of rotation of drive signal with the driving of control step motor (9) for stepper motor (9) through the D/A plate; Stepper motor (9) drives multistage coaxial wave plate (5) rotation that differs; Rotation angle value can be controlled through computing machine (8); The strength information of light intensity detector (7) when obtaining different orientations utilizes computing machine (8) that the strength information that obtains is handled and mathematical analysis, obtains polarization and the birefringent mode and the distribution of optical material.

2. polarization according to claim 1 and birefringence measurement system is characterized in that: light source (1) is the ArF laser instrument of 193.368nm for wavelength.