CN102621072B - Polarization and birefringence measuring system - Google Patents

Abstract

The invention provides a polarization and birefringence measuring system in which measuring accuracy of polarization and stress birefringence is increased by using a multi-wave plate combination. The measuring system comprises a light source module, a light signal modulation module and a data collecting and processing module, wherein the light source module comprises a light source, a collimation and expansion system and a polarizer; the light signal modulation module comprises a multilevel phase difference coaxial wave plate, an analyzer, a stepping motor driving device and a stepping motor; and the data collecting and processing module comprises an image capture card, an luminous intensity detector and a computer. According to the polarization and birefringence measuring system, influences of changes of environments and measurement conditions such as air movement, temperature variation and uncertain vibration modes are effectively avoided; more than one independent phase modulation element is used for suppressing noises and averaging main error sources; and three wave plates are used for measuring the polarization and the birefringence, so that non-linear errors caused by non-linear influence of the system are effectively avoided, and the measuring accuracy is greatly improved.

Description

A kind of polarization and birefringence measurement systems

Technical field

The invention belongs to the contemporary optics technical field of measurement and test, be specifically related to a kind of polarization and birefringence measurement systems, it is accurately measured polarization and birefringence distribution.

Background technology

When photoetching projection objective lens NA＞0.8, the polarization characteristic of light beam can have a strong impact on image quality, and particularly the p polarization can be lowered into image contrast, affects systemic resolution.In addition, the point spread function of polarized light is asymmetric, and light distribution is spread along the polarization direction.For bargraphs, when not parallel, the light intensity transmitance is higher, is conducive to imaging when polarization direction and lines.Therefore, according to mask graph, adopt corresponding light beam to coordinate off-axis illumination, can further improve resolution and the image quality of system.

Simultaneously, increase numerical aperture and can catch the more light of wide-angle of more diffraction sum of series, this has aggravated the negative effect of polarization effect undoubtedly.Due to interacting and interference effect of high angle scattered light, cause the spatial image contrast greatly to be weakened.Therefore the optics of lithographic equipment must use the quartz material of low-birefringence, and birefringent phenomenon is closely connected with polarization effect because optical lens is to birefringent phenomenon is very responsive.

Thereby immersion lithography causes numerical aperture to be increased sharply causes polarization effect, the quartz material of having to use low-birefringence in 193nm ArF immersed photoetching machine, therefore must distribute to the stress birefrin in quartz material and measure, obtain data substitution emulation to guarantee the smooth development of litho machine.Surpass 300mm on considerable part fused quartz lens sizes diameter in the immersed photoetching machine of 193nm ArF simultaneously, the birefringence measurement of material adopts the metering system of sampled point to carry out, and guarantee that measuring accuracy needs sampled point as much as possible.Through simulation analysis, because may causing 4～6nm wave aberration, the birefringence effect of material changes, the performance of 193nm exposure optical system will be had a strong impact on, so must accurately measure polarization effect and the birefringence distribution of material, yet accurately measuring fast of the polarization of material and birefringent mode and distribution is very difficult on room and time, but is necessary for the development of 193nm ArF immersed photoetching machine.

Because the photolithographic exposure system of the reliability of wave aberration quality and precision optics imaging system such as live width 90 nanometers is closely bound up, be the even exposure systems of 16 nanometers of 22 nanometers for live width, the numerical aperture of photoetching projection objective lens is greater than 1, in order to satisfy high-resolution requirement, wavelength is very short, is for example 193 nanometers now, along with the shortening of wavelength, the degree of scattering is quite serious, and the stability of measuring system and the repeatability of measurement result are the huge challenges that we face.So, develop and a kind ofly have more that high precision, polarization birefringence measurement systems more stable, that have higher repeatability could satisfy the demand in engineering research field.

In sum, must explore a kind of new polarization and double refraction detecting method, set up a cover and can carry out the accurately system of measurement to polarization effect and the birefringence distribution of material, being that polarization and birefringence are multistage differs coaxial rotation precision measurement system, the polarization effect of material in the photolithographic exposure system and birefringence are distributed carry out Measurement accuracy, to improve resolution and the image quality of system.

In addition, this measuring system can also be used for jewel, crystal evaluation and medical aspect etc., in a word, along with advancing of photoetching technique, the line thickness that need to expose is more and more thinner, polarization effect and birefringence effect requirement to material are more and more higher, and the optical system of setting up a cover energy Accurate Measurement material polarization effect and birefringence distribution becomes very urgent.

Summary of the invention

The present invention is directed to polarization and birefringence effect is accurately measured, and optical element in measuring system is carried out physical model set up.A kind of polarization and birefringence measurement systems are provided, its core component is the multistage coaxial wave plate that differs, differ coaxial wave plate by rotation multistage light is carried out medelling, obtain corresponding relation between the anglec of rotation and measuring system light intensity, separate out long-pending upper polarization and the stress birefrin through medelling of tested surface by the mathematical operation exact solution and distribute.By adopting many wave plate combined to improve the measuring accuracy of polarization and stress birefrin.

technical scheme of the present invention is: a kind of polarization and birefringence measurement systems, mainly comprise light source module, optical signal modulation module and digital sampling and processing, and wherein light source module comprises light source, collimating and beam expanding system, the polarizer and testing sample, the optical signal modulation module comprises multistage coaxial wave plate, analyzer and the stepper motor of differing, digital sampling and processing comprises light intensity detector and computing machine, wavelength is that the light that the ArF laser light source of 193.368nm sends becomes telecentric beam through collimating and beam expanding system post-concentration light beam, and incide the polarizer, telecentric beam becomes polarized light after the polarizer, polarized light incides on testing sample, light beam enters the light signal modulation module by testing sample is laggard, light beam incides on light intensity detector after differing coaxial wave plate and analyzer through multistage, analyzer and polarizer index Performance Match, multistage to differ coaxial wave plate be the disk that is made of a plurality of wave plates that size does not wait, and multistagely differs coaxial wave plate and driven by stepper motor, realizes modulation to light intensity by driving its rotation, the intensity signal that light intensity detector obtains deposits data in computing machine through data collecting card, and carry out the computing of polarization and birefringence data parameters and store by the computing module of computing machine, simultaneous computer sends to stepper motor by the D/A plate and drives signal with motor-driven beginning and ending time of control step and the anglec of rotation, stepper motor drives the multistage coaxial wave plate rotation that differs, rotation angle value can be controlled by computing machine, strength information when light intensity detector obtains different orientations, utilize computing machine that the strength information that obtains is processed and mathematical analysis, obtain polarization and birefringent mode and the distribution of optical material.

The present invention compared with prior art has the following advantages:

1, this system adopts light path principle altogether, has well avoided environment and measuring condition to change as Air Flow, the impact that temperature variation and some uncertain mode of vibration are brought.

2, this system with surpass one independently a position phase modulator element suppress noise, average main error source.

3, this system adopts three wave plates that polarization and birefringence are measured, and the nonlinearity erron of having avoided well the non-linear effects due to system to cause has improved measuring accuracy greatly.

Description of drawings

Fig. 1 is a kind of polarization and birefringence measurement systems measuring principle schematic diagram;

Fig. 2 is the multistage coaxial wave plate schematic diagram that differs;

Fig. 3 is polarization and birefringence measurement systems mathematical analysis schematic diagram;

Number in the figure explanation: 1-light source, 2-collimating and beam expanding system, the 3-polarizer, 4-testing sample, multistage coaxial wave plate, 6-analyzer, 7-light intensity detector, 8-computing machine, 9-stepper motor, 10-light source module, 11-optical signal modulation module, the 12-digital sampling and processing of differing of 5-.

Embodiment

For objects and advantages of the present invention are described better, the invention will be further described below in conjunction with accompanying drawing.

Fig. 1 is a kind of polarization and birefringence measurement systems measuring principle schematic diagram, comprises light source module 10, optical signal modulation module 11 and digital sampling and processing 12, and described light source module 10 comprises light source 1, collimating and beam expanding system 2, the polarizer 3, testing sample 4; Optical signal modulation module 11 comprises multistage coaxial wave plate 5, the analyzer 6 of differing; Digital sampling and processing 12 comprises light intensity detector 7, computing machine 8 and stepper motor 9.As shown in Figure 1, light source 1 is the ArF laser instrument of 193.368nm for wavelength, the light that the ArF laser instrument sends becomes telecentric beam through collimating and beam expanding system 2 post-concentration light beams, and incide the polarizer 3, light beam becomes polarized light after the polarizer 3, polarized light incides on testing sample 4, and light beam sees through that testing sample 4 is laggard enters light signal modulation module 11, light beam differs coaxial wave plate 5 and analyzer 6 incides on light intensity detector 7 through multistage, analyzer 6 and the polarizer 3 index Performance Match.Multistagely differ the disk that a plurality of wave plates that coaxial wave plate 5 do not wait by size consist of, multistagely differ coaxial wave plate 5 and driven by high precision stepper motor 9, realize modulation to light intensity by driving its rotation.the intensity signal that light intensity detector 7 obtains deposits data in computing machine 8 through data collecting card, and carry out the computing of polarization and birefringence data parameters and store by the computing module of computing machine 8, and can show light spot image simultaneously on display, simultaneous computer 8 sends to stepper motor 9 by the D/A plate and drives signal with beginning and ending time and the anglec of rotation of the driving of control step motor 9, stepper motor 9 drives multistage coaxial wave plate 5 rotations that differ, rotation angle value can be controlled by computing machine, strength information when light intensity detector 7 obtains different orientations, utilize 8 pairs of strength informations that obtain of computing machine to process and mathematical analysis, obtain polarization and birefringent mode and the distribution of optical material.

Multistagely differ coaxial wave plate 5 design tacticses: (1) as shown in Figure 2, in a disk, three kinds of wave plates must have the collimation of height, during wave plate when light beam by rotation, it is identical that their optical axis must keep.Otherwise, just must introduce attitude, thereby being processed, calculating and data become complicated, even bring uncontrollable mistake; (2) about shared angle in three kinds of wave plate one, place circumference, now provide a kind of optimum solution, quarter wave plate 1/2 circle, 1/2 wave plate 1/4 circle, 3/4 wave plate 1/4 circle; (3) connection of every kind of wave plate must be accurate, and each junction is being aligned closely, otherwise, when measuring, when stepper motor goes to the junction, may not have light to pass through.

Three kinds of wave plates shared angular distribution method in circumference: for the linearly polarized light of incident, through quarter wave plate, quarter wave plate is rotated a circle, the position changes 4 π mutually; Through 1/2 wave plate, 1/2 wave plate is rotated a circle, the position changes 8 π mutually; Through 3/4 wave plate, 3/4 wave plate is rotated a circle, the position changes 12 π mutually.Therefore, quarter wave plate rotation half-turn, the position changes one-period 2 π mutually; 1/2 wave plate rotation 1/4 circle, the position changes one-period 2 π mutually; 3/4 wave plate rotation 1/4 circle, the position changes 1.5 cycles 3 π mutually.The rest may be inferred, can also select other wave plate combined: as selecting quarter wave plate 1/2 circle, 1/3 wave plate 3/8 circle, 5/12 wave plate 1/8 circle;

Fig. 3 is polarization and birefringence measurement systems mathematical analysis schematic diagram, mainly comprises following module: light source module 10, optical signal modulation module 11 and digital sampling and processing 12.

Light source module 10: comprise light source 1 and collimating and beam expanding system 2, the polarizer 3, testing sample 4.

Optical signal modulation module 11: comprise multistage coaxial wave plate 5, the analyzer 6 of differing.The core of this module is the multistage coaxial wave plate 5 that differs, and it is that quarter wave plate, 1/2 wave plate, 3/4 wave plate are embedded on same disk, accurately control step angle and rotating.In the wave plate front, the collimator and extender optical system 2 of a standard makes convergent beam become telecentric beam, by being radiated at multistage differing on coaxial wave plate 5 after the polarizer 3 and testing sample 4; In multistage coaxial wave plate 5 back of differing, place an analyzer 6 the same with the polarizer 3 index performances.

In order accurately to measure polarization/birefringence, light source module 10 becomes treats examining system, and position phase and the polarization state of output are as far as possible accurately measured.Can have influence on the precision of measurement due to some uncertain factors: the degree of polarization on specific direction, polarizing angle, the drag angle between two polarization vectors etc., thus adopt a plurality of wave plates to measure, thus measuring accuracy greatly improved to reduce error.

Digital sampling and processing 12: comprise light intensity detector 7, computing machine 8 and stepper motor 9.

after the light that the monochromaticity of being sent by LASER Light Source is good passes through collimating and beam expanding system 2, pass through again the polarizer 3, become polarized light, through testing sample 4, pass through again the multistage coaxial wave plate 5 that differs, multistagely differ 5 rotations of coaxial wave plate when arbitrarily angled when controlling with stepper motor 9, arrive light intensity detectors 7 through the multistage polarized light that differs after coaxial wave plate 5 is modulated through analyzer 6, can read transmitted light intensity at this moment from light intensity detector 7, rotate wave plate with identical angular velocity, thereby obtain many group light intensity values, intensity signal deposits data computing machine 8 in and stores through image pick-up card, the main transmission to stepper motor 9 by the D/A plate drives signal, thereby drive multistage coaxial wave plate 5 rotations that differ, the anglec of rotation is controlled by computing machine 8, strength information during light intensity detector 7 picked-up different orientations, then by the computing module in computing machine 8, image being carried out data processes, just Stokes vector and the degree of polarization of testing sample 4 have been tried to achieve, the test result both that can directly read by computing machine 8.

It is as follows that description light intensity in computing machine-multistage differs the computing module of mathematics physics model of coaxial wave plate:

1. Stokes model description:

S′＝M×S

$\left(\begin{array}{c}{s}_0^{\&prime;}\\ {\mathrm{<figure-callout\; id="1"\; label="s"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">s</figure-callout>}}_1^{\&prime;}\\ {\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">s</figure-callout>}}_2^{\&prime;}\\ {\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">s</figure-callout>}}_3^{\&prime;}\end{array}\right)=\left(\begin{array}{cccc}{m}_{00}& m_{01}& m_{02}& m_{03}\\ m_{10}& m_{11}& m_{12}& m_{13}\\ m_{20}& m_{21}& m_{22}& m_{23}\\ m_{30}& m_{31}& m_{32}& m_{33}\end{array}\right)\left(\begin{array}{c}{s}_0\\ s_1\\ s_2\\ {\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">s</figure-callout>}}_3\end{array}\right)$

2. the Muller matrix under the Stokes model is described:

Rotation matrix:

$R\left(\mathrm{\&theta;}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \sin 2\mathrm{\&theta;}& 0\\ 0& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ 0& 0& 0& 1\end{array}\right)$

The polarization rotation matrix is described:

Be horizontal direction for the polarization direction:

P(θ)＝R(-θ)P(0)R(θ)

$P\left(0\right)=\left(\begin{array}{cccc}1/2& 1/2& 0& 0\\ 1/2& 1/2& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

When being horizontal level, the polarization rotation matrix is expressed as follows when the rotation starting position:

$P(\mathrm{\&theta;},0)=\left(\begin{array}{cccc}1& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \sin 2\mathrm{\&theta;}& 0\\ \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\sin 2\mathrm{\&theta;}& 0\\ \sin 2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\sin 2\mathrm{\&theta;}& {\sin }^{2}2\mathrm{\&theta;}& 0\\ 0& 0& 0& 0\end{array}\right)$

Be the upright position when rotating the starting position, the polarization rotation matrix is expressed as follows:

$P(\mathrm{\&theta;},0)=\left(\begin{array}{cccc}1& -\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\sin 2\mathrm{\&theta;}& 0\\ -\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\sin 2\mathrm{\&theta;}& 0\\ -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\sin 2\mathrm{\&theta;}& {\sin }^{2}2\mathrm{\&theta;}& 0\\ 0& 0& 0& 0\end{array}\right)$

In the upright position:

$P\left(90\right)=\frac{1}{2}\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

3. multistagely differ coaxial wave plate and describe:

A. the wave plate drag angle is Fast axle and X-axis angulation are θ, and Muller matrix is as follows:

$M_c(\mathrm{\&phi;},2\mathrm{\&theta;})=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}+\cos \mathrm{\&phi;}{\sin }^{2}2\mathrm{\&theta;}& (1-\cos \mathrm{\&phi;})\sin 2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\sin \mathrm{\&phi;}\sin 2\mathrm{\&theta;}\\ 0& (1-\cos \mathrm{\&phi;})\sin 2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\sin }^{2}2\mathrm{\&theta;}+\cos \mathrm{\&phi;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \sin \mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\\ 0& \sin \mathrm{\&phi;}\sin 2\mathrm{\&theta;}& -\sin \mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \cos \mathrm{\&phi;}\end{array}\right)$

B.1/4 the fast axle vertical direction of wave plate:

$Q\left(0\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)$

C.1/4 the fast axle horizontal direction of wave plate:

$Q\left(90\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ 0& 0& -1& 0\end{array}\right)$

D.1/4 wave plate, fast axle and X-axis be θ at angle:

$W(\mathrm{\&theta;},90)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \sin 2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \sin 2\mathrm{\&theta;}\\ 0& \sin 2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\sin }^{2}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\\ 0& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\end{array}\right)$

E.1/2 wave plate, fast axle and X-axis are at angle $\mathrm{\&theta;W}(\mathrm{\&theta;},180)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& 2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ 0& 2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\sin }^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& 0\\ 0& 0& 0& -1\end{array}\right)$

F.3/4 wave plate, fast axle and X-axis be θ at angle

$W(\mathrm{\&theta;},270)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \sin 2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\sin 2\mathrm{\&theta;}\\ 0& \sin 2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\sin }^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\\ 0& \sin 2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\end{array}\right)$

4. polarization and birefringence measurement systems mathematical analysis figure as shown in Figure 3:

A. light source assembly 10 refers to LASER Light Source 1 outgoing beam through after collimating and beam expanding system 2, then through a horizontal polarizer 3, and by testing sample 4.If represent the Muller matrix of testing sample 4 with Ms, its phase delay angle is δ, and the position angle is φ,

$\mathrm{Ms}(\mathrm{\&delta;},\mathrm{\&phi;})=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}& (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}& -\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\\ 0& (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}& {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}& \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ 0& \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}& -\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}& 0\end{array}\right)$

The Stokes matrix of the horizontal polarizer 3 is expressed as follows:

$S_0=\left(\begin{array}{c}{s}_0\\ s_1\\ s_2\\ {\mathrm{<figure-callout\; id="3"\; label="s"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">s</figure-callout>}}_3\end{array}\right)=\left(\begin{array}{c}1\\ 1\\ 0\\ 0\end{array}\right)$

The Stokes vector of light source assembly 10:

$S_s=M_s{S}_0$

$=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}& (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}& -\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\\ 0& (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}& {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}& \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ 0& \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}& -\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}& 0\end{array}\right)\left(\begin{array}{c}1\\ 1\\ 0\\ 0\end{array}\right)$

$=\left(\begin{array}{c}1\\ {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}\\ (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\end{array}\right)$

B. modulation component 11 comprises multistage coaxial wave plate 5 and the analyzer 6 of differing, and produces phase angle by the multistage rotation that differs coaxial wave plate 5, and the Muller matrix of quarter-wave plate modulator is:

$M_q=P\left(90\right)W\left(90\right)$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ 0& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\\ 0& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ -1& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

The Muller matrix of 1/2nd wave-plate modulators is:

$M_h=P\left(90\right)W\left(180\right)$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& 2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ 0& 2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& 0\\ 0& 0& 0& -1\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}+{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& -2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ -1& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& 2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

The Muller matrix of 3/4ths wave plates is:

$M_t=P\left(90\right)W\left(270\right)$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ 0& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& {\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}\\ 0& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ -1& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)$

C. probe unit 12 refers to polarization and birefringence are carried out the two-dimensional array camera that signal receives, and by computer control and software analysis, the signal that obtains is processed.

For λ/4 wave plates, light source assembly 10 and modulation component 11 are acted on, obtain the Stokes matrix after the quarter-wave plate effect:

$S_q=M_q{S}_s$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ -1& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}1\\ {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}\\ (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{c}1-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;})-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}(1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\\ -1+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;})+\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}(1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}+\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\\ 0\\ 0\end{array}\right)$

In the following formula Stokes matrix, the first row represents intensity I (θ), by the first row is carried out abbreviation, obtains at last following equation:

$2\&times;I\left(\mathrm{\&theta;}\right)=(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2})+(-\sin \mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}$

$+\frac{(\cos \mathrm{\&delta;}-1)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}}{2}\mathrm{<figure-callout\; id="4"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}4\mathrm{\&theta;}+(-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2})\mathrm{<figure-callout\; id="4"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}4\mathrm{\&theta;}$

Make θ=ω t (ω represents the angular velocity of wave plate rotation, and t represents the time), $a_0=1-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ a ₂＝-sinδsin2φ， $a_4=\frac{(\cos \mathrm{\&delta;}-1)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}}{2},$ ${\mathrm{<figure-callout\; id="4"\; label="b"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">b</figure-callout>}}_4=-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ Following formula can abbreviation be: 2 * I (ω)=a ₀+ a ₂Sin (2 ω t)+a ₄Sin (4 ω t)+b ₄Cos (4 ω t)

For λ/2 wave plates, with light source assembly 10 and modulation device 11 effects, obtain at last the Stokes matrix after 1/2nd wave plate effects:

$S_h=M_h{S}_s$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}+{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& -2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ -1& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;}& 2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}1\\ {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}\\ (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{c}1+(-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}+{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;})({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;})+(-2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;})(1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ -1+({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&theta;})({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;})+2\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}(1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ 0\\ 0\end{array}\right)$

In above matrix, the first row represents light intensity I (θ), and it is carried out abbreviation, obtains at last following equation:

2×I(θ)＝1+(cosδ-1)sin2φcos2φsin4θ-(cos ²2φ+cosδsin ²2φ)cos4θ

Make θ=ω t (ω represents the angular velocity of wave plate rotation, and t represents the time), a ₀=1, a ₂=0, a ₄=(cos δ-1) sin2 φ cos2 φ, b ₄=-(cos ²2 φ+cos δ sin ²2 φ), following formula can abbreviation be: 2 * I (ω)=a ₀+ a ₂Sin (2 ω t)+a ₄Sin (4 ω t)+b ₄Cos (4 ω t)

So following formula can abbreviation be:

2×I(ω)＝a ₀+a ₂sin(2ωt)+a ₄sin(4ωt)+b ₄cos(4ωt)

For 3 λ/4 wave plates, with light source assembly 10 and modulation device 11 effects, obtain at last the Stokes matrix after 3/4ths wave plate effects:

$S_t=M_t{S}_s$

$=\frac{1}{2}\left(\begin{array}{cccc}1& -{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ -1& {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}& \mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}& -\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\left(\begin{array}{c}1\\ {\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}\\ (1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}\\ \sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\end{array}\right)$

$=\frac{1}{2}\left(\begin{array}{c}1-{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;})-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}(1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}+\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\\ -1+{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&theta;}({\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;})+\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&theta;}(1-\cos \mathrm{\&delta;})\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}-\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\\ 0\\ 0\end{array}\right)$

In the following formula Stokes matrix, the first row represents intensity I (θ), by the first row is carried out abbreviation, obtains at last following equation:

$2\&times;I\left(\mathrm{\&theta;}\right)=(1-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2})+\sin \mathrm{\&delta;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&theta;}$

$+\frac{(\cos \mathrm{\&delta;}-1)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}}{2}\mathrm{<figure-callout\; id="4"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}4\mathrm{\&theta;}+(-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2})\mathrm{<figure-callout\; id="4"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}4\mathrm{\&theta;}$

Make θ=ω t (ω represents the angular velocity of wave plate rotation, and t represents the time), $a_0=1-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ a ₂＝sinδsin2φ， $a_4=\frac{(\cos \mathrm{\&delta;}-1)\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}2\mathrm{\&phi;}}{2},$ ${\mathrm{<figure-callout\; id="4"\; label="b"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">b</figure-callout>}}_4=-\frac{{\mathrm{<figure-callout\; id="2"\; label="cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\cos \mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ Following formula can abbreviation be: 2 * I (ω)=a ₀+ a ₂Sin (2 ω t)+a ₄Sin (4 ω t)+b ₄Cos (4 ω t)

In following formula, I (θ) can directly obtain by light intensity detector, and element under test thickness L, lambda1-wavelength are λ, and Δ n represents the difference of the refractive index that non-ordinary light and ordinary light cause because velocity of propagation is different, i.e. birefringence n.By formula δ=2 π (n2-n1) L/ λ, as can be known:

Thereby the birefringence that can obtain testing sample distributes.Differ coaxial wave plate for multistage, the above has carried out the derivation of equation to using every kind of wave plate to carry out birefringence measurement, and by to handle averagely, the birefringence that can obtain at last testing sample distributes.

For polarimetry, can obtain any polarisation of light attitude information from the Stokes vector that obtains:

$\mathrm{tg}2\mathrm{\&theta;}=\frac{S_1}{{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_2},$ $\mathrm{\&xi;}=\frac{1}{2}\arcsin \frac{S_3}{{({\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^2+{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_2^2+{\mathrm{<figure-callout\; id="3"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_3^2)}^{\frac{1}{2}}},$ $e=\mathrm{tg\&xi;}=\frac{b}{a},$ $\mathrm{\&rho;}={({\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_1^2+{\mathrm{<figure-callout\; id="2"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_2^2+{\mathrm{<figure-callout\; id="3"\; label="S"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">S</figure-callout>}}_3^2)}^{\frac{1}{2}}/S_0,$

Wherein θ is oval position angle, the orientation that expression is oval; ξ is oval oval angle, and the positive and negative sign polarized light of ξ is right-handed polarized light and left-hand polarization light; E represents oval ellipticity; ρ describes degree of polarization, changes 1 under it is worth from 0 in the nonpolarized light situation to the full-polarization situation.

Certainly, also can repeatedly repeat above-mentioned measurement, a plurality of measurement results that collect are averaged, can further improve measuring accuracy like this.

Those of ordinary skill in the art will be appreciated that, above embodiment illustrates the present invention, and be not to be used as limitation of the invention, as long as in connotation scope of the present invention, the above embodiment is changed, and modification all will drop in the scope of claims of the present invention.

Claims (2)

1. a polarization and birefringence measurement systems, this measuring system comprises three modules: light source module (10), optical 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); Optical 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 after the polarizer (3), polarized light incides on testing sample (4), light beam sees through that testing sample (4) is laggard enters light signal modulation module (11), light beam differs coaxial wave plate (5) and analyzer (6) incides on light intensity detector (7) through multistage, analyzer (6) and the polarizer (3) index Performance Match; The intensity signal that light intensity detector (7) obtains deposits data in computing machine (8) through data collecting card, simultaneous computer (8) sends to stepper motor (9) by the D/A plate and drives signal with beginning and ending time and the anglec of rotation of the driving of control step motor (9), the strength information of light intensity detector (7) when obtaining different orientations, it is characterized in that: the multistage coaxial wave plate (5) that differs is the disk that is made of a plurality of wave plates that size does not wait, the multistage coaxial wave plate (5) that differs is driven by stepper motor (9), by driving its rotation realization to the modulation of light intensity; Computing module by computing machine (8) carries out the computing of polarization and birefringence data parameters and stores, stepper motor (9) drives multistage coaxial wave plate (5) rotation that differs, rotation angle value can be controlled by computing machine (8), utilize computing machine (8) that the strength information that obtains is processed and mathematical analysis, obtain polarization and birefringent mode and the distribution of optical material.

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

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