CN102621072A - Polarization and birefringence measurement system - Google Patents

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 kind of polarization and birefringence measurement system

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 system, 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, influences 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 polarization direction and lines were not parallel, the light intensity transmitance was higher, helped imaging.Therefore,, adopt corresponding light beam to cooperate off-axis illumination, can further improve the resolution and the image quality of system according to mask graph.

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.Because interacting and interference effect of high angle scattered light causes the spatial image contrast greatly to be weakened.Therefore the optics of lithographic equipment must use the quartz material of low-birefringence, because optical lens is to the unusual sensitivity and birefringent phenomenon is closely connected with polarization effect of birefringent phenomenon.

Thereby immersion lithography causes numerical aperture to be increased sharply causes polarization effect; The quartz material of in 193nm ArF immersed photoetching machine, having to use low-birefringence; Therefore must distribute to the stress birefrin in the quartz material and measure, obtain data substitution emulation to guarantee the smooth development of litho machine.Surpass 300mm on the 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 the birefringence effect of material possibly cause 4～6nm wave aberration and change; With the performance that has a strong impact on the 193nm exposure optical system; Accurately measure so must distribute, yet accurately measuring fast of the polarization of material and birefringent mode and distribution is that ten minutes is difficult on room and time, but is necessary for the development of 193nm ArF immersed photoetching machine to the polarization effect of material and birefringence.

Because the reliability of wave aberration quality and the photolithographic exposure system of precision optics imaging system such as live width 90 nanometers are closely bound up, are the exposure system of 22 nanometers even 16 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, for example is 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 the polarization birefringence measurement system that a kind ofly has high precision more, more stablizes, has higher repeatability and could satisfy the demand in engineering research field.

In sum; Must explore a kind of new polarization and double refraction detecting method; Set up the system that a cover can distribute and accurately measure the polarization effect and the birefringence 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 is distributed accurately measure, to improve the resolution and the image quality of system.

In addition; This measuring system can also be used for jewel, crystal evaluation and medical aspect or the like; In a word, along with advancing of photoetching technique, the line thickness that need expose is more and more thinner; The polarization effect of material and birefringence effect are required increasingly high, set up a cover and can accurately measure the material polarization effect and become very urgently with the optical system that birefringence distributes.

Summary of the invention

The present invention is directed to polarization and birefringence effect is accurately measured, and optical element in the measuring system is carried out physical model set up.A kind of polarization and birefringence measurement system are provided; Its core component is the multistage coaxial wave plate that differs; Multistagely differ coaxial wave plate light is carried out medelling through rotating; Obtain corresponding relation between the anglec of rotation and measuring system light intensity, separate out long-pending polarization and the stress birefrin of going up of tested surface through the mathematical operation exact solution and distribute through medelling.Through 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 system, mainly comprise light source module, light 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 light 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 behind the polarizer, polarized light incides on the testing sample, and light beam is gone into the light signal modulation module through testing sample is laggard; Light beam incides on the light intensity detector analyzer and polarizer index Performance Match through multistage after differing coaxial wave plate and analyzer; Multistage to differ coaxial wave plate be the disk that is made up of a plurality of wave plates that size does not wait, and multistagely differs coaxial wave plate by step motor drive, realizes the modulation to light intensity through driving its rotation; The intensity signal that light intensity detector obtains deposits data in computing machine through data collecting card; And the computing module through computing machine carries out the computing of polarization and birefringence data parameters and stores; Simultaneous computer sends drive signal with the motor-driven beginning and ending time of control step and the anglec of rotation through the D/A plate to stepper motor; Stepper motor drives the multistage coaxial wave plate rotation that differs, and rotation angle value can be controlled through computing machine, the strength information when light intensity detector obtains different orientations; Utilize computing machine that the strength information that obtains is handled and mathematical analysis, obtain polarization and the 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 like air flow the influence that temperature variation and some uncertain mode of vibration are brought.

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

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

Description of drawings

Fig. 1 is a kind of polarization and birefringence measurement systematic survey principle schematic;

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

Fig. 3 is polarization and birefringence measurement system mathematical analysis synoptic diagram;

Label declaration among the figure: 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-light signal modulation module, the 12-digital sampling and processing of differing of 5-.

Embodiment

The present invention is described further objects and advantages of the present invention in order to explain better below in conjunction with accompanying drawing.

Fig. 1 is a kind of polarization and birefringence measurement systematic survey principle schematic, comprises light source module 10, light signal modulation module 11 and digital sampling and processing 12, and said light source module 10 comprises light source 1, collimating and beam expanding system 2, the polarizer 3, testing sample 4; Light 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, and 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 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 incides on the light intensity detector 7 analyzer 6 and the polarizer 3 index Performance Match through the multistage coaxial wave plate 5 that differs with analyzer 6.Multistagely differ the disk that a plurality of wave plates that coaxial wave plate 5 do not wait by size constitute, multistagely differ coaxial wave plate 5 and drive, realize modulation light intensity through driving its rotation by high precision stepper motor 9.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; And can on display, show light spot image simultaneously; 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 rotations that differ, and rotation angle value can be controlled through computing machine, the strength information when light intensity detector 7 obtains different orientations; Utilize 8 pairs of strength informations that obtain of computing machine to handle and mathematical analysis, obtain polarization and the birefringent mode and the distribution of optical material.

Multistagely differ coaxial wave plate 5 design tacticses: (1) is as shown in Figure 2, and in a disk, three kinds of wave plates must have the collimation of height, and when the wave plate of light beam through rotation, it is identical that their optical axis must keep.Otherwise, just must introduce attitude, thereby make calculating and data processing become complicated, even bring uncontrollable mistake; (2) about shared angle in the circumference in three kinds of wave plate places, 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 at present; (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, possibly there is not 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 for use: as selecting quarter wave plate 1/2 circle for use, 1/3 wave plate, 3/8 circle, 5/12 wave plate, 1/8 circle;

Fig. 3 is polarization and birefringence measurement system mathematical analysis synoptic diagram, mainly comprises following module: light source module 10, light 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.

Light 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 to be embedded in quarter wave plate, 1/2 wave plate, 3/4 wave plate on the 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, through being radiated at multistage differing on the coaxial wave plate 5 behind the polarizer 3 and the testing sample 4; In multistage coaxial wave plate 5 back that differ, place one with the same analyzer 6 of the polarizer 3 index performances.

In order accurately to measure polarization/birefringence, light source module 10 becomes treats examining system, and the position phase and the polarization state of output are as far as possible accurately measured.Because some uncertain factors can have influence on the precision of measurement: the degree of polarization on the specific direction, polarizing angle, the drag angle between two polarization vectors etc., thus adopt a plurality of wave plates to measure, thus measuring accuracy improved greatly 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 the polarizer 3 again, become polarized light; Through testing sample 4, differ coaxial wave plate 5 through multistage again, when differing coaxial wave plate 5 and rotate when arbitrarily angled with stepper motor 9 controls are multistage; Arrive light intensity detectors 7 through the multistage polarized light that differs after coaxial wave plate 5 is modulated through analyzer 6; From light intensity detector 7, can read transmitted light intensity at this moment, rotate wave plate, thereby obtain many group light intensity values with unequal angular velocity; Intensity signal deposits data in computing machine 8 and storage through image pick-up card; The main D/A plate that passes through sends drive signals for stepper motor 9, thereby drives multistage coaxial wave plate 5 rotations that differ, and the anglec of rotation is by computing machine 8 controls; Strength information during light intensity detector 7 picked-up different orientations; Through the computing modules in the computing machine 8 image is carried out data processing then, just tried to achieve the Stokes vector and the degree of polarization of testing sample 4, the test result of the two that can directly read through computing machine 8.

It is following that description light intensity in the 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 description under the Stokes model:

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:

For the polarization direction is horizontal 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 the rotation starting position was horizontal level, the polarization rotation matrix was represented as follows:

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

When rotating the starting position is the upright position, and the polarization spin 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. fast axle is θ with X axle angulation to the wave plate drag angle for

, and then the Muller matrix is following:

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

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

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

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 axle 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 axle are at angle $\mathrm{\&theta;\; W}(\mathrm{\&theta;},180)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& {\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}}^{2}2\mathrm{\&theta;}-{\mathrm{Sin}}^{2}2\mathrm{\&theta;}& 2\mathrm{Sin}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}2\mathrm{\&theta;}& 0\\ 0& 2\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}2\mathrm{\&theta;}\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}2\mathrm{\&theta;}& {\mathrm{Sin}}^{2}2\mathrm{\&theta;}-{\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}}^{2}2\mathrm{\&theta;}& 0\\ 0& 0& 0& -1\end{array}\right)$

F.3/4 wave plate, fast axle and X axle 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 as shown in Figure 3 and the mathematical analysis figure of birefringence measurement system:

A. light source assembly 10 is meant LASER Light Source 1 outgoing beam through behind the collimating and beam expanding system 2, again through a horizontal polarizer 3, and through 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, then

$\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 representation of the horizontal polarizer 3 is following:

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

The Stokes vector of light source assembly 10 then:

$S_s=M_s{S}_0$

$=\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 through 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 is meant and is used for polarization and birefringence are carried out the two-dimensional array camera that signal receives, and through computer control and software analysis the signal that is obtained is handled.

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)$

The first line display intensity I (θ) in the following formula Stokes matrix through first row is carried out abbreviation, obtains following equality at last:

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

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

Make θ=ω t (ω representes the angular velocity of wave plate rotation, t express time), $a_0=1-\frac{{\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\mathrm{Cos}\mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ a ₂=-sin δ sin2 φ, $a_4=\frac{(\mathrm{Cos}\mathrm{\&delta;}-1)\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}2\mathrm{\&phi;}}{2},$ ${\mathrm{<figure-callout\; id="4"\; label="b"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">b</figure-callout>}}_4=-\frac{{\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\mathrm{Cos}\mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ Then 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,, obtain the Stokes matrix after 1/2nd wave plate effects at last with light source assembly 10 and modulation device 11 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)$

The first line display light intensity I (θ) carries out abbreviation to it in the above matrix, obtains following equality at last:

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

Make θ=ω t (ω representes the angular velocity of wave plate rotation, t express 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), then 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,, obtain the Stokes matrix after 3/4ths wave plate effects at last with light source assembly 10 and modulation device 11 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)$

The first line display intensity I (θ) in the following formula Stokes matrix through first row is carried out abbreviation, obtains following equality at last:

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

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

Make θ=ω t (ω representes the angular velocity of wave plate rotation, t express time), $a_0=1-\frac{{\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\mathrm{Cos}\mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ a ₂=sin δ sin2 φ, $a_4=\frac{(\mathrm{Cos}\mathrm{\&delta;}-1)\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}2\mathrm{\&phi;}\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}2\mathrm{\&phi;}}{2},$ ${\mathrm{<figure-callout\; id="4"\; label="b"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">b</figure-callout>}}_4=-\frac{{\mathrm{<figure-callout\; id="2"\; label="Cos"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Cos</figure-callout>}}^{2}2\mathrm{\&phi;}+\mathrm{Cos}\mathrm{\&delta;}{\mathrm{<figure-callout\; id="2"\; label="Sin"\; filenames="HDA0000148442460000011.png"\; state="\{\{state\}\}">Sin</figure-callout>}}^{2}2\mathrm{\&phi;}}{2},$ Then 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 through light intensity detector, and element under test thickness L, lambda1-wavelength are λ, and Δ n representes non-ordinary light and the difference of ordinary light owing to the different refractive indexes that cause of velocity of propagation, i.e. birefringence n.By formula δ=2 π (n2-n1) L/ λ, can know:

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

For polarimetry, can obtain any polarization state of light 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 representes oval ellipticity; ρ describes degree of polarization, changes 1 under it is worth from 0 under 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 is used for explaining the present invention; And be not to be used as qualification of the present invention; As long as in connotation scope of the present invention, the above embodiment is changed, modification all will drop in the scope of claims of the present invention.

Claims (2)

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.

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