The Inverse Optimization of Lithographic Source and Mask via GA-APSO Hybrid Algorithm

: Source mask optimization (SMO) is an effective method for improving the image quality of high-node lithography. Reasonable algorithm optimization is the critical issue in SMO. A GA-APSO hybrid algorithm, combining genetic algorithm (GA) and adaptive particle swarm optimization (APSO), was proposed to inversely obtain the global optimal distribution of the pixelated source and mask in the lithographic imaging process. The computational efﬁciency was improved by combining the GA and PSO algorithms. Additionally, the global search and local search were balanced through adaptive strategies, leading to a closer result to the global optimal solution. To verify the performance of GA-APSO, simple symmetric patterns and complex patterns were optimized and compared with GA and APSO, respectively. The results show that the pattern errors (PEs) of the resist image optimized by GA-APSO were reduced by 40.13–52.94% and 10.28–33.31% compared to GA and APSO, respectively. The time cost of GA-APSO was reduced by 75.91–87.00% and 48.43–58.66% compared to GA and APSO, respectively. Moreover, repeated calculation showed that the GA-APSO results were relatively stable. The results demonstrate the superior performance of GA-APSO in efﬁciency, accuracy, and repeatability for source and mask optimization.


Introduction
Lithography is the primary method of manufacturing micromachining. Higher lithography resolution is required according to Moore's Law. Based on the Rayleigh Criterion, shortening wavelength, increasing numerical aperture (NA), and enhancing technology factor are three methods of improving lithography resolution. However, some unfavorable factors, such as diffraction effect and heat accumulation, can cause image quality reduction during lithography. High-frequency information, such as corners or sharp lines in the mask, cannot be effectively transmitted to wafer. These negative factors become more significant as lithography resolution improves.
Resolution-enhancement technology (RET) [1] is introduced to enhance image quality, eliminate these negative factors, and enhance technology factors. Traditional RET includes off-axis illumination (OAI), phase-shifting mask (PSM), and optical proximity correction (OPC). These examples of traditional RET have a low degree of freedom due to optimizing mask or light source modules individually. Source mask optimization (SMO) is proposed, which increases the optimal degree of freedom by optimizing the source and mask at the same time [2].
The input of SMO includes the parameters of the lithography process, such as numerical aperture (NA) and photoresist [3]. Based on the input data, SMO generates an initial light source and mask. Then, the light source and mask are optimized by an intelligent algorithm on an imaging model until the criterion functions are satisfied. Both imaging quality and machinability can be involved in the criterion functions [4].

Partially Coherent Imaging Model
In this section, the lithography imaging process of the partially coherent imaging model is explained. Partially coherent illumination is widely used in lithography to improve resolution. A typical immersion lithography system is shown in Figure 1. The ray emitted by the illumination source forms the Kohler illumination through a condenser lens. Then, the ray passes through the mask and produces a diffracted ray carrying pattern information. Finally, the ray through the objective lenses images the feature on the mask onto the resist. Spatial intensity distribution on the resist is formed by superimposing Photonics 2023, 10, 638 3 of 15 the images of the source at different positions. The images are obtained by moving pupil position. Then, the partial coherence imaging model can be expressed by [21]: Photonics 2023, 10, x FOR PEER REVIEW 3 of 15 images of the source at different positions. The images are obtained by moving pupil position. Then, the partial coherence imaging model can be expressed by [21]: In Equation (1), , represents the intensity distribution of aerial images in optical lithography.
, and , are the frequency-domain coordinates of the pupil and mask, respectively.
, is an extended illumination source. , is the mask frequency spectrum. + , + represent the pupil function characterizing light propagation from a point on the object plane to an image point. To simulate the imaging model, Equation (1) can be approximately expressed as: in which , represents the efficient pixels of the source, • is an inverse Fourier transform operation, and , indicates that the pupil is shifted with the position change of the source point , . In the simulation, the photoresist effect was approximated using a sigmoid function. The resist image * , of the aerial image , on the wafer surface can be displayed after development [22]. * = * , = , where * is the layout distribution of the resist image, • represents a threshold function, is the threshold of the sigmoid function, and controls the steepness of intensity transition.  In Equation (1), I(x, y) represents the intensity distribution of aerial images in optical lithography. ( f , g) and ( f , g ) are the frequency-domain coordinates of the pupil and mask, respectively. S( f , g) is an extended illumination source. M( f , g ) is the mask frequency spectrum. H( f + f , g + g ) represent the pupil function characterizing light propagation from a point on the object plane to an image point. To simulate the imaging model, Equation (1) can be approximately expressed as:

Source and Mask Optimization Using GA-APSO Algorithm
. . , f n , g = g 1 , . . . , g n (2) in which S(f, g) represents the efficient pixels of the source, {·} is an inverse Fourier transform operation, and H(f, g) indicates that the pupil is shifted with the position change of the source point ( f n , g n ). In the simulation, the photoresist effect was approximated using a sigmoid function. The resist image I * (x, y) of the aerial image I(x, y) on the wafer surface can be displayed after development [22].
where I * is the layout distribution of the resist image, sig{·} represents a threshold function, t r is the threshold of the sigmoid function, and α controls the steepness of intensity transition. Figure 2 represents the coding regulations of the source and mask. To reduce computational complexity, half of the symmetrical parts are encoded. In calculation, the complete source and mask are restored symmetrically. For the four-fold symmetrical source and mask shown in the figure, only the first-quadrant variables are encoded.

Source and Mask Optimization Using GA-APSO Algorithm
SMO is a multiparameter with single objective optimization. For a certain system, the imaging result can be calculated using the partially coherent imaging model. The distance between the real imaging result and the ideal image can be evaluated by pattern errors (Pes), which are the L1 norm of the subtraction between I I and I * . The object of SMO is to minimize the Pes. Every pixel of source and mask is limited between 0 and 1 in simulation. Pes are defined as [20]:  SMO is a multiparameter with single objective optimization. For a certain system, the imaging result can be calculated using the partially coherent imaging model. The distance between the real imaging result and the ideal image can be evaluated by pattern errors (PEs), which are the L1 norm of the subtraction between and * . The object of SMO is to minimize the PEs. Every pixel of source and mask is limited between 0 and 1 in simulation. PEs are defined as [20]: where , is the intensity distribution of the ideal image and * , is the intensity distribution of the real image.
The source and mask are optimized by the GA-APSO algorithm. APSO is a bionic algorithm simulating the bird foraging process. In APSO, each particle updates its swarming speed based on memory and current global optimal position. GA is a method to search for the optimal solution by simulating the natural evolution process. In GAs, populations iterate through genetics and mutation. APSO has ascendency in convergence, but can easily fall into local optimum. However, GA is advanced in global optimization, while it has poor convergence. The optimization method in this paper combined GA and APSO to enhance optimization efficiency and retain global optimal search capability.
The steps of the GA-APSO algorithm are as follows: (1) Initialization of the population The parameters are initialized. Additionally, an initial population is randomly generated in the feasible region and the initial fitness is calculated.
(2) Population update by APSO According to the fitness, the population is updated by APSO. Velocity of each individual is as follows: where: is the constriction coefficient; is the inertia weight factor; is the velocity of j-th individual in i-th iteration; and are the individual learning factor and social learning factor, respectively; and are random numbers which range from zero to one; and and are the personal best for the j-th individual, and the global best in the population, respectively. Velocity is limited in a certain range.
Individual learning factor and social learning factor reflect the acceleration of individual learning and social learning, respectively. There are different requirements for and at different stages of the iteration. Hence, adaptive strategies of and are introduced during iterations. During the early iteration, the individual cognitive acceleration is increased to enlarge the search zone of individual particles scattered in the whole search space. During the later period of iteration, is decreased and is The source and mask are optimized by the GA-APSO algorithm. APSO is a bionic algorithm simulating the bird foraging process. In APSO, each particle updates its swarming speed based on memory and current global optimal position. GA is a method to search for the optimal solution by simulating the natural evolution process. In Gas, populations iterate through genetics and mutation. APSO has ascendency in convergence, but can easily fall into local optimum. However, GA is advanced in global optimization, while it has poor convergence. The optimization method in this paper combined GA and APSO to enhance optimization efficiency and retain global optimal search capability.
The steps of the GA-APSO algorithm are as follows: (1) Initialization of the population The parameters are initialized. Additionally, an initial population is randomly generated in the feasible region and the initial fitness is calculated.
(2) Population update by APSO According to the fitness, the population is updated by APSO. Velocity of each individual is as follows: where: f c is the constriction coefficient; w i j is the inertia weight factor; v i j is the velocity of j-th individual in i-th iteration; c 1 and c 2 are the individual learning factor and social learning factor, respectively; r 1 and r 2 are random numbers which range from zero to one; and p i gbest and p i zbest are the personal best for the j-th individual, and the global best in the population, respectively. Velocity is limited in a certain range.
Individual learning factor c 1 and social learning factor c 2 reflect the acceleration of individual learning and social learning, respectively. There are different requirements for c 1 and c 2 at different stages of the iteration. Hence, adaptive strategies of c 1 and c 2 are introduced during iterations. During the early iteration, the individual cognitive acceleration c 1 is increased to enlarge the search zone of individual particles scattered in the whole search space. During the later period of iteration, c 1 is decreased and c 2 is increased properly to make individual particles jump out of the local optimum and improve convergence performance. The adaptive variation rule is as follows: where c 1 i and c 2 i are the parameters for the i-th step.
Additionally, the constriction coefficient [17] f c is introduced to speed up the convergence, and the constriction coefficient f c is computed as: To balance global optimization ability in the early iteration and strong local optimization ability in the late iteration, inertia weight factor w i j requires a relatively large value in initial iteration and decreases in late iteration. Inertia weight factor w i j is controlled by a hyperbolic tangent function which is shown in Figure 3: where: w max and w min are the maximum and minima inertia weight factor; and i max is the maximum iteration number.
increased properly to make individual particles jump out of the local optimum and improve convergence performance. The adaptive variation rule is as follows: where and are the parameters for the i-th step. Additionally, the constriction coefficient [17] is introduced to speed up the convergence, and the constriction coefficient is computed as: where = + .
To balance global optimization ability in the early iteration and strong local optimization ability in the late iteration, inertia weight factor requires a relatively large value in initial iteration and decreases in late iteration. Inertia weight factor is controlled by a hyperbolic tangent function which is shown in Figure 3: where: and are the maximum and minima inertia weight factor; and is the maximum iteration number.    First, whether the individual mutates is determined according to mutation probability p m . Mutation chromosome position is selected randomly. The selected genes change randomly in the feasible region.
Then, the child population is generated. Additionally, the fitness of the child population is calculated and the personal best for each individual and the global best in the population are updated.
Steps (2)~(4) are repeated until the maximum iteration number i max occurs or fitness changes between the set n number of iterations are less than the tolerance tol 1 .
The flow chart of source and mask optimization is as shown in Figure 5. Source and mask optimization steps are as follows: (1) Initialization Input all the parameters of the partially coherent imaging model, including wave length λ, lighting model, mask size, and sampling points of light source and mask. Additionally, input all the parameters which the optimization algorithm needs, including maximum iteration numbers i smax and i mmax , convergence tolerance tol s1 and tol m1 , population size, maximum and minima inertia weight factor maximum w max and w min , maximum and minimal crossover probability p cmax and p cmin , and maximum and minimal mutation probability p mmax and p mmin .

Simulation Parameters
In this paper, the 193 nm immersion lithography model with a 45 nm node was conducted. NA is 1.35. Source optimization was initialized by annular illumination, as shown in Figure 6. The partial coherence factors and were, respectively, set to 0.65 and 0.95. The source was pixeled to a 42 42 matrix. Mask 1 was sampled as a 70 70 matrix. Mask 2 and Mask 3 were sampled as a 250 250 matrix. Three masks with different pattern complexities, as shown in Figure 7, were simulated in this study. Mask 1 was asymmetric and only encoded the first quadrant of the pattern. Mask 2 and Mask 3 were complex asymmetric patterns and encoded the whole patterns. The photoresist parame- (2) Source optimization Initialize the source population randomly selected in the feasible region. Then, optimize the source using the GA-APSO algorithm. The imaging results with source population are calculated according to the partially coherent imaging model, and fitness is evaluated by pattern errors (Pes). Update the source population until the iteration number is greater than the maximum iteration number i smax or the fitness change between the set n number of iterations is less than the tolerance tol s1 . Initialize the mask population randomly selected in the feasible region. Then, optimize the mask using the GA-APSO algorithm. The imaging results with source population are calculated according to the partially coherent imaging model and fitness is evaluated by pattern errors (Pes). Update mask population until the iteration number is greater than the maximum iteration number i mmax or the fitness change between the set n number of iterations is less than the tolerance tol m1 . Then, the optimization of the mask and source is finished and outputs the best mask and source.

Simulation Parameters
In this paper, the 193 nm immersion lithography model with a 45 nm node was conducted. NA is 1.35. Source optimization was initialized by annular illumination, as shown in Figure 6. The partial coherence factors σ in and σ out were, respectively, set to 0.65 and 0.95. The source was pixeled to a 42 × 42 matrix. Mask 1 was sampled as a 70 × 70 matrix. Mask 2 and Mask 3 were sampled as a 250 × 250 matrix. Three masks with different pattern complexities, as shown in Figure 7, were simulated in this study. Mask 1 was asymmetric and only encoded the first quadrant of the pattern. Mask 2 and Mask 3 were complex asymmetric patterns and encoded the whole patterns. The photoresist parameters t r and α were 0.28 and 85, respectively.

Simulation Parameters
In this paper, the 193 nm immersion lithography model with a 45 nm node was con ducted. NA is 1.35. Source optimization was initialized by annular illumination, as shown in Figure 6. The partial coherence factors and were, respectively, set to 0.65 and 0.95. The source was pixeled to a 42 42 matrix. Mask 1 was sampled as a 70 70 ma trix. Mask 2 and Mask 3 were sampled as a 250 250 matrix. Three masks with differen pattern complexities, as shown in Figure 7, were simulated in this study. Mask 1 wa asymmetric and only encoded the first quadrant of the pattern. Mask 2 and Mask 3 wer complex asymmetric patterns and encoded the whole patterns. The photoresist parame ters and were 0.28 and 85, respectively.  Source and mask optimization (SMO) using the GA-APSO algorithm was comp with APSO and GA. In the simulation, the parameters of the algorithm were set as sh in Table 1. The iteration steps were set as 500 for source optimization and 1000 for m optimization. Additionally, the population size was set as 50 in this simulation.

Parameter
Value Maximum and minimum individual learning factor 1.5/2 Source and mask optimization (SMO) using the GA-APSO algorithm was compared with APSO and GA. In the simulation, the parameters of the algorithm were set as shown in Table 1. The iteration steps were set as 500 for source optimization and 1000 for mask optimization. Additionally, the population size was set as 50 in this simulation.

Parameter Value
Maximum and minimum individual learning factor c 1 1.5/2 Maximum and minimum social learning factor c 2 1.5/2 Maximum and minimum inertia weight factor w 0.1/1 Maximum and minimum velocity v −1/1 Crossover probability p c 0.8 Mutation probability p m 0.2

Simulation Results
Initial and optimized resist images are represented in Figure 8.  Source and mask optimization performance of the GA-APSO algorithm was compared with APSO and GA. Sources after optimization are shown in Figure 9. Masks after optimization are shown in Figure 10. After SMO, the source for Mask 1, which consists of only one direction line, was approximate to dipole illumination. Additionally, for Mask 2 and Mask 3 with irregular patterns, sources after SMO were approximate to quadra illumination. The distribution of horizontal and vertical lines affects the final illumination form. The optimized source for patterns with unidirectional lines such as Mask 1 tend to dipole illumination. The angle of dipole illumination is associated with the direction of the lines. The optimized source for patterns with both horizontal and vertical lines such as Mask 2 and Mask 3 tend quadra illumination. Moreover, affected by the different distribution of horizontal and vertical lines, one direction of the quadra illumination is stronger.   Masks after optimization are presented in Figure 10. The binary error R BE and total variation R TV were introduced to evaluate the manufacturability of the mask and the complexity of the mask pattern, which are defined as follows [23]: (10) in which M is the mask matrix, · 1 is the L1 norm operator, and the lower value of binary error R BE and total variation R TV represent the better manufacturability. Masks after optimization by the presented GA-APSO algorithm have obvious advantages in R BE . R BE by GA-APSO decreased by 77.6-99.9% and 28.5-99.8% compared with GA and APSO, respectively. The R BE of masks obtained using GA were the largest among the three. The total variation R TV of masks by three algorithms had slight difference. Masks after optimization are presented in Figure 10. The binary error an variation were introduced to evaluate the manufacturability of the mask a complexity of the mask pattern, which are defined as follows [23]: in which is the mask matrix, ‖•‖ is the L1 norm operator, and the lower valu nary error and total variation represent the better manufacturability. Masks after optimization by the presented GA-APSO algorithm have obvio vantages in . by GA-APSO decreased by 77.6-99.9% and 28.5-99.8% com with GA and APSO, respectively. The of masks obtained using GA were the among the three. The total variation of masks by three algorithms had slight ence. Figure 11 represents the convergence curve of GA, APSO, and GA-APSO. T step converged faster than the MO step because of the smaller number of variabl dropped rapidly at the beginning of the iteration. Then, GA and APSO tended to co to a local optimum fast. GA-APSO achieved better PEs compared to GA and APSO the source optimization (SO) step and mask optimization (MO) step. To obtain an alent PEs with GA and APSO, GA-APSO costs less iteration. Additionally, the re GA-APSO are closer to the global optimal solution.  The time cost of the three algorithms was evaluated under the same calculation conditions (in Figure 12). GA-APSO has obvious advantages in algorithm efficiency. Time cost of GA-APSO reduced by 75.91%, 58.66% for Mask 1, by 87.00%, 50.96% for Mask 2, and by 82.91%, 48.43% for Mask 3. The GA algorithm costed the most time. Especially in complex mask optimization, the time cost for the GA algorithm improved sharply. GA-APSO and PSO costed a slightly higher time for complex mask than for simple mask. The time cost of the three algorithms was evaluated under the same calculation conditions (in Figure 12). GA-APSO has obvious advantages in algorithm efficiency. Time cost of GA-APSO reduced by 75.91%, 58.66% for Mask 1, by 87.00%, 50.96% for Mask 2, and by 82.91%, 48.43% for Mask 3. The GA algorithm costed the most time. Especially in complex mask optimization, the time cost for the GA algorithm improved sharply. GA-APSO and PSO costed a slightly higher time for complex mask than for simple mask.
The time cost of the three algorithms was evaluated under the same calculati ditions (in Figure 12). GA-APSO has obvious advantages in algorithm efficienc cost of GA-APSO reduced by 75.91%, 58.66% for Mask 1, by 87.00%, 50.96% for M and by 82.91%, 48.43% for Mask 3. The GA algorithm costed the most time. Espec complex mask optimization, the time cost for the GA algorithm improved sharp APSO and PSO costed a slightly higher time for complex mask than for simple ma As the initial population is randomly generated near the initial source and ma may affect the accuracy and convergence efficiency of the final results. The three were optimized three times using the GA-APSO to assess differences in convergen cess and results. Figure 13 shows the convergence curve for the three times. For M PEs has a certain difference within 200 iteration steps in the SO step, but finally con to an almost consistent result. For Mask 2, the three iterations differ significan Mask 3, the three iterations show a slight difference. Resist images of the three calcu are presented in Figure 14. As the initial population is randomly generated near the initial source and mask, this may affect the accuracy and convergence efficiency of the final results. The three masks were optimized three times using the GA-APSO to assess differences in convergence process and results. Figure 13 shows the convergence curve for the three times. For Mask 1, Pes has a certain difference within 200 iteration steps in the SO step, but finally converges to an almost consistent result. For Mask 2, the three iterations differ significantly. For Mask 3, the three iterations show a slight difference. Resist images of the three calculations are presented in Figure 14.   Figure 14 shows the resist images of the three repeat calculations. Figures 15 and 16 present the source and mask after SMO of the three calculations, respectively. Despite PE differences, the same pattern of source was obtained for Mask 1 and Mask 3 in the three calculations. Table 2 shows the PEs after optimization. Standard Deviation for the three masks were 0.19, 107.08, and 27.52, respectively. The results show that the algorithm has superior repeatability.   Figure 14 shows the resist images of the three repeat calculations. Figu present the source and mask after SMO of the three calculations, respective differences, the same pattern of source was obtained for Mask 1 and Mask calculations. Table 2 shows the PEs after optimization. Standard Deviation masks were 0.19, 107.08, and 27.52, respectively. The results show that the superior repeatability.

Discussion
Simulation results of SMO by GA, APSO, and GA-APSO show that gorithms can improve the resist image quality. Optimization of source a crease the degree of optimization freedom and achieve better resist ima single source optimization. After the SO step, the PEs decreased significa initial iteration stage in the MO step.
The optimization results by GA, APSO, and GA-APSO reached d which GA-APSO obtained better results and less cost. Although both GA global optimization algorithms, their mechanisms are different. In GA, th tion evolution by randomly crossover and mutation which has not obvi This leads to a slow optimization efficiency for GA in the later iteration s the position and velocity of particles are retained and utilized sim

Discussion
Simulation results of SMO by GA, APSO, and GA-APSO show that all the three algorithms can improve the resist image quality. Optimization of source and mask can increase the degree of optimization freedom and achieve better resist image quality than single source optimization. After the SO step, the PEs decreased significantly during the initial iteration stage in the MO step.
The optimization results by GA, APSO, and GA-APSO reached different PEs, in which GA-APSO obtained better results and less cost. Although both GA and APSO are global optimization algorithms, their mechanisms are different. In GA, the whole population evolution by randomly crossover and mutation which has not obvious orientation. This leads to a slow optimization efficiency for GA in the later iteration stage. However, the position and velocity of particles are retained and utilized simultaneously in optimization by APSO. Personal best and global best information are shared for each particle, which has obvious orientation in optimization. This leads to fast optimization efficiency for APSO but may fall into local optima. GA-APSO combined GA and APSO. In the algorithm mechanism of GA-APSO, population is updated by APSO first and then crossover and mutation, which effectively improve the algorithm efficiency and obtain optimization result closer to the global optimal. Moreover, adaptive strategies for APSO particle updates are utilized to further improve the algorithm efficiency and global optimization capability.

Conclusions
In this paper, a GA-APSO algorithm was proposed to inversely obtain the global optimal distribution of the pixelated source and mask in the lithographic imaging process. The computational efficiency was improved by combining the GA and APSO algorithms. Additionally, the global search and local search were balanced through adaptive strategies, leading to a result closer to the global optimal solution. To verify the performance of GA-APSO, horizontal lines and two different complex patterns were optimized. Moreover, the simulation results are compared with optimization by GA and APSO. The results show that the PEs of the resist image optimized by GA-APSO were reduced by 40.13-52.94% and 10.28%-33.31% compared to GA and APSO, respectively. The time cost of GA-APSO was reduced by 75.91-87.00% and 48.43-58.66% compared to GA and APSO, respectively. Moreover, repeated calculation showed that the GA-APSO results were relatively stable. The results demonstrate the superior performance of GA-APSO in efficiency, accuracy, source repeatability, and mask optimization.