CN107644131A - Intersection transmission function quick decomposition method based on polar coordinates sampling - Google Patents

Intersection transmission function quick decomposition method based on polar coordinates sampling Download PDF

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CN107644131A
CN107644131A CN201710820290.8A CN201710820290A CN107644131A CN 107644131 A CN107644131 A CN 107644131A CN 201710820290 A CN201710820290 A CN 201710820290A CN 107644131 A CN107644131 A CN 107644131A
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matrix
function
sampling
polar coordinates
transmission function
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贡顶
沈忱
崔绍春
陈雪莲
鄢丽萍
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SUZHOU COGENDA ELECTRONIC CO Ltd
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Abstract

The invention discloses a kind of intersection transmission function quick decomposition method based on polar coordinates sampling, it comprises the following steps：1) optical parametric of imaging system is obtained；2) coordinate (r of spatial domain up-sampling point is obtained using the polar coordinates method of samplingi,θj)；3) light source interaural crosscorrelation function corresponding to sampled point is calculated by non-homogeneous inverse Fourier transformAnd pupil function4) the intersection transfer function values that sampled point corresponds to spatial domain are calculatedAnd establish sampling matrix5) one group of orthogonal basis function is established in spatial domainCalculate orthogonal basis functionFunctional value on corresponding polar coordinates sampling locationAnd establish matrix Q=[q1,q2,…qk]；6) QR matrix decompositions are carried out to matrix Q7) projection matrix is calculatedAnd singular value decomposition is carried out to projection matrix P and obtains P=UU*；8) obtain in spatial domain and intersect transmission functionKernel function beThe present invention can fast resolving obtain TCC kernel functions so that light distribution calculate it is quick and efficient, so as to meet actual photoetching process design requirement.

Description

Intersection transmission function quick decomposition method based on polar coordinates sampling
【Technical field】
Photoetching resolution in being emulated the invention belongs to semiconductor device technology strengthens technical field, more particularly to one kind Intersection transmission function quick decomposition method based on polar coordinates sampling.
【Background technology】
In semiconductor devices production process, the photoetching process of traditional top-down pattern transfer technology, there is production The advantages such as efficiency high, relative cost be low, for current industrial Main Means.Photoetching process emulation technology mainly includes photoetching System imaging model and reverse optimization process, wherein reverse optimization are a processes to iterate, and each iteration is required for Optical patterning model is called, in the image under the conditions of specific photoetching on correct estimation silicon chip.Optical imagery model is photoetching process Key, at present the general principle of optical patterning mainly include Hopkins image-forming principles and Abbe image-forming principles, based on this two kinds The Imaging fast algorithm of image-forming principle obtains long-term development, and the essence of optical patterning model is a series of Fourier optics computings Process.Increasing with the scale of integrated circuit, the device integrated on single semiconductor chip is more and more, corresponding photoetching The mask plate exposure figure that process needs becomes increasingly complex, and this requires that the calculating of optical patterning model must be rapidly and efficiently , so that whole optical near-correction process circulation time is reduced, production efficiency is improved, reduces production cost.
There are document (A.K.-K.Wong, Resolution enhancement techniques in the prior art Optical lithography, vol.47.SPIE press, 2001.) a kind of photoetching resolution enhancing technology is disclosed, its profit Four-dimensional intersection transmission function (abbreviation TCC, Transmission Cross is established with Hopkins image-forming principles Coefficient) optical parametric of imaging system (light source, numerical aperture, difference, out of focus etc.) is characterized, for identical optical The imaging system of parameter, TCC only needs to calculate once, and can be recycled.There is document in the prior art in addition (N.B.Cobb, Fast Optical and Process Proximity Correction Algorithms for Integrated Circuit Manufacturing, Ph.D.dissertation, University of California, Berkeley, 1998.) fast optical and technique near-correction algorithm of a kind of IC manufacturing are proposed, utilizes characteristic value Analysis method, TCC characteristic value and characteristic vector are extracted, by retaining the characteristic value and characteristic vector larger to Imaging, The number for calculating the Fourier transform needed for light distribution can then be greatly reduced so as to realizing quick calculating.But according to Hopkins Optical imagery is theoretical, establishes four-dimensional intersection transmission function TCC, and it intersects transmission function TCC calculating and is related to quad-slope integration computing, It is quite time-consuming.If respective optical parameter changes, if it has to recalculates TCC, it is calculated according to normal computational methods TCC will have a strong impact on the efficiency of light distribution calculating, so as to influence the design efficiency of photoetching process.
The final purpose of optical patterning is to obtain the light distribution of imaging plane, and most of ways are to pass through Fourier Conversion, tcc=F is described by the intersection transmission function TCC on frequency domain in spatial domain-1[TCC], and Eigenvalues analysis is done to it. At present, Fast Fourier Transform (FFT) (FFT, Fast Fourier Transform), computation complexity are O [n log (n)], often quilt For Fourier transformation.On the other hand, there is frequency domain discrete sampling and spatial domain discrete sampling coupled relation in FFT, and discrete adopt Sample can only be quadrature sampling.
Therefore, it is necessary to a kind of new intersection transmission function quick decomposition method based on polar coordinates sampling is provided to solve Above mentioned problem.
【The content of the invention】
It is a primary object of the present invention to provide a kind of intersection transmission function quick decomposition method based on polar coordinates sampling, Can fast resolving obtain TCC kernel functions so that light distribution calculate it is quick and efficient, so as to which the photoetching process for meeting actual is set Meter demand.
The present invention is achieved through the following technical solutions above-mentioned purpose：A kind of intersection transmission function based on polar coordinates sampling is fast Fast decomposition method, it comprises the following steps,
Step S101：Obtain the light source function J (f, g) and pupil function P (f, g) of imaging system；
Step S102：In effective coverage, using the polar coordinates method of sampling, the coordinate of some sampled points in spatial domain is obtained (ri,θj)；
Step S103：Light source interaural crosscorrelation function corresponding to all sampled points is calculated by non-homogeneous inverse Fourier transformAnd pupil function
Step S104：Calculate the intersection transfer function values that the sampled point corresponds to spatial domain
And establish sampling matrix
Step S105：One group of orthogonal basis function is established in spatial domainCalculate the orthogonal basis function Functional value on corresponding polar coordinates sampling locationAnd establish matrix Q=[q1,q2,…qk], wherein k is selection The number of orthogonal basis function, column vector qlEach element qlkThe functional value CB of a corresponding sampled pointn,s(rk,θk)；
Step S106：QR matrix decompositions are carried out to the matrix Q
Step S107：Calculate projection matrixAnd singular value decomposition is carried out to projection matrix P and obtains P= UU*；
Step S108：Obtain and intersect transmission function in spatial domainKernel function be
Wherein uijFor corresponding element in matrix U.
Further, the polar coordinates method of sampling used in the step S102 for
1) r is equidistantly sampled in radial directioni=iL/n, i=0,1 ..., n, wherein L are the effective district of sampling；
2) in angular equidistantly sampling θj=j2 π/m, j=0,1 ..., m；
Further, the formula of the non-homogeneous inverse Fourier transform in the step S103 is：
Wherein, F (n) is uniform orthogonal sampling position (f on frequency domaini,gi) respective function value, f (ωm) in spatial domain Polar coordinates sampled point (ri,θj) inverse Fourier transform calculated value.
Further, the orthogonal basis function in the step S105For the orthogonal basis letter of Bessel functional dependences Number, its expression formula are
Wherein, λn,sFor s-th of zero root of n rank Bessel functions.
Further, the matrix R in the step S106 is upper triangular matrix, matrixFor orthonormal matrix.
Further, the step S107 comprises the following steps,
1) transmission function will be intersectedProject to orthogonal basis functionOn obtain：
Wherein pns,mtTo intersect transmission functionIn orthogonal basis functionOn projection coefficient；
2) the matrix Q obtained according to the step S104 sampling matrix T obtained and step S105, will
It is expressed as follows using matrix form：
T=QPQ*
3) the QR matrix decompositions in step S106, to T=QPQ*Matrix operation is carried out to solve to obtain projection matrix
Further, the orthogonal basis function selected in the step S105Energy concentrate on its peak value λn,s/ (2 π) position, according to regular λn+3,s/ (2 π) ＜ L select basic function.
Further, the angular number of samples m=2nmax+ 1, wherein nmaxTo select orthogonal basis functionIn The most high-order of Bessel functions.
Compared with prior art, a kind of intersection transmission function quick decomposition method based on polar coordinates sampling of the present invention has Beneficial effect is：One group of sampled point is established using polar coordinates, by Nonuniform fast Fourier transform, establishes the sampling square of respective function Battle array；By establishing the orthogonal basis function of one group of Bessel functional dependence, and pass through the corresponding orthogonal basis of polar coordinates sampled point Functional value establishes an auxiliary solution matrix, is decomposed with reference to the QR of the companion matrix so that be more prone to efficiently pass through matrix Decomposition operation establishes projection matrix；Projection matrix dimension be less than intersect transfer function matrix dimension so that projection matrix it is strange Different value, which is decomposed, to be more prone to；According to singular value decomposition, the decomposition for intersecting transmission function is realized, that is, establishes kernel function, intersected Transmission functionKernel functionSo that light distribution calculating is quick and efficient, so as to meet actual photoetching Technological design demand.
【Brief description of the drawings】
Fig. 1 is the calculation process schematic diagram of the embodiment of the present invention；
Fig. 2 is simple polar coordinates method of sampling schematic diagram in the embodiment of the present invention；
Fig. 3 is to intersect polar coordinates method of sampling schematic diagram in the embodiment of the present invention；
Fig. 4 is Nonuniform fast Fourier transform result of the present invention and the contrast schematic diagram of continuous fourier transform；
Fig. 5 is orthogonal basis function in the embodiment of the present inventionSchematic diagram；
Fig. 6 is orthogonal basis function in the embodiment of the present inventionSchematic diagram；
Fig. 7 is the light distribution schematic diagram of mask pattern in the embodiment of the present invention；
Fig. 8 is outline drawing of the light distribution in specific threshold of mask pattern in the embodiment of the present invention.
【Embodiment】
Embodiment：
It is as follows based on the imaging theory of Hopkins diffraction optics, imaging intensity distribution function formula：
Wherein, i is imaginary unit, and M (f, g)=F [m (x, y)] is the two-dimension fourier transform of mask plate spatial distribution (FFT, Fast Fourier Transform), TCC are corresponding four-dimensional intersection transmission function, and it is defined as：
TCC(f1,g1；f2,g2)=∫ ∫ J (f, g) P (f+f1,g+g1)·P*(f+f2,g+g2)dfdg (2)
Wherein, J (f, g) is light source function, and P (f, g) is the pupil function of imaging system, P*(f, g) is the P of pupil function The complex conjugate of (f, g), state the optical parametric of optical imaging system.It is in domain space that above-mentioned intersection transmission function, which defines, Description, then it is as follows the intersection transmission function expression formula in spatial domain can be obtained by Fourier transformation：
tcc(x1,y1；x2,y2)=F-1[TCC(f1,g1；f2,g2)]
=j (- x1-x2,-y1-y2)p(x1,y1)p*(-x2,-y2) (3)
Wherein, j (x, y)=F-1[J (f, g)], p (x, y)=F-1[P(f,g)].According to formula (3), formula (1) is convertible For：
According to Cobb decomposition algorithm, then the singular value decomposition that tcc be present is as follows：
Wherein, keri(x, y) is tcc kernel function, then can pass through the related quick calculating imaging system of one group of convolution Light distribution is as follows：
Fig. 1 is refer to, the present embodiment is the intersection transmission function quick decomposition method based on polar coordinates sampling, i.e., to formula (3) fast decoupled calculating is carried out, it comprises the following steps：
Step S101：The optical parametric of imaging system, specially light source function J (f, g) and pupil function P (f, g) are obtained, Optical parametric in the present embodiment in imaging system includes：The two poles of the earth sector light source σin=0.4, σout=0.8, λ=248nm, NA= 0.53。
Step S102：In effective coverage, using the polar coordinates method of sampling, the coordinate of some sampled points in spatial domain is obtained Information (ri,θj).Specifically, the polar coordinates method of sampling includes：
1) r is equidistantly sampled in radial directioni=iL/n, i=0,1 ..., n, wherein L are the effective district of sampling；
2) in angular equidistantly sampling θj=j2 π/m, j=0,1 ..., m.
The polar coordinates method of sampling is of equal value with quadrature sampling, and the polar coordinates method of sampling has the small great advantage of amount of calculation, this The polar coordinates method of sampling that invention proposes includes the sampling of simple polar coordinates, intersects polar coordinates sampling, the signal of two kinds of method of samplings Figure difference is as shown in Figure 2 and Figure 3.
Step S103：Light source interaural crosscorrelation function corresponding to all sampled points is calculated by non-homogeneous inverse Fourier transformAnd pupil function
Specifically, the formula of non-homogeneous inverse Fourier transform is：
Wherein, ωmFor the relevant position in spatial domain, F (n) is uniform orthogonal sampling position (f on frequency domaini,gi) it is corresponding Functional value, f (ωm) it is polar coordinates sampled point (r in spatial domaini,θj) the inverse Fourier that is calculated according to above-mentioned formula (7) becomes Change.Compared with continuous fourier transform, the Nonuniform fast Fourier transform fast method of the present embodiment sampling, it can reach higher Required precision, its comparing result are as shown in Figure 4.
Step S104：Calculate the intersection transfer function values that polar coordinates sampled point corresponds to spatial domain
And establish sampling matrix
Specifically, step S103 obtains the functional value on polar coordinates sampling location by non-homogeneous inverse Fourier transform, lead to Crossing simple multiplication computing can obtain
Step S105：One group of orthogonal basis function is established in spatial domainCalculate the orthogonal basis function Functional value on corresponding polar coordinates sampling locationAnd establish matrix Q=[q1,q2,…qk], wherein k is selection The number of orthogonal basis function, column vector qlEach element qlkThe functional value CB of a corresponding sampled pointn,s(rk,θk)。
Orthogonal basis function described in the present embodimentUsing the orthogonal basis function of Bessel functional dependences, it is expressed Formula is
λ in formula (8)n,sFor s-th of zero root of n rank Bessel functions；Orthogonal basis function n and s are fixed, calculation procedure All sampled point (r in S102i,θj) corresponding to orthogonal basis function value, then column vector qlEach element qlkA corresponding sampling The orthogonal basis function value CB of pointn,s(rk,θk).Due to orthogonal basis functionEnergy all concentrate on peak value λn,s/ (2 π) is attached Closely, thus only need consider peak positioned at sampling effective coverage in basic function, i.e., according to regular λn+3,s/ (2 π) ＜ L are selected The required orthogonal basis function of the present invention.And according to sampling thheorem, pole in step S102 is can determine that with reference to the orthogonal basis function of selection The angular number of samples of coordinate sampling meets m=max (4,4nmax), wherein nmaxFor Bessel functions in selection orthogonal basis function Most high-order.Orthogonal basis function value in the present embodimentAs shown in figure 5,As shown in Figure 6.
Step S106：QR matrix decompositions are carried out to matrix QMatrix R is upper triangular matrix, thus can be held very much Its inverse matrix of easy calculating, matrixFor orthonormal matrix.
Step S107：Calculate projection matrixAnd singular value decomposition is carried out to projection matrix P and obtains P= UU*。
Specifically, comprise the following steps：
1) in order that projection matrix P solution is more prone to and fast, then needs to intersect transmission functionIn the presence of Orthogonal basis functionVariables separation, therefore transmission function will be intersectedProject to orthogonal basis functionOn obtain：
Wherein pns,mtTo intersect transmission functionIn orthogonal basis functionOn projection coefficient.
2) the matrix Q obtained according to the step S104 sampling matrix T obtained and step S105, matrix is used by formula (9) Form is expressed as follows：
T=QPQ* (10)
3) the QR matrix decompositions in step S106, matrix operation is carried out to formula (10) and solved so as to can more hold Easily it is quickly obtained projection matrixIn addition, projection matrix P size is less than sampling matrix T, So that projection matrix P singular value decomposition is easier.
Step S108：Obtain and intersect transmission function in spatial domainKernel function be
Wherein uijFor corresponding element in matrix U.
Specifically, in step S107, projection matrix P singular value decomposition so that formula (9) can be described as a matrix and oneself The product of body conjugation, so as to can very easily calculate the analytic solutions for obtaining kernel function.
The content of the invention of the present embodiment is mainly to intersect the quick decomposition method of transmission function, for changing for technological parameter Become, can again quick calculating optical imaging system intersection transmit kernel function, so as to quickly calculate mask pattern in image plane On light distribution, to meet photoetching process design requirement.Fig. 1 is refer to, the intersection being quickly calculated according to the present embodiment The kernel function of transmission functionAnd mask plate spatial distributionAccording to the convolution algorithm of formula (6), image plane is readily available On light distribution.In addition, convolution algorithm can be calculated according to Fourier transformation by simple multiplication, the present invention is no longer described in detail.Pin Light distribution to simple mask pattern m (x, y) and in image plane is as shown in Figure 7, Figure 8.
A kind of intersection transmission function fast decoupled based on polar coordinates sampling of the present embodiment based on Hopkins image-forming principles The beneficial effect of method is：One group of sampled point is established using polar coordinates, by Nonuniform fast Fourier transform, establishes respective function Sampling matrix；By establishing the orthogonal basis function of one group of Bessel functional dependence, and it is corresponding by polar coordinates sampled point Orthogonal basis function value establish one auxiliary solution matrix, with reference to the companion matrix QR decompose so that be more prone to efficiently Projection matrix is established by matrix decomposition computing；Projection matrix dimension is less than the dimension for intersecting transfer function matrix so that projection Matrix singular value decomposition is more prone to；According to singular value decomposition, the decomposition for intersecting transmission function is realized, that is, establishes core letter Number, intersect transmission functionKernel functionSo that light distribution calculating is quick and efficient, so as to meet reality Photoetching process design requirement.
Above-described is only some embodiments of the present invention.For the person of ordinary skill of the art, not On the premise of departing from the invention design, various modifications and improvements can be made, these belong to the protection model of the present invention Enclose.

Claims (8)

  1. A kind of 1. intersection transmission function quick decomposition method based on polar coordinates sampling, it is characterised in that：It comprises the following steps,
    Step S101：Obtain the light source function J (f, g) and pupil function P (f, g) of imaging system；
    Step S102：In effective coverage, using the polar coordinates method of sampling, the coordinate (r of some sampled points in spatial domain is obtainedi, θj)；
    Step S103：Light source interaural crosscorrelation function corresponding to all sampled points is calculated by non-homogeneous inverse Fourier transformAnd pupil function
    Step S104：Calculate the intersection transfer function values that the sampled point corresponds to spatial domain And establish sampling matrix
    Step S105：One group of orthogonal basis function is established in spatial domainCalculate the orthogonal basis functionIn phase Answer the functional value on polar coordinates sampling locationAnd establish matrix Q=[q1,q2,…qk], wherein k is the orthogonal of selection The number of basic function, column vector qlEach element qlkThe functional value CB of a corresponding sampled pointn,s(rk,θk)；
    Step S106：QR matrix decompositions are carried out to the matrix Q
    Step S107：Calculate projection matrixAnd singular value decomposition is carried out to projection matrix P and obtains P=UU*；
    Step S108：Obtain and intersect transmission function in spatial domainKernel function be Wherein uijFor corresponding element in matrix U.
  2. 2. the intersection transmission function quick decomposition method as claimed in claim 1 based on polar coordinates sampling, it is characterised in that：Institute State the polar coordinates method of sampling that is used in step S102 for
    1) r is equidistantly sampled in radial directioni=iL/n, i=0,1 ..., n, wherein L are the effective district of sampling；
    2) in angular equidistantly sampling θj=j2 π/m, j=0,1 ..., m.
  3. 3. the intersection transmission function quick decomposition method as claimed in claim 1 based on polar coordinates sampling, it is characterised in that：Institute The formula for stating the non-homogeneous inverse Fourier transform in step S103 is：
    Wherein, F (n) is uniform orthogonal sampling position (f on frequency domaini,gi) respective function value, f (ωm) sat for pole in spatial domain Mark sampled point (ri,θj) inverse Fourier transform calculated value.
  4. 4. the intersection transmission function quick decomposition method as claimed in claim 2 based on polar coordinates sampling, it is characterised in that：Institute State the orthogonal basis function in step S105For the orthogonal basis function of Bessel functional dependences, its expression formula is
    Wherein, λn,sFor s-th of zero root of n rank Bessel functions.
  5. 5. the intersection transmission function quick decomposition method as claimed in claim 1 based on polar coordinates sampling, it is characterised in that：Institute It is upper triangular matrix to state the matrix R in step S106, matrixFor orthonormal matrix.
  6. 6. the intersection transmission function quick decomposition method as claimed in claim 4 based on polar coordinates sampling, it is characterised in that：Institute Step S107 is stated to comprise the following steps,
    1) transmission function will be intersectedProject to orthogonal basis functionOn obtain：
    Wherein pns,mtTo intersect transmission functionIn orthogonal basis functionOn projection coefficient；
    2) the matrix Q obtained according to the step S104 sampling matrix T obtained and step S105, will It is expressed as follows using matrix form：
    T=QPQ*
    3) the QR matrix decompositions in step S106, to T=QPQ*Matrix operation is carried out to solve to obtain projection matrix
  7. 7. the intersection transmission function quick decomposition method as claimed in claim 4 based on polar coordinates sampling, it is characterised in that：Institute State the orthogonal basis function selected in step S105Energy concentrate on its peak value λn,s/ (2 π) positions, according to rule λn+3,s/ (2 π) ＜ L select basic function.
  8. 8. the intersection transmission function quick decomposition method as claimed in claim 4 based on polar coordinates sampling, it is characterised in that：Institute State angular number of samples m=2nmax+ 1, wherein nmaxTo select orthogonal basis functionThe most high-order of middle Bessel functions.
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Cited By (8)

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Publication number Priority date Publication date Assignee Title
CN109164683A (en) * 2018-09-30 2019-01-08 墨研计算科学（南京）有限公司 Light distribution fast determination method and device based on mask graph processing
CN109212913A (en) * 2018-09-30 2019-01-15 墨研计算科学（南京）有限公司 Light distribution acquisition methods and device based on non-homogeneous calculating
CN109270802A (en) * 2018-09-30 2019-01-25 墨研计算科学（南京）有限公司 A kind of fast acquiring method and device of crystal column surface light distribution
CN110232438A (en) * 2019-06-06 2019-09-13 北京致远慧图科技有限公司 The image processing method and device of convolutional neural networks under a kind of polar coordinate system
CN110703438A (en) * 2019-12-16 2020-01-17 墨研计算科学（南京）有限公司 Method and device for calculating photoetching model based on polar coordinate vector
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CN110929366A (en) * 2018-08-31 2020-03-27 台湾积体电路制造股份有限公司 Method for computing a feature kernel for optical model simulation
CN110929366B (en) * 2018-08-31 2024-05-28 台湾积体电路制造股份有限公司 Method for calculating feature kernels for optical model simulation
CN109164683A (en) * 2018-09-30 2019-01-08 墨研计算科学（南京）有限公司 Light distribution fast determination method and device based on mask graph processing
CN109212913A (en) * 2018-09-30 2019-01-15 墨研计算科学（南京）有限公司 Light distribution acquisition methods and device based on non-homogeneous calculating
CN109270802A (en) * 2018-09-30 2019-01-25 墨研计算科学（南京）有限公司 A kind of fast acquiring method and device of crystal column surface light distribution
CN110232438A (en) * 2019-06-06 2019-09-13 北京致远慧图科技有限公司 The image processing method and device of convolutional neural networks under a kind of polar coordinate system
CN110703438A (en) * 2019-12-16 2020-01-17 墨研计算科学（南京）有限公司 Method and device for calculating photoetching model based on polar coordinate vector
WO2021121272A1 (en) * 2019-12-16 2021-06-24 墨研计算科学（南京）有限公司 Method and device for vector calculation photoetching model based on polar coordinates
CN112767296A (en) * 2021-01-20 2021-05-07 山东大学 Multi-sensor data fusion method and system based on time series imaging
CN117452779A (en) * 2023-12-04 2024-01-26 武汉宇微光学软件有限公司 Modeling and calibrating method and device for calculating photoetching nonlinear system
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