CN114740557A - Method for designing linear density of fringe by eliminating aberration and changing grating pitch in raster scanning photoetching - Google Patents

Method for designing linear density of fringe by eliminating aberration and changing grating pitch in raster scanning photoetching Download PDF

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CN114740557A
CN114740557A CN202210041105.6A CN202210041105A CN114740557A CN 114740557 A CN114740557 A CN 114740557A CN 202210041105 A CN202210041105 A CN 202210041105A CN 114740557 A CN114740557 A CN 114740557A
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CN114740557B (en
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宋�莹
张刘
刘玉娟
朱杨
章家保
王文华
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Jilin University
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Abstract

The invention provides a fringe line density design method for eliminating aberration and variable-pitch grating scanning photoetching, relates to the technical field of holographic grating manufacture, and provides an interference fringe line density design method for eliminating aberration and variable-pitch grating scanning photoetching. The line density change rule of the photoetching interference fringes can be designed according to the method. The precision of the groove density of the variable-pitch grating is improved, the controllability of the exposure contrast process parameters is ensured, and the method has important significance for improving the scanning photoetching manufacturing level of the variable-pitch grating and improving the success rate of grating manufacturing.

Description

Method for designing linear density of fringe by eliminating aberration and changing grating pitch in raster scanning photoetching
Technical Field
The invention relates to the technical field of holographic grating manufacture, in particular to a design method of interference fringe line density of aberration-eliminating variable-pitch grating scanning photoetching.
Background
The aberration-eliminating variable-pitch grating is a plane grating with grating groove density changing according to a certain rule, and optical aberrations such as defocusing, spherical aberration and the like are corrected through the change of the grating groove density. Compared with the curved surface grating, the grating substrate is a plane, and the processing difficulty of the substrate is reduced. The meridian light incident to the grating with the variable grating pitch can form a spectrum line, extra collimating and focusing optical elements are not needed in a spectrum instrument, the volume and the weight of the instrument are reduced, the light energy utilization rate is improved, a high laser damage threshold value is achieved, and the method has important application in the fields of synchronous radiation light source devices, high-energy laser devices and the like.
The variable pitch grating is usually fabricated by mechanical scribing, electron beam direct writing, laser direct writing, holographic exposure and the like. The manufacturing modes of mechanical scribing, electron beams, laser direct writing and the like belong to ultra-precision machining, the machining of grating grooves is completed line by line, the manufacturing efficiency is low, and the change of adjacent grating pitches of the variable-pitch grating is generally not more than the nanometer level, so that the requirements on corresponding ultra-precision machining equipment and machining conditions are high, and the manufacturing difficulty and the manufacturing cost are high. The conventional holographic exposure method for manufacturing the grating pitch-variable grating can adopt a spherical wave or aspheric wave exposure system, but the adjustable freedom of the spherical wave exposure system is less, and the problem of groove bending exists, which can cause the reduction of the resolution capability of the grating. Aspheric wave exposure system design, processing and debugging difficulty are large, the process is not easy to realize, and large error exists between the actual grating groove density and the expected value.
The variable period scanning photoetching is another important method for manufacturing the variable grating pitch grating, an interference optical system forms an interference pattern with a small caliber (micrometer-millimeter magnitude), a two-dimensional worktable bears a grating substrate to carry out step scanning movement, relative movement is generated between the interference pattern and the grating substrate, and interference fringes are recorded in photoresist coated on the grating substrate until the exposure of the whole effective area of the grating substrate is completed. In order to realize the manufacture of the variable-pitch grating, in the exposure process, the interference included angle of a coherent light beam is changed through precise photoelectric control according to the change function of the line density of the interference fringe for photoetching, and the line density of the interference fringe is continuously and precisely adjusted, so that the groove density of the manufactured grating meets the requirement of a design index. The variable-pitch grating manufactured by the manufacturing method has no problem of groove bending, hundreds of interference fringes exist in the interference pattern, the manufacturing efficiency is greatly improved, and a large-caliber optical system is not needed when large-area grating manufacturing is carried out.
However, if the density variation function of the interference fringe lines is equal to the target groove density function of the variable-pitch grating, the density of the interference fringe lines is adjusted, and the groove density of the manufactured variable-pitch grating has larger deviation from a design value. This is mainly due to the following two reasons that when the line density of the interference fringes is changed by the variable-period scanning lithography system, the line densities of all the interference fringes in the interference pattern change equally, and the precise adjustment of the interference fringes from one line pitch to another cannot be realized. And the intensity distribution of the interference patterns is Gaussian distribution, in order to ensure the uniformity of the exposure, a certain overlap exists between the interference patterns of adjacent scanning sections, and the manufactured grating groove distribution has a homogenization effect.
The establishment of a design method of interference fringe line density is a key problem in the application of the variable period scanning lithography technology. The invention provides a method for designing the linear density of interference fringes for aberration-eliminating variable-pitch grating scanning lithography, according to the linear density change function of the interference fringes designed by the method, the linear density of the interference fringes in the lithography process is changed, and the groove density of the variable-pitch grating finally obtained can meet the design index requirement.
Disclosure of Invention
The invention provides an interference fringe line density design method for aberration-eliminating variable-pitch grating scanning photoetching, which can be used for designing an interference fringe line density function in a scanning photoetching process according to a variable-pitch grating groove density target function, designing the function according to the interference fringe line density, changing the line density of interference fringes, and finally obtaining the variable-period grating groove density which meets the requirements of the grating groove density target function.
The method for designing the linear density of the fringe of the optical grating scanning photoetching with the aberration elimination and the variable pitch comprises the following steps:
firstly, determining a scribed line density function of a variable-pitch grating and general photoetching process manufacturing parameters;
step one, according to the aberration eliminating characteristic of the variable-pitch grating and the application requirement of the variable-pitch grating in an instrument, designing a target function g (x) of the groove density of the variable-pitch grating as follows:
g(x)＝ng0+ng1(x-Wg/2)+ng2(x-Wg/2)2+ng3(x-Wg/2)3
ideal phase distribution phi of the gratingg(x) Expressed as:
Φg(x)＝2πg(x)·x
where x is the coordinate of the vector direction of the grating, x is 0 at the grating boundary, and W isgIs the total width in the direction of the grating vector, ng0Is the groove density at the center of the grating, ng1Coefficient of first order of grating groove density, ng2Coefficient of quadratic term, n, for grating groove densityg3The coefficient of the cubic term of the grating groove density; n is saidg0、 ng1、ng2And ng3Determining according to the variable-pitch grating aberration correction principle and the use parameters of a spectroscopic instrument or a laser device;
determining the following manufacturing parameters when the variable-period grating is manufactured according to the design and the debugging parameters of the variable-period scanning photoetching system and the target function of the groove density of the variable-pitch grating obtained in the step one by one:
setting the radius of Gaussian beam waist of interference pattern to be RhoThe overlapping width of the interference patterns of the adjacent scanning segments accounts for the radius R of the beam waisthoThe ratio StepRatio, the width of the interference pattern overlap is StepRatio Rho；
Setting the total step number of step scanning as N, wherein N is more than or equal to Wg/(RhoStepRatio) +1, making the width of the exposure area greater than the effective width of the grating;
the step number of each step of step scanning is Nsteps，Nsteps＝round(Rho·StepRatio·ng0) (ii) a round () is four shed fiveInputting an integer taking function;
step two, calculating the grating phase distribution error phi when the interference fringe line density change function f (x) is equal to the grating groove density function g (x) of the variable-pitch grating according to the calculation method of the total exposure of the variable-period scanning photoetchinge(x)；
The method for calculating the total exposure of the variable-period scanning photoetching comprises the following steps:
setting the linear density variation function f (x) of the interference fringe and the target groove density function g (x) of the variable-pitch grating to have the same form, and expressing the following conditions:
f(x)＝m0+m1(x-Wg/2)+m2(x-Wg/2)2+m3(x-Wg/2)3
in the formula, m0Coefficient of constant term being a function of variation of the linear density of the interference fringes, m1Coefficient of first order of variation function of linear density of interference fringe, m2Coefficient of quadratic term, m, being a function of the variation of the linear density of the interference fringes3Coefficient of cubic term of density variation function of interference fringe line; the initial scanning segment of scanning photoetching starts from x being 0, the corresponding step number k being 0 when x being 0, the initial scanning being the 1 st scanning, the corresponding exposure being D0(x)，SkStep distance of the k step;
S0＝0，
Figure BDA0003470252210000041
is the total distance from step 0 to step k,
Figure BDA0003470252210000042
after k steps are carried out, the line density of interference fringes of the (k + 1) th scanning is obtained; deltak＝fk-fk-1Is the difference between the interference fringe line density of k +1 scanning and k scanning, when k is equal to 0, delta0When k is equal to 0>At the time of 0, the number of the first,
Figure BDA0003470252210000043
after the step of the k-th step,exposure D of k +1 th scank(x) And phase difference between the (k + 1) th scan and the initial scan
Figure BDA0003470252210000044
Comprises the following steps:
Figure BDA0003470252210000045
Figure BDA0003470252210000046
wherein B (x) is the background component of the single scan exposure, and A (x) is the Gaussian distribution exposure intensity envelope of the single scan exposure;
Figure BDA0003470252210000047
when photoetching is finished, the total exposure on the grating is the superposition D of the exposure of N +1 times of scanning after the total steps of step scanning are N stepstot(x) Namely:
Dtot(x)＝D0(x)+D1(x)+…DN(x)
＝Btot(x)+Atot(x)sin(Ψtot(x))
in the formula, Btot(x) Background component of total exposure, Atot(x) The amplitude of the AC component of the total exposure;
Figure BDA0003470252210000048
Figure BDA0003470252210000049
Figure BDA0003470252210000051
Figure BDA0003470252210000052
Ψtot(x)＝2πxf0+Ψ(x)
Ψ(x)＝arctan[F(x)/E(x)]
in the formula, Ψtot(x) Psi (x) is the phase increment between the total exposure and the 1 st scan, psitot(x) Equal to the actual phase distribution of the manufactured variable pitch grating; gamma (x) ═ Atot(x)/Btot(x) Is the total exposure contrast;
setting f (x) g (x), i.e., m0＝ng0，m1＝ng1，m2＝ng2，m3＝ng3Calculating the phase variation Ψ of the exposure by using the method for calculating the total exposure of variable period scanning lithographytot(x) The error of the grating phase distribution between the actual phase distribution of the manufactured variable pitch grating and the ideal phase distribution of the grating is phie(x)＝ Ψtot(x)-Φg(x)；
Step three, designing a three-order term coefficient m of an interference fringe line density change function through a data fitting and iterative optimization method3Optimized design value of (m)3_optimal；
Step four, designing a secondary term coefficient m of the interference fringe linear density variation function through a data fitting and iterative optimization method2Optimized design value of (m)2_optimal；
Step five, designing an optimized design value m of a first-order coefficient m1 of the interference fringe linear density change function through a data fitting and iterative optimization method1_optimal；
Step six, designing a constant term coefficient m of a linear density change function of the interference fringe0Optimized design value of (m)0_optimal；
Step seven, optimizing m according to the step three to the step six3_optimal、m2_optimal、m1_optimalAnd m0_optimalChecking whether the exposure contrast satisfies the requirements of the exposure processSolving;
the optimized and designed interference fringe line density change function is as follows:
foptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3；
calculating to obtain an exposure contrast gamma (x) according to the method for calculating the total exposure of the variable-period scanning photoetching given in the second step, judging whether the exposure contrast gamma (x) meets the requirement of the exposure contrast within the whole range of x, if not, reducing the ratio StepRatio of the overlapping width of the interference patterns in the second step to the beam waist radius, and executing the second step to the seventh step again until the gamma (x) meets the requirement of the exposure contrast;
if so, finishing the whole optimization process and designing the interference fringe line density change function f according to the optimizationoptimal(x) And changing the linear density of interference fringes in the photoetching process to obtain the variable-pitch grating with the target groove density.
The invention has the following positive effects: the invention relates to an interference fringe line density design method for aberration-eliminating variable-pitch grating scanning photoetching, which is used for manufacturing a variable-pitch grating by a variable-period scanning interference photoetching system. According to the known groove density of the variable-pitch grating, the line density variation rule of the photoetching interference fringes can be designed according to the method. The precision of the groove density of the variable-pitch grating is improved, the controllability of the exposure contrast process parameters is ensured, and the method has important significance for improving the scanning photoetching manufacturing level of the variable-pitch grating and improving the success rate of grating manufacturing.
Drawings
FIG. 1 is a simplified schematic diagram of a variable period scanning lithography apparatus used in the method for designing the linear density of interference fringes for aberration-reducing variable pitch grating scanning lithography according to the present invention.
FIG. 2 is a schematic diagram of coordinate system definition.
FIG. 3 is a schematic diagram of interference pattern beam waist radius and overlay lithography.
FIG. 4 is a diagram showing the phase distribution of a variable pitch gratingg(x) And as a function of the grating groove densityThe linear density of the interference fringe is changed to obtain the phase distribution psi of the gratingtot(x)(ng0＝1200gr/mm，ng1＝-0.7783gr/mm2， ng2＝1.865×10-4gr/mm3，ng3＝-8.1336×10-8gr/mm4，W g30 mm).
FIG. 5 shows the grating phase distribution error phi obtained by varying the density of interference fringe lines according to the grating groove density functione(x)＝Ψtot(x)-Φg(x) The grating parameter effect diagram is the same as that in FIG. 4.
FIG. 6 shows the cubic coefficient m of the interference fringe line density3_optimalThe optimization design flow chart of (1).
FIG. 7 is a phase distribution Φ of a variable pitch gratingg(x) And changing the density of interference fringe lines according to the density of interference fringe lines obtained by the design method to obtain a grating phase distribution psitot(x)(ng0＝1200gr/mm， ng1＝-0.7783gr/mm2，ng2＝1.865×10-4gr/mm3，ng3＝-8.1336×10-8gr/mm4，Wg＝30mm，Rho＝0.1mm，ξ1order＝ξ2order＝ξ3order＝ξ4order1e-4 rad).
FIG. 8 shows the phase distribution error Φ of the grating obtained according to the design valuee(x)＝Ψtot(x)-Φg(x) Effect graphs;
fig. 9 is a graph showing the effect of the exposure contrast γ (x) obtained according to the design value.
Detailed Description
The method for designing the line density of the interference fringes for the aberration-eliminating variable-pitch grating scanning lithography according to the embodiment is described with reference to fig. 1 to 9, and the method is mainly applied to a variable-period scanning lithography system, and is composed as shown in fig. 1, wherein a plurality of measurement and control elements are removed. In the figure, 1 and 2 are two coherent light beams, 3 and 4 are interference light beam adjusting mirrors, 5 is a semi-reflecting and semi-transparent mirror, 6 and 7 are lenses, and the interference light beam adjusting mirrors are used for forming a 4f optical system, so that interference of 1 and 2 is realized to form an interference pattern 8, and the intensity of the interference pattern 8 is in Gaussian distribution. 3 and 4 are located at the front focal plane position of 6, and 11 is an interference fringe linear density control system, and the linear density of interference fringes in the interference pattern can be adjusted by adjusting the interference angles of the coherent light beams 1 and 2 through 3 and 4 according to an interference fringe linear density function. The grating substrate 9 is coated with photoresist, and the two-dimensional motion worktable 10 is used for bearing the grating substrate 9 to carry out step-and-scan motion.
The method proposed by the present embodiment comprises the following steps:
step one, determining a groove density function of the variable-pitch grating and manufacturing parameters of a general photoetching process.
According to the aberration eliminating characteristic of the variable-pitch grating and the application requirement of the variable-pitch grating in an instrument, designing a groove density objective function of the variable-pitch grating as follows:
g(x)＝ng0+ng1(x-Wg/2)+ng2(x-Wg/2)2+ng3(x-Wg/2)3
the grating coordinate system is defined as shown in FIG. 2, and the ideal phase distribution phi of the gratingg(x) Can be expressed as
Φg(x)＝2πg(x)·x
Wherein g (x) is the target groove density of the grating with variable grating pitch, x is the coordinate of the vector direction of the grating, x is 0 and is positioned at the boundary of the grating, and W is the distance between the grating and the edge of the gratinggIs the total width of the vector direction of the grating, ng0The groove density at the center of the grating is expressed in gr/mm (the number of grooves contained in each mm), ng1Is the first order coefficient of the grating groove density, and the unit is gr/mm2，ng2Is the quadratic coefficient of the grating groove density, and the unit is gr/mm3，ng3Coefficient of cubic term of grating groove density in unit of gr/mm4。ng0、ng1、ng2、ng3The method is determined according to the aberration correction principle of the variable-pitch grating and the use parameters of a spectroscopic instrument or a laser device.
In view of the basic composition and operation of a variable period scanning lithography system, as shown in FIG. 1, to ensure uniformity of exposure, there is overlap between the interference patterns of adjacent scan segments. According to the design, the installation and adjustment parameters of the system and the groove density function of the variable-pitch grating, the following manufacturing parameters are determined when the variable-period grating is manufactured:
the radius of the Gaussian beam waist of the interference pattern is Rho。
The overlapping width of the interference patterns of adjacent scanning segments accounts for the ratio StepRatio of the beam waist radius, so that the overlapping width of the interference patterns is StepRatio × RhoIn order to ensure the uniformity of the exposure, StepRatio is required to be less than 0.9, and the smaller StepRatio is, the higher the uniformity of the exposure is, but the larger the total number of scanning steps is, the lower the manufacturing efficiency is. As shown in fig. 3.
The total step number N of step scanning is more than or equal to Wg/(RhoStepRatio) +1, making the width of the exposed area larger than the effective width of the grating.
Step number N of each step of step-by-step scanningsteps，Nsteps＝round(Rho·StepRatio·ng0)。
And step two, establishing a variable period scanning photoetching total exposure calculation method, and calculating the exposure phase change psi (x) and the exposure contrast gamma when the grating groove density function is used as an interference fringe line density change function f (x).
The method for calculating the total exposure of the variable-period scanning photoetching comprises the following steps:
the interference fringe line density variation function and the grating groove density function have the same form and can be expressed as:
f(x)＝m0+m1(x-Wg/2)+m2(x-Wg/2)2+m3(x-Wg/2)3
in the formula, m0Coefficient of constant term, m, being a function of the variation of the linear density of the interference fringes1Coefficient of first order of variation function of linear density of interference fringe, m2Coefficient of quadratic term, m, being a function of the variation of the linear density of the interference fringes3Coefficient of cubic term of density variation function of interference fringe line; the scanning photoetching initial scanning segment starts when x is 0, the scanning step number k is 0, and SkStep distance of the k-th step, S0＝0，
Figure BDA0003470252210000081
The total distance from x to step k is 0,
Figure BDA0003470252210000082
interference fringe line density for the k +1 th scan after stepping for k steps. Deltak＝fk-fk-1Is the difference between the interference fringe line density of k +1 scanning and k scanning, when k is equal to 0, delta0When k is equal to 0>At the time of 0, the number of the first,
Figure BDA0003470252210000083
exposure D for the k +1 th scan after the k stepk(x) And phase difference between the (k + 1) th scan and the initial scan
Figure BDA0003470252210000091
Comprises the following steps:
Figure BDA0003470252210000092
Figure BDA0003470252210000093
where B (x) is the background intensity of the exposure, and A (x) is the intensity envelope of the Gaussian distribution interference pattern.
Figure BDA0003470252210000094
When photoetching is finished, the total exposure on the grating is the superposition D of the exposure of N +1 times of scanning after the total steps of step scanning are N stepstot(x) Namely:
Dtot(x)＝D0(x)+D1(x)+…DN(x)
＝Btot(x)+Atot(x)sin(2πxf0+Ψ)
Btot(x) Is composed ofBackground component of total exposure, Atot(x) The amplitude of the alternating current component of the total exposure;
Figure BDA0003470252210000095
Figure BDA0003470252210000096
Figure BDA0003470252210000097
Figure BDA0003470252210000098
Ψtot(x)＝2πxf0+Ψ(x)
Ψ(x)＝arctan[F(x)/E(x)]
in this embodiment, in addition to using the general photolithography process parameters in the first step, the following design parameters are set:
(1) setting the density function of interference fringe lines to be equal to the groove density function of the variable-pitch grating, namely m0＝ng0，m1＝ng1，m2＝ng2，m3＝ng3。
(2) The change in b (x) does not affect the design of the interference fringe line density, and b (x) is set to a (x) and the contrast of a single scanning exposure is set to 1.
(3) Step distance S of the kth stepkOne of the following three stepping modes can be selected, different interference fringe design values can be obtained according to the method of the embodiment, but the precision requirement of the groove density of the variable-pitch grating can be met:
①Sk＝Nsteps/fk-1。
②Sk＝2Nsteps/(fk+fk-1)，
Figure BDA0003470252210000101
③Sk＝Nsteps/fk，
Figure BDA0003470252210000102
according to the parameters of the step one and the parameters, by adopting a numerical calculation method and a method for calculating the total exposure of variable-period scanning photoetching, a psi (x) changing curve along with x, psi (x) and phi can be obtainedg(x) As shown in fig. 4. The phase error between the two is phie(x)＝Ψ(x)-Φg(x) As shown in FIG. 5, the density of the interference fringe lines is adjusted according to the groove density function of the grating with variable grating pitche(x) Not equal to 0, the variable pitch grating obtained by photoetching has a large difference from the groove density of a designed value.
Step three, designing a three-order term coefficient m of an interference fringe line density change function through a data fitting and iterative optimization method3Optimized design value of (m)3_optimal。
Adopting a polynomial curve fitting algorithm to correct the phase distribution error phi obtained in the step twoe(x) Performing a fourth-order polynomial curve fitting of phie(x) Has a fitting polynomial of phiep(x)＝2π(a40x4+a30x3+a20x2+a10x+a00)，a40、 a30、a20、a10And a00Respectively fitting a polynomial of four times phiep(x) The coefficient of (a);
fourth phase error threshold xi of optimized design for setting coefficient m3 of interference fringe linear density change function f (x)4orderM is obtained by calculation through a one-dimensional search iterative optimization method3_optimalThe iterative process is as follows, and the flow chart is shown in fig. 6.
(1) Setting search range [ a ] of iterative optimizationm3 (0),bm3 (0)]，am3 (0)＝m3_nearby-Hm3，bm3 (0)＝ m3_nearby+Hm3，2Hm3Is m3Breadth of initial search range of optimum value, number of iterations im3＝0，m3_nearbyIs the optimal solution m3_optimalInitial approximation of, m3_nearby＝ng3 2/(ng3+a40) The maximum phase error of the initial quartic term is phie_4order_m3 (0)＝abs(2πa40Wg 4)；
(2) If it is
Figure BDA0003470252210000111
If the threshold requirement is not met, executing the steps (3) to (7), otherwise, exiting the loop and executing the step (8);
(3) by using a one-dimensional search method (e.g. golden section, Fibonacci, etc.) in
Figure BDA00034702522100001126
Internally determined optimization variables
Figure BDA00034702522100001127
And
Figure BDA00034702522100001128
Figure BDA00034702522100001129
(4) according to the optimization variables
Figure BDA00034702522100001130
Value setting of linear density variation function of interference fringe
Figure BDA0003470252210000112
Calculating according to the variable period scanning photoetching total exposure quantity calculation method given in the step two
Figure BDA0003470252210000113
Figure BDA0003470252210000114
Corresponding phase error
Figure BDA0003470252210000115
To pair
Figure BDA0003470252210000116
A fourth-order polynomial fitting is performed,
Figure BDA0003470252210000117
is a fitting polynomial of
Figure BDA0003470252210000118
Recording the coefficient of four items
Figure BDA0003470252210000119
(5) According to the optimization variables
Figure BDA00034702522100001110
Value setting of the linear density variation function of the interference fringes
Figure BDA00034702522100001111
Calculating according to the variable period scanning photoetching total exposure quantity calculation method given in the step two
Figure BDA00034702522100001112
Figure BDA00034702522100001113
Corresponding phase error
Figure BDA00034702522100001114
For is to
Figure BDA00034702522100001115
A fourth-order polynomial fitting is performed,
Figure BDA00034702522100001116
is a fitting polynomial of
Figure BDA00034702522100001117
Recording the coefficient of four items
Figure BDA00034702522100001118
(6) If it is
Figure BDA00034702522100001119
Then the
Figure BDA00034702522100001120
Figure BDA00034702522100001121
If not, then,
Figure BDA00034702522100001122
then
Figure BDA00034702522100001123
Figure BDA00034702522100001124
(7)im3＝im3+1, return to step (2);
(8)
Figure BDA00034702522100001125
step four, designing a secondary term coefficient m of the interference fringe linear density variation function through a data fitting and iterative optimization method2Optimized design value of (m)2_optimalThe iterative process is as follows:
(1) m obtained according to step three3_optimalSetting the linear density variation function f of the interference fringe3optimal＝ng0+ng1·(x-Wg/2)+ng2·(x-Wg/2)2+m3_optimal·(x-Wg/2)3Calculating to obtain psi according to the calculation method of the total exposure of the variable-period scanning photoetching given in the step twotot_3optimalError of phase phie_3optimal＝Ψtot_3optimal-ΦgTo phie_3optimalPerforming a fourth degree polynomial fitting of phie_3optimalHas a fitting polynomial of phiep_3optimal(x)＝2π(a4_ 3optimalx4+a3_3optimalx3+a2_3optimalx2+a1_3optimalx+a0_3optimal) Recording the coefficient of cubic term a3_3optimal；
Setting a coefficient m2 of an interference fringe linear density variation function f (x) to optimize a third-order phase error threshold xi of design3order；
(2) Setting search range [ a ] of iterative optimizationm2 (0),bm2 (0)]，am2 (0)＝m2_nearby-Hm2，bm2 (0)＝ m2_nearby+Hm2，2Hm2Is m2Width of initial search range of optimum value, number of iterations im2＝0，m2_nearbyIs the optimal solution m2_optimalTo an initial approximation of the value of (a),
Figure BDA0003470252210000121
the maximum phase error of the initial cubic phase is phie_3order_m2 (0)＝abs(2πa3_3optimalWg 3)；
(3) If it is
Figure BDA0003470252210000122
If the threshold requirement is not met, executing the steps (4) to (8), otherwise, exiting the loop and executing the step (9);
(4) by using a one-dimensional search method (e.g. golden section, Fibonacci, etc.) in
Figure BDA0003470252210000123
Internally determined optimization variables
Figure BDA0003470252210000124
And
Figure BDA0003470252210000125
Figure BDA0003470252210000126
(5) according to the optimization variables
Figure BDA0003470252210000127
Value setting of the linear density variation function of the interference fringes
Figure BDA0003470252210000128
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure BDA0003470252210000129
Figure BDA00034702522100001210
Corresponding phase error
Figure BDA00034702522100001211
To pair
Figure BDA00034702522100001212
A fourth-order polynomial fitting is performed,
Figure BDA00034702522100001213
is a fitting polynomial of
Figure BDA00034702522100001214
Recording cubic item coefficients
Figure BDA00034702522100001215
(6) According to the optimization variables
Figure BDA00034702522100001216
Value setting of linear density variation function of interference fringe
Figure BDA00034702522100001217
According to the variable period scanning photoetching total exposure given in the step twoLight amount calculating method of obtaining
Figure BDA0003470252210000131
Figure BDA0003470252210000132
Corresponding phase error
Figure BDA0003470252210000133
To pair
Figure BDA0003470252210000134
A fourth-order polynomial fitting is performed,
Figure BDA0003470252210000135
is a fitting polynomial of
Figure BDA0003470252210000136
Recording cubic item coefficients
Figure BDA0003470252210000137
(7) If it is
Figure BDA0003470252210000138
Then
Figure BDA0003470252210000139
Figure BDA00034702522100001310
If not, then the mobile terminal can be switched to the normal mode,
Figure BDA00034702522100001311
then
Figure BDA00034702522100001312
Figure BDA00034702522100001313
(8)im2＝im2+1, returning to the step (3);
(9)
Figure BDA00034702522100001314
step five, designing an optimized design value m of a first-order coefficient m1 of the interference fringe linear density change function through a data fitting and iterative optimization method1_optimalThe iterative process is as follows:
(1) m obtained according to step four2_optimalSetting the linear density variation function f of the interference fringe2optimal＝ng0+ng1·(x-Wg/2)+m2_optimal·(x-Wg/2)2+m3_optimal·(x-Wg/2)3Calculating to obtain psi according to the calculation method of the total exposure of the variable-period scanning photoetching given in the step twotot_2optimalError of phase phie_2optimal＝Ψtot_2optimal-ΦgTo phie_2optimalPerforming a fourth order polynomial fit of phie_2optimalHas a fitting polynomial of phiep_2optimal(x)＝2π(a4_ 2optimalx4+a3_2optimalx3+a2_2optimalx2+a1_2optimalx+a0_2optimal) Recording the coefficient of quadratic term a2_2optimal；
Setting coefficient m1 optimized design quadratic term phase error threshold xi of interference fringe line density change function f (x)2order；
(2) Setting search range [ a ] of iterative optimizationm1 (0),bm1 (0)]，am1 (0)＝m1_nearby-Hm1，bm1 (0)＝ m1_nearby+Hm1，2Hm1Is m1Breadth of initial search range of optimum value, number of iterations im1＝0，m1_nearbyIs the optimal solution m1_optimalTo an initial approximation of the first,
Figure BDA00034702522100001315
initial quadratic phase error maximum of phie_2order_m1 (0)＝abs(2πa2_2optimalWg 2)；
(3) If it is
Figure BDA0003470252210000141
If the threshold requirement is not met, executing the steps (4) to (8), otherwise, exiting the loop and executing the step (9);
(4) using a one-dimensional search method (e.g., golden section, Fibonacci, etc.) at [ am1 (im1),bm1 (im1)]Internally determined optimization variable m1L (im1)And m1R (im1)，am1 (im1)≤m1L (im1)<m1R (im1)≤ bm1 (im1)；
(5) According to the optimization variable m1L (im1)Value setting of linear density variation function of interference fringe
Figure BDA0003470252210000142
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure BDA0003470252210000143
m1L (im1)Corresponding phase error
Figure BDA0003470252210000144
To pair
Figure BDA0003470252210000145
A fourth-order polynomial fitting is performed,
Figure BDA0003470252210000146
is a fitting polynomial of
Figure BDA0003470252210000147
Recording the coefficient of quadratic term
Figure BDA0003470252210000148
(6) According to the optimization variable m1R (im1)Value setting of linear density variation function of interference fringe
Figure BDA0003470252210000149
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure BDA00034702522100001410
m1R (im1)Corresponding phase error
Figure BDA00034702522100001411
To pair
Figure BDA00034702522100001412
A fourth-order polynomial fitting is performed,
Figure BDA00034702522100001413
is a fitting polynomial of
Figure BDA00034702522100001414
Recording secondary item coefficients
Figure BDA00034702522100001415
(7) If it is
Figure BDA00034702522100001416
Then am1 (im1+1)＝m1L (im1)，bm1 (im1+1)＝bm1 (im1)，
Figure BDA00034702522100001417
If not, then,
Figure BDA00034702522100001418
then am1 (im1+1)＝am1 (im1)， bm1 (im1+1)＝m1R (im1)，
Figure BDA00034702522100001419
(8)im1＝im1+1, returning to the third step;
(9)m1_optimal＝(m1L (im1)+m1R (im1))/2。
step six, designing a constant term coefficient m of a linear density change function of the interference fringe0Optimized design value of (m)0_optimalThe iterative process is as follows:
(1) m obtained according to step five1_optimalSetting the linear density variation function f of the interference fringe1optimal＝ng0+m1_optimal·(x-Wg/2)+m2_optimal·(x-Wg/2)2+m3_optimal·(x-Wg/2)3Calculating to obtain psi according to the calculation method of the total exposure of the variable-period scanning photoetching given in the step twotot_1optimalError of phase phie_1optimal＝Ψtot_1optimal-ΦgTo phie_1optimalPerforming a fourth order polynomial fit of phie_1optimalHas a fitting polynomial of phiep_1optimal(x)＝2π(a4_1optimalx4+a3_1optimalx3+a2_1optimalx2+a1_1optimalx+a0_1optimal) Recording the coefficient of a primary term1_1optimal；
Setting coefficient m0 of interference fringe linear density change function f (x) to optimize primary term phase error threshold xi of design1order；
(2) Setting search range [ a ] of iterative optimizationm0 (0),bm0 (0)]，am0 (0)＝m0_nearby-Hm0，bm0 (0)＝ m0_nearby+Hm0，2Hm0Is m0Breadth of initial search range of optimum value, number of iterations im0＝0，m0_nearbyIs the optimal solution m0_optimalInitial approximation of, m0_nearby＝ng0-a1_1optimalThe maximum value of the initial primary phase error is phie_1order_m0 (0)＝abs(2πa1_1optimalWg)；
(3) If it is
Figure BDA0003470252210000151
If the threshold requirement is not met, executing the steps (4) to (8), otherwise, exiting the loop and executing the step (9);
(4) by using a one-dimensional search method (e.g. golden section, Fibonacci, etc.) in
Figure BDA0003470252210000152
Internally determined optimization variables
Figure BDA0003470252210000153
And
Figure BDA0003470252210000154
Figure BDA0003470252210000155
(5) according to the optimization variables
Figure BDA0003470252210000156
Value setting of the linear density variation function of the interference fringes
Figure BDA0003470252210000157
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure BDA0003470252210000158
Figure BDA0003470252210000159
Corresponding phase error
Figure BDA00034702522100001510
To pair
Figure BDA00034702522100001511
A fourth-order polynomial fitting is performed,
Figure BDA00034702522100001512
is a fitting polynomial of
Figure BDA00034702522100001513
Recording one-time item coefficient
Figure BDA00034702522100001514
(6) According to the optimization variables
Figure BDA00034702522100001515
Value setting of linear density variation function of interference fringe
Figure BDA0003470252210000161
Calculating according to the variable period scanning photoetching total exposure given in the step two
Figure BDA0003470252210000162
Figure BDA0003470252210000163
Corresponding phase error
Figure BDA0003470252210000164
To pair
Figure BDA0003470252210000165
A fourth-order polynomial fitting is performed,
Figure BDA0003470252210000166
is a fitting polynomial of
Figure BDA0003470252210000167
Recording one-time item coefficient
Figure BDA0003470252210000168
(7) If it is
Figure BDA0003470252210000169
Then
Figure BDA00034702522100001610
Figure BDA00034702522100001611
If not, then,
Figure BDA00034702522100001612
then
Figure BDA00034702522100001613
Figure BDA00034702522100001614
(8)im0＝im0+1, return to step (3);
(9)
Figure BDA00034702522100001615
step seven, m optimized according to the third step to the seventh step3_optimal、m2_optimal、m1_optimalAnd m0_optimalAnd checking whether the exposure contrast meets the requirements of the exposure process.
Setting the interference fringe linear density to foptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3And calculating according to the variable-period scanning photoetching total exposure amount calculation method given in the step two to obtain an exposure contrast ratio gamma (x), and judging whether the exposure contrast ratio gamma meets the exposure contrast ratio requirement in the whole range of x, wherein the exposure contrast ratio requirement is determined by the manufacturing process parameters, and if the exposure contrast ratio is required to be more than 0.95.
If gamma does not meet the requirement of exposure contrast, the ratio StepRatio of the overlapping width of the interference pattern in the step one to the radius of the beam waist needs to be reduced, and the steps two, three, four, five, six and seven are carried out again until gamma meets the requirement of exposure contrast.
If gamma meets the exposure contrast requirement, the whole optimization process is ended according to foptimalThe linear density of the interference fringes in the photoetching process is changed to obtain the variable-pitch grating with the target groove density, and the optimization result according to the method is shown in fig. 7-9.
In the second embodiment, the present embodiment is an example of the interference fringe line density design method for aberration-reducing variable-pitch grating scan lithography described in the first embodiment:
the embodiment is implemented by the method of the first step, the second step, the third step, the fourth step, the fifth step, the sixth step and the seventh step set in the first embodiment. Wherein the lithographically coherent light beams 1 and 2 are emitted for laser splitting that satisfies the coherence length and exposure wavelength requirements, here from Kr+Laser light was generated at 413.1 nm. The opto- mechanical elements 3, 4, 5, 6, 7 are mounted vertically on a stationary optical platform and remain stationary, resulting in a small-sized circular interference pattern.
In step one, the interference pattern Gaussian beam waist radius RhoThe design parameters of the variable-pitch grating are as follows, wherein the parameters are 100 mu m and StepRatio is 0.8: n isg0＝1200gr/mm，ng1＝-0.7783gr/mm2，ng2＝1.865×10-4gr/mm3， ng3＝-8.1336×10-8gr/mm4，Wg30mm, total number of scanning steps is 400, and the number of steps N of each stepsteps＝96。
Step two, step three and step four, can adopt Matlab or Visual studio platform to finish the design process of numerical value calculation. Step two, setting m0＝ng0＝1200gr/mm，m1＝ng1＝-0.7783gr/mm2， m2＝ng2＝1.865×10-4gr/mm3，m3＝ng3＝-8.1336×10-8gr/mm4Step by Sk＝Nsteps/fk-1。
In step three, xi is set4order1e-4rad, and obtaining m after iterative optimization design by adopting a golden section method3_optimal＝-3.2461×10-7gr/mm4。
In the fourth step, xi is set3order1e-4rad, and obtaining m after iterative optimization design by adopting a golden section method2_optimal＝5.5583×10-4gr/mm3。
In step five, xi is set2order1e-4rad, and obtaining m after iterative optimization design by adopting a golden section method1_optimal＝-1.5510gr/mm2。
In the sixth step, xi is set1orderAfter iterative optimization design is carried out by adopting a golden section method, m is obtained0_optimal1188.2629gr/mm, the rule of the change of the interference fringe line density is foptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3。
And seventhly, calculating to obtain an exposure contrast gamma (x) which is better than 0.99 within the whole grating range of 0-30mm, meeting the process requirement of the exposure contrast of 0.95, and completing the design of the line density of the interference fringes.
The technical features of the embodiments described above may be arbitrarily combined, and for the sake of brevity, all possible combinations of the technical features in the embodiments described above are not described, but should be considered as being within the scope of the present specification as long as there is no contradiction between the combinations of the technical features.
The above-mentioned embodiments only express several embodiments of the present invention, and the description thereof is more specific and detailed, but not construed as limiting the scope of the invention. It should be noted that, for a person skilled in the art, several variations and modifications can be made without departing from the inventive concept, which falls within the scope of the present invention. Therefore, the protection scope of the present patent shall be subject to the appended claims.

Claims (7)

1. The design method of the linear density of the aberration-eliminating variable-pitch grating scanning photoetching stripes is characterized by comprising the following steps of: the method is realized by the following steps:
firstly, determining a scribed line density function of a variable-pitch grating and general photoetching process manufacturing parameters;
step one, according to the aberration eliminating characteristic of the variable-pitch grating and the application requirement of the variable-pitch grating in an instrument, designing a target function g (x) of the groove density of the variable-pitch grating as follows:
g(x)＝ng0+ng1(x-Wg/2)+ng2(x-Wg/2)2+ng3(x-Wg/2)3
ideal phase distribution phi of the gratingg(x) Expressed as:
Φg(x)＝2πg(x)·x
where x is the coordinate of the vector direction of the grating, x is 0 at the grating boundary, and W isgIs the total width of the vector direction of the grating, ng0Is the groove density at the center of the grating, ng1Coefficient of first order of grating groove density, ng2Coefficient of quadratic term, n, for grating groove densityg3The cubic term coefficient of the grating groove density; n isg0、ng1、ng2And ng3Determining according to the variable-pitch grating aberration correction principle and the use parameters of a spectroscopic instrument or a laser device;
determining the following manufacturing parameters when the variable-pitch grating is manufactured according to the design and the debugging parameters of the variable-period scanning photoetching system and the target function of the groove density of the variable-pitch grating obtained in the first step:
setting the radius of Gaussian beam waist of interference pattern to be RhoThe overlapping width of the interference patterns of the adjacent scanning sections accounts for the radius R of the beam waisthoThe ratio StepRatio, the width of the interference pattern overlap is StepRatio Rho；
Setting the total step number of step scanning as N, wherein N is more than or equal to Wg/(RhoStepRatio) +1, making the width of the exposure area larger than the effective width of the grating;
the step number of each step of step scanning is Nsteps，Nsteps＝round(Rho·StepRatio·ng0) (ii) a round () is a round-off fetchAn integer function;
step two, calculating the grating phase distribution error phi when the density change function f (x) of the interference fringe lines is equal to the groove density function g (x) of the grating with the variable grating pitch according to the calculation method of the total exposure of the variable period scanning photoetchinge(x)；
The method for calculating the total exposure of the variable-period scanning photoetching comprises the following steps:
setting the linear density variation function f (x) of the interference fringe and the target function g (x) of the groove density of the grating with the variable grating pitch to have the same form, and expressing the form as follows:
f(x)＝m0+m1(x-Wg/2)+m2(x-Wg/2)2+m3(x-Wg/2)3
in the formula, m0Coefficient of constant term being a function of variation of the linear density of the interference fringes, m1Coefficient of first order term, m, being a function of variation of line density of interference fringes2Coefficient of quadratic term, m, being a function of the variation of the linear density of the interference fringes3Coefficient of cubic term of linear density variation function of interference fringe; the initial scanning segment of scanning photoetching starts from x being 0, the corresponding step number k being 0 when x being 0, the initial scanning being the 1 st scanning, the corresponding exposure being D0(x)，SkStep distance of the k step;
S0＝0，
Figure FDA0003470252200000021
is the total distance from step 0 to step k,
Figure FDA0003470252200000022
interference fringe line density, Δ, for the k +1 th scan after k stepsk＝fk-fk-1Is the difference between the line density of interference fringe of k +1 times scanning and k times scanning, when k is 0, delta0When k is equal to 0>At the time of 0, the number of the first,
Figure FDA0003470252200000023
exposure D of the k +1 th scan after the k stepk(x) And phase difference between the (k + 1) th scan and the initial scan
Figure FDA0003470252200000024
Comprises the following steps:
Figure FDA0003470252200000025
Figure FDA0003470252200000026
wherein B (x) is the background component of the single scan exposure, and A (x) is the Gaussian distribution exposure intensity envelope of the single scan exposure;
Figure FDA0003470252200000027
when the photoetching is finished, the total exposure on the grating is the superposition D of the exposure of N +1 times of scanning after the total steps of step scanning are N stepstot(x) Namely:
Dtot(x)＝D0(x)+D1(x)+…DN(x)
＝Btot(x)+Atot(x)sin(Ψtot(x))
in the formula, Btot(x) Background component of total exposure, Atot(x) The amplitude of the AC component of the total exposure;
Figure FDA0003470252200000031
Figure FDA0003470252200000032
Figure FDA0003470252200000033
Figure FDA0003470252200000034
Ψtot(x)＝2πxf0+Ψ(x)
Ψ(x)＝arctan[F(x)/E(x)]
in the formula, Ψtot(x) Psi (x) is the phase increment between the total exposure and the 1 st scan, psitot(x) Equal to the actual phase distribution of the manufactured variable-pitch grating; gamma (x) ═ Atot(x)/Btot(x) Is the total exposure contrast;
setting f (x) g (x), i.e., m0＝ng0，m1＝ng1，m2＝ng2，m3＝ng3Calculating the phase variation Ψ of the exposure by using the method for calculating the total exposure of variable period scanning lithographytot(x) The error of the grating phase distribution between the actual phase distribution of the manufactured variable pitch grating and the ideal phase distribution of the grating is phie(x)＝Ψtot(x)-Φg(x)；
Step three, designing a cubic term coefficient m of an interference fringe linear density change function through a data fitting and iterative optimization method3Optimized design value of (m)3_optimal；
Step four, designing quadratic term coefficient m of interference fringe linear density change function through data fitting and iterative optimization method2Optimized design value of (m)2_optimal；
Step five, designing an optimized design value m1 of a first-order coefficient m1 of the linear density change function of the interference fringe through a data fitting and iterative optimization method1_optimal；
Step six, designing a constant term coefficient m of a linear density change function of the interference fringe0Optimized design value of (m)0_optimal；
Step seven, optimizing m according to the step three to the step six3_optimal、m2_optimal、m1_optimalAnd m0_optimalChecking whether the exposure contrast meets the requirements of the exposure process;
the optimized and designed interference fringe line density change function is as follows: f. ofoptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3；
Calculating to obtain an exposure contrast gamma (x) according to the method for calculating the total exposure of the variable-period scanning photoetching given in the second step, judging whether the exposure contrast gamma (x) meets the requirement of the exposure contrast within the whole range of x, if not, reducing the ratio StepRatio of the overlapping width of the interference patterns in the second step to the beam waist radius, and executing the second step to the seventh step again until the gamma (x) meets the requirement of the exposure contrast;
if so, ending the whole optimization process and carrying out the optimization design according to the interference fringe linear density change function foptimal(x) And changing the linear density of interference fringes in the photoetching process to obtain the variable-pitch grating with the target groove density.
2. The method for designing the linear density of the aberration-eliminating variable-pitch grating scanning photoetching stripes according to claim 1, characterized in that:
the change in b (x) does not affect the design of the interference fringe line density, and b (x) is set to a (x) and the contrast of a single scanning exposure is set to 1.
3. The variable pitch grating scanning lithography fringe line density design method of claim 1, characterized in that:
step distance S of the kth stepkOne of the following three stepping modes is selected to obtain different interference fringe design values, and the different interference fringe design values all meet the precision requirement of the groove density of the variable-pitch grating:
one, Sk＝Nsteps/fk-1；
II, Sk＝2Nsteps/(fk+fk-1)，
Figure FDA0003470252200000041
III, Sk＝Nsteps/fk，
Figure FDA0003470252200000042
4. The method for designing the linear density of the aberration-eliminating variable-pitch grating scanning photoetching stripes according to claim 1, characterized in that: the concrete process of the third step is as follows:
adopting a polynomial curve fitting algorithm to correct the phase distribution error phi obtained in the step twoe(x) Performing a fourth order polynomial curve fitting of phie(x) Has a fitting polynomial of phiep(x)＝2π(a40x4+a30x3+a20x2+a10x+a00)，a40、a30、a20、a10And a00Respectively fitting a polynomial of four times phiep(x) The coefficients of (c);
setting the coefficient m of the interference fringe linear density variation function f (x)3Quartic phase error threshold xi of optimal design4orderM is obtained by calculation through a one-dimensional search iterative optimization method3_optimalThe iterative process is as follows;
step three, setting a search range [ a ] of iterative optimizationm3 (0),bm3 (0)]，am3 (0)＝m3_nearby-Hm3，bm3 (0)＝m3_nearby+Hm3，2Hm3Is m3Width of initial search range of optimum value, number of iterations im3＝0，m3_nearbyIs the optimal solution m3_optimalInitial approximation of (c), m3_nearby＝ng3 2/(ng3+a40) The initial quartic phase error has a maximum value of phie_4order_m3 (0)＝abs(2πa40Wg 4)；
Step three, two, if
Figure FDA0003470252200000051
If the requirement of the threshold value is not met, executing the third step to the pseudo-ginseng, otherwise, exiting the circulation and executing the third step and the eighth step;
step three, adopting a one-dimensional searching method (such as golden section method, Fibonacci method, bisection method and the like) to search the space between the two adjacent images
Figure FDA00034702522000000522
Internally determined optimization variables
Figure FDA00034702522000000523
And
Figure FDA00034702522000000524
Figure FDA00034702522000000525
Figure FDA00034702522000000526
step three and four, according to the optimization variables
Figure FDA00034702522000000527
Value setting of linear density variation function of interference fringe
Figure FDA0003470252200000052
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA0003470252200000053
Figure FDA00034702522000000529
Corresponding phase error
Figure FDA0003470252200000054
For is to
Figure FDA0003470252200000055
A fourth-order polynomial fitting is performed,
Figure FDA0003470252200000056
is a fitting polynomial of
Figure FDA0003470252200000057
Recording the coefficient of four items
Figure FDA0003470252200000058
Step three and five, according to the optimization variables
Figure FDA0003470252200000059
Value setting of linear density variation function of interference fringe
Figure FDA00034702522000000510
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA00034702522000000511
Figure FDA00034702522000000528
Corresponding phase error
Figure FDA00034702522000000512
To pair
Figure FDA00034702522000000513
A fourth-order polynomial fitting is performed,
Figure FDA00034702522000000514
is a fitting polynomial of
Figure FDA00034702522000000515
Recording the coefficient of four items
Figure FDA00034702522000000516
Step three and six, if
Figure FDA00034702522000000517
Then the
Figure FDA00034702522000000518
Figure FDA00034702522000000519
If not, then,
Figure FDA00034702522000000520
Figure FDA00034702522000000521
step of Notoginseng, im3＝im3+1, returning to the third step;
the third step and the eighth step,
Figure FDA00034702522000000614
5. The method for designing the linear density of the aberration-eliminating variable-pitch grating scanning photoetching stripes according to claim 1, characterized in that: the concrete process of the step four is as follows:
step four, obtaining m according to step three3_optimalSetting the linear density variation function f of the interference fringe3optimal＝ng0+ng1·(x-Wg/2)+ng2·(x-Wg/2)2+m3_optimal·(x-Wg/2)3Calculating to obtain psi according to the calculation method of the total exposure of the variable-period scanning photoetching given in the step twotot_3optimalPhase error phie_3optimal＝Ψtot_3optimal-ΦgTo phie_3optimalPerforming a fourth order polynomial fit of phie_3optimalHas a fitting polynomial of phiep_3optimal(x)＝2π(a4_3optimalx4+a3_3optimalx3+a2_3optimalx2+a1_3optimalx+a0_3optimal) Recording the coefficient of cubic term a3_3optimal；
Setting a coefficient m2 optimized design cubic term phase error threshold xi of interference fringe linear density change function f (x)3order；
Step four and two, setting the search range [ a ] of iterative optimizationm2 (0),bm2 (0)]，am2 (0)＝m2_nearby-Hm2，bm2 (0)＝m2_nearby+Hm2，2Hm2Is m2Breadth of initial search range of optimum value, number of iterations im2＝0，m2_nearbyIs an optimal solution m2_optimalTo an initial approximation of the first,
Figure FDA0003470252200000061
initial cubic phase error maximum of phie_3order_m2 (0)＝abs(2πa3_3optimalWg 3)；
Step four, three, if
Figure FDA0003470252200000062
If the requirement of the threshold value is not met, executing the step four to the step four eight, otherwise, exiting the circulation and executing the step four nine;
step four, adopting a one-dimensional searching method in
Figure FDA0003470252200000063
Internally determined optimization variables
Figure FDA0003470252200000064
And
Figure FDA0003470252200000065
Figure FDA0003470252200000066
step four and five, according to the optimization variables
Figure FDA00034702522000000613
Value setting of linear density variation function of interference fringe
Figure FDA0003470252200000067
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA0003470252200000068
Figure FDA00034702522000000615
Corresponding phase error
Figure FDA0003470252200000069
To pair
Figure FDA00034702522000000610
A fourth-order polynomial fitting is performed,
Figure FDA00034702522000000611
is a fitting polynomial of
Figure FDA00034702522000000612
Recording cubic item coefficients
Figure FDA0003470252200000071
Step four and six, according to the optimization variables
Figure FDA0003470252200000072
Value setting of the linear density variation function of the interference fringes
Figure FDA0003470252200000073
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA0003470252200000074
Figure FDA00034702522000000716
Corresponding phase error
Figure FDA0003470252200000075
To pair
Figure FDA0003470252200000076
A fourth-order polynomial fitting is performed,
Figure FDA0003470252200000077
is a fitting polynomial of
Figure FDA0003470252200000078
Recording cubic item coefficients
Figure FDA0003470252200000079
Step four and seven, if
Figure FDA00034702522000000710
Then
Figure FDA00034702522000000711
Figure FDA00034702522000000712
If not, then the mobile terminal can be switched to the normal mode,
Figure FDA00034702522000000713
Figure FDA00034702522000000714
step four eight, im2＝im2+1, returning to the fourth step and the third step;
step four nine,
Figure FDA00034702522000000715
6. The method for designing the linear density of the aberration-eliminating variable-pitch grating scanning photoetching stripes according to claim 1, characterized in that: the concrete process of the fifth step is as follows:
step five, obtaining m according to step four2_optimalSetting the linear density variation function f of the interference fringe2optimal＝ng0+ng1·(x-Wg/2)+m2_optimal·(x-Wg/2)2+m3_optimal·(x-Wg/2)3Calculating to obtain psi according to the calculation method of the total exposure of the variable-period scanning photoetching given in the step twotot_2optimalError of phase phie_2optimal＝Ψtot_2optimal-ΦgTo phie_2optimalPerforming a fourth order polynomial fit of phie_2optimalHas a fitting polynomial of phiep_2optimal(x)＝2π(a4_2optimalx4+a3_2optimalx3+a2_2optimalx2+a1_2optimalx+a0_2optimal) Recording the coefficient of quadratic term a2_2optimal；
Setting coefficient m1 optimized design quadratic term phase error threshold xi of interference fringe linear density change function f (x)2order；
Step five and step two, setting the search range [ a ] of iterative optimizationm1 (0),bm1 (0)]，am1 (0)＝m1_nearby-Hm1，bm1 (0)＝m1_nearby+Hm1，2Hm1Is m1Breadth of initial search range of optimum value, number of iterations im1＝0，m1_nearbyIs the optimal solution m1_optimalTo an initial approximation of the first,
Figure FDA0003470252200000081
initial quadratic phase error maximum of phie_2order_m1 (0)＝abs(2πa2_2optimalWg 2)；
Step five and three, if
Figure FDA0003470252200000082
If the requirement of the threshold value is not met, executing the step five four to five eight, otherwise, exiting the circulation and executing the step five nine;
step five and four, adopting a one-dimensional searching method in
Figure FDA0003470252200000083
Internally determined optimization variables
Figure FDA0003470252200000084
And
Figure FDA0003470252200000085
Figure FDA0003470252200000086
step five, according to the optimization variables
Figure FDA0003470252200000087
Value setting of linear density variation function of interference fringe
Figure FDA0003470252200000088
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA0003470252200000089
Figure FDA00034702522000000829
Corresponding phase error
Figure FDA00034702522000000810
To pair
Figure FDA00034702522000000811
A fourth-order polynomial fitting is performed,
Figure FDA00034702522000000812
is a fitting polynomial of
Figure FDA00034702522000000813
Recording secondary item coefficients
Figure FDA00034702522000000814
Step five and six, according to the optimization variables
Figure FDA00034702522000000815
Value setting of the linear density variation function of the interference fringes
Figure FDA00034702522000000816
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA00034702522000000817
Figure FDA00034702522000000830
Corresponding phase error
Figure FDA00034702522000000818
To pair
Figure FDA00034702522000000819
A fourth-order polynomial fitting is performed,
Figure FDA00034702522000000820
is a fitting polynomial of
Figure FDA00034702522000000821
Recording secondary item coefficients
Figure FDA00034702522000000822
Step five and seven, if
Figure FDA00034702522000000823
Then
Figure FDA00034702522000000824
Figure FDA00034702522000000825
If not, then,
Figure FDA00034702522000000826
Figure FDA00034702522000000827
step five eight, im1＝im1+1, returning to the third step;
the fifth step,
Figure FDA00034702522000000828
7. The method for designing the linear density of the aberration-eliminating variable-pitch grating scanning photoetching stripes according to claim 1, characterized in that: the concrete process of the step six is as follows:
step six, obtaining m according to step five1_optimalSetting the linear density variation function f of the interference fringe1optimal＝ng0+m1_optimal·(x-Wg/2)+m2_optimal·(x-Wg/2)2+m3_optimal·(x-Wg/2)3Calculating to obtain psi according to the calculation method of the total exposure of the variable-period scanning photoetching given in the step twotot_1optimalError of phase phie_1optimal＝Ψtot_1optimal-ΦgTo phie_1optimalPerforming a fourth order polynomial fit of phie_1optimalHas a fitting polynomial of phiep_1optimal(x)＝2π(a4_ 1optimalx4+a3_1optimalx3+a2_1optimalx2+a1_1optimalx+a0_1optimal) Recording the coefficient of a primary term1_1optimal；
Setting coefficient m0 optimized design first order phase error threshold xi of interference fringe linear density change function f (x)1order；
Sixthly, setting a search range [ a ] of iterative optimizationm0 (0),bm0 (0)]，am0 (0)＝m0_nearby-Hm0，bm0 (0)＝m0_nearby+Hm0，2Hm0Is m0Breadth of initial search range of optimum value, number of iterations im0＝0，m0_nearbyIs the optimal solution m0_optimalInitial approximation of, m0_nearby＝ng0-a1_1optimalThe initial first order phase error maximum is phie_1order_m0 (0)＝abs(2πa1_1optimalWg)；
Step six and three, if
Figure FDA0003470252200000091
If the threshold requirement is not met, executing the steps six four to six eight, otherwise, exiting the circulation and executing the step six nine;
sixthly, adopting a one-dimensional searching method to perform
Figure FDA0003470252200000092
Internally determined optimization variables
Figure FDA0003470252200000093
And
Figure FDA0003470252200000094
Figure FDA0003470252200000095
step six and five, according to the optimization variables
Figure FDA0003470252200000096
Value setting of the linear density variation function of the interference fringes
Figure FDA0003470252200000097
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA0003470252200000098
Figure FDA00034702522000000916
Corresponding phase error
Figure FDA0003470252200000099
To pair
Figure FDA00034702522000000910
A fourth-order polynomial fitting is performed,
Figure FDA00034702522000000911
is a fitting polynomial of
Figure FDA00034702522000000912
Recording one-time item coefficient
Figure FDA00034702522000000913
Step six, according to the optimization variables
Figure FDA00034702522000000914
Value setting of linear density variation function of interference fringe
Figure FDA00034702522000000915
Calculating according to the method for calculating the total exposure of the variable-period scanning photoetching given in the step two
Figure FDA0003470252200000101
Figure FDA00034702522000001014
Corresponding phase error
Figure FDA0003470252200000102
To pair
Figure FDA0003470252200000103
A fourth-order polynomial fitting is performed,
Figure FDA0003470252200000104
is a fitting polynomial of
Figure FDA0003470252200000105
Recording one-time item coefficient
Figure FDA0003470252200000106
Step six and seven, if
Figure FDA0003470252200000107
Then the
Figure FDA0003470252200000108
Figure FDA0003470252200000109
If not, then,
Figure FDA00034702522000001010
then
Figure FDA00034702522000001011
Figure FDA00034702522000001012
Step six, eight, im0＝im0+1, returning to the step five;
sixty nine steps,
Figure FDA00034702522000001013
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