CN102495535B - Method for obtaining mask three-dimensional vector space image based on Abbe vector imaging model - Google Patents

Abstract

The invention provides a method for obtaining a mask three-dimensional vector space image based on an Abbe vector imaging model. The method comprises the following steps of: rasterizing a mask graph M to be N*N sub-areas; rasterizing a light source surface grid to be multiple point light sources, and representing coordinates of a point light source relative to a grid area by a central point coordinate (xs, ys) of each grid area; calculating space image I (alphas, betas) at a wafer position in a non-ideal photoetching system when each point light source illuminates; overlaying the space image I (alphas, betas) relative to each point light source according to an Abbe method to obtain the space image I at the wafer position in the non-ideal photoetchcing system when partial relative light sources illuminate; further obtaining a space image Iclear on the wafer position in the photo-etching system when a mask does not have any graphs, and executing a normalization process on the Itotal by the Iclear to obtain relative light intensity distribution Irelative. According to the method for obtaining the mask three-dimensional vector space image based on the Abbe vector imaging model, disclosed by the invention, a three-dimensional incidence electric field on the mask and three-dimensional distribution of a diffraction frequency spectrum of a mask at a pupil entrance of a projection system are considered so that relative intensity of the space image is obtained, thus the method disclosed by the invention has larger applicability.

Description

Obtain the method for mask trivector aerial image based on Abbe vector imaging model

Technical field

The present invention relates to a kind ofly obtain the method for mask trivector aerial image, belong to photoetching resolution enhancement techniques field based on Abbe (Abbe) vector imaging model.

Background technology

Current large scale integrated circuit generally adopts etching system manufacturing.Etching system mainly is divided into: four parts such as illuminator (comprising light source and condenser), mask, optical projection system and wafer.The light that light source sends is incident to mask, the opening portion printing opacity of mask after focusing on through condenser; Through behind the mask, light is incident on the wafer that scribbles photoresist via optical projection system, so just mask graph is replicated on the wafer.

Reach with lower node along with photoetching technique enters 45nm, the critical size of circuit has been far smaller than the wavelength of exposure light source.Interference of light this moment and diffraction phenomena are more remarkable, cause optical patterning to produce distortion and fuzzy.Therefore etching system must adopt resolution enhance technology, in order to improve image quality.

Industry generally adopts immersion lithographic system at present.Immersion lithographic system is: added refractive index greater than 1 light transmission medium between the lower surface of last lens of projection objective and photoresist, enlarged numerical aperture (numerical aperture NA), improves the purpose of imaging resolution thereby play.Because immersion lithographic system has the characteristic of high NA (NA＞1), and when NA＞0.6, the vector imaging characteristic of electromagnetic field can not be ignored the influence of optical patterning.Therefore for immersion lithographic system, the scalar imaging model of optical patterning is no longer suitable.

Even but in the liquid immersion lithography imaging field, the part that still comprises in the vector imaging model of current most of commercial lithography simulation softwares in the scalar model is approximate.This comprises that mainly the two dimension of incident electric field on the mask is similar to and the paraxial of optical projection system entrance pupil place mask diffraction spectrum is similar to, these two kinds are similar in little NA etching system and have been proved to be enough accurately, but under the liquid immersion lithography imaging, these approximate meetings bring bigger error for the imaging of simulation photoetching vector.

Simultaneously, since the absolute strength of photoetching aerial image can be along with the difference of etching system exposure difference, the relative intensity of aerial image more highlights the imaging features of mask graph, and therefore relatively the relative intensity of aerial image has bigger reference value for analyzing lithographic results.

In order to describe the imaging characteristic of immersion lithographic system more accurately, resolution enhance technology in the research immersion lithographic system, must foundation accurately obtain the vector imaging model of etching system aerial image, so be necessary that removal is similar to for above-mentioned two kinds in the imaging model of setting up.

Pertinent literature (Proc.of SPIE 2010.7640:76402Y1-76402Y9.) has proposed a kind of method of calculating the photoetching aerial image at the partial coherence imaging system.But above method expression on the mask the incident electric field and all adopted two-dimentional approximate form during the diffraction spectrum of optical projection system place mask, what utilize that this method obtains is the absolute strength of aerial image, and this method does not provide the analytical expression of the matrix form between the etching system aerial image and mask graph under the vector imaging model yet, therefore is not suitable for the research of the etching system intermediate-resolution enhancement techniques optimization method of high NA.

Summary of the invention

The invention provides a kind of method of obtaining mask trivector aerial image based on Abbe vector imaging model, this method is in the process of computer memory picture, consider the three-dimensional incident electric field on the mask and the distributed in three dimensions of optical projection system entrance pupil place mask diffraction spectrum, made the aerial image that calculates have higher accuracy.

Realize that technical scheme of the present invention is as follows:

A kind ofly obtain the method for mask trivector aerial image based on Abbe vector imaging model, concrete steps are:

The setting world coordinates is: the direction with optical axis is the z axle, and according to the left-handed coordinate system principle with the z axle set up global coordinate system (x, y, z);

Step 101, the mask graph grid is turned to N * N sub regions;

Step 102, according to the shape of partial coherence light source surface of light source is tiled into a plurality of grid region, each grid region is as a pointolite, with each grid region center point coordinate (x _s, y _s) represent the pairing pointolite coordinate of this grid region;

Step 103, obtain expression etching system optical path difference scalar aberration matrix W (α ', β '), the Polarization aberration matrix J of expression etching system Polarization aberration (α ', β ') and the light phase changing capacity ξ that causes of etching system defocusing amount δ (α ', β '), wherein (α ', β ', γ ') be that global coordinate system carries out coordinate system behind the Fourier transform on the wafer;

Step 104, at a single point light source, utilize its coordinate (x _s, y _s), optical projection system object space numerical aperture NA _m, transmittance function, scalar aberration matrix W (α ', β '), the Polarization aberration matrix J (α ', β ') of mask and incident light phase place variable quantity ξ (α ', β '), when obtaining this spot light, the aerial image I (α in the etching system on the wafer position _s, β _s);

Step 105, when judging whether to calculate all a single point light illuminations, the aerial image in the imperfect etching system on the wafer position is if then enter step 106, otherwise return step 104;

Step 106, according to Abbe Abbe method, to the aerial image I (α of each pointolite correspondence _s, β _s) superpose, thereby obtain aerial image I on partial coherence when illumination wafer position _Total

Step 107, at the partial coherence light source of integral body, obtain when on the mask during the aerial image I in the etching system on the wafer position without any figure _Clear, utilize I _ClearTo the aerial image I on the wafer position in the step 106 _TotalCarry out normalized, obtain relative light intensity distribution I _Relative

The detailed process of step 104 of the present invention is:

Step 201, according to pointolite coordinate (x _s, y _s) and optical projection system object space numerical aperture NA _m, the three-dimensional incident electric field E of the light wave that the calculation level light source sends on mask _Incident

Step 202, according to the three-dimensional incident electric field E that obtains in the step 201 _Incident, utilize the amplitude transmittance function of mask, calculate near field distribution E through N * N sub regions on the mask, wherein, E is the vector matrix of N * N, and its each element is one 3 * 1 vector, 3 components of the diffraction near field distribution of mask subregion in the expression global coordinate system;

Step 203, obtain the Electric Field Distribution in optical projection system entrance pupil the place ahead according near field distribution E

Wherein Be the vector matrix of N * N, its each element is one 3 * 1 vector, 3 components of the Electric Field Distribution in entrance pupil the place ahead in the expression global coordinate system;

Step 204, according to the Electric Field Distribution in entrance pupil the place ahead With the electric field transformation matrix at entrance pupil place, obtain the Electric Field Distribution of light wave at optical projection system entrance pupil rear

Wherein,

Be the vector matrix of N * N, its each element is one 2 * 1 vector, the x of the Electric Field Distribution at entrance pupil rear and the component of y direction in the expression global coordinate system, and it axially is that the z durection component equals 0;

Step 205, to establish light wave direction of propagation in optical projection system approximate parallel with optical axis, further according to the Electric Field Distribution at entrance pupil rear

Scalar aberration matrix W (α ', β ') and the Polarization aberration matrix J (α ', β '), obtain the Electric Field Distribution of light wave in optical projection system emergent pupil the place ahead

Step 206, according to the Electric Field Distribution in optical projection system emergent pupil the place ahead

With the transformation matrix at optical projection system emergent pupil place, obtain the Electric Field Distribution at optical projection system emergent pupil rear

Step 207, utilize Wolf Wolf optical imagery theory, according to the Electric Field Distribution at emergent pupil rear And the variable quantity ξ of incident light phase place (α ', β '), obtain the Electric Field Distribution E on the wafer position ^Wafer, and according to E ^WaferAerial image I (α on the wafer position of acquisition point light source correspondence _s, β _s).

The detailed process of step 201 of the present invention is:

When the incident light on the mask is TE or TM polarized light:

Set the first local coordinate system (e _⊥, e _P), e wherein _⊥The direction of vibration of axle middle TE polarized light for light source emits beam, e _pThe direction of vibration of axle middle TM polarized light for light source emits beam; Then

$E_{\mathrm{incident}}=\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\·\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]---(1-A)$

Wherein, E _x, E _yAnd E _zBe respectively the component of electric field in global coordinate system that light source sends light wave, E _⊥And E _PBe the component of electric field in first local coordinate system that light source sends light wave, transition matrix T is:

$T=\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ 0& {\mathrm{\ρ}}_s\end{array}\right]$

Wherein, ${\mathrm{\ρ}}_s=\sqrt{{\mathrm{\α}}_s^2+{\mathrm{\β}}_s^2}.$

When the incident light on the mask is X or Y polarized light:

Set the second local coordinate system (e _i, e _j), e _iThe emit beam direction of vibration of middle X polarized light of axle and light source is consistent, e _jThe emit beam direction of vibration of middle Y polarized light of axle and light source is consistent, then

$E_{\mathrm{incident}}=\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\·\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]=\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ 0& {\mathrm{\ρ}}_s\end{array}\right]\begin{array}{c}\end{array}\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}\\ -\frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}\end{array}\right]\·\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]$ (1-B)

$=\left[\begin{array}{cc}1-\frac{{\mathrm{\α}}_s^2}{1+{\mathrm{\γ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\β}}_s}{1+{\mathrm{\γ}}_s}\\ -\frac{{\mathrm{\α}}_s{\mathrm{\β}}_s}{1+{\mathrm{\γ}}_s}& 1-\frac{{\mathrm{\β}}_s^2}{1+{\mathrm{\γ}}_s}\\ -{\mathrm{\α}}_s& -{\mathrm{\β}}_s\end{array}\right]\·\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]$

Wherein, E _iAnd E _jBe the component of electric field in second local coordinate system that light source sends light wave.

Beneficial effect

At first, the present invention has considered the three-dimensional incident electric field on the mask in the process of utilizing Abbe Model Calculation aerial image, and the distributed in three dimensions of optical projection system entrance pupil place mask diffraction spectrum.Therefore the inventive method can be obtained in the etching system aerial image of image planes position arbitrarily exactly, satisfies 45nm and with the lithography simulation requirement of lower node.

Secondly, the present invention has not only obtained the absolute strength of the aerial image of etching system when calculating the aerial image of mask, also by choosing suitable reference conditions, obtained the relative intensity of aerial image, makes the present invention have bigger applicability.

Once more, the present invention has set up the analytical expression of the matrix form of imperfect etching system aerial image under the vector imaging model, helps the sequencing processing of optical patterning model and the research of high NA etching system intermediate-resolution enhancement techniques optimization method.

Description of drawings

The process flow diagram of Fig. 1 imperfect etching system aerial image method for the present invention calculates.

Fig. 2 for pointolite send light wave after mask, optical projection system on wafer position the synoptic diagram of imaging.

Fig. 3 is the decomposing schematic representation of incident polarized light on the mask.

Fig. 4 is the deflection synoptic diagram of big NA projection lithography system entrance pupil place light.

Fig. 5 is the two-dimentional intensive linear mask aerial image that utilizes method among the present invention to obtain, and adopt the inventive method and the comparison synoptic diagram that adopts existing method simulation result.

Fig. 6 is the intensive linear mask of the one dimension aerial image that utilizes method among the present invention to obtain, and adopt the inventive method and the comparison synoptic diagram that adopts existing method simulation result.

Fig. 7 is the intensive contact hole structure mask aerial image that utilizes method among the present invention to obtain, and adopt the inventive method and the comparison synoptic diagram that adopts existing method simulation result.

Embodiment

Further the present invention is described in detail below in conjunction with accompanying drawing.

Variable predefine

As shown in Figure 2, the direction of setting optical axis is the z axle, and according to the left-handed coordinate system principle with the z axle set up global coordinate system (x, y, z).If the world coordinates of any point light source is (x on the partial coherence light source face _s, y _s, z _s), the direction cosine of being sent and be incident to the plane wave of mask by this pointolite are (α _s, β _s, γ _s), then the pass between world coordinates and the direction cosine is:

α _s＝x _s·NA _m，β _s＝y _s·NA _m， ${\mathrm{\γ}}_s=\cos \left[{\sin }^{-1}({\mathrm{NA}}_m\·\sqrt{x_s^2+y_s^2})\right]$

Wherein, NA _mBe optical projection system object space numerical aperture.

If the world coordinates of any point is (x on the mask, y, z), based on diffraction principle, the direction cosine that are incident to the plane wave of optical projection system entrance pupil from mask are (α, beta, gamma), (α wherein, β, be that mask (object plane) goes up that (x, y z) carry out coordinate system behind the Fourier transform to global coordinate system γ).

If it is (x that wafer (image planes) is gone up the world coordinates of any point _w, y _w, z _w), the direction cosine that are incident to the plane wave of image planes from the optical projection system emergent pupil are (α ', β ', γ '), and wherein (α ', β ', γ ') be that wafer (image planes) is gone up global coordinate system (x _w, y _w, z _w) carry out the coordinate system behind the Fourier transform.

As shown in Figure 1, obtain the method for mask trivector aerial image based on Abbe vector imaging model:

Step 101, the mask graph grid is turned to N * N sub regions.

Step 102, according to the shape of partial coherence light source surface of light source is tiled into a plurality of grid region, each grid region is as an electric light source (promptly approximate with pointolite), each grid region center point coordinate (x _s, y _s) represent the pairing pointolite coordinate of this grid region.

Step 103, obtain the scalar aberration matrix W (α ', β ') of expression etching system optical path difference and represent the Polarization aberration matrix J (α ', β ') of etching system Polarization aberration; Again according to the defocusing amount δ of etching system, obtain the phase changing capacity ξ (α ', β ') of propagating light in the etching system that causes by described defocusing amount δ.

W (α ', β ') and J (α ', β ') are the matrix of N * N; Each element is a numerical value in W (α ', the β ') matrix, the actual corrugated at its expression emergent pupil place and the wavelength number that desirable corrugated differs; J (α ', β ') be the vector matrix of one N * N, each matrix element is a Jones matrix, because TE and TM polarized light by transition matrix, all be expressed as the form of xy component, so Jones matrix concrete form are:

$J({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)=\left(\begin{array}{cc}{J}_{\mathrm{xy}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)& J_{\mathrm{xy}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)\\ J_{\mathrm{xy}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)& J_{\mathrm{yy}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)\end{array}\right)$ m，n＝1，2，...，N

J _{I ', j '}(α ', β ', m, n) (i '=x, y, j '=x, y) expression incident i ' polarized light is through becoming the ratio of j ' polarized light after the optical projection system.

The phase changing capacity of propagating light can be expressed as in the caused etching system of defocusing amount δ:

$\mathrm{\ξ}=k^{\′}\·n_w\·\mathrm{\δ}\·(1-{\mathrm{\γ}}^{\′})=k^{\′}\·n_w\·\mathrm{\δ}\·(1-\sqrt{1-{\mathrm{\α}}^{\′2}-{\mathrm{\β}}^{\′2}})$

Wherein,

Be wave number, ξ is the scalar matrix of a N * N, and each element representation is through the phase change of light wave in etching system of certain point on the pupil in the matrix.

Step 104, at a single point light source, utilize its coordinate (x _s, y _s), optical projection system object space numerical aperture NA _m, the variable quantity ξ (α ', β '), scalar aberration matrix W (α ', β ') of transmittance function, incident light phase place of mask and Polarization aberration matrix J (α ', β '), the aerial image I (α when obtaining this spot light on the corresponding wafer position _s, β _s).

Step 105, when judging whether to calculate all a single point light illuminations, the aerial image in the imperfect etching system on the wafer position is if then enter step 106, otherwise return step 104.

Step 106, according to the Abbe method, to the aerial image I (α of each pointolite correspondence _s, β _s) superpose, thereby the aerial image I on the wafer position when obtaining the partial coherence light illumination _Total

Step 107, at the partial coherence light source of integral body, obtain when on the mask during the aerial image I in the etching system on the wafer position without any figure _Clear, utilize I _ClearTo the aerial image I on the wafer position in the step 106 _TotalCarry out normalized, obtain relative light intensity distribution I _Relative

The I here _ClearAcquisition methods and step 104 I in the step 106 _TotalThe acquisition methods basically identical, the step 201 to 207 that idiographic flow can vide infra.The difference of the two is to obtain I _ClearThe time mask difference that adopted, here above the mask of Cai Yonging without any graphic structure, promptly the matrix element of the mask matrix of N * N is equal to 1, and is identical in all the other steps and 104.Obtain I like this _ClearAfter, can obtain relative light intensity distribution I _Relative

Below to step 104 at a single point light source, when obtaining this spot light, the aerial image I (α when obtaining this spot light on the corresponding wafer position _s, β _s) process be further elaborated.

Step 201, according to pointolite coordinate (x _s, y _s) and optical projection system object space numerical aperture NA _m, the three-dimensional incident field E of the light wave that the calculation level light source sends on mask _Incident

When the incident light on the mask is TE or TM polarized light, set up the first local coordinate system (e _⊥, e _P), e _⊥The direction of vibration of axle middle TE polarized light for light source emits beam, e _PThe direction of vibration of axle middle TM polarized light for light source emits beam is shown among Fig. 3 301.Wave vector is And

α _s＝x _s·NA _m，β _s＝y _s·NA _m， ${\mathrm{\γ}}_s=\cos \left[{\sin }^{-1}({\mathrm{NA}}_m\·\sqrt{x_s^2+y_s^2})\right]$

Wherein, (x _s, y _s) be the world coordinates of any point light source on the partial coherence light source face, NA _mBe optical projection system object space numerical aperture.The plane that is made of wave vector and optical axis is called the plane of incidence, and the direction of vibration of TM polarized light is in the plane of incidence, and the direction of vibration of TE polarized light is perpendicular to the plane of incidence.TE and TM polarized light can be expressed as under first local coordinate system like this

$E_{\mathrm{TE}}=\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right],$ $E_{\mathrm{TM}}=\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]=\left[\begin{array}{c}0\\ 1\end{array}\right]$

Then under global coordinate system, this incident electric field can be expressed as:

$E_{\mathrm{incident}}=\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\·\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]---(1-A)$

Wherein, E _x, E _yAnd E _zBe respectively the component of electric field in global coordinate system that light source sends light wave, E _⊥And E _PBe the component of electric field in first local coordinate system that light source sends light wave, transition matrix T is:

$T=\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ 0& {\mathrm{\ρ}}_s\end{array}\right]$

Wherein, ${\mathrm{\ρ}}_s=\sqrt{{\mathrm{\α}}_s^2+{\mathrm{\β}}_s^2}.$

When the incident light on the mask is X or Y polarized light, set up the second local coordinate system (e _i, e _j), e _iThe emit beam direction of vibration of middle X polarized light of axle and light source is consistent, e _jThe emit beam direction of vibration of middle Y polarized light of axle and light source is consistent, shown among Fig. 3 302.Wave vector is

When wave vector is parallel with optical axis, e _iAxle overlaps e with X-axis in the global coordinate system _jAxle overlaps with Y-axis in the global coordinate system, and e _iAxle and e _jAxle, e ⊥ axle and eP axle coplane are all perpendicular to the direction of wave vector.X and Y polarized light can be expressed as under second local coordinate system like this:

$E_X=\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]=\left[\begin{array}{c}1\\ 0\end{array}\right],$ $E_Y=\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]=\left[\begin{array}{c}0\\ 1\end{array}\right]$

Because e _iAxle and e _jAxle, e _⊥Axle and e _PThe axle coplane is so can at first be illustrated in coordinate system one (e to X or Y polarized light _⊥, e _P) under, and then utilize the transition matrix T of front that X or Y polarized light are illustrated under the global coordinate system, that is:

$E_{\mathrm{incident}}=\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\·\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]=\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ 0& {\mathrm{\ρ}}_s\end{array}\right]\begin{array}{c}\end{array}\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}\\ -\frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}\end{array}\right]\·\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]$ (1-B)

$=\left[\begin{array}{cc}1-\frac{{\mathrm{\α}}_s^2}{1+{\mathrm{\γ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\β}}_s}{1+{\mathrm{\γ}}_s}\\ -\frac{{\mathrm{\α}}_s{\mathrm{\β}}_s}{1+{\mathrm{\γ}}_s}& 1-\frac{{\mathrm{\β}}_s^2}{1+{\mathrm{\γ}}_s}\\ -{\mathrm{\α}}_s& -{\mathrm{\β}}_s\end{array}\right]\·\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]$

Wherein, E _x, E _yAnd E _zBe respectively the component of electric field in global coordinate system that light source sends light wave, E _iAnd E _jBe the component of electric field in second local coordinate system that light source sends light wave.

Step 202, according to the three-dimensional incident electric field that obtains in the step 201, utilize the amplitude transmittance function of mask, calculate near field distribution E through N * N sub regions on the mask;

Wherein, E is that the vector matrix of N * N is (if all elements of a matrix is matrix or vector, then be called vector matrix), each element in this vector matrix is one 3 * 1 vector, 3 components of the diffraction near field distribution of mask subregion in the expression global coordinate system.M is the amplitude transmittance function of mask.E represents that two matrix corresponding elements multiply each other. Be the vector matrix of one N * N, each element is the electric field intensity of electric field in global coordinate system that pointolite sends light wave, shown in (1) formula.

The diffraction matrices B of mask is the scalar matrix of one N * N, and each element is single numerical value in the scalar matrix.Approximate according to Hopkins (Thelma Hopkins), each element of B can be expressed as:

$B(m,n)=\exp \left(\frac{j2\mathrm{\π}{\mathrm{\β}}_{s}x}{\mathrm{\λ}}\right)\exp \left(\frac{j2\mathrm{\π}{\mathrm{\α}}_{s}y}{\mathrm{\λ}}\right)$

$=\exp \left(\frac{j2\mathrm{\π}{\mathrm{\β}}_{s}m\×\mathrm{pixel}}{\mathrm{\λ}}\right)\exp \left(\frac{j2\mathrm{\π}{\mathrm{\α}}_{s}n\×\mathrm{pixel}}{\mathrm{\λ}}\right),$ m，n＝1，2，...，N

Wherein, pixel represents the length of side of all subregion on the mask graph.

Step 203, obtain the Electric Field Distribution in optical projection system entrance pupil the place ahead according near field distribution E

Each subregion on the mask is all regarded as a secondary sub-light source, with the center of subregion coordinate as this sub-light source.According to the Fourier optics theory, the Electric Field Distribution in optical projection system entrance pupil the place ahead can be expressed as the function of α and β:

$E_I^{\mathrm{ent}}(\mathrm{\α},\mathrm{\β})=\frac{\mathrm{\γ}}{\mathrm{j\λ}}\frac{e^{-\mathrm{jkr}}}{r}F\left\{E\right\}---\left(3\right)$

Wherein, owing to have N * N sub regions on the mask, so the Electric Field Distribution in entrance pupil the place ahead

Be the vector matrix of N * N, each element in this vector matrix is one 3 * 1 vector, 3 components of the Electric Field Distribution in entrance pupil the place ahead in the expression global coordinate system.F{} represents Fourier transform, and r is the entrance pupil radius,

Be wave number, λ is the wavelength that pointolite sends light wave, n _mBe the object space medium refraction index.

Step 204, according to the Electric Field Distribution in entrance pupil the place ahead

With the electric field transformation matrix at entrance pupil place, obtain the Electric Field Distribution of light wave at optical projection system entrance pupil rear

As shown in Figure 4, when the incident angle of optical projection system entrance pupil the place ahead light wave increased along with the increase of NA, the axial component of ignoring the mask diffraction spectrum was inaccurate.At this moment, light wave is through optical projection system entrance pupil sphere the time, and its wave vector changes, and promptly at entrance pupil the place ahead wave vector and optical axis certain included angle is arranged, and parallel with optical axis at entrance pupil rear wave vector, this causes electric field component to change.

The electric field at optical projection system entrance pupil rear is:

$E_b^{\mathrm{ent}}(\mathrm{\α},\mathrm{\β})={\mathrm{\Ψ}}_{\mathrm{entrance}}\·E_I^{\mathrm{ent}}(\mathrm{\α},\mathrm{\β})---\left(4\right)$

Wherein,

Be the vector matrix of N * N, its each element is one 2 * 1 vector, the X and the Y component of Electric Field Distribution at entrance pupil rear in the expression global coordinate system, its axially (Z to) component equal 0.Ψ _EntranceBe the electric field transformation matrix at optical projection system entrance pupil place, it is the vector matrix of N * N, represents light wave through the optical projection system entrance pupil time, the variation of the electric field component of mask diffraction spectrum, and each element is one 2 * 3 vector in this matrix:

${\mathrm{\Ψ}}_{\mathrm{entrance}}=\left[\begin{array}{ccc}\frac{{\mathrm{\β}}^2+{\mathrm{\α}}^2\mathrm{\γ}}{1-{\mathrm{\γ}}^2}& -\frac{\mathrm{\α\β}}{1+\mathrm{\γ}}& -\mathrm{\α}\\ -\frac{\mathrm{\α\β}}{1+\mathrm{\γ}}& \frac{{\mathrm{\α}}^2+{\mathrm{\β}}^2\mathrm{\γ}}{1-{\mathrm{\γ}}^2}& -\mathrm{\β}\end{array}\right]$

Step 205, to establish light wave direction of propagation in optical projection system approximate parallel with optical axis, further according to the Electric Field Distribution at entrance pupil rear

Scalar aberration matrix W (α ', β ') and the Polarization aberration matrix J (α ', β '), obtain the Electric Field Distribution of light wave in optical projection system emergent pupil the place ahead

The detailed process of this step is:

For aberrationless preferred view system, the mapping process of entrance pupil rear and emergent pupil the place ahead Electric Field Distribution can be expressed as the form of a low-pass filter function and a modifying factor product, that is:

${\hat{E}}_I^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})=\mathrm{cUe}{E}_b^{\mathrm{ent}}(\mathrm{\α},\mathrm{\β})$

Wherein, c is a modifying factor, and low-pass filter function U is the scalar matrix of N * N, and the numerical aperture of expression optical projection system is to the limited receiving ability of diffraction spectrum, and promptly the value in pupil inside is 1, and the value of pupil outside is 0, specifically is expressed as follows:

$U=\left\{\begin{array}{cc}1& \sqrt{f^2+g^2}\≤1\\ 0& \mathrm{elsewhere}\end{array}\right.$

Wherein, (f g) is normalized world coordinates on the entrance pupil.

Modifying factor c can be expressed as:

$c=\frac{r}{r^{\′}}\sqrt{\frac{{\mathrm{\γ}}^{\′}}{\mathrm{\γ}}}\frac{n_w}{R}$

Wherein, r and r ' are respectively optical projection system entrance pupil and emergent pupil radius, n _wBe the refractive index of optical projection system picture side immersion liquid, R is the reduction magnification of preferred view system, is generally 4.

Because the approximate optical axis that is parallel in the direction of propagation of light wave between optical projection system entrance pupil and emergent pupil, therefore for arbitrarily (α ', β '), the entrance pupil rear is identical with phase differential between emergent pupil the place ahead.Owing to finally will find the solution the aerial image (being light distribution) on the wafer, so the constant phase difference in entrance pupil rear and emergent pupil the place ahead can be ignored.Thereby the Electric Field Distribution that can obtain emergent pupil the place ahead is:

${\hat{E}}_I^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})=\frac{1}{\mathrm{\λ}{r}^{\′}}\sqrt{{\mathrm{\γ}}^{\′}\mathrm{\γ}}\frac{n_w}{R}\mathrm{Ue}{\mathrm{\Ψ}}_{\mathrm{entrance}}\mathrm{eF}\left\{E\right\}$

For the imperfect etching system of reality, consider the influence of its scalar aberration matrix W (α ', β ') and Polarization aberration matrix J (α ', β '), obtain the Electric Field Distribution in imperfect etching system emergent pupil the place ahead:

$E_I^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})=\frac{1}{\mathrm{\λ}{r}^{\′}}\sqrt{{\mathrm{\γ}}^{\′}\mathrm{\γ}}\frac{n_w}{R}\mathrm{UeJ}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})e{\mathrm{\Ψ}}_{\mathrm{entrance}}\mathrm{eF}\left\{E\right\}e{e}^{j2\mathrm{\πW}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})}---\left(5\right)$

Step 206, according to the Electric Field Distribution in optical projection system emergent pupil the place ahead

With the transformation matrix at optical projection system emergent pupil place, obtain the Electric Field Distribution at optical projection system emergent pupil rear

Similar with entrance pupil, according to the rotation effect of TM component between emergent pupil the place ahead and rear of electromagnetic field, to establish in the global coordinate system, the forward and backward side's of emergent pupil electric field is expressed as: the vector matrix of N * N With

With

Each element as follows:

$E_I^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)={[E_{\mathrm{lx}}^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n);E_{\mathrm{ly}}^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)]}^T$

$E_b^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)={[E_{\mathrm{bx}}^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n);E_{\mathrm{by}}^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n);E_{\mathrm{bz}}^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′},m,n)]}^T$

Wherein, m, n=1,2 ..., N, α '=cos φ ' sin θ ', β '=sin φ ' sin θ ', γ '=cos θ ', promptly the optical projection system emergent pupil is incident to the direction cosine (wave vector) of the plane wave of image planes and is

φ ' and θ ' are respectively the position angle and the elevations angle of wave vector, then

With

Relational expression be:

$E_b^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})={\mathrm{\Ψ}}_{\mathrm{exit}}e{E}_I^{\mathrm{ext}}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})---\left(6\right)$

Wherein, Ψ _ExitBe the vector matrix of a N * N, each element is one 3 * 2 matrix:

${\mathrm{\Ψ}}_{\mathrm{exit}}=\left[\begin{array}{cc}\frac{{\mathrm{\β}}^{\′2}+{\mathrm{\α}}^{\′2}{\mathrm{\γ}}^{\′}}{1-{\mathrm{\γ}}^{\′2}}& -\frac{{\mathrm{\α}}^{\′}{\mathrm{\β}}^{\′}}{1+{\mathrm{\γ}}^{\′}}\\ -\frac{{\mathrm{\α}}^{\′}{\mathrm{\β}}^{\′}}{1+{\mathrm{\γ}}^{\′}}& \frac{{\mathrm{\α}}^{\′2}+{\mathrm{\β}}^{\′2}{\mathrm{\γ}}^{\′}}{1-{\mathrm{\γ}}^{\′2}}\\ -{\mathrm{\α}}^{\′}& -{\mathrm{\β}}^{\′}\end{array}\right]$

Step 207, utilize the optical imagery theory of Wolf, according to the Electric Field Distribution at emergent pupil rear

And the variable quantity ξ of incident light phase place (α ', β '), obtain the Electric Field Distribution E on the wafer position ^Wafer, and the aerial image I (α on the corresponding wafer position of further acquisition point light source _s, β _s).

The detailed process of this step is:

Electric Field Distribution on the imperfect etching system wafer position is shown in (7) formula:

$E^{\mathrm{wafer}}=\frac{2\mathrm{\π\λ}{r}^{\′}}{j{n}_w^2}{e}^{{\mathrm{jk}}^{\′}{r}^{\′}}{F}^{-1}\left\{\frac{e^{\mathrm{j\ξ}}}{{\mathrm{\γ}}^{\′}}e{E}_b^{\mathrm{ext}}\right\}---\left(7\right)$

Wherein,

F ^-1{ } is inverse Fourier transform.(1) formula, (5) formula and (6) formula are updated in (7) formula, and ignore the constant phase item, can obtain pointolite (α _s, β _s) Electric Field Distribution of image planes when throwing light on, that is:

$E^{\mathrm{wafer}}({\mathrm{\α}}_s,{\mathrm{\β}}_s)=\frac{2\mathrm{\π}}{n_{w}R}{F}^{-1}\{\sqrt{\frac{\mathrm{\γ}}{{\mathrm{\γ}}^{\′}}}e{e}^{\mathrm{j\ξ}}e{\mathrm{\Ψ}}_{\mathrm{exit}}\mathrm{eUeJ}({\mathrm{\α}}^{\′},{\mathrm{\β}}^{\′})---\left(8\right)$

Because E _i' middle element value and mask coordinate are irrelevant, so following formula can also be write as:

$E^{\mathrm{wafer}}({\mathrm{\α}}_s,{\mathrm{\β}}_s)=\frac{2\mathrm{\π}}{n_{w}R}{F}^{-1}\left\{V^{\′}\right\}\⊗\left(\mathrm{Be\; M}\right)$

Wherein,

The expression convolution,

Wherein, V ' is the vector matrix of N * N, and each matrix element is 3 * 1 vector (v _x', v _y', v _z') ^T, v wherein _x', v _y', v _z' be the function of α ' and β '.

E then ^Wafer(α _s, β _s) three components in global coordinate system are

$E_P^{\mathrm{wafer}}({\mathrm{\α}}_s,{\mathrm{\β}}_s)=H_p\⊗\left(\mathrm{Be\; M}\right)$

Wherein,

P=x, y, z, wherein V _p' be the scalar matrix of N * N, formed by the x component of each element of vector matrix V '.

$I({\mathrm{\α}}_s,{\mathrm{\β}}_s)=\underset{p=x,y,z}{\mathrm{\Σ}}{\left|\right|H_p\⊗\left(\mathrm{Be\; M}\right)\left|\right|}_2^2$

Wherein

Expression is to the matrix delivery and ask square.H wherein _pBe (α with B _s, β _s) function, be designated as respectively

With Therefore following formula can be designated as:

$I({\mathrm{\α}}_s,{\mathrm{\β}}_s)=\underset{p=x,y,z}{\mathrm{\Σ}}{\left|\right|H_p^{{\mathrm{\α}}_s{\mathrm{\β}}_s}\⊗\left(B^{{\mathrm{\α}}_s{\mathrm{\β}}_s}\mathrm{e\; M}\right)\left|\right|}_2^2$

Following formula obtains when being spot light in the imperfect etching system, and the aerial image on the wafer position distributes, according to the Abbe principle, then in the step 106 under the partial coherence light illumination in the imperfect etching system aerial image on the wafer position can be expressed as:

$I_{\mathrm{total}}=\frac{1}{N}\underset{{\mathrm{\α}}_s}{\mathrm{\Σ}}\underset{{\mathrm{\β}}_s}{\mathrm{\Σ}}\underset{p=x,y,z}{\mathrm{\Σ}}{\left|\right|H_p^{{\mathrm{\α}}_s{\mathrm{\β}}_s}\⊗\left(B^{{\mathrm{\α}}_s{\mathrm{\β}}_s}\mathrm{eM}\right)\left|\right|}_2^2---\left(9\right)$

Wherein, N _sIt is the sampling number of partial coherence light source.

According to described in the step 107, can obtain reference light intensity I under the partial coherence illumination according to top flow process _ClearAfter, it also is the scalar matrix of a N * N.Then relative light intensity is:

I _relative＝I _total./I _clear (10)

In the following formula, " ./" represent that the corresponding element in two matrixes is divided by.

Embodiment of the present invention:

As shown in Figure 5, the aerial image result of the 501 two-dimentional intensive linear masks that obtain for emulation.502 is the comparison that utilizes the resulting simulation result of method among the present invention and adopt the simulation result of existing method.Wherein 503 for only considering the two-dimentional incident field on the mask, and the aerial image result who obtains when ignoring mask diffraction spectrum axial component.504 is the aerial image result who utilizes method of the present invention to obtain.Root-mean-square error between 503 and 504 is 0.97%, and the error that is embodied on the optical patterning result is 1.5nm.

As shown in Figure 6, the aerial image result of the 601 intensive linear masks of one dimension that obtain for emulation.602 for utilizing the resulting simulation result of method and the comparison of adopting existing method simulation result among the present invention.Wherein 603 for only considering the two-dimentional incident field on the mask, and the aerial image result who obtains when ignoring mask diffraction spectrum axial component.604 is the aerial image result who utilizes method of the present invention to obtain.Root-mean-square error between 603 and 604 is 0.91%, and the error that is embodied on the optical patterning result is 0.9nm.

As shown in Figure 7, the aerial image result of the 701 contact hole structure masks that obtain for emulation.702 is the comparison that utilizes the resulting simulation result of method among the present invention and adopt the simulation result of existing method.Wherein 703 for only considering the two-dimentional incident field on the mask, and the aerial image result who obtains when ignoring mask diffraction spectrum axial component.704 is the aerial image result who utilizes method of the present invention to obtain.Root-mean-square error between 703 and 704 is 3.78%, and the error that is embodied on the optical patterning result is 6.4nm.

Respectively comparison diagram 5, Fig. 6 and Fig. 7 as can be known, the actual three-dimensional incident field on the mask distributes and the axial component of mask diffraction spectrum has considerable influence to the optical patterning result, especially for the mask of contact hole structure, its error even reached near 10%.Above the result proved exist in the model that has used forefathers to set up than mistake and meaning that the present invention possessed.Because method of the present invention is that the weak point in the existing vector imaging model is improved, so can realize imperfect partial coherence is photo-etched into the accurate simulation of picture, make the more approaching actual photolithographic exposure result of simulation result, therefore can reduce image error greatly, accurately foretell lithography performance.

Though described the specific embodiment of the present invention in conjunction with the accompanying drawings, for those skilled in the art, under the premise of not departing from the present invention, can also do some distortion, replacement and improvement, these also are considered as belonging to protection scope of the present invention.

Claims (3)

1. one kind is obtained the method for mask trivector aerial image based on Abbe vector imaging model, it is characterized in that concrete steps are:

The setting world coordinates is: the direction with optical axis is the z axle, and according to the left-handed coordinate system principle with the z axle set up global coordinate system (x, y, z);

Step 101, the mask pattern grid is turned to N * N sub regions;

Step 102, according to the shape of partial coherence light source surface of light source is tiled into a plurality of grid region, each grid region is as a pointolite, with each grid region center point coordinate (x _s, y _s) represent the pairing pointolite coordinate of this grid region;

Step 103, obtain expression etching system optical path difference scalar aberration matrix W (α ', β '), the Polarization aberration matrix J of expression etching system Polarization aberration (α ', β ') and the light phase changing capacity ξ that causes of etching system defocusing amount δ (α ', β '), wherein (α ', β ', γ ') be that global coordinate system carries out coordinate system behind the Fourier transform on the wafer;

Step 104, at a single point light source, utilize its coordinate (x _s, y _s), optical projection system object space numerical aperture NA _m, transmittance function, scalar aberration matrix W (α ', β '), the Polarization aberration matrix J (α ', β ') of mask and incident light phase place variable quantity ξ (α ', β '), when obtaining this spot light, the aerial image I (α in the etching system on the wafer position _s, β _s);

Step 105, when judging whether to calculate all a single point light illuminations, the aerial image in the imperfect etching system on the wafer position is if then enter step 106, otherwise return step 104;

Step 106, according to Abbe Abbe method, to the aerial image I (α of each pointolite correspondence _s, β _s) superpose, thereby obtain aerial image I on partial coherence when illumination wafer position _Total

Step 107, at the partial coherence light source of integral body, obtain when on the mask during the aerial image I in the etching system on the wafer position without any figure _Clear, utilize I _ClearTo the aerial image I on the wafer position in the step 106 _TotalCarry out normalized, obtain relative light intensity distribution I _Relative

2. obtain the method for mask trivector aerial image according to claim 1 is described based on Abbe vector imaging model, it is characterized in that the detailed process of described step 104 is:

Step 201, according to pointolite coordinate (x _s, y _s) and optical projection system object space numerical aperture NA _m, the three-dimensional incident electric field E of the light wave that the calculation level light source sends on mask _Incident

Step 202, according to the three-dimensional incident electric field E that obtains in the step 201 _Incident, utilize the amplitude transmittance function of mask, calculate near field distribution E through N * N sub regions on the mask, wherein, E is the vector matrix of N * N, and its each element is one 3 * 1 vector, 3 components of the diffraction near field distribution of mask subregion in the expression global coordinate system;

Step 203, obtain the Electric Field Distribution in optical projection system entrance pupil the place ahead according near field distribution E

Wherein

Be the vector matrix of N * N, its each element is one 3 * 1 vector, 3 components of the Electric Field Distribution in entrance pupil the place ahead in the expression global coordinate system;

Step 204, according to the Electric Field Distribution in entrance pupil the place ahead

With the electric field transformation matrix at entrance pupil place, obtain the Electric Field Distribution of light wave at optical projection system entrance pupil rear

Wherein,

Be the vector matrix of N * N, its each element is one 2 * 1 vector, the x of the Electric Field Distribution at entrance pupil rear and the component of y direction in the expression global coordinate system, and it axially is that the z durection component equals 0;

Step 205, to establish light wave direction of propagation in optical projection system approximate parallel with optical axis, further according to the Electric Field Distribution at entrance pupil rear

Scalar aberration matrix W (α ', β ') and the Polarization aberration matrix J (α ', β '), obtain the Electric Field Distribution of light wave in optical projection system emergent pupil the place ahead

Step 206, according to the Electric Field Distribution in optical projection system emergent pupil the place ahead

With the transformation matrix at optical projection system emergent pupil place, obtain the Electric Field Distribution at optical projection system emergent pupil rear

Step 207, utilize Wolf Wolf optical imagery theory, according to the Electric Field Distribution at emergent pupil rear

And the variable quantity ξ of incident light phase place (α ', β '), obtain the Electric Field Distribution E on the wafer position ^Wafer, and according to E ^WaferAerial image I (α on the wafer position of acquisition point light source correspondence _s, β _s).

3. obtain the method for mask trivector aerial image according to claim 2 is described based on Abbe vector imaging model, it is characterized in that the detailed process of described step 201 is:

When the incident light on the mask is TE or TM polarized light:

Set the first local coordinate system (e _⊥, e _P), e wherein _⊥The direction of vibration of axle middle TE polarized light for light source emits beam, e _PThe direction of vibration of axle middle TM polarized light for light source emits beam; Then

$E_{\mathrm{incident}}=\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\·\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]---(1-A)$

Wherein, E _x, E _yAnd E _zBe respectively the component of electric field in global coordinate system that light source sends light wave, E _⊥And E _PBe the component of electric field in first local coordinate system that light source sends light wave, transition matrix T is:

$T=\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ 0& {\mathrm{\ρ}}_s\end{array}\right]$

Wherein, ${\mathrm{\ρ}}_s=\sqrt{{\mathrm{\α}}_s^2+{\mathrm{\β}}_s^2}.$

When the incident light on the mask is X or Y polarized light:

Set the second local coordinate system (e _i, e _j), e _iThe emit beam direction of vibration of middle X polarized light of axle and light source is consistent, e _jThe emit beam direction of vibration of middle Y polarized light of axle and light source is consistent, then

$E_{\mathrm{incident}}=\left[\begin{array}{c}{E}_x\\ E_y\\ E_z\end{array}\right]=T\·\left[\begin{array}{c}{E}_{\⊥}\\ E_P\end{array}\right]=\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s{\mathrm{\γ}}_s}{{\mathrm{\ρ}}_s}\\ 0& {\mathrm{\ρ}}_s\end{array}\right]\begin{array}{c}\end{array}\left[\begin{array}{cc}-\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}& \frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}\\ -\frac{{\mathrm{\α}}_s}{{\mathrm{\ρ}}_s}& -\frac{{\mathrm{\β}}_s}{{\mathrm{\ρ}}_s}\end{array}\right]\·\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]$ (1-B)

$=\left[\begin{array}{cc}1-\frac{{\mathrm{\α}}_s^2}{1+{\mathrm{\γ}}_s}& -\frac{{\mathrm{\α}}_s{\mathrm{\β}}_s}{1+{\mathrm{\γ}}_s}\\ -\frac{{\mathrm{\α}}_s{\mathrm{\β}}_s}{1+{\mathrm{\γ}}_s}& 1-\frac{{\mathrm{\β}}_s^2}{1+{\mathrm{\γ}}_s}\\ -{\mathrm{\α}}_s& -{\mathrm{\β}}_s\end{array}\right]\·\left[\begin{array}{c}{E}_i\\ E_j\end{array}\right]$

Wherein, E _iAnd E _jBe the component of electric field in second local coordinate system that light source sends light wave.

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