CN114740557A - Design method of stripe density for aberration variable grating-pitch raster scanning lithography - Google Patents

Design method of stripe density for aberration variable grating-pitch raster scanning lithography 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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宋�莹
张刘
刘玉娟
朱杨
章家保
王文华
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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

消像差变栅距光栅扫描光刻条纹线密度设计方法Design method of stripe density for aberration variable grating-pitch raster scanning lithography

技术领域technical field

本发明涉及全息光栅制作的技术领域，具体涉及一种消像差变栅距光栅扫 描光刻的干涉条纹线密度的设计方法。The invention relates to the technical field of holographic grating fabrication, in particular to a method for designing the line density of interference fringes in aberration variable grating-pitch grating scanning lithography.

背景技术Background technique

消像差变栅距光栅是指光栅刻槽密度按照一定规律变化的平面光栅，通过 光栅刻槽密度的变化校正离焦、球差等光学像差。与曲面光栅相比，光栅基底 为平面，降低了基底的加工难度。入射到变栅距光栅的子午光线可以自形成谱 线，光谱仪器中无需额外的准直及聚焦光学元件，减小了仪器的体积重量，提 高了光能利用率，具有较高激光损伤阈值，在同步辐射光源装置、高能激光装 置等领域具有重要应用。Aberration-variable grating is a plane grating whose groove density changes according to a certain law, and optical aberrations such as defocus and spherical aberration are corrected by the change of grating groove density. Compared with the curved grating, the grating substrate is flat, which reduces the processing difficulty of the substrate. The meridional light incident on the variable pitch grating can form a spectral line by itself, and no additional collimating and focusing optical elements are required in the spectrometer, which reduces the volume and weight of the instrument, improves the utilization rate of light energy, and has a high laser damage threshold. It has important applications in the fields of synchrotron radiation light source devices and high-energy laser devices.

变栅距光栅的制作方式通常采用机械刻划、电子束直写、激光直写、全息 曝光等方式制作。机械刻划、电子束及激光直写等制作方式属于超精密加工， 逐线完成光栅刻槽的加工，制作效率低，且由于变栅距光栅相邻栅距的变化一 般不超过纳米量级，对相应的超精密加工设备及加工条件要求很高，制作难度 和成本高。传统全息曝光方式制作变栅距光栅，可采用球面波或非球面波曝光 系统，但球面波曝光系统可调整的自由度较少，且存在刻槽弯曲的问题，会导 致光栅的分辨能力下降。非球面波曝光系统设计、加工及调试难度大，工艺上 不易实现，往往导致实际的光栅刻槽密度与期望值存在较大误差。The production methods of variable pitch gratings are usually made by mechanical scribing, electron beam direct writing, laser direct writing, and holographic exposure. The production methods such as mechanical scribing, electron beam and laser direct writing belong to ultra-precision processing. The processing of grating grooves is completed line by line, and the production efficiency is low. Since the change of adjacent grating pitch of variable pitch grating generally does not exceed the nanometer level, The requirements for the corresponding ultra-precision processing equipment and processing conditions are very high, and the manufacturing difficulty and cost are high. The traditional holographic exposure method is used to make a variable pitch grating, and a spherical wave or aspheric wave exposure system can be used. However, the spherical wave exposure system has fewer degrees of freedom to adjust, and there is the problem of groove bending, which will reduce the resolution of the grating. The aspheric wave exposure system is difficult to design, process and debug, and it is not easy to realize in the process, which often leads to a large error between the actual grating groove density and the expected value.

变周期扫描光刻是制作变栅距光栅的另一种重要方法，干涉光学系统形成 小口径(微米～毫米量级)的干涉图样，由二维工作台承载光栅基底进行步进扫描 运动，使干涉图样与光栅基底之间产生相对运动，将干涉条纹记录在光栅基底 涂覆的光刻胶中，直至完成整块光栅基底有效面积的曝光。为实现变栅距光栅 的制作，在曝光过程中，需要根据用于光刻的干涉条纹线密度变化函数，通过 精密光电控制改变相干光束的干涉夹角，不断精密调整干涉条纹线密度，使制 作的光栅刻槽密度满足设计指标要求。这种制作方式所制作的变栅距光栅不存 在刻槽弯曲的问题，干涉图样中存在数百条干涉条纹，大大提高了制作效率， 在进行大面积光栅制作时，无需大口径的光学系统。Variable period scanning lithography is another important method for making variable grating pitch gratings. The interference optical system forms an interference pattern with a small diameter (micrometers to millimeters), and the two-dimensional worktable supports the grating substrate to perform step-by-step scanning motion, so that the Relative motion is generated between the interference pattern and the grating substrate, and the interference fringes are recorded in the photoresist coated on the grating substrate until the exposure of the effective area of the entire grating substrate is completed. In order to realize the production of variable grating pitch grating, in the exposure process, it is necessary to change the interference angle of the coherent beam through precise photoelectric control according to the linear density change function of the interference fringe used for lithography, and continuously and precisely adjust the linear density of the interference fringe, so that the production The grating groove density meets the design requirements. The variable grating pitch grating produced by this production method does not have the problem of curved grooves, and there are hundreds of interference fringes in the interference pattern, which greatly improves the production efficiency. When producing large-area gratings, a large-diameter optical system is not required.

但若令干涉条纹线密度变化函数等于变栅距光栅的目标刻槽密度函数，进 行干涉条纹线密度的调整，制作得到的变栅距光栅刻槽密度与设计值存在较大 偏差。这主要是由于以下两点原因，变周期扫描光刻系统在改变干涉条纹的线 密度时，干涉图样中全部干涉条纹的线密度发生相同的变化，无法实现干涉条 纹逐条线距的精密调整。且干涉图样的强度分布为高斯分布，为了保证曝光量 的均匀性，相邻扫描段的干涉图样之间存在一定的重叠，对制作出的光栅的刻 槽分布具有均化效应。However, if the linear density variation function of the interference fringes is equal to the target groove density function of the variable pitch grating, and the linear density of the interference fringes is adjusted, there is a large deviation between the groove density and the design value of the obtained variable pitch grating. This is mainly due to the following two reasons: when the variable period scanning lithography system changes the line density of the interference fringes, the line density of all the interference fringes in the interference pattern changes the same, and the precise adjustment of the line spacing of the interference fringes cannot be realized. In addition, the intensity distribution of the interference pattern is a Gaussian distribution. In order to ensure the uniformity of the exposure amount, there is a certain overlap between the interference patterns of adjacent scanning segments, which has a homogenizing effect on the groove distribution of the fabricated grating.

建立干涉条纹线密度的设计方法，是变周期扫描光刻技术应用的关键问题。 本发明提出一种用于消像差变栅距光栅扫描光刻的干涉条纹线密度设计方法， 按照此方法设计的干涉条纹线密度变化函数，改变光刻过程中干涉条纹的线密 度，可使最终得到变栅距光栅刻槽密度满足设计指标要求。Establishing a design method for the line density of interference fringes is a key issue in the application of variable period scanning lithography. The present invention proposes a method for designing interference fringe line density for aberration variable grating-pitch grating scanning lithography. According to the interference fringe line density variation function designed by this method, the line density of interference fringes in the lithography process is changed, so that the Finally, the groove density of the variable pitch grating can meet the design requirements.

发明内容SUMMARY OF THE INVENTION

本发明提出消像差变栅距光栅扫描光刻的干涉条纹线密度设计方法，利用 该方法可根据变栅距光栅刻槽密度目标函数，完成扫描光刻过程中干涉条纹线 密度函数的设计，按此干涉条纹线密度设计函数，变化干涉条纹的线密度，最 终得到的变周期光栅刻槽密度满足光栅刻槽密度目标函数的要求。The invention proposes an interference fringe line density design method for aberration variable grating pitch grating scanning lithography. By using the method, the design of the interference fringe line density function in the scanning lithography process can be completed according to the variable grating pitch grating groove density objective function. The function is designed according to the linear density of the interference fringes, and the linear density of the interference fringes is changed, and the finally obtained groove density of the variable period grating meets the requirements of the objective function of the groove density of the grating.

消像差变栅距光栅扫描光刻条纹线密度设计方法，该方法由以下步骤实现：An aberration variable grating pitch raster scanning lithography fringe density design method, which is realized by the following steps:

步骤一、确定变栅距光栅的刻线密度函数及通用光刻过程制作参数；Step 1. Determine the line density function of the variable pitch grating and the general photolithography process manufacturing parameters;

步骤一一、根据变栅距光栅的消像差特性及其在仪器中的应用需求，设计 变栅距光栅刻槽密度目标函数g(x)为：Step 11. According to the aberration characteristics of the variable pitch grating and its application requirements in the instrument, design the variable pitch grating groove density objective function g(x) as:

g(x)＝ng0+ng1(x-Wg/2)+ng2(x-Wg/2)2+ng3(x-Wg/2)3 g(x)=n g0 +n g1 (xW g /2)+n g2 (xW g /2) 2 +n g3 (xW g /2) 3

光栅的理想相位分布Φg(x)表示为：The ideal phase distribution of the grating Φ g (x) is expressed as:

Φg(x)＝2πg(x)·xΦ g (x)=2πg(x) x

式中，x为光栅矢量方向的坐标，x＝0位于光栅边界处，Wg为光栅矢量方 向的总宽度，ng0为光栅中心处的刻槽密度，ng1为光栅刻槽密度的一次项系数， ng2为光栅刻槽密度的二次项系数，ng3为光栅刻槽密度的三次项系数；所述ng0、 ng1、ng2和ng3根据变栅距光栅像差校正原理及光谱仪器或激光装置的使用参数 确定；In the formula, x is the coordinate of the grating vector direction, x=0 is located at the grating boundary, W g is the total width of the grating vector direction, n g0 is the groove density at the center of the grating, and n g1 is the first-order term of the grating groove density coefficient, n g2 is the quadratic term coefficient of the grating groove density, and n g3 is the cubic term coefficient of the grating groove density; the n g0 , n g1 , n g2 and n g3 are based on the principle of variable grating pitch grating aberration correction and Determination of operating parameters of spectroscopic instruments or laser devices;

步骤一二、根据变周期扫描光刻系统的设计、装调参数及步骤一一获得的 变栅距光栅刻槽密度目标函数，确定在进行该变周期光栅制作时的以下制作参 数：Step 12, according to the design of the variable period scanning lithography system, the adjustment parameters and the variable grating pitch grating groove density objective function obtained in step 11, determine the following production parameters when making this variable period grating:

设定干涉图样高斯束腰半径为Rho，相邻扫描段的干涉图样重叠宽度占所述 束腰半径Rho的比例StepRatio，则干涉图样重叠宽度为StepRatio×Rho；Set the Gaussian beam waist radius of the interference pattern to be R ho , and the overlapping width of the interference pattern of adjacent scanning segments accounts for the ratio StepRatio of the beam waist radius R ho , then the overlapping width of the interference pattern is StepRatio×R ho ;

设定步进扫描的总步数为N，所述N≥Wg/(Rho·StepRatio)+1，使曝光区域的宽 度大于光栅的有效宽度；The total number of steps of the step-by-step scanning is set as N, the N≥W g /(R ho ·StepRatio)+1, so that the width of the exposure area is greater than the effective width of the grating;

所述步进扫描每一步的步数为Nsteps，Nsteps＝round(Rho·StepRatio·ng0)；round() 为四舍五入取整数函数；The number of steps in each step of the step scan is N steps , where N steps =round(R ho ·StepRatio ·n g0 ); round( ) is a rounding integer function;

步骤二、根据变周期扫描光刻总曝光量计算方法，计算干涉条纹线密度变 化函数f(x)等于变栅距光栅刻槽密度函数g(x)时的光栅相位分布误差Φe(x)；Step 2: Calculate the grating phase distribution error Φ e (x) when the interference fringe line density change function f(x) is equal to the variable grating pitch grating groove density function g(x) according to the calculation method of the total exposure amount of variable-period scanning lithography ;

所述变周期扫描光刻总曝光量的计算方法为：The calculation method of the total exposure amount of the variable period scanning lithography is:

设定干涉条纹线密度变化函数f(x)与变栅距光栅刻槽密度目标函数g(x)具 有相同的形式，表示为：It is assumed that the variation function f(x) of the line density of the interference fringes has the same form as the objective function g(x) of the groove density of the variable pitch grating, which is expressed as:

f(x)＝m0+m1(x-Wg/2)+m2(x-Wg/2)2+m3(x-Wg/2)3 f(x)=m 0 +m 1 (xW g /2)+m 2 (xW g /2) 2 +m 3 (xW g /2) 3

式中，m0为干涉条纹线密度变化函数的常数项系数，m1为干涉条纹线密度 变化函数的一次项系数，m2为干涉条纹线密度变化函数的二次项系数，m3为干 涉条纹线密度变化函数的三次项系数；扫描光刻起始扫描段从x＝0时开始，x＝0 时对应的步进个数k＝0，起始扫描为第1次扫描，其对应的曝光量为D0(x)，Sk为第k步的步进距离；In the formula, m 0 is the constant term coefficient of the interference fringe line density change function, m 1 is the first order coefficient of the interference fringe line density change function, m 2 is the quadratic term coefficient of the interference fringe line density change function, and m 3 is the interference fringe line density change function. The cubic term coefficient of the stripe line density variation function; the initial scan segment of scanning lithography starts from x=0, the corresponding step number k=0 when x=0, the initial scan is the first scan, and its corresponding The exposure amount is D 0 (x), and Sk is the stepping distance of the kth step;

S0＝0，

Figure BDA0003470252210000041
为从第0步至第k步的总距离，
Figure BDA0003470252210000042
为k步步进后，第 k+1次扫描的干涉条纹线密度；Δk＝fk-fk-1为k+1次扫描与k次扫描干涉条纹线 密度的差值，当k＝0时，Δ0＝0，当k>0时,S 0 =0,
Figure BDA0003470252210000041
is the total distance from step 0 to step k,
Figure BDA0003470252210000042
is the interference fringe line density of the k+1th scan after k steps; Δ k = f k -f k-1 is the difference between the interference fringe line density of the k+1 scan and the k scan, when k = When 0, Δ 0 =0, when k>0,

Figure BDA0003470252210000043
Figure BDA0003470252210000043

第k步步进后，第k+1次扫描的曝光量Dk(x)及第k+1次扫描与初始扫描的 相位差

Figure BDA0003470252210000044
为：After the kth step, the exposure amount D k (x) of the k+1th scan and the phase difference between the k+1th scan and the initial scan
Figure BDA0003470252210000044
for:

Figure BDA0003470252210000045
Figure BDA0003470252210000045

Figure BDA0003470252210000046
Figure BDA0003470252210000046

式中，B(x)为单次扫描曝光量的背景分量，A(x)为单次扫描曝光量中高斯分 布的曝光量强度包络；

Figure BDA0003470252210000047
In the formula, B(x) is the background component of the single-scan exposure, and A(x) is the exposure intensity envelope of the Gaussian distribution in the single-scan exposure;
Figure BDA0003470252210000047

光刻结束时，光栅上的总曝光量为步进扫描总步数N步后N+1次扫描曝光 量的叠加Dtot(x)，即：At the end of lithography, the total exposure amount on the grating is the superposition D tot (x) of the exposure amount of N+1 scans after the total number of step scanning steps N steps, namely:

Dtot(x)＝D0(x)+D1(x)+…DN(x)D tot (x)=D 0 (x)+D 1 (x)+…D N (x)

＝Btot(x)+Atot(x)sin(Ψtot(x))＝B tot (x)+A tot (x) sin(Ψ tot (x))

式中，Btot(x)为总曝光量的背景分量，Atot(x)为总曝光量交流分量幅值；In the formula, B tot (x) is the background component of the total exposure, and A tot (x) is the amplitude of the AC component of the total exposure;

Figure BDA0003470252210000048
Figure BDA0003470252210000048

Figure BDA0003470252210000049
Figure BDA0003470252210000049

Figure BDA0003470252210000051
Figure BDA0003470252210000051

Figure BDA0003470252210000052
Figure BDA0003470252210000052

Ψtot(x)＝2πxf0+Ψ(x)Ψ tot (x)=2πxf 0 +Ψ(x)

Ψ(x)＝arctan[F(x)/E(x)]Ψ(x)=arctan[F(x)/E(x)]

式中，Ψtot(x)为总曝光量的相位变化量，Ψ(x)为总曝光量与第1次扫描之间 的相位增量，Ψtot(x)等于所制作的变栅距光栅的实际相位分布； γ(x)＝Atot(x)/Btot(x)为总曝光对比度；In the formula, Ψ tot (x) is the phase change of the total exposure amount, Ψ (x) is the phase increment between the total exposure amount and the first scan, and Ψ tot (x) is equal to the produced variable grating pitch grating The actual phase distribution of ; γ(x)=A tot (x)/B tot (x) is the total exposure contrast;

设定f(x)＝g(x)，即m0＝ng0，m1＝ng1，m2＝ng2，m3＝ng3，利用上述变周期扫描 光刻总曝光量计算方法，计算曝光量相位变化量Ψtot(x)，所制作的变栅距光栅的 实际相位分布与光栅的理想相位分布之间的光栅相位分布误差为Φe(x)＝ Ψtot(x)-Φg(x)；Set f(x)=g(x), that is, m 0 =n g0 , m 1 =n g1 , m 2 =n g2 , m 3 =n g3 , using the above-mentioned method for calculating the total exposure of variable period scanning lithography, Calculate the amount of exposure phase change Ψ tot (x), the grating phase distribution error between the actual phase distribution of the variable grating and the ideal phase distribution of the grating is Φ e (x) = Ψ tot (x)-Φ g (x);

步骤三、通过数据拟合与迭代寻优方法设计干涉条纹线密度变化函数的三 次项系数m3的优化设计值m3_optimal；Step 3, designing the optimal design value m 3_optimal of the cubic term coefficient m 3 of the interference fringe line density variation function through data fitting and iterative optimization method;

步骤四、通过数据拟合与迭代寻优方法设计干涉条纹线密度变化函数的二 次项系数m2的优化设计值m2_optimal；Step 4, designing the optimal design value m 2_optimal of the quadratic term coefficient m 2 of the interference fringe line density variation function through data fitting and iterative optimization method;

步骤五、通过数据拟合与迭代寻优方法设计干涉条纹线密度变化函数的一 次项系数m1的优化设计值m1_optimal；Step 5, designing the optimal design value m 1_optimal of the linear coefficient m 1 of the linear density variation function of the interference fringe by data fitting and iterative optimization method;

步骤六、设计干涉条纹线密度变化函数的常数项系数m0的优化设计值 m0_optimal；Step 6, designing the optimal design value m 0_optimal of the constant term coefficient m 0 of the interference fringe line density variation function;

步骤七、根据步骤三至步骤六优化的m3_optimal、m2_optimal、m1_optimal和m0_optimal， 核对曝光对比度是否满足曝光工艺需求；Step 7: Check whether the exposure contrast meets the requirements of the exposure process according to m 3_optimal , m 2_optimal , m 1_optimal and m 0_optimal optimized in steps 3 to 6;

优化设计后的干涉条纹线密度变化函数：The linear density change function of the interference fringes after the optimized design:

foptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3；f optimal (x)=m 0_optimal +m 1_optimal (xW g /2)+m 2_optimal (xW g /2) 2 +m 3_optimal (xW g /2) 3 ;

按照步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到曝光对比 度γ(x)，判断在整个x范围内，曝光对比度γ(x)是否满足曝光对比度要求， 如果否，则减小步骤一二中的干涉图样重叠宽度占束腰半径的比例StepRatio， 重新执行步骤二至七，直至γ(x)满足曝光对比度要求；According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the exposure contrast γ(x) is obtained by calculation, and it is judged whether the exposure contrast γ(x) meets the exposure contrast requirement in the entire x range, if not, reduce the The ratio of the overlap width of the interference pattern to the beam waist radius in the small steps 1 and 2 is StepRatio, and steps 2 to 7 are performed again until γ(x) meets the exposure contrast requirement;

如果是，则整个优化过程结束，按照优化设计后的干涉条纹线密度变化函 数foptimal(x)改变光刻过程中干涉条纹的线密度，获得目标刻槽密度的变栅距光栅。If yes, the entire optimization process is over, and the linear density of the interference fringes in the lithography process is changed according to the optimally designed interference fringe linear density variation function f optimal (x) to obtain a variable pitch grating with the target groove density.

本发明的积极效果：本发明所述的消像差变栅距光栅扫描光刻的干涉条纹 线密度设计方法，用于变周期扫描干涉光刻系统进行变栅距光栅制作。根据已 知的变栅距光栅的刻槽密度，按照本方法可设计光刻干涉条纹的线密度变化规 律。提高变栅距光栅的刻槽密度的精度，保证曝光对比度工艺参数的可控性， 对提升变栅距光栅扫描光刻制作水平，提高光栅制作成功率具有重要意义。Positive effects of the present invention: the method for designing the line density of interference fringes for aberration variable grating-pitch grating scanning lithography according to the present invention is used for variable-period scanning interference lithography system to manufacture variable grating-pitch gratings. According to the known groove density of the variable pitch grating, the linear density variation rule of the lithographic interference fringes can be designed according to this method. Improving the precision of the groove density of the variable pitch grating and ensuring the controllability of the exposure contrast process parameters is of great significance for improving the production level of the variable pitch grating scanning lithography and improving the success rate of the grating production.

附图说明Description of drawings

图1为本发明所述的用于消像差变栅距光栅扫描光刻的干涉条纹线密度设 计方法所应用的变周期扫描光刻装置简化示意图。Fig. 1 is a simplified schematic diagram of a variable period scanning lithography device applied to the interference fringe line density design method for aberration variable grating-pitch grating scanning lithography according to the present invention.

图2为坐标系定义示意图。Figure 2 is a schematic diagram of the definition of the coordinate system.

图3为干涉图样束腰半径及重叠光刻示意图。FIG. 3 is a schematic diagram of the beam waist radius of the interference pattern and the overlapping lithography.

图4为变栅距光栅的相位分布Φg(x)和按照光栅刻槽密度函数进行干涉条纹 线密度变化，得到的光栅相位分布Ψtot(x)(ng0＝1200gr/mm，ng1＝-0.7783gr/mm2， ng2＝1.865×10-4gr/mm3，ng3＝-8.1336×10-8gr/mm4，Wg＝30mm)的效果图。Figure 4 shows the phase distribution Φ g (x) of the variable grating pitch grating and the line density change of the interference fringes according to the grating groove density function, the obtained grating phase distribution Ψ tot (x) (n g0 =1200gr/mm, n g1 = -0.7783gr/mm 2 , n g2 =1.865×10 -4 gr/mm 3 , n g3 =-8.1336×10 -8 gr/mm 4 , W g =30mm).

图5为按照光栅刻槽密度函数进行干涉条纹线密度变化，得到的光栅相位 分布误差Φe(x)＝Ψtot(x)-Φg(x)，光栅参数效果图，同图4。Figure 5 is the grating phase distribution error Φ e (x) = Ψ tot (x)-Φ g (x) obtained by changing the line density of the interference fringes according to the grating groove density function, the same as Figure 4.

图6为干涉条纹线密度三次项系数m3_optimal的优化设计流程图。Fig. 6 is a flow chart of the optimal design of the cubic term coefficient m 3_optimal of the linear density of the interference fringes.

图7为变栅距光栅的相位分布Φg(x)，及按照本设计方法得到的干涉条纹线 密度进行干涉条纹线密度变化，得到的光栅相位分布Ψtot(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＝ξ4order＝1e-4rad)的效果图。Figure 7 shows the phase distribution Φ g (x) of the variable grating pitch grating, and the interference fringe line density obtained according to the design method. The interference fringe line density is changed, and the obtained grating phase distribution Ψ tot (x) (n g0 =1200gr/ mm, n g1 =-0.7783gr/mm 2 , n g2 =1.865×10 -4 gr/mm 3 , n g3 =-8.1336×10 -8 gr/mm 4 , W g =30mm, Rho=0.1mm,ξ 1order =ξ 2order =ξ 3order =ξ 4order =1e-4rad).

图8为按照设计值得到的光栅相位分布误差Φe(x)＝Ψtot(x)-Φg(x)效果图；Fig. 8 is the effect diagram of the grating phase distribution error Φ e (x)=Ψ tot (x)-Φ g (x) obtained according to the design value;

图9为按照设计值得到的曝光对比度γ(x)的效果图。FIG. 9 is an effect diagram of the exposure contrast γ(x) obtained according to the design value.

具体实施方式Detailed ways

结合图1至图9说明本实施方式，用于消像差变栅距光栅扫描光刻的干涉 条纹线密度设计方法，该方法主要应用于变周期扫描光刻系统，其组成如图1 所示，去掉其中若干测控元件。图中1和2为两束相干光束，3和4为干涉光束 调整镜，5为半反半透镜，6和7为透镜，用于构成4f光学系统，实现1和2 的干涉形成干涉图样8，干涉图样8的强度为高斯分布。3和4位于6的前焦面 位置，11为干涉条纹线密度控制系统，根据干涉条纹线密度函数，通过3和4 调整相干光束1和2的干涉角度，可调整干涉图样中干涉条纹的线密度。9为涂 有光刻胶的光栅基底，10为二维运动工作台，用于承载9进行步进扫描运动。1 to 9 , the method for designing the line density of interference fringes for aberration variable grating-pitch raster scanning lithography is described in conjunction with FIGS. 1 to 9 . , remove some of the measurement and control components. In the figure, 1 and 2 are two coherent beams, 3 and 4 are interference beam adjustment mirrors, 5 is a half mirror, 6 and 7 are lenses, which are used to form a 4f optical system, and realize the interference of 1 and 2 to form an interference pattern 8 , the intensity of the interference pattern 8 is a Gaussian distribution. 3 and 4 are located at the front focal plane position of 6, and 11 is the interference fringe line density control system. According to the interference fringe line density function, the interference angles of coherent beams 1 and 2 can be adjusted through 3 and 4, and the lines of the interference fringes in the interference pattern can be adjusted. density. 9 is a grating substrate coated with photoresist, and 10 is a two-dimensional motion table, which is used for carrying 9 to perform step-and-scan motion.

可用于本实施方式提出的方法包括以下步骤：The proposed method that can be used in this embodiment includes the following steps:

步骤一、确定变栅距光栅的刻线密度函数及通用光刻过程制作参数。Step 1: Determine the line density function of the variable pitch grating and the general photolithography process manufacturing parameters.

根据变栅距光栅的消像差特性及其在仪器中的应用需求，设计变栅距光栅 刻槽密度目标函数为：According to the aberration characteristics of the variable pitch grating and its application requirements in the instrument, the objective function of the groove density of the variable pitch grating is designed as:

g(x)＝ng0+ng1(x-Wg/2)+ng2(x-Wg/2)2+ng3(x-Wg/2)3 g(x)=n g0 +n g1 (xW g /2)+n g2 (xW g /2) 2 +n g3 (xW g /2) 3

光栅坐标系的定义如图2所示，光栅的理想相位分布Φg(x)可以表示为The definition of the grating coordinate system is shown in Figure 2, and the ideal phase distribution Φ g (x) of the grating can be expressed as

Φg(x)＝2πg(x)·xΦ g (x)=2πg(x) x

其中，g(x)为变栅距光栅的目标刻槽密度，x为光栅矢量方向的坐标，x＝0 位于光栅边界处，Wg为光栅矢量方向的总宽度，ng0为光栅中心处的刻槽密度， 单位为gr/mm(每mm内包含的刻槽数)，ng1为光栅刻槽密度的一次项系数，单 位为gr/mm2，ng2为光栅刻槽密度的二次项系数，单位为gr/mm3，ng3为光栅刻 槽密度的三次项系数，单位为gr/mm4。ng0、ng1、ng2、ng3根据变栅距光栅像差 校正原理及光谱仪器或激光装置的使用参数确定。Among them, g(x) is the target groove density of the variable pitch grating, x is the coordinate of the grating vector direction, x=0 is located at the grating boundary, W g is the total width of the grating vector direction, n g0 is the grating center Groove density, the unit is gr/mm (the number of grooves contained in each mm), n g1 is the linear coefficient of the grating groove density, the unit is gr/mm 2 , n g2 is the quadratic term of the grating groove density coefficient, the unit is gr/mm 3 , n g3 is the cubic term coefficient of the grating groove density, and the unit is gr/mm 4 . n g0 , n g1 , n g2 , and n g3 are determined according to the principle of aberration correction of variable grating pitch gratings and the use parameters of spectroscopic instruments or laser devices.

根据变周期扫描光刻系统的基本组成与工作原理，如图1所示，为保证曝 光量的均匀性，相邻扫描段的干涉图样之间存在重叠。根据系统的设计、装调 参数及变栅距光栅的刻槽密度函数，确定在进行该变周期光栅制作时的以下制 作参数：According to the basic composition and working principle of the variable period scanning lithography system, as shown in Figure 1, in order to ensure the uniformity of exposure, there is overlap between the interference patterns of adjacent scanning segments. According to the design of the system, the adjustment parameters and the groove density function of the variable pitch grating, the following production parameters are determined in the production of the variable period grating:

干涉图样高斯束腰半径为Rho。The Gaussian beam waist radius of the interference pattern is R ho .

相邻扫描段的干涉图样重叠宽度占束腰半径的比例StepRatio，则干涉图样 重叠宽度为StepRatio×Rho，为保证曝光量的均匀性，要求StepRatio小于0.9， StepRatio越小，曝光量均匀性越高，但总的扫描步数越多，制作效率越低。如 图3所示。The ratio of the overlap width of the interference pattern of adjacent scanning segments to the beam waist radius StepRatio, then the overlap width of the interference pattern is StepRatio×R ho . In order to ensure the uniformity of exposure, StepRatio is required to be less than 0.9. The smaller the StepRatio, the better the uniformity of the exposure. High, but the more total scanning steps, the lower the production efficiency. As shown in Figure 3.

步进扫描的总步数N，N≥Wg/(Rho·StepRatio)+1，使曝光区域的宽度大于光 栅的有效宽度。The total number of steps N of the step-by-step scanning, N≥W g /(R ho ·StepRatio)+1, makes the width of the exposure area larger than the effective width of the grating.

步进扫描每一步的步数Nsteps，Nsteps＝round(Rho·StepRatio·ng0)。The number of steps in each step of the step scan is N steps , N steps =round(R ho ·StepRatio· ng0 ).

步骤二、建立变周期扫描光刻总曝光量计算方法，计算光栅刻槽密度函数 作为干涉条纹线密度变化函数f(x)时的曝光量相位变化Ψ(x)及曝光对比度γ。Step 2: Establish a method for calculating the total exposure of variable period scanning lithography, and calculate the exposure phase change Ψ(x) and exposure contrast γ when the grating groove density function is used as the interference fringe linear density change function f(x).

变周期扫描光刻总曝光量的计算方法为：The calculation method of the total exposure of variable period scanning lithography is:

干涉条纹线密度变化函数与光栅刻槽密度函数具有相同的形式，可以表示 为：The linear density variation function of the interference fringes has the same form as the grating groove density function, and can be expressed as:

f(x)＝m0+m1(x-Wg/2)+m2(x-Wg/2)2+m3(x-Wg/2)3 f(x)=m 0 +m 1 (xW g /2)+m 2 (xW g /2) 2 +m 3 (xW g /2) 3

式中，m0为干涉条纹线密度变化函数的常数项系数，m1为干涉条纹线密度 变化函数的一次项系数，m2为干涉条纹线密度变化函数的二次项系数，m3为干 涉条纹线密度变化函数的三次项系数；扫描光刻起始扫描段从x＝0时开始，其 扫描步数序号k＝0，Sk为第k步的步进距离，S0＝0，

Figure BDA0003470252210000081
从x＝0至第k步的总距 离，
Figure BDA0003470252210000082
为k步步进后，第k+1次扫描的干涉条纹线密度。Δk＝fk-fk-1为 k+1次扫描与k次扫描干涉条纹线密度的差值，当k＝0时，Δ0＝0，当k>0时,In the formula, m 0 is the constant term coefficient of the interference fringe line density change function, m 1 is the first order coefficient of the interference fringe line density change function, m 2 is the quadratic term coefficient of the interference fringe line density change function, and m 3 is the interference fringe line density change function. The cubic term coefficient of the stripe line density variation function; the initial scanning segment of scanning lithography starts from x=0, the scanning step number k=0, Sk is the stepping distance of the kth step, S 0 =0,
Figure BDA0003470252210000081
The total distance from x=0 to the kth step,
Figure BDA0003470252210000082
After k steps, the interference fringe line density of the k+1th scan. Δ k =f k -f k-1 is the difference between the line density of interference fringes between k+1 scans and k scans, when k=0, Δ 0 =0, when k>0,

Figure BDA0003470252210000083
Figure BDA0003470252210000083

第k步步进后，第k+1次扫描的曝光量Dk(x)及第k+1次扫描与初始扫描的 相位差

Figure BDA0003470252210000091
为：After the kth step, the exposure amount D k (x) of the k+1th scan and the phase difference between the k+1th scan and the initial scan
Figure BDA0003470252210000091
for:

Figure BDA0003470252210000092
Figure BDA0003470252210000092

Figure BDA0003470252210000093
Figure BDA0003470252210000093

式中，B(x)为曝光量的背景光强，A(x)为高斯分布的干涉图样强度包络。In the formula, B(x) is the background light intensity of the exposure amount, and A(x) is the intensity envelope of the interference pattern of the Gaussian distribution.

Figure BDA0003470252210000094
Figure BDA0003470252210000094

光刻结束时，光栅上的总曝光量为步进扫描总步数N步后N+1次扫描曝光 量的叠加Dtot(x)，即：At the end of lithography, the total exposure amount on the grating is the superposition D tot (x) of the exposure amount of N+1 scans after the total number of step scanning steps N steps, namely:

Dtot(x)＝D0(x)+D1(x)+…DN(x)D tot (x)=D 0 (x)+D 1 (x)+…D N (x)

＝Btot(x)+Atot(x)sin(2πxf0+Ψ)=Btot(x)+ Atot (x)sin( 2πxf 0 +Ψ)

Btot(x)为总曝光量的背景分量，Atot(x)为总曝光量交流分量幅值；B tot (x) is the background component of the total exposure, A tot (x) is the amplitude of the AC component of the total exposure;

Figure BDA0003470252210000095
Figure BDA0003470252210000095

Figure BDA0003470252210000096
Figure BDA0003470252210000096

Figure BDA0003470252210000097
Figure BDA0003470252210000097

Figure BDA0003470252210000098
Figure BDA0003470252210000098

Ψtot(x)＝2πxf0+Ψ(x)Ψ tot (x)=2πxf 0 +Ψ(x)

Ψ(x)＝arctan[F(x)/E(x)]Ψ(x)=arctan[F(x)/E(x)]

本实施方式中，利用步骤一中的通用光刻过程参数外，还包括设定以下设 计参数：In the present embodiment, in addition to the general photolithography process parameters in step 1, the following design parameters are also included:

(1)设定干涉条纹线密度函数与变栅距光栅的刻槽密度函数完全相等，即 m0＝ng0，m1＝ng1，m2＝ng2，m3＝ng3。(1) The line density function of the interference fringes is set to be exactly equal to the groove density function of the variable pitch grating, that is, m 0 =n g0 , m 1 =n g1 , m 2 =n g2 , and m 3 =n g3 .

(2)B(x)的变化不影响干涉条纹线密度的设计，设定B(x)＝A(x)，单次扫描 曝光的对比度为1。(2) The change of B(x) does not affect the design of the line density of the interference fringes, set B(x)=A(x), and the contrast of single scanning exposure is 1.

(3)第k步的步进距离Sk可以选用以下三种步进方式中的一种，按本实施 方式的方法均可得到不同的干涉条纹设计值，但均可满足变栅距光栅刻槽密度 的精度要求：(3) The stepping distance Sk of the kth step can be selected from one of the following three stepping methods. According to the method of this embodiment, different design values of interference fringes can be obtained, but all of them can meet the requirements of variable grating pitch grating. Accuracy requirements for groove density:

①Sk＝Nsteps/fk-1。①S k =N steps /f k-1 .

②Sk＝2Nsteps/(fk+fk-1)，

Figure BDA0003470252210000101
②S k =2N steps /(f k +f k-1 ),
Figure BDA0003470252210000101

③Sk＝Nsteps/fk，

Figure BDA0003470252210000102
③S k =N steps /f k ,
Figure BDA0003470252210000102

按照步骤一的参数和上述参数，通过数值计算的方法，采用变周期扫描光 刻总曝光量计算方法，可以得到Ψ(x)随x变化曲线，Ψ(x)与Φg(x)如图4所示。 二者之间的相位误差为Φe(x)＝Ψ(x)-Φg(x)，如图5所示，按照变栅距光栅刻槽密 度函数进行干涉条纹线密度调整，Φe(x)≠0，光刻得到的变栅距光栅与设计值的 刻槽密度存在较大差别。According to the parameters of step 1 and the above parameters, through the numerical calculation method and the total exposure calculation method of variable period scanning lithography, the curve of Ψ(x) with x can be obtained, Ψ(x) and Φ g (x) are shown in the figure 4 shown. The phase error between the two is Φ e (x)=Ψ(x)-Φ g (x), as shown in Fig. 5, the line density of interference fringes is adjusted according to the grating groove density function of variable pitch, Φ e ( x)≠0, there is a big difference in the groove density between the variable pitch grating obtained by photolithography and the design value.

步骤三、通过数据拟合与迭代寻优方法设计干涉条纹线密度变化函数的三 次项系数m3的优化设计值m3_optimal。Step 3: Design the optimal design value m 3_optimal of the cubic term coefficient m 3 of the interference fringe line density change function through data fitting and iterative optimization method.

采用多项式曲线拟合算法，对步骤二得到的相位分布误差Φe(x)进行四次多 项式曲线拟合，Φe(x)的拟合多项式为Φep(x)＝2π(a40x4+a30x3+a20x2+a10x+a00)，a40、 a30、a20、a10和a00分别为四次拟合多项式Φep(x)的系数；The polynomial curve fitting algorithm is used to fit the phase distribution error Φ e (x) obtained in step 2 with a quartic polynomial curve. The fitting polynomial of Φ e (x) is Φ ep (x)=2π(a 40 x 4 +a 30 x 3 +a 20 x 2 +a 10 x+a 00 ), a 40 , a 30 , a 20 , a 10 and a 00 are the coefficients of the quartic fitting polynomial Φ ep (x);

设定干涉条纹线密度变化函数f(x)的系数m3优化设计的四次相位误差阈值ξ4order，通过一维搜索迭代寻优的方法，计算得到m3_optimal，迭代过程如下，流程 图见图6。Set the fourth-order phase error threshold ξ 4order for the optimal design of the coefficient m3 of the linear density variation function f(x) of the interference fringes, and obtain m 3_optimal by one-dimensional search iterative optimization method. The iterative process is as follows, and the flowchart is shown in Figure 6 .

(1)设定迭代寻优的搜索范围[am3 (0),bm3 (0)]，am3 (0)＝m3_nearby-Hm3，bm3 (0)＝ m3_nearby+Hm3，2Hm3为m3最优值初始搜索范围的宽度，迭代次数im3＝0，m3_nearby为最优解m3_optimal的初始近似值，m3_nearby＝ng3 2/(ng3+a40)，初始四次项相位误差最大 值为Φe_4order_m3 (0)＝abs(2πa40Wg 4)；(1) Set the search range of iterative optimization [a m3 (0) , b m3 (0) ], a m3 (0) = m 3_nearby -H m3 , b m3 (0) = m 3_nearby +H m3 , 2H m3 is the width of the initial search range for the optimal value of m3 , the number of iterations i m3 =0, m3_nearby is the initial approximate value of the optimal solution m3_optimal , m3_nearby =n g3 2 /(n g3 +a 40 ), the initial four times The maximum value of the term phase error is Φ e_4order_m3 (0) =abs(2πa 40 W g 4 );

(2)若

Figure BDA0003470252210000111
不满足阈值要求，执行步骤(3)至(7)，否则，退出循环，执行 步骤(8)；(2) If
Figure BDA0003470252210000111
If the threshold requirement is not met, execute steps (3) to (7), otherwise, exit the loop and execute step (8);

(3)采用一维搜索方法(如黄金分割法、斐波那契法、等分法等方法)， 在

Figure BDA00034702522100001126
内确定寻优变量
Figure BDA00034702522100001127
和
Figure BDA00034702522100001128
Figure BDA00034702522100001129
(3) Using a one-dimensional search method (such as the golden section method, the Fibonacci method, the equal division method, etc.), in
Figure BDA00034702522100001126
Internally determined optimization variables
Figure BDA00034702522100001127
and
Figure BDA00034702522100001128
Figure BDA00034702522100001129

(4)根据寻优变量

Figure BDA00034702522100001130
值设定干涉条纹的线密度变化函数
Figure BDA0003470252210000112
根据步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA0003470252210000113
Figure BDA0003470252210000114
对应的相位误差
Figure BDA0003470252210000115
对
Figure BDA0003470252210000116
进行四次多项式拟合，
Figure BDA0003470252210000117
的拟合多项式为
Figure BDA0003470252210000118
记录四次项系数
Figure BDA0003470252210000119
(4) According to the optimization variables
Figure BDA00034702522100001130
The value sets the linear density variation function of the interference fringes
Figure BDA0003470252210000112
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA0003470252210000113
Figure BDA0003470252210000114
Corresponding phase error
Figure BDA0003470252210000115
right
Figure BDA0003470252210000116
Perform a quartic polynomial fit,
Figure BDA0003470252210000117
The fitting polynomial of is
Figure BDA0003470252210000118
Record the quartic coefficients
Figure BDA0003470252210000119

(5)根据寻优变量

Figure BDA00034702522100001110
值设定干涉条纹的线密度变化函数
Figure BDA00034702522100001111
根据步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA00034702522100001112
Figure BDA00034702522100001113
对应的相位误差
Figure BDA00034702522100001114
对
Figure BDA00034702522100001115
进行四次多项式拟合，
Figure BDA00034702522100001116
的拟合多项式 为
Figure BDA00034702522100001117
记录四次项系数
Figure BDA00034702522100001118
(5) According to the optimization variables
Figure BDA00034702522100001110
The value sets the linear density variation function of the interference fringes
Figure BDA00034702522100001111
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA00034702522100001112
Figure BDA00034702522100001113
Corresponding phase error
Figure BDA00034702522100001114
right
Figure BDA00034702522100001115
Perform a quartic polynomial fit,
Figure BDA00034702522100001116
The fitting polynomial of is
Figure BDA00034702522100001117
Record the quartic coefficients
Figure BDA00034702522100001118

(6)若

Figure BDA00034702522100001119
则
Figure BDA00034702522100001120
Figure BDA00034702522100001121
否则，
Figure BDA00034702522100001122
则
Figure BDA00034702522100001123
Figure BDA00034702522100001124
(6) If
Figure BDA00034702522100001119
but
Figure BDA00034702522100001120
Figure BDA00034702522100001121
otherwise,
Figure BDA00034702522100001122
but
Figure BDA00034702522100001123
Figure BDA00034702522100001124

(7)im3＝im3+1，返回步骤(2)；(7) i m3 =i m3 +1, return to step (2);

(8)

Figure BDA00034702522100001125
(8)
Figure BDA00034702522100001125

步骤四、通过数据拟合与迭代寻优方法设计干涉条纹线密度变化函数的二 次项系数m2的优化设计值m2_optimal，迭代过程如下：Step 4: Design the optimal design value m 2_optimal of the quadratic term coefficient m 2 of the linear density variation function of the interference fringe by data fitting and iterative optimization method. The iterative process is as follows:

(1)根据步骤三得到的m3_optimal，设定干涉条纹的线密度变化函数f3optimal＝ng0+ng1·(x-Wg/2)+ng2·(x-Wg/2)2+m3_optimal·(x-Wg/2)3，根据步骤二给出的 变周期扫描光刻总曝光量计算方法，计算得到Ψtot_3optimal，相位误差 Φe_3optimal＝Ψtot_3optimal-Φg，对Φe_3optimal进行四次多项式拟合，Φe_3optimal的拟合多项式 为Φep_3optimal(x)＝2π(a4_ 3optimalx4+a3_3optimalx3+a2_3optimalx2+a1_3optimalx+a0_3optimal)，记录三次项 系数a3_3optimal；(1) According to m 3_optimal obtained in step 3, set the linear density variation function f 3optimal of interference fringes f 3optimal =n g0 +n g1 ·(xW g /2)+n g2 ·(xW g /2) 2 +m 3_optimal · (xW g /2) 3 , according to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, Ψ tot_3optimal is calculated, and the phase error Φ e_3optimal =Ψ tot_3optimal -Φ g , and Φ e_3optimal is fitted by a quartic polynomial , the fitting polynomial of Φ e_3optimal is Φ ep_3optimal (x)=2π(a 4_ 3optimal x 4 +a 3_3optimal x 3 +a 2_3optimal x 2 +a 1_3optimal x+a 0_3optimal ), record the cubic coefficient a 3_3optimal ;

设定干涉条纹线密度变化函数f(x)的系数m2优化设计的三次项相位误差阈 值ξ3order；Set the cubic phase error threshold ξ 3order of the optimal design of the coefficient m2 of the interference fringe line density variation function f(x);

(2)设定迭代寻优的搜索范围[am2 (0),bm2 (0)]，am2 (0)＝m2_nearby-Hm2，bm2 (0)＝ m2_nearby+Hm2，2Hm2为m2最优值初始搜索范围的宽度，迭代次数im2＝0，m2_nearby为最优解m2_optimal的初始近似值，

Figure BDA0003470252210000121
初始三次项相位误 差最大值为Φe_3order_m2 (0)＝abs(2πa3_3optimalWg 3)；(2) Set the search range of iterative optimization [a m2 (0) , b m2 (0) ], a m2 (0) = m 2_nearby -H m2 , b m2 (0) = m 2_nearby +H m2 , 2H m2 is the width of the initial search range for the optimal value of m2, the number of iterations i m2 = 0 , m2_nearby is the initial approximate value of the optimal solution m2_optimal ,
Figure BDA0003470252210000121
The maximum value of the initial cubic phase error is Φ e_3order_m2 (0) =abs(2πa 3_3optimal W g 3 );

(3)若

Figure BDA0003470252210000122
不满足阈值要求，执行步骤(4)至(8)，否则， 退出循环，执行步骤(9)；(3) If
Figure BDA0003470252210000122
If the threshold requirement is not met, execute steps (4) to (8), otherwise, exit the loop and execute step (9);

(4)采用一维搜索方法(如黄金分割法、斐波那契法、等分法等方法)， 在

Figure BDA0003470252210000123
内确定寻优变量
Figure BDA0003470252210000124
和
Figure BDA0003470252210000125
Figure BDA0003470252210000126
(4) Using a one-dimensional search method (such as golden section method, Fibonacci method, equal division method, etc.), in
Figure BDA0003470252210000123
Internally determined optimization variables
Figure BDA0003470252210000124
and
Figure BDA0003470252210000125
Figure BDA0003470252210000126

(5)根据寻优变量

Figure BDA0003470252210000127
值设定干涉条纹的线密度变化函数
Figure BDA0003470252210000128
根据步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA0003470252210000129
Figure BDA00034702522100001210
对应的相位误 差
Figure BDA00034702522100001211
对
Figure BDA00034702522100001212
进行四次多项式拟合，
Figure BDA00034702522100001213
的拟合多项 式为
Figure BDA00034702522100001214
记录三次 项系数
Figure BDA00034702522100001215
(5) According to the optimization variables
Figure BDA0003470252210000127
The value sets the linear density variation function of the interference fringes
Figure BDA0003470252210000128
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA0003470252210000129
Figure BDA00034702522100001210
Corresponding phase error
Figure BDA00034702522100001211
right
Figure BDA00034702522100001212
Perform a quartic polynomial fit,
Figure BDA00034702522100001213
The fitting polynomial of is
Figure BDA00034702522100001214
record cubic coefficients
Figure BDA00034702522100001215

(6)根据寻优变量

Figure BDA00034702522100001216
值设定干涉条纹的线密度变化函数
Figure BDA00034702522100001217
根据步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA0003470252210000131
Figure BDA0003470252210000132
对应的相位误 差
Figure BDA0003470252210000133
对
Figure BDA0003470252210000134
进行四次多项式拟合，
Figure BDA0003470252210000135
的拟合多项 式为
Figure BDA0003470252210000136
记录三次 项系数
Figure BDA0003470252210000137
(6) According to the optimization variables
Figure BDA00034702522100001216
The value sets the linear density variation function of the interference fringes
Figure BDA00034702522100001217
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA0003470252210000131
Figure BDA0003470252210000132
Corresponding phase error
Figure BDA0003470252210000133
right
Figure BDA0003470252210000134
Perform a quartic polynomial fit,
Figure BDA0003470252210000135
The fitting polynomial of is
Figure BDA0003470252210000136
record cubic coefficients
Figure BDA0003470252210000137

(7)若

Figure BDA0003470252210000138
则
Figure BDA0003470252210000139
Figure BDA00034702522100001310
否则，
Figure BDA00034702522100001311
则
Figure BDA00034702522100001312
Figure BDA00034702522100001313
(7) If
Figure BDA0003470252210000138
but
Figure BDA0003470252210000139
Figure BDA00034702522100001310
otherwise,
Figure BDA00034702522100001311
but
Figure BDA00034702522100001312
Figure BDA00034702522100001313

(8)im2＝im2+1，返回步骤(3)；(8) i m2 =i m2 +1, return to step (3);

(9)

Figure BDA00034702522100001314
(9)
Figure BDA00034702522100001314

步骤五、通过数据拟合与迭代寻优方法设计干涉条纹线密度变化函数的一 次项系数m1的优化设计值m1_optimal，迭代过程如下：Step 5: Design the optimal design value m 1_optimal of the linear coefficient m1 of the linear density variation function of the interference fringe by data fitting and iterative optimization method. The iterative process is as follows:

(1)根据步骤四得到的m2_optimal，设定干涉条纹的线密度变化函数 f2optimal＝ng0+ng1·(x-Wg/2)+m2_optimal·(x-Wg/2)2+m3_optimal·(x-Wg/2)3，根据步骤二给 出的变周期扫描光刻总曝光量计算方法，计算得到Ψtot_2optimal，相位误差 Φe_2optimal＝Ψtot_2optimal-Φg，对Φe_2optimal进行四次多项式拟合，Φe_2optimal的拟合多项式 为Φep_2optimal(x)＝2π(a4_ 2optimalx4+a3_2optimalx3+a2_2optimalx2+a1_2optimalx+a0_2optimal)，记录二次项 系数a2_2optimal；(1) According to m 2_optimal obtained in step 4, set the linear density variation function f 2optimal of interference fringes f 2optimal =n g0 +n g1 ·(xW g /2)+m 2_optimal ·(xW g /2) 2 +m 3_optimal · (xW g /2) 3 , according to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, Ψ tot_2optimal is obtained by calculation, the phase error Φ e_2optimal =Ψ tot_2optimal -Φ g , and Φ e_2optimal is fitted by a quartic polynomial , the fitting polynomial of Φ e_2optimal is Φ ep_2optimal (x)=2π(a 4_ 2optimal x 4 +a 3_2optimal x 3 +a 2_2optimal x 2 +a 1_2optimal x+a 0_2optimal ), record the quadratic term coefficient a 2_2optimal ;

设定干涉条纹线密度变化函数f(x)的系数m1优化设计的二次项相位误差阈 值ξ2order；Set the quadratic term phase error threshold ξ 2order of the optimal design of the coefficient m1 of the interference fringe line density variation function f(x);

(2)设定迭代寻优的搜索范围[am1 (0),bm1 (0)]，am1 (0)＝m1_nearby-Hm1，bm1 (0)＝ m1_nearby+Hm1，2Hm1为m1最优值初始搜索范围的宽度，迭代次数im1＝0，m1_nearby为最优解m1_optimal的初始近似值，

Figure BDA00034702522100001315
初始二次项 相位误差最大值为Φe_2order_m1 (0)＝abs(2πa2_2optimalWg 2)；(2) Set the search range of iterative optimization [a m1 (0) , b m1 (0) ], a m1 (0) = m 1_nearby -H m1 , b m1 (0) = m 1_nearby +H m1 , 2H m1 is the width of the initial search range for the optimal value of m 1 , the number of iterations i m1 =0, m 1_nearby is the initial approximate value of the optimal solution m 1_optimal ,
Figure BDA00034702522100001315
The maximum value of the initial quadratic phase error is Φ e_2order_m1 (0) =abs(2πa 2_2optimal W g 2 );

(3)若

Figure BDA0003470252210000141
不满足阈值要求，执行步骤(4)至(8)，否则， 退出循环，执行步骤(9)；(3) If
Figure BDA0003470252210000141
If the threshold requirement is not met, execute steps (4) to (8), otherwise, exit the loop and execute step (9);

(4)采用一维搜索方法(如黄金分割法、斐波那契法、等分法等方法)， 在[am1 (im1),bm1 (im1)]内确定寻优变量m1L (im1)和m1R (im1)，am1 (im1)≤m1L (im1)<m1R (im1)≤ bm1 (im1)；(4) Using a one-dimensional search method (such as the golden section method, the Fibonacci method, the equal division method, etc. ) , determine the optimization variable m 1L ( im1 ) and m 1R (im1) , a m1 (im1) ≤ m 1L (im1) <m 1R (im1) ≤ b m1 (im1) ;

(5)根据寻优变量m1L (im1)值设定干涉条纹的线密度变化函数

Figure BDA0003470252210000142
根据步骤二给 出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA0003470252210000143
m1L (im1)对应的相位 误差
Figure BDA0003470252210000144
对
Figure BDA0003470252210000145
进行四次多项式拟合，
Figure BDA0003470252210000146
的拟合多项 式为
Figure BDA0003470252210000147
记录二次项 系数
Figure BDA0003470252210000148
(5) Set the linear density change function of the interference fringes according to the value of the optimization variable m 1L (im1)
Figure BDA0003470252210000142
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA0003470252210000143
Phase error corresponding to m 1L (im1)
Figure BDA0003470252210000144
right
Figure BDA0003470252210000145
Perform a quartic polynomial fit,
Figure BDA0003470252210000146
The fitting polynomial of is
Figure BDA0003470252210000147
record quadratic coefficients
Figure BDA0003470252210000148

(6)根据寻优变量m1R (im1)值设定干涉条纹的线密度变化函数

Figure BDA0003470252210000149
根据步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA00034702522100001410
m1R (im1)对应的相 位误差
Figure BDA00034702522100001411
对
Figure BDA00034702522100001412
进行四次多项式拟合，
Figure BDA00034702522100001413
的拟合多 项式为
Figure BDA00034702522100001414
记录二次 项系数
Figure BDA00034702522100001415
(6) Set the linear density change function of the interference fringes according to the value of the optimization variable m 1R (im1)
Figure BDA0003470252210000149
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA00034702522100001410
Phase error corresponding to m 1R (im1)
Figure BDA00034702522100001411
right
Figure BDA00034702522100001412
Perform a quartic polynomial fit,
Figure BDA00034702522100001413
The fitting polynomial of is
Figure BDA00034702522100001414
record quadratic coefficients
Figure BDA00034702522100001415

(7)若

Figure BDA00034702522100001416
则am1 (im1+1)＝m1L (im1)，bm1 (im1+1)＝bm1 (im1)，
Figure BDA00034702522100001417
否则，
Figure BDA00034702522100001418
则am1 (im1+1)＝am1 (im1)，bm1 (im1+1)＝m1R (im1)，
Figure BDA00034702522100001419
(7) If
Figure BDA00034702522100001416
Then a m1 (im1+1) = m 1L (im1) , b m1 (im1+1) = b m1 (im1) ,
Figure BDA00034702522100001417
otherwise,
Figure BDA00034702522100001418
Then a m1 (im1+1) = a m1 (im1) , b m1 (im1+1) = m 1R (im1) ,
Figure BDA00034702522100001419

(8)im1＝im1+1，返回步骤五三；(8) i m1 =i m1 +1, return to step 53;

(9)m1_optimal＝(m1L (im1)+m1R (im1))/2。(9) m 1_optimal =(m 1L (im1) +m 1R (im1) )/2.

步骤六、设计干涉条纹线密度变化函数的常数项系数m0的优化设计值m0_optimal，迭代过程如下：Step 6: Design the optimal design value m 0_optimal of the constant term coefficient m 0 of the linear density change function of the interference fringe, and the iterative process is as follows:

(1)根据步骤五得到的m1_optimal，设定干涉条纹的线密度变化函数 f1optimal＝ng0+m1_optimal·(x-Wg/2)+m2_optimal·(x-Wg/2)2+m3_optimal·(x-Wg/2)3，根据步骤二 给出的变周期扫描光刻总曝光量计算方法，计算得到Ψtot_1optimal，相位误差 Φe_1optimal＝Ψtot_1optimal-Φg，对Φe_1optimal进行四次多项式拟合，Φe_1optimal的拟合多项式 为Φep_1optimal(x)＝2π(a4_1optimalx4+a3_1optimalx3+a2_1optimalx2+a1_1optimalx+a0_1optimal)，记录一次项 系数a1_1optimal；(1) According to m 1_optimal obtained in step 5, set the linear density variation function f 1optimal =n g0 +m 1_optimal ·(xW g /2)+m 2_optimal ·(xW g /2) 2 +m 3_optimal · (xW g /2) 3 , according to the calculation method of the total exposure of variable-period scanning lithography given in step 2, Ψ tot_1optimal is calculated, and the phase error Φ e_1optimal = Ψ tot_1optimal -Φ g , and Φ e_1optimal is fitted by a quartic polynomial , the fitting polynomial of Φ e_1optimal is Φ ep_1optimal (x)=2π(a 4_1optimal x 4 +a 3_1optimal x 3 +a 2_1optimal x 2 +a 1_1optimal x+a 0_1optimal ), record the first-order coefficient a 1_1optimal ;

设定干涉条纹线密度变化函数f(x)的系数m0优化设计的一次项相位误差阈 值ξ1order；Set the first-order phase error threshold ξ 1order of the optimal design of the coefficient m0 of the interference fringe line density variation function f(x);

(2)设定迭代寻优的搜索范围[am0 (0),bm0 (0)]，am0 (0)＝m0_nearby-Hm0，bm0 (0)＝ m0_nearby+Hm0，2Hm0为m0最优值初始搜索范围的宽度，迭代次数im0＝0，m0_nearby为最优解m0_optimal的初始近似值，m0_nearby＝ng0-a1_1optimal，初始一次项相位误差最大 值为Φe_1order_m0 (0)＝abs(2πa1_1optimalWg)；(2) Set the search range of iterative optimization [a m0 (0) , b m0 (0) ], a m0 (0) = m 0_nearby -H m0 , b m0 (0) = m 0_nearby +H m0 , 2H m0 is the width of the initial search range for the optimal value of m 0 , the number of iterations i m0 =0, m 0_nearby is the initial approximate value of the optimal solution m 0_optimal , m 0_nearby =n g0 -a 1_1optimal , the maximum value of the initial first-order phase error is Φ e_1order_m0 (0) =abs(2πa 1_1optimal W g );

(3)若

Figure BDA0003470252210000151
不满足阈值要求，执行步骤(4)至(8)，否则，退出循环，执行 步骤(9)；(3) If
Figure BDA0003470252210000151
Do not meet the threshold requirements, execute steps (4) to (8), otherwise, exit the loop and execute step (9);

(4)采用一维搜索方法(如黄金分割法、斐波那契法、等分法等方法)， 在

Figure BDA0003470252210000152
内确定寻优变量
Figure BDA0003470252210000153
和
Figure BDA0003470252210000154
Figure BDA0003470252210000155
(4) Using a one-dimensional search method (such as golden section method, Fibonacci method, equal division method, etc.), in
Figure BDA0003470252210000152
Internally determined optimization variables
Figure BDA0003470252210000153
and
Figure BDA0003470252210000154
Figure BDA0003470252210000155

(5)根据寻优变量

Figure BDA0003470252210000156
值设定干涉条纹的线密度变化函数
Figure BDA0003470252210000157
根据步骤 二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA0003470252210000158
Figure BDA0003470252210000159
对应的 相位误差
Figure BDA00034702522100001510
对
Figure BDA00034702522100001511
进行四次多项式拟合，
Figure BDA00034702522100001512
的拟合 多项式为
Figure BDA00034702522100001513
记录 一次项系数
Figure BDA00034702522100001514
(5) According to the optimization variables
Figure BDA0003470252210000156
The value sets the linear density variation function of the interference fringes
Figure BDA0003470252210000157
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA0003470252210000158
Figure BDA0003470252210000159
Corresponding phase error
Figure BDA00034702522100001510
right
Figure BDA00034702522100001511
Perform a quartic polynomial fit,
Figure BDA00034702522100001512
The fitting polynomial of is
Figure BDA00034702522100001513
record primary coefficients
Figure BDA00034702522100001514

(6)根据寻优变量

Figure BDA00034702522100001515
值设定干涉条纹的线密度变化函数
Figure BDA0003470252210000161
根据步骤二给出的变周期扫描光刻总曝光量计算方法，计算得到
Figure BDA0003470252210000162
Figure BDA0003470252210000163
对应 的相位误差
Figure BDA0003470252210000164
对
Figure BDA0003470252210000165
进行四次多项式拟合，
Figure BDA0003470252210000166
的 拟合多项式为
Figure BDA0003470252210000167
记录一次项系数
Figure BDA0003470252210000168
(6) According to the optimization variables
Figure BDA00034702522100001515
The value sets the linear density variation function of the interference fringes
Figure BDA0003470252210000161
According to the calculation method of the total exposure amount of variable period scanning lithography given in step 2, the calculation is obtained
Figure BDA0003470252210000162
Figure BDA0003470252210000163
Corresponding phase error
Figure BDA0003470252210000164
right
Figure BDA0003470252210000165
Perform a quartic polynomial fit,
Figure BDA0003470252210000166
The fitting polynomial of is
Figure BDA0003470252210000167
record primary coefficients
Figure BDA0003470252210000168

(7)若

Figure BDA0003470252210000169
则
Figure BDA00034702522100001610
Figure BDA00034702522100001611
否则，
Figure BDA00034702522100001612
则
Figure BDA00034702522100001613
Figure BDA00034702522100001614
(7) If
Figure BDA0003470252210000169
but
Figure BDA00034702522100001610
Figure BDA00034702522100001611
otherwise,
Figure BDA00034702522100001612
but
Figure BDA00034702522100001613
Figure BDA00034702522100001614

(8)im0＝im0+1，返回步骤(3)；(8) i m0 =i m0 +1, return to step (3);

(9)

Figure BDA00034702522100001615
(9)
Figure BDA00034702522100001615

步骤七、根据第三步至第七步优化的m3_optimal、m2_optimal、m1_optimal和m0_optimal， 核对曝光对比度是否满足曝光工艺需求。Step 7: Check whether the exposure contrast meets the requirements of the exposure process according to m 3_optimal , m 2_optimal , m 1_optimal and m 0_optimal optimized in the third step to the seventh step.

设定干涉条纹线密度为 foptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3，按照步 骤二给出的变周期扫描光刻总曝光量计算方法计算得到曝光对比度γ(x)，判断 在整个x范围内，曝光对比度γ是否满足曝光对比度要求，曝光对比度要求由制 作工艺参数决定，如要求曝光对比度大于0.95。Set the line density of interference fringes as f optimal (x)=m 0_optimal +m 1_optimal (xW g /2)+m 2_optimal (xW g /2) 2 +m 3_optimal (xW g /2) 3 , which is given according to step 2 The exposure contrast γ(x) is calculated by the variable period scanning lithography total exposure calculation method, and it is judged whether the exposure contrast γ meets the exposure contrast requirement in the entire x range. The exposure contrast requirement is determined by the manufacturing process parameters. If the exposure contrast is required to be greater than 0.95.

若γ不满足曝光对比度要求，需要减小步骤一中的干涉图样重叠宽度占束腰 半径的比例StepRatio，重新进行步骤二、三、四、五、六及七，直至γ满足曝光 对比度要求。If γ does not meet the exposure contrast requirement, it is necessary to reduce the ratio of the overlap width of the interference pattern to the beam waist radius StepRatio in step 1, and repeat steps 2, 3, 4, 5, 6 and 7 until γ meets the exposure contrast requirement.

若γ满足曝光对比度要求，则整个优化过程结束，按照foptimal改变光刻过程 中干涉条纹的线密度，可以得到目标刻槽密度的变栅距光栅，按此方法的优化 结果如图7-图9所示。If γ satisfies the exposure contrast requirement, the entire optimization process ends, and the linear density of the interference fringes in the lithography process is changed according to f optimal , and the variable pitch grating with the target groove density can be obtained. The optimization result according to this method is shown in Fig. 7-Fig. 9 shown.

具体实施方式二、本实施方式为具体实施方式一所述的用于消像差变栅距 光栅扫描光刻的干涉条纹线密度设计方法的实施例：Specific embodiment 2, this embodiment is an example of the interference fringe line density design method for aberration variable grating pitch raster scanning lithography described in specific embodiment 1:

本实施例按实施方式一中设定的步骤一、步骤二、步骤三、步骤四、步骤 五、步骤六、步骤七的方法实施。其中光刻相干光束1和2为满足相干长度和 曝光波长要求的激光器分光得到发出，此处由Kr+激光器产生，波长为413.1nm。 光机元件3、4、5、6、7固定在静态的光学平台上垂直放置，保持静止，最终 产生小尺寸的圆形干涉图样。This embodiment is implemented according to the methods of step 1, step 2, step 3, step 4, step 5, step 6, and step 7 set in the first embodiment. The lithography coherent beams 1 and 2 are emitted by lasers that meet the requirements of coherence length and exposure wavelength, and are generated by Kr + laser here, with a wavelength of 413.1nm. The opto-mechanical elements 3, 4, 5, 6, and 7 are fixed on a static optical platform and placed vertically and remain stationary, eventually producing a small-sized circular interference pattern.

在步骤一中，干涉图样高斯束腰半径Rho＝100μm，StepRatio＝0.8，设计某变 栅距光栅参数为：ng0＝1200gr/mm，ng1＝-0.7783gr/mm2，ng2＝1.865×10-4gr/mm3， ng3＝-8.1336×10-8gr/mm4，Wg＝30mm，扫描总步数为400，每一步的步数Nsteps＝96。In step 1, the Gaussian beam waist radius of the interference pattern R ho = 100μm, StepRatio = 0.8, and the design parameters of a variable grating pitch grating are: n g0 =1200gr/mm, n g1 =-0.7783gr/mm 2 , n g2 =1.865 ×10 -4 gr/mm 3 , n g3 =-8.1336×10 -8 gr/mm 4 , W g =30mm, the total number of scanning steps is 400, and the number of steps in each step is N steps =96.

步骤二、步骤三及步骤四，可采用Matlab或Visual studio平台完成数值计 算的设计过程。步骤二种，设定m0＝ng0＝1200gr/mm，m1＝ng1＝-0.7783gr/mm2， m2＝ng2＝1.865×10-4gr/mm3，m3＝ng3＝-8.1336×10-8gr/mm4步进方式采用 Sk＝Nsteps/fk-1。In step 2, step 3 and step 4, the design process of numerical calculation can be completed by using Matlab or Visual studio platform. Step two, set m 0 =n g0 =1200gr/mm, m 1 =n g1 =-0.7783gr/mm 2 , m 2 =n g2 =1.865×10 -4 gr/mm 3 , m 3 =n g3 =-8.1336×10 -8 gr/mm 4 The step method adopts S k =N steps /f k-1 .

步骤三中，设定ξ4order＝1e-4rad，采用黄金分割法迭代优化设计后，得到 m3_optimal＝-3.2461×10-7gr/mm4。In step 3, set ξ 4order = 1e-4rad, after using the golden section method to iteratively optimize the design, m 3_optimal =-3.2461×10 -7 gr/mm 4 is obtained.

步骤四中，设定ξ3order＝1e-4rad，采用黄金分割法迭代优化设计后，得到 m2_optimal＝5.5583×10-4gr/mm3。In step 4, set ξ 3order = 1e-4rad, and after adopting the golden section method to iteratively optimize the design, m 2_optimal =5.5583×10 -4 gr/mm 3 is obtained.

步骤五中，设定ξ2order＝1e-4rad，采用黄金分割法迭代优化设计后，得到 m1_optimal＝-1.5510gr/mm2。In step 5, set ξ 2order = 1e-4rad, after adopting the golden section method to iteratively optimize the design, m 1_optimal =-1.5510gr/mm 2 is obtained.

步骤六中，设定ξ1order＝1e-4rad，采用黄金分割法迭代优化设计后，得到 m0_optimal＝1188.2629gr/mm，则干涉条纹线密度变化的规律为 foptimal(x)＝m0_optimal+m1_optimal(x-Wg/2)+m2_optimal(x-Wg/2)2+m3_optimal(x-Wg/2)3。In step 6, set ξ 1order = 1e-4rad, after using the golden section method to iteratively optimize the design, get m 0_optimal = 1188.2629gr/mm, then the law of interference fringe line density change is f optimal (x)=m 0_optimal +m 1_optimal (xW g /2)+m 2_optimal (xW g /2) 2 +m 3_optimal (xW g /2) 3 .

步骤七中，计算得到曝光对比度γ(x)在x在0-30mm的整个光栅范围内， 均优于0.99，满足曝光对比度0.95的工艺要求，自此完成了干涉条纹线密度的 设计。In step 7, the exposure contrast γ(x) is calculated to be better than 0.99 in the entire grating range of x in the range of 0-30mm, which meets the technological requirement of exposure contrast of 0.95. Since then, the design of the line density of the interference fringes has been completed.

以上所述实施例的各技术特征可以进行任意的组合，为使描述简洁，未对 上述实施例中的各个技术特征所有可能的组合都进行描述，然而，只要这些技 术特征的组合不存在矛盾，都应当认为是本说明书记载的范围。The technical features of the above-described embodiments can be combined arbitrarily. For the sake of brevity, all possible combinations of the technical features in the above-described embodiments are not described. However, as long as there is no contradiction between the combinations of these technical features, All should be regarded as the scope described in this specification.

以上所述实施例仅表达了本发明的几种实施方式，其描述较为具体和详细， 但并不能因此而理解为对发明专利范围的限制。应当指出的是，对于本领域的 普通技术人员来说，在不脱离本发明构思的前提下，还可以做出若干变形和改 进，这些都属于本发明的保护范围。因此，本发明专利的保护范围应以所附权 利要求为准。The above-mentioned embodiments only represent several embodiments of the present invention, and the descriptions thereof are specific and detailed, but should not be construed as a limitation on the scope of the invention patent. It should be pointed out that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can be made, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the patent of the present invention 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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