CN102636951B - Computing method for diffraction field of double-absorbing-layer alternating phase shift contact hole mask - Google Patents

Abstract

The invention provides a computing method for a diffraction field of a double-absorbing-layer alternating phase shift contact hole mask. According to the method, diffraction of the double-absorbing-layer alternating phase shift contact hole mask in photo-etching can be quickly computed. The computing method specifically comprises the following steps of: 1, setting harmonic number reserved in the x direction and setting harmonic number reserved in the y direction; 2, solving components of wave vector of each diffraction order in the tangential direction and the normal direction according to a Floquet condition; 3, performing Fourier series expansion on dielectric constant of a two-dimensional grating on each layer and a reciprocal of the dielectric constant; and 4, solving the diffraction field of an emission region by using an enhanced transmission matrix method. In two orthogonal directions, the Fourier series expansion is performed by selecting the minimum common multiple of cycles of three grating layers in the corresponding orthogonal direction, so that the diffraction of a plurality of layers of two-dimensional mask gratings with different cycles in the two orthogonal directions can be analyzed, and meanwhile, the diffraction field of the double-absorbing-layer alternating phase shift contact hole mask can be quickly solved.

Description

The computing method of double absorption layer alternating phase-shift contact hole mask diffractional field

Technical field

The present invention relates to a kind of computing method of double absorption layer alternating phase-shift contact hole mask diffractional field, belong to photoetching resolution and strengthen technical field.

Background technology

The develop rapidly of semiconductor industry, mainly has benefited from the progress of the Micrometer-Nanometer Processing Technology of microelectric technique, and photoetching technique is one of manufacturing technology of most critical in chip preparation.Due to constantly bringing forth new ideas of optical lithography techniques, it breaks through the optical exposure limit that people expect again and again, makes it to become the mainstream technology when prior exposure.

Etching system is mainly divided into: illuminator (light source), mask, optical projection system and wafer four parts.Light incides diffraction occurs on mask, and diffraction light enters after optical projection system interference imaging on wafer, then through development and etch processes after, just figure is transferred on wafer.

Complicated structure on mask, according to the periodicity in all directions, mask can be divided into one dimension, X-Y scheme.One dimension figure only has periodically in one direction, and fairly simple, common lines/space (Line/Space) structure is exactly one dimension figure.X-Y scheme all has periodically on both direction, is some more complicated geometric figures, more approaching with practical devices structure.Contact hole (Contact Hole), L figure, splicing figure and H figure are all two-dimensional structures.In addition, can be divided into again intensive figure, half intensive figure and isolation pattern three classes according to pattern density.

The Physical Mechanism occurring in order to understand better said process, need to set up model, and the propagation therein of analog simulation light.And lithography simulation has become development, has optimized the important tool of photoetching process.Here we introduce the computing method of mask diffraction.

Analog simulation mask diffraction mainly contains two kinds of methods: kirchhoff method (Kirchhoff approach) and strict electromagnetic method (Rgorous electromagnetic field).Kirchhoff method is by mask as unlimited thin, and the amplitude, the phase place that see through electric field are directly determined by mask layout (mask layout).For example, in binary mask (binary masks, BIM), the light intensity of transmission region is 1, and phase place is 0, and light tight region light intensity is 0.For example, in alternating phase-shift mask (alternating phase shift masks, Alt.PSM), it is 1 that the etched area of transmission region sees through light intensity, phase place is π, the non-etched area of transmission region see through light intensity be 1, phase place is 0, light tight region see through light intensity be all 0.The principal feature of Kirchhoff method is that intensity, the phase place of mask zones of different changes very steep.

When mask feature size is much larger than wavelength and thickness much smaller than wavelength time, polarisation of light characteristic is not obvious, and now Kirchhoff is approximate is very accurate.While developing into 45nm along with photoetching technique, the characteristic dimension of mask approaches optical source wavelength (ArF), and mask thickness also reaches wavelength magnitude, add and adopt large-numerical aperture (Numerical Aperture, NA) liquid immersion lithography, polarisation of light effect is fairly obvious, must adopt strict electromagnetic field model to simulate the diffraction of mask.

Strict electromagnetic field model has been considered 3D (Three Dimensional) effect of mask and the impact of material completely.The numerical method adopting mainly comprises: Finite-Difference Time-Domain Method (finite-difference time domain method, FDTD), rigorous coupled wave method (rigorous coupled wave analysis, ), waveguide method (the waveguide method, ) and finite element method (finite element methods, FEM).In FDTD, Maxwell (Maxwell) equation is carried out to discretize in space, on the time, the equation of these discretizes carries out integration to the time and has just obtained mask diffractional field, the size of step-length when the precision of solution depends on discretize. and be mask electromagnetic field, specific inductive capacity to be carried out to Fourier Fourier series expansion obtain eigenvalue equation, then obtain the solution of problem by solving eigenvalue equation, the precision of solution depends on exponent number when Fourier launches.FEM more complicated, understanding is got up also very difficult, not all the fashion.By these strict electromagnetic field models, or obtain amplitude, the phase place in mask near field, or directly obtain amplitude, the phase place of far field construction light.Strict electromagnetic field model shows, mask sees through region and do not see through that region sees through electric field magnitude, phase place changes no longer so steep.

Prior art (J.Opt.Soc.Am.A, 1994,11,9:2494-2502; JOURNAL OF MUDANJIANG COLLEGE OF EDUCATION, 2009,6:57-59) a kind of utilization disclosed the diffraction characteristic of analysis of two-dimensional sub-wave length grating.But the method has following deficiency, it can only the identical multilayer two-dimension grating of analytical cycle; The method analysis be dielectric diffraction properties, and convergence is poor; The method has only been analyzed the diffraction of one deck two-dimensional grating simultaneously, and in alternating phase-shift contact hole mask, mask has three grating layers, in substrate of glass, in two orthogonal directions, (x, cycle y) are two times of corresponding cycle of mask absorption layer to etch areas, cycle on two orthogonal directionss is not identical, and substrate phase shift district presents crossed grating characteristic.Therefore adopt said method can not calculate the diffraction of double absorption layer alternating phase-shift contact hole mask.

Summary of the invention

The computing method that the invention provides a kind of double absorption layer alternating phase-shift contact hole mask diffraction, the method can be calculated the diffraction of double absorption layer alternating phase-shift contact hole mask in photoetching fast.

Realize technical scheme of the present invention as follows:

Computing method for double absorption layer alternating phase-shift contact hole mask diffraction, concrete steps are:

The harmonic number retaining in step 1, setting x direction is L _x, setting the harmonic number retaining in y direction is L _y;

Step 2, according to Bu Luokai (Floquet) condition, solve the wave vector of (m, n) individual order of diffraction time along component tangential, normal direction, wherein m is for getting time [D _x, D _x] between integer, n is for getting all over [D _y, D _y] between integer, L _x=2D _x+ 1, L _y=2D _y+ 1;

Wave vector is along the component α that is tangentially x, y axle _m, β _nfor:

$\left\{\begin{array}{c}{\mathrm{\α}}_m={\mathrm{\α}}_0-2\mathrm{\πm}/{\mathrm{\Λ}}_x\\ {\mathrm{\β}}_n={\mathrm{\β}}_0-2\mathrm{\πn}/{\mathrm{\Λ}}_y\end{array}\right.---\left(1\right)$

α ₀＝n _Iksinθcosδ，β ₀＝n _Iksinθsinδ (2)

Wherein k is incident light wave wave vector in a vacuum, n _ibe the refractive index of incidence zone, θ is the incident angle of light wave, the position angle that δ is light wave, Λ _xfor the mask lowest common multiple in three layers of grating cycle in the x-direction, Λ _yfor the mask lowest common multiple in three layers of grating cycle in the y-direction;

Wave vector is the component r of z axle along the normal direction of grating planar _mn, t _mnfor:

$r_{\mathrm{mn}}=\left\{\begin{array}{cc}{[{\left(n_{I}k\right)}^2-{\mathrm{\α}}_m^2-{\mathrm{\β}}_n^2]}^{1/2}& {\mathrm{\α}}_m^2+{\mathrm{\β}}_n^2\≤{\left(n_{I}k\right)}^2\\ -j{[{\mathrm{\α}}_m^2+{\mathrm{\β}}_n^2-{\left(n_{I}k\right)}^2]}^{1/2}& {\mathrm{\α}}_m^2+{\mathrm{\β}}_n^2>{\left(n_{I}k\right)}^2\end{array}\right.---\left(3\right)$

$t_{\mathrm{mn}}=\left\{\begin{array}{cc}{[{\left(n_{\mathrm{II}}k\right)}^2-{\mathrm{\α}}_m^2-{\mathrm{\β}}_n^2]}^{1/2}& {\mathrm{\α}}_m^2+{\mathrm{\β}}_n^2\≤{\left(n_{\mathrm{II}}k\right)}^2\\ -j{[{\mathrm{\α}}_m^2+{\mathrm{\β}}_n^2-{\left(n_{\mathrm{II}}k\right)}^2]}^2& {\mathrm{\α}}_m^2+{\mathrm{\β}}_n^2>{\left(n_{\mathrm{II}}k\right)}^2\end{array}\right.---\left(4\right)$

Wherein subscript I represents incidence zone, n _ithe refractive index that represents incidence zone, subscript II represents outgoing district, n _iIthe refractive index that represents outgoing district, j represents imaginary unit;

Step 3, the specific inductive capacity of every one deck two-dimensional grating and elastivity are carried out to Fourier Fourier series expansion;

Specific inductive capacity can be expressed as fourier expansion:

${\mathrm{\ϵ}}_l(x,y)=\underset{p=-{\∂}_x}{\overset{{\∂}_x}{\mathrm{\Σ}}}\underset{q=-{\∂}_y}{\overset{{\∂}_y}{\mathrm{\Σ}}}{\mathrm{\ϵ}}_{l,(p,q)}\exp \left[j2\mathrm{\π}(\mathrm{px}/{\mathrm{\Λ}}_x+\mathrm{qy}/{\mathrm{\Λ}}_y)\right](l=\mathrm{1,2})---\left(5\right)$

${\mathrm{\ϵ}}_{l^{\′}}(x,y)=\underset{p=-{\∂}_x}{\overset{{\∂}_x}{\mathrm{\Σ}}}\underset{q=-{\∂}_y}{\overset{{\∂}_y}{\mathrm{\Σ}}}{\mathrm{\ϵ}}_{l^{\′},(p,q)}\exp \left[j2\mathrm{\π}(\mathrm{px}/{\mathrm{\Λ}}_x+\mathrm{qy}/{\mathrm{\Λ}}_y)\right]{(}^{l^{\′}}=3)---\left(6\right)$

Wherein ε _{l, (p, q)}the individual Fourier component of l layer grating relative dielectric constant (p, q), ε _{l ', (p, q)}it is the individual Fourier component of l ' layer grating relative dielectric constant (p, q);

Elastivity can be expressed as fourier expansion:

$1/{\mathrm{\ϵ}}_l(x,y)=\underset{p=-{\∂}_x}{\overset{{\∂}_x}{\mathrm{\Σ}}}\underset{q=-{\∂}_y}{\overset{{\∂}_y}{\mathrm{\Σ}}}{\mathrm{\ξ}}_{l,(p,q)}\exp \left[j2\mathrm{\π}(\mathrm{px}/{\mathrm{\Λ}}_x+\mathrm{qy}/{\mathrm{\Λ}}_y)\right](l=\mathrm{1,2})---\left(7\right)$

$1/{\mathrm{\ϵ}}_{l^{\′}}(x,y)=\underset{p=-{\∂}_x}{\overset{{\∂}_x}{\mathrm{\Σ}}}\underset{q=-{\∂}_y}{\overset{{\∂}_y}{\mathrm{\Σ}}}{\mathrm{\ξ}}_{l^{\′},(p,q)}\exp \left[j2\mathrm{\π}(\mathrm{px}/{\mathrm{\Λ}}_x+\mathrm{qy}/{\mathrm{\Λ}}_y)\right]{(}^{l^{\′}}=3)---\left(8\right)$

Wherein ξ _{l, (p, q)}(p, q) individual Fourier component of l layer grating relative dielectric constant inverse, ξ _{l ', pq}it is (p, q) individual Fourier component of l ' layer grating relative dielectric constant inverse;

Step 4, according to the α calculating in step 2 _m, β _n, r _{(m, n)}, t _{(m, n)}and the ε calculating in step 3 _{l, (p, q)}, ε _{l ' (p, q)}, ξ _{l, (p, q)}and ξ _{l ', (p, q)}solve the eigenmatrix of every layer of grating, according to the tangential continuity boundary conditions of electromagnetic field, utilize enhancing transmission matrix method to solve the diffractional field in outgoing region.

Beneficial effect

In the present invention at two orthogonal direction (x, y) on, by choosing the lowest common multiple in three grating layer cycles on correspondence direction, carry out Fourier series expansion, can analyze two orthogonal directions (x, y) all different multilayer two-dimension mask grating diffrations of upper cycle, and diffraction that can as analysed basis base area crossed grating; The present invention strengthens the situation of three layers of grating taper incident of transmission matrix method analysis by employing, energy rapid solving obtains the diffractional field of double absorption layer alternating phase-shift contact hole mask.

Brief description of the drawings

Fig. 1 is double absorption layer alternating phase-shift contact hole mask schematic diagram.

Fig. 2 is for solving double absorption layer alternating phase-shift contact hole mask diffraction process flow diagram.

Fig. 3 is matrix E of the present invention _lschematic diagram.

Fig. 4 is matrix E of the present invention _l' schematic diagram.

Fig. 5 is matrix A of the present invention _lschematic diagram.

Fig. 6 is matrix A of the present invention _l' schematic diagram.

Fig. 7 is matrix K of the present invention _xschematic diagram.

Fig. 8 is matrix K of the present invention _yschematic diagram.

Fig. 9 is the schematic diagram of unit matrix I of the present invention.

Figure 10 is matrix Y of the present invention _ischematic diagram.

Figure 11 is matrix Z of the present invention _ischematic diagram.

Figure 12 is matrix Y of the present invention _iIschematic diagram.

Figure 13 is matrix Z of the present invention _iIschematic diagram.

Figure 14 is matrix F of the present invention _cschematic diagram.

Figure 15 is matrix F of the present invention _sschematic diagram.

Figure 16 be TE polarized light taper incident (θ=10 °, λ=193nm) when double absorption layer (CrO/Cr) alternating phase-shift contact hole mask, (0,0), (0,2), (1,1), (2,0) diffraction efficiency of the order of diffraction time is along with the variation relation figure of characteristic dimension (wafer yardstick, nm).

Embodiment

Below in conjunction with accompanying drawing, the present invention is further elaborated.

Double absorption layer alternating phase-shift contact hole mask schematic diagram as shown in Figure 1, below describes the mask relating in the present embodiment.

The present invention, taking the normal direction of mask plane (grating planar) as z axle, according to right-handed coordinate system principle, determines x axle and y axle, sets up coordinate system (x, y, z) as shown in Figure 1.

Double absorption layer alternating phase-shift contact hole mask is divided into three layers along z direction of principal axis, two-layer absorption layer and one deck phase shift layer; The first absorption layer (z ₀< z < z ₁) being generally CrO, thickness is d ₁=z ₁-z ₀, the second absorption layer (z ₁< z < z ₂) being generally Cr, thickness is d ₂=z ₂-z ₁, third phase moves layer, and its etching depth is d=λ/2 (n-1), to realize the phase shift of 180 °.The first absorption layer is periods lambda along x axle _1xdistribute, dutycycle is f _1x, the first absorption layer is periods lambda along y axle _1ydistribute, dutycycle is f _1y.Second layer absorption layer is periods lambda along x axle _2xdistribute, dutycycle is f _2x, the second absorption layer is periods lambda along y axle _2ydistribute, dutycycle is f _2y.And the front two-layer cycle on x, y axle is respectively all identical with dutycycle, i.e. Λ _1x=Λ _2x, f _1x=f _2x, Λ _1y=Λ _2y, f _1y=f _2y, but f _1x≠ f _1y, f _2x≠ f _2y.The 3rd layer is dielectric, is periods lambda along x axle _3xdistribute, dutycycle is f _3x, be periods lambda along y axle _3ydistribute, dutycycle is f _3y, and Λ _3y=2 Λ _1y, Λ _3x=2 Λ _1x.Top layer (L '=0), bottom (L '=4) are to be respectively incidence zone, outgoing district, and are infinite expanding along negative sense, the forward of z axle, and the refractive index of top layer is n _i, the refractive index of bottom is n _iI.

A branch of linearly polarized light incides diffraction occurs on two-dimensional grating, incident angle is θ, position angle (plane of incidence and x axle clamp angle) is δ, polarization angle (angle of incident electric field intensity and plane of incidence) is ψ, ψ=90 ° are when corresponding to TE polarized light, and ψ=0 is ° corresponding to TM polarized light.

As shown in Figure 2, the process flow diagram of double absorption layer alternating phase-shift contact hole mask diffractional field computing method of the present invention; Concrete steps are:

The harmonic number retaining in step 1, setting x direction is L _x, setting the harmonic number retaining in y direction is L _y; Above-mentioned two harmonic numbers can be set as required, have speed faster if wish solving electric field energy, can be arranged littlely, if wish, the electric field solving has higher precision, can be arranged greatlyr, and the while also can make L _x=L _y.

Step 2, according to Bu Luokai (Floquet) condition, solve the wave vector of (m, n) individual order of diffraction time along component tangential, normal direction, wherein m is for getting time [D _x, D _x] between integer, n is for getting all over [D _y, D _y] between integer, L _x=2D _x+ 1, L _y=2D _y+ 1.

Wave vector is along the component α that is tangentially x, y axle _m, β _nfor:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_m={\mathrm{\&alpha;}}_0-2\mathrm{\&pi;m}/{\mathrm{\&Lambda;}}_x\\ {\mathrm{\&beta;}}_n={\mathrm{\&beta;}}_0-2\mathrm{\&pi;n}/{\mathrm{\&Lambda;}}_y\end{array}\right.---\left(1\right)$

α ₀＝n _Iksinθcosδ，β ₀＝n _Iksinθsinδ (2)

Wherein k is incident light wave wave vector in a vacuum, n _ibe the refractive index of incidence zone, θ is the incident angle of light wave, the position angle that δ is light wave, Λ _xfor the mask lowest common multiple in three layers of grating cycle in the x-direction, due to Λ _3y=2 Λ _1yso, Λ herein _y=Λ _3y, Λ _yfor the mask lowest common multiple in three layers of grating cycle in the y-direction, due to Λ _3x=2 Λ _1xso, Λ herein _x=Λ _3x.

Wave vector is the component r of z axle along the normal direction of grating planar _mn, t _mnfor:

$r_{\mathrm{mn}}=\left\{\begin{array}{cc}{[{\left(n_{I}k\right)}^2-{\mathrm{\&alpha;}}_m^2-{\mathrm{\&beta;}}_n^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2\&le;{\left(n_{I}k\right)}^2\\ -j{[{\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2-{\left(n_{I}k\right)}^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2>{\left(n_{I}k\right)}^2\end{array}\right.---\left(3\right)$

$t_{\mathrm{mn}}=\left\{\begin{array}{cc}{[{\left(n_{\mathrm{II}}k\right)}^2-{\mathrm{\&alpha;}}_m^2-{\mathrm{\&beta;}}_n^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2\&le;{\left(n_{\mathrm{II}}k\right)}^2\\ -j{[{\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2-{\left(n_{\mathrm{II}}k\right)}^2]}^2& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2>{\left(n_{\mathrm{II}}k\right)}^2\end{array}\right.---\left(4\right)$

Wherein subscript I represents incidence zone, n _ithe refractive index that represents incidence zone, subscript II represents outgoing district, n _iIthe refractive index that represents outgoing district, j represents imaginary unit.

Step 3, the specific inductive capacity of every one deck two-dimensional grating and elastivity are carried out to Fourier Fourier series expansion.Due to the cycle difference of three layers of two-dimensional grating in x, y direction, when series expansion, should choose the lowest common multiple in three layers of grating cycle on correspondence direction.In the time doing Fourier series expansion, selected unit area is as shown in dotted line in Fig. 1 (a).

Specific inductive capacity can be expressed as fourier expansion:

${\mathrm{\&epsiv;}}_l(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l,(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right](l=\mathrm{1,2})---\left(5\right)$

${\mathrm{\&epsiv;}}_{l^{\&prime;}}(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l^{\&prime;},(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right]{(}^{l^{\&prime;}}=3)---\left(6\right)$

Wherein ε _{l, (p, q)}the individual Fourier component of l layer grating relative dielectric constant (p, q), ε _{l ', (p, q)}it is the individual Fourier component of l ' layer grating relative dielectric constant (p, q).

Elastivity can be expressed as fourier expansion:

$1/{\mathrm{\&epsiv;}}_l(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_{l,(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right](l=\mathrm{1,2})---\left(7\right)$

$1/{\mathrm{\&epsiv;}}_{l^{\&prime;}}(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&xi;}}_{l^{\&prime;},(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right]{(}^{l^{\&prime;}}=3)---\left(8\right)$

Wherein ξ _{l, (p, q)}it is (p, q) individual Fourier component of l layer grating relative dielectric constant inverse.Wherein ξ _{l ', pq}it is (p, q) individual Fourier component of l ' layer grating relative dielectric constant inverse.

Step 4, according to the α calculating in step 2 _m, β _n, r _{(m, n)}, t _{(m, n)}and the ε calculating in step 3 _{l, (p, q)}, ε _{l ', (p, q)}, ξ _{l, (p, q)}and ξ _{l ', (p, q)}solve the eigenmatrix of every layer of grating, according to the tangential continuity boundary conditions of electromagnetic field, utilize enhancing transmission matrix method to solve the diffractional field in outgoing region.

Step 401, solve the eigenmatrix of each two-dimensional grating layer;

The eigenmatrix of two-dimensional grating is:

M _l＝F _lG _l(l＝1，2) (9)

M _l′＝F _l′G _l′(l′＝3) (10)

Wherein

$F_l=\left[\begin{array}{cc}{K}_y{A}_l{K}_x& I-K_y{A}_l{K}_y\\ K_x{A}_l{K}_x-I& -K_x{A}_l{K}_y\end{array}\right](l=\mathrm{1,2})---\left(11\right)$

$G_l=\left[\begin{array}{cc}{K}_x{K}_y& {\mathrm{\&alpha;A}}_l^{-1}+(1-\mathrm{\&alpha;})E_l-K_y^2\\ K_x^2-{\mathrm{\&alpha;E}}_l-(1-\mathrm{\&alpha;})A_l^{-1}& -K_x{K}_y\end{array}\right](l=\mathrm{1,2})---\left(12\right)$

$\mathrm{\&alpha;}=\frac{f_{1y}{\mathrm{\&Lambda;}}_{1y}}{f_{1x}{\mathrm{\&Lambda;}}_{1x}+f_{1y}{\mathrm{\&Lambda;}}_{1y}}---\left(13\right)$

$F_{l^{\&prime;}}=\left[\begin{array}{cc}{K}_y{E}_{l^{\&prime;}}^{-1}{K}_x& I-K_y{E}_{l^{\&prime;}}^{-1}{K}_y\\ K_x{E}_{l^{\&prime;}}^{-1}{K}_x-I& -K_x{E}_{l^{\&prime;}}^{-1}{K}_y\end{array}\right](l^{\&prime;}=3)---\left(14\right)$

$G_{l^{\&prime;}}=\left[\begin{array}{cc}{K}_x{K}_y& A_{l^{\&prime;}}^{-1}-K_y^2\\ K_x^2-A_{l^{\&prime;}}^{-1}& -K_x{K}_y\end{array}\right](l^{\&prime;}=3)---\left(15\right)$

Wherein E _l, E _l', A _l, A _l', K _x, K _y, I is (L _t× L _t) rank matrix, L _t=L _x× L _y, l and l ' all represent the number of plies.E _lin element be ε _{l, (p, q)}, E _l' in element be ε _{l ', (p, q)}, A _lin element be ξ _{l, (p, q)}, A _l' in element be ξ _{l ', (p, q)}.

For example the present invention sets L _x=3, L _y=3, due to p=[-2 ,-1,0,1,2], q=[-2 ,-1,0,1,2]; For l layer grating, the ε generating according to step 3 _{l, (p, q)}for individual, be respectively: ε _{l, (2 ,-2)}, ε _{l, (2 ,-1)}... ε _{l, (2,2)}.

The E that every one deck grating pair is answered _lallocation rule identical, below ignore the consideration to the number of plies, be (L to size _t× L _t) i.e. 9 × 9 matrix E _lthe distribution rule of upper element describes:

By E _lbe divided into L _y× L _y(9) individual L _x× L _xthe minor matrix of (3 × 3), and by each minor matrix e _{(i, j)}be used as an element, wherein i=[1,2,3], j=[1,2,3], e _(1,1)for coordinate equals the minor matrix of (1,1), e _(1,2)for coordinate equals the minor matrix of (1,2), and the like.Be directed to each minor matrix e _{(i, j)}in it, comprise 9 element e ' _{(i, j), (i ', j ')}, wherein i '=[1,2,3], j '=[1,2,3], e ' _{(i, j), (1,1)}for minor matrix e _{(i, j)}internal coordinate equals the minor matrix element of (1,1), e ' _{(i, j), (1,2)}for minor matrix e _{(i, j)}internal coordinate equals the minor matrix of (1,2), and the like.

Distribute rule to be: at minor matrix e _{(i, j)}in, its (i ', j ') individual element e ' _{(i, j) (i ', j ')}=ε _{l, (i '-j ', i-j)};

For example, to minor matrix e _(1,1)in (1,1) individual element e ' _{(1,1), (1,1)}, due to i-j=0, i '-j '=0, so e ' _{(1,1), (1,1)}(be namely equivalent to E _lin (1,1) individual element) equal ε _{l, (0,0)}.

For example, to minor matrix e _(1,1)in (1,2) individual element e ' _{(1,1), (1,2)}, due to i-j=0, i '-j '=-1, so e ' _{(1,1), (1,2)}(be namely equivalent to E _lin (1,2) individual element) equal ε _{l, (1,0)}.

For example, to minor matrix e _(2,1)in (1,2) individual element e ' _{(2,1), (1,2)}, due to i-j=1, i '-j '=-1, so e ' _{(2,1), (1,2)}(be namely equivalent to E _lin (4,2) individual element) equal ε _{l, (1,1)}.

For example, to minor matrix e _(2,1)in (3,3) individual element e ' _{(2,1), (3,3)}, due to i-j=1, i '-j '=0, so e ' _{(2,1), (3,3)}(be namely equivalent to E _lin (6,3) individual element) equal ε _{l, (0,1)}.

For example, to minor matrix e _(3,3)in (1,3) individual element e ' _{(3,3), (1,3),}due to i-j=0, i '-j '=-2, so e ' _{(3,3), (1,3)}(be namely equivalent to E _lin (7,9) individual element) equal ε _{l, (2,0)}.

For example, to minor matrix e _(3,3)in (3,3) individual element e ' _{(3,3), (3,3)}, due to i-j=0, i '-j '=0, so e ' _{(3,3), (3,3)}namely be equivalent to E _lin (9,9) individual element equal ε _{l, (0,0)}.

The E obtaining according to above-mentioned rule _las shown in Figure 3.

E _l', A _land A _ldistribution rule and the E of ' upper element _lidentical, as Figure 4-Figure 6.

K _xfor diagonal matrix, its diagonal element is α _m.

For example the present invention sets L _x=3, L _y=3, due to D _x=(L _x-1)/2, D _x=1, m=[-1,0,1]; The α generating according to step 2 _mbe 3, be respectively: α _-1, α ₀, α ₁.

Be (L to size below _t× L _t) i.e. 9 × 9 diagonal matrix K _xthe distribution rule of diagonal element describes:

By K _xbe divided into L _y× L _y(9) individual L _x× L _xthe minor matrix of (3 × 3), and by each minor matrix be used as an element, wherein i=[1,2,3], j=[1,2,3], for coordinate equals the minor matrix of (1,1), for coordinate equals the minor matrix of (1,2), and the like.Be directed to each minor matrix in it, comprise 9 elements wherein i '=[1,2,3], j '=[1,2,3], for minor matrix internal coordinate equals the minor matrix element of (1,1), for minor matrix internal coordinate equals the minor matrix of (1,2), and the like.

Due to K _xfor diagonal matrix, only exist and diagonal position on there is element value, the element value of all the other minor matrixs is all 0.

Distribute rule to be: at minor matrix in (being j=i), its (i ', j ') (being j '=i ') individual element because this distribution rule is irrelevant with (i, j), so and identical.

For example, to minor matrix in (1,1) individual element due to i '=1, D _x=1, i '-(D _x+ 1)=-1, so (be namely equivalent to K _xin (1,1) individual element) equal a _-1.

For example, to minor matrix in (2,2) individual element due to i '=2, D _x=1, i '-(D _x+ 1)=0, so (be namely equivalent to K _xin (2,2) individual element) equal a ₀.

The K obtaining according to above-mentioned rule _xas shown in Figure 7.

K _yfor diagonal matrix, its diagonal element is β _n.

For example the present invention sets L _x=3, L _y=3, due to D _y=(L _y-1)/2, D _y=1, n=[-1,0,1]; , the β generating according to step 2 _nbe 3, be respectively: β _-1, β ₀, β ₁.

Be (L to size below _t× L _t) i.e. 9 × 9 diagonal matrix K _ythe distribution rule of diagonal element describes:

By K _ybe divided into L _y× L _y(9) individual L _x× L _xthe minor matrix of (3 × 3), and by each minor matrix be used as an element, wherein i=[1,2,3], j=[1,2,3], for coordinate equals the minor matrix of (1,1), for coordinate equals the minor matrix of (1,2), and the like.Be directed to each minor matrix in it, comprise 9 elements wherein i '=[1,2,3], j '=[1,2,3], for minor matrix internal coordinate equals the minor matrix element of (1,1), for minor matrix internal coordinate equals the minor matrix of (1,2), and the like.

Distribute rule to be: at minor matrix in (being j=i), its diagonal element because this distribution rule is irrelevant with (i ', j '), so in each minor matrix diagonal element identical.

For example, to minor matrix in (1,1) individual element due to i=1, D _y=1, i-(D _y+ 1)=-1, so (be namely equivalent to K _yin (1,1) individual element) equal β _-1.

For example, to minor matrix in (2,2) individual element due to i=2, D _y=1, i-(D _y+ 1)=0, so (be namely equivalent to K _yin (5,5) individual element) equal β ₀.

The K obtaining according to above-mentioned rule _yas shown in Figure 8.

I is unit matrix, as shown in Figure 9.

Step 402, solve the matrix Y of incidence zone _i, Z _i, and transmission area matrix Y _iI, Z _iI.

Wherein Y _i, Z _ifor diagonal matrix, diagonal element is respectively y _iI, Z _iIalso be diagonal matrix, diagonal element is respectively

Matrix Y _i, Z _iy _iIand Z _iIthe distribution rule of upper element is identical, below chooses Y _ias analytic target, because k is constant, therefore ignore k the distribution rule of element on it is elaborated.

For example the present invention sets L _x=3, L _y=3, due to D _x=(L _x-1)/2, D _x=1, m=[-1,0,1], due to D _y=(L _y-1)/2, D _y=1, n=[-1,0,1], the r generating according to step 2 _{(m, n)}be 3 × 3=9, be respectively: r _{(1 ,-1)}, r _(1,0), r _(1,1)... r _{(1 ,-1)}, r _(1,0), r _(1,1).

Y _ifor (L _t× L _t) i.e. 9 × 9 diagonal matrix, below to Y _ithe distribution rule of diagonal element describes:

By Y _ibe divided into L _y× L _y(9) individual L _x× L _xthe minor matrix of (3 × 3), and by each minor matrix y _{(i, j)}be used as an element, wherein i=[1,2,3], j=[1,2,3], y _(1,1)for coordinate equals the minor matrix of (1,1), y _(1,2)for coordinate equals the minor matrix of (1,2), and the like.Be directed to each minor matrix y _{(i, j)}in it, comprise 9 element y ' _{(i, j), (i ', j ')}, wherein i '=[1,2,3], j '=[1,2,3], y ' _{(i, j), (1,1)}for minor matrix y _{(i, j)}internal coordinate equals the minor matrix element of (1,1), y ' _{(i, j), (1,2)}for minor matrix y _{(i, j)}internal coordinate equals the minor matrix of (1,2), and the like.

Due to Y _ifor diagonal matrix, only at y _(1,1), y _(2,2)and y _(3,3)diagonal position on there is element value, the element value of all the other minor matrixs is all 0.

Distribute rule to be: at minor matrix y _{(i, j)}in (being j=i), its (i ', j ') (being j '=i ') individual element y ' _{(i, j), (i ', j ')}=r ((i '-D _x-1), (i-D _y-1)).

For example, to minor matrix y _(1,1)in (1,1) individual element y ' _{(1,1), (1,1)}, due to i '=1, D _x=1, i=1, D _y=1, i '-(D _x+ 1)=-1, i-(D _y+ 1)=-1, so y ' _{(1,1), (1,1)}(be namely equivalent to Y _iin (1,1) individual element) equal r _{(1 ,-1)}.

For example, to minor matrix y _(1,1)in (2,2) individual element y ' _{(1,1), (2,2)}, due to i '=2, D _x=1, i=1, D _y=1, i '-(D _x+ 1)=0, i-(D _y+ 1)=-1, so y ' _{(1,1), (2,2)}(be namely equivalent to Y _iin (2,2) individual element) equal r _{(0 ,-1)}.

For example, to minor matrix y _(2,2)in (2,2) individual element y ' _{(2,2), (2,2)}, due to i '=2, D _x=1, i=2, D _y=1, i '-(D _x+ 1)=0, i-(D _y+ 1)=0, so y ' _{(2,2), (2,2)}(be namely equivalent to Y _iin (5,5) individual element) equal r _(0,0).

The Y obtaining according to above-mentioned rule _i, Z _iy _iIand Z _iIas shown in Figure 10-13.

Step 403, utilize the tangential continuous boundary condition of electromagnetic field, obtain the expression formula between incidence zone and outgoing district electromagnetic field;

$\left[\begin{array}{c}\sin \mathrm{\&psi;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\\ j\sin \mathrm{\&psi;}{n}_I\cos \mathrm{\&theta;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\\ -j{n}_I\cos \mathrm{\&psi;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\\ \cos \mathrm{\&psi;}\cos \mathrm{\&theta;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\end{array}\right]+\left[\begin{array}{cc}I& 0\\ -j{Y}_I& 0\\ 0& I\\ 0& -j{Z}_I\end{array}\right]R=$

$\underset{L=1}{\overset{N=3}{\mathrm{\&Pi;}}}\left[\begin{array}{cc}{V}_{L,1}& V_{L,1}{X}_L\\ W_{L,1}& -W_{L,1}{X}_L\\ w_{L,2}& -W_{L,2}{X}_L\\ V_{L,2}& V_{L,2}{X}_L\end{array}\right]{\left[\begin{array}{cc}{V}_{L,1}{X}_L& V_{L,1}\\ W_{L,1}{X}_L& -W_{L,1}\\ W_{L,2}{X}_L& -W_{L,2}\\ V_{L,2}{X}_L& V_{L,2}\end{array}\right]}^{-1}\left[\begin{array}{cc}I& 0\\ j{Y}_{\mathrm{II}}& 0\\ 0& I\\ 0& j{Z}_{\mathrm{II}}\end{array}\right]T$ (L＝1，2，3) (16)

Wherein

V _L，1＝F _cW _L，y-F _sW _L，x V _L，2＝F _cW _L，x+F _sW _L，y

W _L，1＝F _cV _L，x+F _sV _L，y W _L，2＝F _cV _L，y-F _sV _L，x

(17)

W _L，x＝[w _L，x] W _L，y＝[w _L，y]

V _L，x＝[v _L，x] V _L，y＝[v _L，y]

L represents L layer two-dimensional grating;

$W_L=\left[\begin{array}{c}{w}_{L,y}\\ w_{L,x}\end{array}\right]$ Be L layer two-dimensional grating eigenmatrix M _leigenvector matrix;

Q _{l, l}be L layer two-dimensional grating eigenmatrix M _leigenvalue matrix in the positive square root l=[1 of (l, l) individual element, 2,3 ..., 2L _t];

X _lrepresent the diagonal matrix in L layer two-dimensional grating, diagonal element (l, l) is exp (kq _{l, l}d _l);

D _lrepresent the thickness of L layer two-dimensional grating;

$V_L=\left[\begin{array}{c}{v}_{L,y}\\ v_{L,x}\end{array}\right]=F_L^{-1}{Q}_L{W}_L;$

Q _lthat diagonal element (l, l) is q _{l, l}diagonal matrix;

F _cthat diagonal element is diagonal matrix;

F _sthat diagonal element is diagonal matrix;

F _cand F _sallocation rule and the Y of diagonal element _iidentical, as shown in Figure 14-15.

δ _m0for L _x× 1 matrix, wherein in the time of m=0, δ (m+D _x+ 1,1)=1; In the time of m ≠ 0, δ (m+D _x+ 1,1)=0;

δ ' _n0for L _y× 1 matrix, wherein in the time of n=0, δ ' (n+D _y+ 1,1)=1; In the time of n ≠ 0, δ ' (n+D _y+ 1,1)=0;

R is intermediate variable;

T is the inferior amplitude of each order of diffraction of transmitted field to be solved;

Step 404, utilization strengthen transmission matrix method, solve the inferior amplitude T of each order of diffraction of transmitted field; Wherein T is 2L _t× 1 matrix, the form that each element in T is plural a+bj, wherein the amplitude of diffractional field is obtain the diffractional field in polarized light outgoing district.

Utilize and strengthen transmission matrix method, the expression formula between incidence zone and outgoing district electromagnetic field is:

$\left[\begin{array}{c}\sin \mathrm{\&psi;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\\ j\sin \mathrm{\&psi;}{n}_I\cos \mathrm{\&theta;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\\ -j{n}_I\cos \mathrm{\&psi;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\\ \cos \mathrm{\&psi;}\cos \mathrm{\&theta;}{\mathrm{\&delta;}}_{m0}{\mathrm{\&delta;}}_{n0}\end{array}\right]+\left[\begin{array}{cc}I& 0\\ -j{Y}_I& 0\\ 0& I\\ 0& -j{Z}_I\end{array}\right]R=\begin{array}{}\end{array}\left[\begin{array}{c}{f}_1\\ g_1\end{array}\right]T_1---\left(18\right)$

Wherein

$\left[\begin{array}{c}{f}_L\\ g_L\end{array}\right]T_L=\left[\begin{array}{cc}{V}_{L,1}& V_{L,1}{X}_L\\ W_{L,1}& -W_{L,1}{X}_L\\ W_{L,2}& -W_{L,2}{X}_L\\ V_{L,2}& V_{L,2}{X}_L\end{array}\right]\left[\begin{array}{c}I\\ b_L{a}_L^{-1}{X}_L\end{array}\right]T_L---\left(19\right)$

$\left[\begin{array}{c}{a}_L\\ b_L\end{array}\right]{\left[\begin{array}{cc}{V}_{L,1}& V_{L,1}\\ W_{L,1}& -W_{L,1}\\ W_{L,2}& -W_{L,2}\\ V_{L,2}& V_{L,2}\end{array}\right]}^{-1}\left[\begin{array}{c}{f}_{L+1}\\ g_{L+1}\end{array}\right]---\left(20\right)$

$\left[\begin{array}{c}{f}_4\\ g_4\end{array}\right]=\left[\begin{array}{cc}I& 0\\ j{Y}_{\mathrm{II}}& 0\\ 0& I\\ 0& {\mathrm{jZ}}_{\mathrm{II}}\end{array}\right]---\left(21\right)$

$T=a_3^{-1}{X}_3{a}_2^{-1}{X}_2{a}_1^{-1}{X}_1{T}_1---\left(22\right)$

Further, the present invention also can solve the inferior diffraction efficiency of each order of diffraction;

The inferior diffraction efficiency of (m, n) level is:

${\mathrm{\&eta;}}_{(m,n)}={\left|T_{s,(m,n)}\right|}^2\mathrm{Re}\left(\frac{t_{(m,n)}}{{\mathrm{kn}}_I\cos \mathrm{\&theta;}}\right)+{\left|T_{p,(m,n)}\right|}^2\mathrm{Re}\left(\frac{t_{(m,n)}/n_{\mathrm{II}}^2}{{\mathrm{kn}}_I\cos \mathrm{\&theta;}}\right)---\left(23\right)$

Wherein T _sfor the matrix of the first half element composition in T, T _pfor the matrix of the latter half element composition in T.T _{s, (m, n)}for T _sin ((m+D _x+ 1)+(n+D _y) L _x) individual element, T _{p, (m, n)}for T _pin ((m+D _x+ 1)+(n+D _y) L _x) individual element.

Further, the present invention can also solve the inferior degree of polarization (Degree of Polarization, DoP) of each order of diffraction

${\mathrm{DoP}}_{(m,n)}=\frac{{\mathrm{\&eta;}}_{(m,n)}^{\mathrm{TE}}-{\mathrm{\&eta;}}_{(m,n)}^{\mathrm{TM}}}{{\mathrm{\&eta;}}_{(m,n)}^{\mathrm{TE}}+{\mathrm{\&eta;}}_{(m,n)}^{\mathrm{TM}}}\&CenterDot;100\%---\left(24\right)$

Wherein in the time that incident light is TE polarized light, by η _{(m, n)}be defined as in the time that incident light is TM polarized light, by η _{(m, n)}be defined as doP, for just, represents the similar TE polaroid of mask, and DoP, for negative, represents the similar TM polaroid of mask.

Invention example one:

Here calculated in double absorption layer (CrO/Cr) alternating phase-shift contact hole mask, TE taper incident (θ=10 °, λ=193nm) time, when different mask linewidths (wafer yardstick), (0,0), (0,2), (1,1), the inferior diffraction efficiency of (2,0) level.Wherein CrO refractive index, extinction coefficient and thickness be respectively 1.965,1.201 and 18nm.Cr refractive index, extinction coefficient and thickness be respectively 1.477,1.762 and the dutycycle of 55nm. mask on x axle be 0.5, the dutycycle on y axle is 0.6.

Figure 16 be TE polarized light taper incident (θ=10 °, λ=193nm) when double absorption layer (CrO/Cr) alternating phase-shift contact hole mask, (0,0), (0,2), (1,1), (2,0) diffraction efficiency of the order of diffraction time is along with the variation relation figure of characteristic dimension (wafer yardstick, nm).(a) (0,0) level diffraction of light efficiency is with the graph of a relation of line width variation, (b) (0,2) level diffraction of light efficiency is with the graph of a relation of line width variation, (c) (1,1) level diffraction of light efficiency is with the graph of a relation of line width variation, and (d) (2,0) level diffraction of light efficiency is with the graph of a relation of line width variation.

In sum, these are only preferred embodiment of the present invention, be not intended to limit protection scope of the present invention.Within the spirit and principles in the present invention all, any amendment of doing, be equal to replacement, improvement etc., within all should being included in protection scope of the present invention.

Claims (1)

1. computing method for double absorption layer alternating phase-shift contact hole mask diffractional field, is characterized in that, concrete steps are:

The harmonic number retaining in step 1, setting x direction is L _x, setting the harmonic number retaining in y direction is L _y;

Step 2, according to Bu Luokai (Floquet) condition, solve the wave vector of (m, n) individual order of diffraction time along component tangential, normal direction, wherein m is for getting time [D _x, D _x] between integer, n is for getting all over [D _y, D _y] between integer, L _x=2D _x+ 1, L _y=2D _y+ 1;

Wave vector is along the component α that is tangentially x, y axle _m, β _nfor:

$\left\{\begin{array}{c}{\mathrm{\&alpha;}}_m={\mathrm{\&alpha;}}_0-2\mathrm{\&pi;m}/{\mathrm{\&Lambda;}}_x\\ {\mathrm{\&beta;}}_n={\mathrm{\&beta;}}_0-2\mathrm{\&pi;n}/{\mathrm{\&Lambda;}}_y\end{array}\right.---\left(1\right)$

α ₀＝n _Iksinθcosδ，β ₀＝n _Iksinθsinδ (2)

Wherein k is incident light wave wave vector in a vacuum, n _ibe the refractive index of incidence zone, θ is the incident angle of light wave, the position angle that δ is light wave, Λ _xfor the mask lowest common multiple in three layers of grating cycle in the x-direction, Λ _yfor the mask lowest common multiple in three layers of grating cycle in the y-direction;

Wave vector is the component r of z axle along the normal direction of grating planar _mn, t _mnfor:

$r_{\mathrm{mn}}=\left\{\begin{array}{cc}{[{\left(n_{I}k\right)}^2-{\mathrm{\&alpha;}}_m^2-{\mathrm{\&beta;}}_n^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2\&le;{\left(n_{I}k\right)}^2\\ -j{[{\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2-{\left(n_{I}k\right)}^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2>{\left(n_{I}k\right)}^2\end{array}\right.---\left(3\right)$

$t_{\mathrm{mn}}=\left\{\begin{array}{cc}{[{\left(n_{\mathrm{II}}k\right)}^2-{\mathrm{\&alpha;}}_m^2-{\mathrm{\&beta;}}_n^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2\&le;{\left(n_{\mathrm{II}}k\right)}^2\\ -j{[{\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2-{\left(n_{\mathrm{II}}k\right)}^2]}^{1/2}& {\mathrm{\&alpha;}}_m^2+{\mathrm{\&beta;}}_n^2>{\left(n_{\mathrm{II}}k\right)}^2\end{array}\right.---\left(4\right)$

Wherein subscript I represents incidence zone, n _ithe refractive index that represents incidence zone, subscript II represents outgoing district, n _iIthe refractive index that represents outgoing district, j represents imaginary unit;

Step 3, the specific inductive capacity of every one deck two-dimensional grating and elastivity are carried out to Fourier Fourier series expansion;

Specific inductive capacity can be expressed as fourier expansion:

${\mathrm{\&epsiv;}}_l(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l,(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right](l=\mathrm{1,2})---\left(5\right)$

${\mathrm{\&epsiv;}}_{l^{\&prime;}}(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l^{\&prime;},(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right](l^{\&prime;}=3)---\left(6\right)$

Wherein ε _{l, (p}, _q)the individual Fourier component of l layer grating relative dielectric constant (p, q),

In the time of l=1, ε _{l, (p, q)}=ε _{1, (p, q)}the 1st layer of individual Fourier component of grating relative dielectric constant (p, q),

In the time of l=2, ε _{l, (p, q)}=ε _{2, (p, q)}the 2nd layer of individual Fourier component of grating relative dielectric constant (p, q);

ε _{l, (p, q)}the individual Fourier component of l' layer grating relative dielectric constant (p, q),

In l '=3 o'clock, ε _{l ', (p, q)}=ε _{3, (p, q)}the 3rd layer of individual Fourier component of grating relative dielectric constant (p, q);

${\&PartialD;}_x=L_x-1,{\&PartialD;}_y=L_y-1;$

Elastivity is expressed as fourier expansion:

$1/{\mathrm{\&epsiv;}}_l(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l,(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right](l=\mathrm{1,2})---\left(7\right)$

$1/{\mathrm{\&epsiv;}}_{l^{\&prime;}}(x,y)=\underset{p=-{\&PartialD;}_x}{\overset{{\&PartialD;}_x}{\mathrm{\&Sigma;}}}\underset{q=-{\&PartialD;}_y}{\overset{{\&PartialD;}_y}{\mathrm{\&Sigma;}}}{\mathrm{\&epsiv;}}_{l^{\&prime;},(p,q)}\exp \left[j2\mathrm{\&pi;}(\mathrm{px}/{\mathrm{\&Lambda;}}_x+\mathrm{qy}/{\mathrm{\&Lambda;}}_y)\right](l^{\&prime;}=3)---\left(8\right)$

Wherein ξ _{l, (p, q)}(p, q) individual Fourier component of l layer grating relative dielectric constant inverse, ξ _{l', pq}it is (p, q) individual Fourier component of l' layer grating relative dielectric constant inverse;

Step 4, according to the α calculating in step 2 _m, β _n, r _{(m, n)}, t _{(m, n)}and the ε calculating in step 3 _{l, (p, q)}, ε _{l ', (p, q)}, ξ _{l, (p, q)}and ξ _{l ', (p, q)}solve the eigenmatrix of every layer of grating, according to the tangential continuity boundary conditions of electromagnetic field, utilize enhancing transmission matrix method to solve the diffractional field in outgoing region.