WU Yue，WANG Li-ping，YU Jie，ZHANG Xu，JIN Chun-shui

（1.Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences,Changchun 130033, China；2.University of Chinese Academy of Sciences, Beijing 100039, China）

Abstract: An initial construction satisfying aberration balance and multi-constraint control is essential for the design of an off-axis multi-reflective optical system with minimal aberration.In this paper, a mathematical model for calculating the initial structure of off-axis multi-reflective is established based on the grouping design method combining spatial ray tracing and aberration correction, and an improved Particle Swarm Optimization (PSO) is proposed to solve the initial structure problem of an off-axis multi-reflective optical system.The PSO of natural selection with shrinkage factor is applied to improve calculation accuracy and design efficiency, so as to obtain the initial structure of the off-axis multi-reflection optical system.In the last part of this paper, taking an Extreme UltraViolet (EUV) lithography projection objective with six-mirror reflective aspheric mirrors as an example, the reliability and effectiveness of this method are verified.A 0.33 numerical aperture EUV lithographic objective with wave-front error better than 1/80λ (λ=13.5 nm) RMS is achieved.

Key words: optical design; geometric optics; aberration theory; particle swarm optimization

1 Introduction

Compared with refractive optical systems, reflective optical systems have the following advantages: reflective optical systems do not produce chromatic aberration; their optical path can be folded,which facilitates the shortening of the barrel length to make the structure compact; they are relatively insensitive to changes in temperature and air pressure; they effectively solve the obstruction constraint by off-axis eccentric tilt, which is conducive to increasing the field of view and improving the image quality.Therefore, the design of off-axis reflective optical systems has received extensive attention, and is often used in space cameras, telescopes, and infrared/ultraviolet fields[1-5].In recent years, off-axis six-mirror reflective EUV lithography projection objectives have become a typical application of off-axis multi-reflection optical systems.EUV lithography projection objective has extremely high imaging requirements, which needs to achieve super diffraction limit resolution and the wave aberration better than 1/50λ[6-9].The design of optical systems to achieve ultra-diffraction-limited minimum aberrations depends heavily on aberration balance.It needs to solve the contradiction among the multi-constraint, multi-objective and few degrees of freedom, which brings great challenges to optical designers.The current optical design software mainly solves the optimization problem of optical systems by using the damped least squares local optimization algorithm to calculate the minimum of the error function in the multi-dimensional variable space, and the final optimization result is in most cases a local optimal solution close to the initial structure, which has great limitations[10], and its global optimization function is difficult to achieve aberration balance and constraint control simultaneously in the case of poor initial structure.Optimal design using optical design software relies heavily on the selection of the initial structure, and the initial structure construction is the key to optical system design[11].Especially for minimal aberration optical systems, which are more sensitive to the aberration balance, the construction of the initial structure to satisfy the aberration balance and multi-constraint control is a key issue in order to balance the constraint and aberration and to avoid the optical system perturbation too large, which makes it difficult to realize the minimal aberration optical system.

At present, the common initial structure construction methods of reflective optical systems include paraxial search method, Y, Y-bar method and group design method.Among them, the paraxial search method was proposed by M.F.Bal, which uses a paraxial model to exhaustively enumerate the first-order aberrations of the optical system.The number of constraints that can be met is too small,which greatly affects the efficiency[12-13].Scott A.Lerner et al.applied the Y, Y-bar method to the solution of the optical system structure, which uses the height of the marginal rays and chief rays on the surface of the optical element to solve the radius of curvature and the distance between the mirrors to construct the initial structure.The height of chief ray and marginal ray of the optical surface of each structure is not easy to determine, and it is not universal.When the number of off-axis reflective optical system components is large, the calculation amount of the above method will increase greatly,which affects the design efficiency[14-15].The grouping design method was applied to the design of offaxis six-mirror reflective optical system by Hudyma.The optical system was divided into two groups, but he did not give a specific design method[16].The research group of Professor Li Yanqiu of Beijing Institute of Technology proposed a real ray-tracing grouping design method applied to off-axis six-reflector and more multi-part reflector systems, by which the off-axis reflector optical system is divided into three groups, the structural parameters of each mirror group are determined by real ray-tracing calculation based on constraint control,and finally the structural parameters of the three mirror groups are spliced[17-18].The above methods have not yet solved the following problems: (1) the optimization process may produce large disturbances, and the structure deviates too much from the initial structure, making the constraints difficult to control; (2) optical design software optimization in the aberration balance process of low-order aberration and high-order aberration may appear large residuals, thus making the optical system design residuals large, which is not conducive to the realization of minimal aberration system.

In order to solve the problem of aberration balance and multi-constraint control in the process of initial structure construction, a mathematical model of off-axis multi-reflection initial structure calculation is established in this paper based on the grouping design method combining spatial ray tracing and aberration correction[19].The Particle Swarm Algorithm (PSO) is used to calculate and solve the high-dimensional nonlinear mathematical model.Several common hybrid PSOs are compared.The proposed PSO with natural selection and shrinkage factor improves the computational accuracy and design efficiency, and provides a design fundation for the design of off-axis multi-reflection optical systems with minimal aberration optimization potential.In this paper, the above method is applied to the EUV lithography projection objective optical system to realize the design of the off-axis six-mirror reflective minimal aberration optical system,whose main design process includes: (1) divide the optical system into two groups, namely the objectside mirror group and the image-side mirror group;(2) based on the aberration theory and space ray tracing theory, parameterize the structural parameters of the optical system, establish mathematical models for the calculation of the corresponding parameters for the front and rear mirror groups, and join the front and rear mirror groups to obtain the mathematical models of the initial structure; (3) the traditional PSOs are compared horizontally, a natural selection PSO with shrinkage factor is proposed to calculate and solve the initial structure parameters; (4)optimize the initial structure.Use the incremental optimization method to avoid excessive changes in the system structure during the optimization process and deviation from the initial structure.Finally,an off-axis six-mirror reflective optical system with engineering feasibility is realized, and its integrated system wave aberration is close to 1/80λ RMS.

2 Mathematical model for initial structure calculation

2.1 Grouping design method

As shown in Figure 1, in this paper, the off-axis six-mirror reflective EUV lithography objectives are divided into two groups, the object-side mirror group and the image-side mirror group, and the optical paths of the first near-axis ray and the second near-axis ray are calculated.The parameters of the optical system mainly include the miniaturization magnification of the system asβ, the center height of the object-side field of view asy, the center height of the image-side field of view asyim, the numerical aperture of the object-side asNAO, and the numerical aperture of the image-side asNA.Then we can getyim=y·β,NAO=NA·β.The object-side mirror group and the image-side mirror group are spliced at the intermediate image, and the splicing should ensure that the object image matches, the pupil matches and the magnification matches.Based on the above principles, the structural parameters of the object-side mirror and image-side mirror of the optical system are parameterized, the spatial ray tracing is introduced, and the constraint parameters of the optical system, such as blocking, mirror spacing,image apocenter, aperture and incident angle, are quantified to establish a mathematical model for the calculation of the initial structural parameters of the minimal aberration off-axis six-mirror reflective EUV lithography objective.

Fig.1 Schematic diagram of the Off-axis six-mirror reflective optical system structure圖1 離軸六反光學(xué)系統(tǒng)結(jié)構(gòu)示意圖

2.2 Aberration analysis of the object-side mirror group (Group 1)

As shown in Figure 2, the height of the center of the field of view on the object side isy, the aperture angle of the object side isu1, and the diaphragm is located at M2.The light starts from the object field of view and reaches the intermediate image point through M1, M2, M3, M4. Among them,d1,d2,d3,d4respectively represent mirror pitch from M1to M2, mirror pitch from M2to M3, mirror pitch from M3to M4, and the distance from M4to the middle the image point IM;h1,h2,h3,h4denote the heights of the first auxiliary rays on M1, M2, M3, M4,respectively;r1,r2,r3,r4andk1,k2,k3,k4denote the radius of curvature and quadratic surface coefficients of M1, M2, M3, M4, respectively;l1,l2,l3,l4andare the object and image distances of M1, M2,M3, and M4, respectively.

Fig.2 Group 1 structure schematic diagram圖 2 Group 1結(jié)構(gòu)示意圖

The third-order monochromatic aberrations mainly include spherical aberration, comatic aberration, astigmatism, curvature of field, and distortion,which are represented bySI,SII,SIII,SIV,SVrespectively, and the formula is[4,20]:

Based on paraxial approximation conditions,following parameters are introduced:

For this reflection system,

For the ray tracing of the paraxial chief ray and the paraxial marginal ray, the following formula is obtained:

The distance between the exit pupil position and M4in Group 1 is

Substituting formula (2?4) into (1), the G1 aberration coefficient of the front lens group can be calculated.

2.3 Aberration analysis of the image-side mirror group (Group 2)

As shown in Figure 3, in the image-side mirror group, the height of the object-side field of view center isy5, the height of the image-side field of view center isyim, the object-side aperture angle isu5.The light starts from the intermediate image point and reaches the image point through M5and M6, whered5andd6denote the mirror distance from M5to M6and the distance from M6to the image plane, respectively;h5andh6denote the height of the first auxiliary ray on M5and M6, respectively;r5,r6andk5,k6denote the radius of curvature and quadratic surface coefficient of M5and M6, respectively;l5,l6, andl′5,l′6are the object distance and image distance of M5and M6, respectively.

Fig.3 Schematic diagram of the Group 2圖 3 Group 2示意圖

Similarly, parameters are introduced based on paraxial approximation conditions:

For this reflection system,

Perform ray tracing on the paraxial chief ray and the marginal ray to obtain the following formula:

The distance between the entrance pupil position and M5in Group2 is

According to the optical pupil matching condition, namely, the outgoing pupil of G1 coincides with the incoming pupil of G2.Through calculation,we can get the distance between the entrance pupil position and M5in Group1 is

Substituting the formula (7?8) into (1) to obtain the aberration coefficient of the rear lens group G2.

2.4 Mirror group splicing and mathematical model establishment

According to the splicing principle, which mainly includes object image matching, magnification matching and optical pupil matching, the following function can be derived after splicing the object-side mirror group and the image-side mirror group:

The aberration of the coaxial optical system satisfies the linear superposition, that is, the aberration of the entire optical system is the sum of the aberration contributions of each element in the optical system.Therefore, the aberration coefficient of the off-axis six-mirror reflective optical system is the sum of the aberration coefficients of Group 1 and Group 2, and the aberration coefficient of the optical system can be obtained by solving the aberration coefficients of the two lens groups.

Through the above aberration theory, the thirdorder aberration coefficient and structural parameters of the EUV lithography objective optical system are calculated.In order to meet some special requirements of the optical system, spatial optical tracing is also introduced, and the special requirements of the optical system are taken as constraints which can effectively control the constraints such as occlusion, mirror spacing, aperture, image side telecentricity, and incident angle during the construction of the initial structure.Based on the above constraints and according to the objective, i.e., the thirdorder aberration coefficient is as small as possible,the evaluation function can therefore be written as

where constraints represent the above mentioned constraints.The evaluation functionFreflects the size of the primary aberration of the optical system and the constrained control ability.The smaller the value, the smaller the primary aberration of the initial structure, the better the aberration control of the structure, and the bigger the potential for achieving high imaging quality.In this paper, through aberration theory and constraint control, a mathematical model of off-axis six-mirror reflective parameter design with minimal aberration is established, and the physical model is transformed into a mathematical model, which is the problem of solving high-dimensional nonlinear parameter equations.

3 Using PSO to calculate and solve the mathematical model of off-axis six-mirror reflective initial structure

At present, the main algorithm for calculating the initial structure of the optical system structure is the genetic algorithm[11,21-22].The genetic algorithm based on biological evolution has good global search capabilities.Because of its inherent parallelism, multiple individuals comparison can be carried out at the same time, and its scalability makes it easy to combine with other algorithms.However,the poor local search ability of genetic algorithm results in low search efficiency of genetic algorithm and a certain dependence on the selection of the initial population.PSO was proposed by Eberhart and Kennedy in 1995[23].The advantage of this algorithm lies in the simplicity, ease of implementation, versatility, and speed of calculation, and does not require gradient information.It is an effective optimization tool for nonlinear optimization problems, combinatorial optimization problems, and mixed-integer nonlinear optimization problems.It is currently widely used in application fields such as function optimization, neural network training, and practical engineering.The basic PSO algorithm also has certain shortcomings.When dealing with highdimensional complex problems, the algorithm tends to fall into local values; when the problem scale is large, the algorithm convergence speed is slower and the accuracy is limited.In order to overcome the deficiencies of the basic PSO algorithm, the current related improvements mainly include parameter improvement and hybrid algorithm.Parameter improvement is mainly through the introduction of some new parameters, but this also increases the complexity of the algorithm to a certain extent while improving the algorithm, including the PSO algorithm with improved weight, the PSO algorithm with shrinkage factor, and the PSO algorithm with variable learning factor.Hybrid algorithm is the hotspot of PSO algorithm improvement.Combining other algorithms in PSO algorithm improves the global search ability and search accuracy of PSO algorithm, including PSO algorithm based on natural selection, PSO algorithm based on hybridization,PSO algorithm based on simulated annealing, etc.[24-25].

Because the learning factorsc1andc2determine the influence of the particle's own experience information and the experience information of other particles on the particle trajectory, they reflect the information exchange between particle swarms.Whenc1is set larger, the particles will linger too much in the local area, whenc2is set larger, the particles will converge to the local minimum prematurely.In order to effectively control the flying speed of particles so that the algorithm achieves an effective balance between global detection and local mining, Clerc constructed a PSO algorithm that introduces a shrinkage factor to ensure the convergence of the PSO algorithm, and can cancel the boundary limit on the speed, which can effectively control the flight speed of constrained particles and enhance the local search capability of the algorithm.

In this paper, a natural selection PSO with shrinkage factor is proposed to construct the mathematical model of the initial structure of the off-axis six-mirror reflective optical system.The above three hybrid PSO algorithms are compared horizontally, and the shrinkage factor are introduced into the simulated annealing PSO algorithm and the hybrid PSO algorithm, and the natural selection PSO algorithm with inertia weight is introduced for comparison.Since the condition for introducing shrinkage factor isc1+c2> 4.Setc1+c2=4.1 in Clerc's method with shrinkage factor.Figure 4 shows the convergence curves of the four algorithms under 6 groups of different learning factors, wherec1+c2=4.1 in the group 1?5, andc1+c2= 4.3 in the sixth group, the number of particlesN=1000, the maximum number of iterations is 100, the annealing constant is 0.42, the hybridization probability is 0.9, the size ratio of the hybridization pool is 0.2, and the inertia weight factor is 0.7.The minimum evaluation function values calculated by the four algorithms with six groups of learning factors are given in Table 1, respectively.From Figure 4 and Table 1, it can be seen that for the above 6 groups of different learning factors, the natural selection PSO with shrinkage factors has a great improvement in the calculation accuracy and the convergence speed of the mathematical model of the initial structure of the off-axis six-mirror reflective optical system and which provides a design fundation for the design of the off-axis six-mirror reflective optical system with minimal aberration optimization potential.

The main process of the natural selection PSO with shrinkage factor is shown in Figure 5: Step 1:establish a mathematical model for calculating the initial structure of the off-axis six-mirror reflective optical system, and initialize the position and velocity of the particles in the population randomly.Step 2: evaluate the fitness of each particle, store the current position and fitness value of each particle in the individual extreme value of each particle, calculate the individual extreme value and the global optimal value, where for each particle the individual extreme value is found, from which a global value is to found, which is called the global optimal solution.Step 3: based on the shrinkage factor, update the speed and position of each particle.Step 4: according to the fitness value, update the individual extreme value and the global optimal solution among the particles.Step 5: sort the entire particle swarm by fitness value, replace the position and velocity of the worst half with the velocity and position of the best half of the particles in the swarm, and keep the individual extreme value and the global optimal solution unchanged.Step 6: if the termination condition is met (the error is good enough or the maximum number of cycles is reached), the cycle ends,otherwise it returns to Step 3.Step 7: solve the offaxis six-mirror reflective initial structure with minimal aberration according to the calculation result of the algorithm.

Fig.4 Convergence curves of evaluation functions calculated by four different algorithms for 6 groups of learning factors圖4 針對6組學(xué)習(xí)因子4種不同算法計算的評價函數(shù)的收斂曲線圖

Tab.1 Evaluation function values calculated by four different algorithms for 6 groups of learning factors表1 6組學(xué)習(xí)因子4種不同算法計算的評價函數(shù)值

Fig.5 Flow chart of the proposed algorithm圖5 算法流程圖

The initial structure of the off-axis six-mirror reflective optical system is solved by the natural selection PSO with shrinkage factor.The initial structure diagram is shown in Figure 6.

Fig.6 Schematic diagram of initial structure for the off-axis six-mirror reflective optical system圖6 離軸六反光學(xué)系統(tǒng)初始結(jié)構(gòu)示意圖

4 Optimal design

We optimize the design of the above initial structure.At the beginning of the optimization, a high-order aspheric coefficient is added to the quadric surface shape to obtain high imaging quality.The constraints in the solution of the initial structure (mirror spacing, no obstruction, incident angle,mirror aperture, telecentricity, asphericity, etc.) are not destroyed, so that the optimized result has a small deviation from the initial structure, and the disturbance is controlled during the optimization process (If it is too large, the extreme value will be skipped, if it is too small, the result will fall into a local minimum).

Based on the above-mentioned initial structure solution and optimization principles, the design of an off-axis aspheric six-mirror reflective optical system with minimal aberration is realized.Figure 7 shows the design results of the off-axis six-mirror reflective optical system, and the specific parameters are shown in Table 2.The total working distance is 1371 mm, the rear working distance is 38 mm, and the image-side field of view is an arc field of view of 26 mm×2 mm.The maximum asphericity of the component is 60 μm, and the maximum distortion of the full field of view of the off-axis sixmirror reflective optical system is 1 nm, and the distortion distribution of the full field of view is shown in Figure 8 (Color online).The comprehensive wave aberration of the full field of view is 0.011λ RMS,its full field of view wave aberration distribution is shown in Figure 9 (Color online).According to the above design results, the design satisfies the constrained control while achieving extremely low aberration, making the EUV lithography objective lens system more engineering realizable.

Fig.7 Schematic diagram of the optimized structure for the off-axis six-mirror reflective optical system圖7 極小像差離軸六反光學(xué)系統(tǒng)優(yōu)化后結(jié)構(gòu)示意圖

Fig.8 Distortion on full image field for the off-axis sixmirror reflective optical system圖8 離軸六反光學(xué)系統(tǒng)全視場畸變分布圖

Fig.9 Wavefront error RMS on full image for the off-axis six-mirror reflective optical system圖9 離軸六反光學(xué)系統(tǒng)全視場波像差分布圖

Tab.2 Specifications of the off-axis six-mirror reflective optical system表2 離軸六反光學(xué)系統(tǒng)設(shè)計主要參數(shù)指標(biāo)

5 Conclusion

In order to solve the problem of aberration balance and multi-constraint control in the process of initial structure construction, a mathematical model of the initial structure calculation of off-axis sixmirror reflective is established based on a grouping design method combining spatial ray tracing and aberration correction.In this paper, the solution accuracy and design efficiency are improved by using the natural selection PSO with shrinkage factor, and a design fundation for off-axis six-mirror reflective optical system with minimal aberration optimization potential is provided.Using this method, the design of an off-axis six-mirror reflective optical system with minimal aberration is realized, and its comprehensive wave aberration of the full field of view is 0.011λ RMS.

The method proposed in this paper can be extended to off-axis multi-reflective optical systems,which can be used to design the off-axis multi-reflective initial structures with aberration balance and multi-constraint control capabilities, providing a design fundation for off-axis multi-mirror reflective optical systems with the potential for optimization of extremely low aberration.

——中文對照版——

1 引言

與折射光學(xué)系統(tǒng)相比，反射光學(xué)系統(tǒng)具有以下優(yōu)勢：反射光學(xué)系統(tǒng)不產(chǎn)生色差；光路可折疊，便于縮短筒長使結(jié)構(gòu)緊湊；對溫度和氣壓的變化相對不敏感；通過離軸偏心傾斜有效解決遮攔約束，有利于增大視場，提升像質(zhì)，因此離軸反射式光學(xué)系統(tǒng)的設(shè)計受到了廣泛關(guān)注，常應(yīng)用于空間相機(jī)、望遠(yuǎn)鏡以及紅外/紫外等領(lǐng)域[1-5]。近年來，離軸六反極紫外光刻投影物鏡是離軸多反光學(xué)系統(tǒng)的一種典型應(yīng)用。極紫外光刻投影物鏡成像要求極高，需要實現(xiàn)超衍射極限分辨率，波像差優(yōu)于1/50λ[6-9]。實現(xiàn)超衍射極限極小像差光學(xué)系統(tǒng)設(shè)計，非常依賴于像差平衡，需要解決多約束、多目標(biāo)與少自由度矛盾，這給光學(xué)設(shè)計人員帶來極大挑戰(zhàn)。目前光學(xué)設(shè)計軟件主要解決光學(xué)系統(tǒng)優(yōu)化問題，采用阻尼最小二乘法的局部優(yōu)化算法，計算多維變量空間誤差函數(shù)的最小值，最終優(yōu)化結(jié)果大多數(shù)情況下是離初始結(jié)構(gòu)較近的局部最優(yōu)解，有著很大的局限性[10]，其全局優(yōu)化功能在初始結(jié)構(gòu)不佳的情況下，難以同時實現(xiàn)像差平衡與約束控制，利用光學(xué)設(shè)計軟件優(yōu)化設(shè)計嚴(yán)重依賴于初始結(jié)構(gòu)的選擇，初始結(jié)構(gòu)構(gòu)建是光學(xué)系統(tǒng)設(shè)計的關(guān)鍵[11]。尤其是對于像差平衡更為敏感的極小像差光學(xué)系統(tǒng)，為了在優(yōu)化過程中平衡約束與像差使得光學(xué)系統(tǒng)擾動過大，難以實現(xiàn)極小像差光學(xué)系統(tǒng)，滿足像差平衡和多約束控制的初始結(jié)構(gòu)構(gòu)建是關(guān)鍵問題。

目前，常見的反射式光學(xué)系統(tǒng)初始結(jié)構(gòu)構(gòu)造方法有近軸搜索法和Y，Y-bar方法以及分組設(shè)計方法。其中，近軸搜索法是由M.F.Bal提出的，該方法采用近軸模型，對光學(xué)系統(tǒng)的一階像差進(jìn)行窮舉，能夠滿足約束要求的數(shù)量太少，極大影響效率[12-13]。Scott A.Lerner等人將Y，Y-bar方法應(yīng)用到光學(xué)系統(tǒng)結(jié)構(gòu)求解上，該方法是利用邊緣光線和主光線在光學(xué)元件表面的高度求解曲率半徑和鏡間距來構(gòu)建初始結(jié)構(gòu)。每一種結(jié)構(gòu)的光學(xué)表面的主光線和邊緣光線高度并不容易確定，不具備普適性。對于離軸反射光學(xué)系統(tǒng)元件數(shù)目大時，上述方法的計算量將大幅增加，影響設(shè)計效率[14-15]。分組設(shè)計方法由Hudyma應(yīng)用于離軸六反光學(xué)系統(tǒng)設(shè)計，他將光學(xué)系統(tǒng)分為兩組，但并沒有給出具體設(shè)計方法[16]。北京理工大學(xué)李艷秋教授課題組提出真實光線追跡的分組設(shè)計方法應(yīng)用于離軸六反及更多元件數(shù)的反射系統(tǒng)，該方法將離軸反射光學(xué)系統(tǒng)分成3組，基于約束控制通過真實光線追跡計算確定每個鏡組的結(jié)構(gòu)參數(shù)，最后將3個鏡組結(jié)構(gòu)參數(shù)進(jìn)行拼接[17-18]。上述方法尚未解決以下問題：（1）優(yōu)化過程可能會產(chǎn)生較大的擾動，結(jié)構(gòu)偏離初始結(jié)構(gòu)過大，使得約束難以控制；（2）光學(xué)設(shè)計軟件優(yōu)化中像差平衡過程低階像差與高階像差可能出現(xiàn)較大的殘余量，使得光學(xué)系統(tǒng)設(shè)計殘差大，不利于實現(xiàn)極小像差系統(tǒng)。

為了解決初始結(jié)構(gòu)構(gòu)建過程中像差平衡和多約束控制問題，本文基于空間光線追跡與像差矯正相結(jié)合的分組設(shè)計方法，建立了離軸多反的初始結(jié)構(gòu)計算的數(shù)學(xué)模型[19]，采用粒子群算法對該高維非線性數(shù)學(xué)模型進(jìn)行計算求解，橫向?qū)Ρ攘藥追N常見的混合粒子群算法，提出帶收縮因子的自然選擇的粒子群算法提高了計算精度，提升了設(shè)計效率，為離軸多反光學(xué)系統(tǒng)設(shè)計具備極小像差優(yōu)化潛力提供設(shè)計起點。本文將上述方法應(yīng)用于極紫外光刻投影物鏡光學(xué)系統(tǒng)中，實現(xiàn)離軸六反極小像差光學(xué)系統(tǒng)設(shè)計，其主要設(shè)計流程為：（1）將光學(xué)系統(tǒng)分為兩組，即物方鏡組與像方鏡組。（2）基于像差理論與空間光線追跡理論，將光學(xué)系統(tǒng)的結(jié)構(gòu)參量參數(shù)化，針對前后鏡組，分別建立相應(yīng)的用于參數(shù)計算的數(shù)學(xué)模型，將前后鏡組拼接，獲取初始結(jié)構(gòu)計算數(shù)學(xué)模型。（3）橫向?qū)Ρ葌鹘y(tǒng)的粒子群算法，提出了帶收縮因子的自然選擇的粒子群算法計算求解初始結(jié)構(gòu)參數(shù)。（4）優(yōu)化初始結(jié)構(gòu)。利用漸進(jìn)式優(yōu)化方法，避免優(yōu)化過程系統(tǒng)結(jié)構(gòu)變化過大，偏離初始結(jié)構(gòu)。最終實現(xiàn)了具備工程可實現(xiàn)的離軸六反光學(xué)系統(tǒng)，其綜合系統(tǒng)波像差接近1 /80λRMS。

2 初始結(jié)構(gòu)計算的數(shù)學(xué)模型

2.1 分組設(shè)計方法

如圖1所示，將離軸六反極紫外光刻物鏡分成兩組，即物方鏡組和像方鏡組，分別計算第一近軸光線和第二近軸光線的光路。光學(xué)系統(tǒng)參數(shù)主要包括系統(tǒng)的微縮倍率β，物方視場中心高度y，像方視場中心高度yim, 物方數(shù)值孔徑NAO，像方數(shù)值孔徑NA。則可以得出yim=y?β，NAO=NA?β。物方鏡組與像方鏡組在中間像處拼接，拼接時應(yīng)保證物像匹配，光瞳匹配以及倍率匹配?；谏鲜鲈瓌t，將光學(xué)系統(tǒng)的物方鏡組和像方鏡組的結(jié)構(gòu)參量參數(shù)化，引入空間光線追跡，將光學(xué)系統(tǒng)中的遮攔、鏡間距、像方遠(yuǎn)心、口徑以及入射角等約束參量化，建立極小像差離軸六反極紫外光刻物鏡 初始結(jié)構(gòu)參數(shù)計算的數(shù)學(xué)模型。

2.2 物方鏡組（Group 1）像差分析

如圖2（彩圖見期刊電子版）所示，物方視場中心高度為y，物方孔徑角為u1， 光闌位于 M2處。光線從物方視場出發(fā)，經(jīng) M1、 M2、M3、M4到達(dá)中間像點。其中d1、d2、d3、d4分 別表示M1到 M2的鏡間距、 M2到 M3的 鏡間距、 M3到M4的鏡間距、M4到 中間像點IM的距離；h1、h2、h3、h4分別表示第一輔助光線在 M1、 M2、 M3、 M4上 的高度;r1、r2、r3、r4和k1、k2、k3、k4分 別 表 示 M1、 M2、 M3、M4的 曲 率 半 徑 和 二 次 曲 面 系 數(shù)；l1、l2、l3、l4、分別為M1、M2、M3、M4的物距和像距。

三階單色像差主要包括：球差、慧差、像散、場曲、畸變，分別用SⅠ、SⅡ、SⅢ、SⅣ、SⅤ表示，其公式為[4,20]：

其中，

基于近軸近似條件，引入?yún)?shù)：

對于該反射系統(tǒng)，有

對近軸主光線和近軸邊緣光線進(jìn)行光線追跡，得出

Group1中出瞳位置與M4的距離為

將式（2）～式（4）帶入式（1）中即可計算出前鏡組G1像差系數(shù)。

2.3 像方鏡組（Group 2）的像差分析

如圖3所示，像方鏡組物方視場中心高度為y5， 像方視場中心高度為yim, 物方孔徑角為u5，像方遠(yuǎn)心。光線從中間像點出發(fā)，經(jīng) M5、 M6到達(dá)像點。其中d5、d6分 別表示 M5到M6的鏡間距， M6到像面的距離；h5、h6分 別表示第一輔助光線在M5、M6上的高度；r5、r6和k5、k6分 別表示 M5、 M6的曲率半徑和二次曲面系數(shù)；別 為 M5、M6的物距和像距。

同理，基于近軸近似條件，引入?yún)?shù)：

對于該反射系統(tǒng)，有

對近軸主光線和邊緣光線進(jìn)行光線追跡，得出

Group2中入瞳位置與M5的距離為

由于光瞳匹配條件，G1的出瞳與G2的入瞳重合，可以算出Group1中入瞳位置與M5的距離為

將式（7）～式（8）帶入式（1）中即可計算出后鏡組G2像差系數(shù)。

2.4 鏡組拼接與數(shù)學(xué)模型建立

對物方鏡組與像方鏡組進(jìn)行拼接，拼接原則主要包括物像匹配、倍率匹配以及光瞳匹配?？梢缘贸?/p>

共軸光學(xué)系統(tǒng)的像差滿足線性疊加，即整個光學(xué)系統(tǒng)的像差是光學(xué)系統(tǒng)中各個元件像差貢獻(xiàn)之和。因此離軸六反光學(xué)系統(tǒng)的像差系數(shù)是Group1與Group 2像差系數(shù)之和，利用兩個鏡組求解的像差系數(shù)即可得到光學(xué)系統(tǒng)的像差系數(shù)。

通過上述像差理論，將極紫外光刻物鏡光學(xué)系統(tǒng)的三階像差系數(shù)和結(jié)構(gòu)參量參數(shù)化。同時，為了滿足光學(xué)系統(tǒng)的一些特殊要求，引入空間光學(xué)追跡方法，將光學(xué)系統(tǒng)的特殊要求作為約束將其參數(shù)化，在初始結(jié)構(gòu)構(gòu)建過程中能夠有效對遮攔、鏡間距、口徑、像方遠(yuǎn)心以及入射角等約束進(jìn)行控制?；谏鲜黾s束條件，由于目標(biāo)是三階像差系數(shù)盡可能小，因此其評價函數(shù)可以寫為：

其中constraints表示上述約束。評價函數(shù)F反映了光學(xué)系統(tǒng)初級像差大小以及約束控制能力，其值越小，則初始結(jié)構(gòu)的初階像差越小，故該結(jié)構(gòu)的像差控制就越好，實現(xiàn)高成像質(zhì)量的潛力就越大。本文通過像差理論以及約束控制，建立了極小像差離軸六反參數(shù)設(shè)計數(shù)學(xué)模型，將物理模型轉(zhuǎn)化為數(shù)學(xué)模型，即為高維非線性參數(shù)方程求解問題。

3 粒子群算法計算求解離軸六反初始結(jié)構(gòu)數(shù)學(xué)模型

目前，光學(xué)系統(tǒng)結(jié)構(gòu)初始結(jié)構(gòu)計算算法主要是遺傳算法[11,21-22]，遺傳算法以生物進(jìn)化為原型，具有良好的全局搜索能力，并因其內(nèi)在并行性，可以同時進(jìn)行多個個體的比較，具有可擴(kuò)展性，易于與其他算法結(jié)合。但是遺傳算法的局部搜索能力較差，搜索效率低，并且對于初始種群的選擇有一定依賴性。粒子群算法（Partical Swarm Optimization，PSO）由Eberhart和Kennedy于1995年提出[23]，該算法的優(yōu)勢在于算法簡潔、通用、易于實現(xiàn)、而且計算快速，且不需要梯度信息，是解決非線性優(yōu)化問題、組合優(yōu)化問題和混合整數(shù)非線性優(yōu)化問題的有效工具，目前廣泛應(yīng)用于函數(shù)優(yōu)化、神經(jīng)網(wǎng)絡(luò)訓(xùn)練、實踐工程等應(yīng)用領(lǐng)域?；A(chǔ)PSO算法也存在一定缺點，在處理高維復(fù)雜問題時，算法易于陷入局部值；問題規(guī)模較大時，算法收斂速度較慢，精度有限。為了克服基礎(chǔ)PSO算法的不足，相關(guān)學(xué)者提出很多改進(jìn)方法，主要包括：參數(shù)改進(jìn)與混合算法。參數(shù)改進(jìn)主要通過引入一些新的參數(shù)，在改進(jìn)算法的同時會在一定程度上增加算法的復(fù)雜性，包括權(quán)重改進(jìn)的PSO算法、帶收縮因子的PSO算法、變學(xué)習(xí)因子的PSO算法等。混合算法是PSO算法改進(jìn)的熱點方向，在PSO算法中結(jié)合其他算法，以提高PSO算法的全局搜索能力和搜索精度，包括基于自然選擇的PSO算法、基于雜交的PSO算法、基于模擬退火的PSO算法等[24-25]。

由于學(xué)習(xí)因子c1和c2決定了粒子本身的經(jīng)驗信息和其他粒子的經(jīng)驗信息對粒子運行軌跡的影響，反映了粒子群之間的信息交流。當(dāng)c1較大時，會使粒子過多的在局部范圍徘徊，當(dāng)c2較大時，會使粒子過早收斂到局部最小值。為了有效控制粒子的飛行速度以實現(xiàn)全局探測與局部開采兩者間的有效平衡，Clerc構(gòu)造了引入收縮因子的PSO算法，確保PSO算法的收斂性，并可取消對速度的邊界限制，能更有效地控制約束粒子的飛行速度，增強(qiáng)算法局部搜索能力。

本文在其基礎(chǔ)上，提出了帶收縮因子的自然選擇的粒子群算法用于離軸六反光學(xué)系統(tǒng)初始結(jié)構(gòu)數(shù)學(xué)模型求解，通過橫向?qū)Ρ壬鲜?種混合粒子群算法，并且將模擬退火PSO算法與雜交PSO算法引入收縮因子，同時引入帶慣性權(quán)重的自然選擇PSO算法作為對比。由于收縮因子的引入，其條件為c1+c2>4，Clerc的帶收縮因子方法中設(shè)c1+c2=4.1，本文圖4給出6組不同學(xué)習(xí)因子下4種算法的收斂曲線，其中1～5組中c1+c2=4.1，第6組c1+c2=4.3，粒子數(shù)目N=1000，最大迭代次數(shù)為100，退火常數(shù)為0.42，雜交概率為0.9，雜交池的大小比例為0.2，慣性權(quán)重因子為0.7。表1分別給出了6組學(xué)習(xí)因子4種算法計算的最小評價函數(shù)值，從圖4（彩圖見期刊電子版）以及表1可以看出，針對以上6組不同學(xué)習(xí)因子，帶收縮因子的自然選擇的粒子群算法提升了離軸六反光學(xué)系統(tǒng)初始結(jié)構(gòu)數(shù)學(xué)模型的求解計算精度，對于收斂速度也有很大程度的提升，為離軸六反光學(xué)系統(tǒng)設(shè)計具備極小像差優(yōu)化潛力提供設(shè)計起點。

帶收縮因子的自然選擇的粒子群算法的主要流程如圖5所示。Step 1：建立離軸六反光學(xué)系統(tǒng)初始結(jié)構(gòu)計算數(shù)學(xué)模型，并隨機(jī)初始化種群中的粒子位置和速度。Step 2：評價每個粒子的適應(yīng)度，將當(dāng)前各粒子的位置和適應(yīng)度值存于各粒子的個體極值中，計算個體極值和全局最優(yōu)值，其中個體極值為每個粒子找到的最優(yōu)解，從這些最優(yōu)解找到一個全局值，叫做本次全局最優(yōu)解。Step 3：基于收縮因子，更新每個粒子的速度和位置。Step 4：根據(jù)適應(yīng)度值，更新各粒子間的個體極值與全局最優(yōu)解。Step 5：將整個粒子群按適應(yīng)度值排序，用群體中最好的一半的粒子的速度和位置替換最差的一半的位置和速度，保持個體極值和全局最優(yōu)解不變。Step 6：若滿足終止條件（誤差足夠好或到達(dá)最大循環(huán)次數(shù)）退出，否則返回Step 3。Step 7：根據(jù)算法計算結(jié)果求解極小像差離軸六反初始結(jié)構(gòu)。

通過帶收縮因子的自然選擇粒子群算法實現(xiàn)了離軸六反光學(xué)系統(tǒng)初始結(jié)構(gòu)求解，其初始結(jié)構(gòu)示意圖如圖6所示。

4 優(yōu)化設(shè)計

針對以上初始結(jié)構(gòu)進(jìn)行優(yōu)化設(shè)計，首先在表面面型上對二次曲面面形加入高階非球面系數(shù)以獲得高成像質(zhì)量，同時在優(yōu)化過程中不破壞初始結(jié)構(gòu)求解中的約束(鏡間距、無遮攔、入射角、鏡子口徑、遠(yuǎn)心、非球面度等)，使得優(yōu)化后的結(jié)果與初始結(jié)構(gòu)偏差較小，同時在優(yōu)化過程中控制擾動量(過大會跳過極值，過小會陷入局部極小值)。

基于上述初始結(jié)構(gòu)求解和優(yōu)化原則，實現(xiàn)了極小像差離軸非球面六反光學(xué)系統(tǒng)設(shè)計，如圖7所示。具體參數(shù)指標(biāo)如表2所示。其總工作距為1371 mm，后工作距為38 mm，像方視場為26 mm×2 mm的弧形視場。元件最大非球面度為60 μm，離軸六反光學(xué)系統(tǒng)全視場的最大畸變?yōu)?nm，其全視場畸變分布如圖8（彩圖見期刊電子版）所示。全視場綜合波像差為0.011λRMS，其全視場波像差分布如圖9（彩圖見期刊電子版）所示。根據(jù)上述設(shè)計結(jié)果可知，該設(shè)計在實現(xiàn)極小像差的同時滿足約束控制，使得極紫外光刻物鏡系統(tǒng)更具備工程可實現(xiàn)性。

5 結(jié)論

為了解決初始結(jié)構(gòu)構(gòu)建過程中像差平衡和多約束控制問題，本文基于空間光線追跡與像差矯正相結(jié)合的分組設(shè)計方法，建立了離軸六反初始結(jié)構(gòu)計算數(shù)學(xué)模型。提出使用帶收縮因子的自然選擇的粒子群算法以提高求解精度，提升了設(shè)計效率，為離軸六反光學(xué)系統(tǒng)設(shè)計具備極小像差優(yōu)化潛力提供設(shè)計起點。利用該方法實現(xiàn)了極小像差離軸六反光學(xué)系統(tǒng)的設(shè)計，其全視場綜合波像差為0.011λRMS。

離軸多反光學(xué)系統(tǒng)，依然可以利用該方法進(jìn)行初始結(jié)構(gòu)設(shè)計，實現(xiàn)具備像差平衡和多約束控制能力的離軸多反初始結(jié)構(gòu)，為離軸多反光學(xué)系統(tǒng)提供具備極小像差優(yōu)化潛力的設(shè)計起點。