Inverse Lithography Source Optimization via Particle Swarm Optimization and Genetic Combined Algorithm

Inverse lithography technologies (ILTs) are critical for improving the imaging performance of lithography in advanced technology nodes. Pixel-based source optimization (SO), as an efficient part of ILTs, can be implemented via heuristic approaches to achieve high-performance lithographic imaging. In this paper, a SO approach based on a combination of the particle-swarm optimization and genetic algorithms (PSO–GA) is proposed to determine the optimal intensity distribution of the source via iterations. The pixelated source can be decoded into the optimized variables of the merit functions in the SO model. The proposed PSO–GA algorithm, as a high-efficiency hybrid algorithm, can transform the discrete SO problem into the optimal search solution for the merit function, thereby inversely enhancing the lithographic-imaging performance. In the forward-imaging model in the lithography, the extraction of the mask's effective diffraction spectrum is implemented to calculate the layout of resist patterns. The simulation results highlight the superior performance of the proposed approach in achieving pixelated SO over the traditional GA and PSO algorithm in terms of convergence capacity.

continually shrinking toward the sizes of advanced technology nodes, image fidelity is becoming an increasingly important parameter in improving the performance of optical lithography [1], [2], [3], [4]. Inverse lithography technology (ILT), as a part of resolution-enhancement technologies, is essential to overcoming the optical proximity effect and promoting image fidelity [5], [6], [7]. As a significant ILT approach, pixelated source optimization (SO) has been proven to be necessary for improving the imaging performance of lithography in advanced technology nodes [8], [9]. Furthermore, it has been successfully applied by several institutions, such as ASML [10], [11], [12] and IBM [13], [14], to modulate the intensity distribution and incident angles of lithography-illumination sources in industrial applications [15], [16]. However, the highly complex representation of the source impacts the performance of the pixelated SO method. Furthermore, the high computation efficiency required to achieve further enhancements is a source of concern.
To enhance the performance of the pixelated SO methods, a set of algorithms have been proposed, including gradient-based [17], [18] and heuristic algorithms [19], [20]. In these methods, the pattern errors, as a generally utilized merit function in the iteration procedure, can be defined by calculating the cumulative sum of the difference between the resist pattern (RP) and the desired image with point-by-point. Moreover, multifarious merit functions have been employed to evaluate the simulation results and convergence effect in SO methods, such as the edge-placement errors, normalized-image log slope, and maskerror enhancement factor [19], [21], [22]. Thus, the threshold and the sigmoid functions can be employed to approximately represent the layout pattern after the resist effect is exerted on the wafer surface [23], [24]. Regarding the application of the gradient-based SO method, many studies have proven that the imaging performance of lithography can be improved by optimizing the source's intensity distribution [18], [25]. Peng et al. employed a gray-level pixel to represent the lithographic source and utilized the gradient-based SO method to improve the image fidelity and depth of focus [18]. Ding et al. employed the gradient-descent method to optimize the lithographic source and mask in the hybrid Hopkins-Abbe imaging model to enhance the optimization performance [25]. However, for a lithographic imaging process using a complicated resist model, it is not recommended to calculate the gradients of a highly complex merit function in the gradient-based SO model. Moreover, for the typical, local optimization methods, the convergence capability This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ of gradient-based approaches is restricted by the high-dimension variate matrices of the pixelated source.
Heuristic algorithms, such as the genetic algorithm (GA) [26], [27], particle-swarm optimization (PSO) [28], [29], and differential evolution [30], are commonly applied to handle optimization challenges because they are free of complex optimization structures and tedious gradient calculations that search for the global suboptimal solution to the merit function. Moreover, it is convenient to simplify the complex challenges; for instance, the optimization of the lithographic source can be simplified to the calculation of the optimal value of the merit function [31]. Additionally, these approaches have been widely employed to improve the performance of lithographic imaging simulations. Tian et al. simulated the global optimization of the lithographic source over multiple patterns to emphasize the predominant convergence capability of SO methods based on heuristic algorithms [32]. Yang et al. proposed a multipole source representation to denote the low pupil-filling-ratio freeform, which was combined with a GA to significantly enhance the global SO performance [33]. Fuhner et al. utilized a GA to optimize mask and illumination geometries, and the simulation results demonstrated the successful application and future potential of the proposed approach [34]. Chen et al. utilized the covariance matrix adaptation evolution strategy with a new sourcerepresentation method to inversely optimize the lithographic source and mask, satisfying the high optimization-capacity and convergence-efficiency requirements in lithographic-imaging simulations [31]. Wang et al. employed PSO to evaluate the intensity distribution of the source, in which pattern fidelity was adopted as the fitness function to evaluate the simulation results [20]. In research works, heuristic algorithms have been proven to exhibit considerable potential for improving the performance of lithographic imaging [30], [35], [36], [37]. Nevertheless, the existing heuristic algorithms that are generally applied in optimization models make it difficult to deviate from the local optimum in the case of a complex merit function [38], [39].
To improve the search capability of these standard global methods at the optimization stage, it is more efficient for their hybrid form. [40], [41], [42], [43]. Particularly, the hybrid PSO algorithm has been generally applied to solve the optimization problem of complex models. Cao et al. embedded the local search method into PSO to significantly enhance the global search capability of this method [42]. Tian et al. utilized the hybrid PSO based on the multi-objective differential-Evolution approach to optimize the real-word emergency scheduling problem of the forest fire [43]. Therefore, for a complex model, it is essential to employ a hybrid algorithm to search the optimum solution.
In this paper, a highly efficient hybrid algorithm combining particle-swarm optimization with genetic algorithm (PSO-GA), is proposed to inversely optimize the intensity distribution of the pixelated source in lithographic-imaging simulations. Although the PSO model shows promise in the global search for the complicated lithographic inverse optimization model, the convergence to global optimality is not accomplished directly. To solve this challenge, crossover and mutation operations involving GA are employed to modify the particle variables in PSO. In these simulations, two merit functions are utilized to evaluate the convergence results: the pattern errors (PEs) and the edge placement errors (EPEs). Moreover, we employ a typical lithographic-imaging model based on the effective spectrum extraction of the mask to improve the imaging performance. The simulation results confirm that SO approach based on PSO-GA exhibits better convergence performance than the standard GA and PSO algorithm.
The organization of this paper is as follows. The forward lithographic imaging model is described in Section II-A. The proposed PSO-GA approach utilized to achieve SO is introduced in Section II-B. Section III provides the simulation results and discussions, followed by the summary in Section V

A. Forward Lithographic Imaging Model
In this section, the imaging process of the lithography system is explained using mathematical formulas. For a typical lithography simulation, the lithographic imaging model is composed of two indispensable units: the illumination source and the projection objective. The former can produce incident light in the form of Kohler illumination. The rays are transmitted through the mask to generate the diffraction light with the feature pattern information. The latter collects the diffraction information of the feature patterns. However, due to the optical diffraction limitation of lithography, the projection objective, which can be regarded as a low-pass filter, can limit the participation of higherorder diffraction during the formation of the desired aerial image in the traditional illumination methods. This phenomenon results in a loss of the lithographic-imaging fidelity. Therefore, it is essential to improve the imaging performance of lithography via SO. Furthermore, the intensity distribution of the aerial image can be calculated via the Abbe theory, as follows [44], [45]: Here, I represents the intensity distribution of the aerial image; (x i , y i ) denotes the spatial coordinates on the image plane; (f, g) and (f , g ) represent the normalized-frequency-domain coordinates of the pupil and the mask, respectively; S is the lithographic illumination source shape; P is the optical-transfer function of the projection objective; M is the frequency spectrum of the mask pattern, which can be obtained via 2D fast Fourier transform (FFT).
The forward lithographic imaging model, as a typical partially coherent imaging (PCI) model, can be dissected into a set of coherent imaging processes. All the spectrums can be established under the same spectral coordinate system using the mask's frequency spectrum as the standard. Assuming that the quadrate mask's length is 2L, the coordinate range can be set to [−L, L]. Therefore, the coordinate range of the frequency spectrum is  Position relationship between the source and the pupil in the frequency spectrum of the mask. The blue and red circles represent the pupil and source, respectively, the green shaded part represents the transmission cross-coefficient (TCC) matrix; r s represents the source radius; r p is the pupil's radius representing the cut-off frequency; P and P * are the pupil matrix and the pupil conjugate matrix, respectively; R is the extreme shifting length of the pupil matrices' center relative to the source's center.
the number of pixels for the lateral dimension. According to the theory of lithographic imaging, the cut-off frequency of the pupil is NA/λ, where NA and λ are the numerical aperture and illumination wavelength, respectively. The partial coherent factor (σ) of the illumination source can be defined as the ratio of the source radius to the pupil radius, σ = r s /r p , 0 < σ < 1. Therefore, the radii of the source and pupil matrices can be respectively represented by r s = σN A/λ and r p = NA/λ. Fig. 1 illustrates the change in the pupil's position relative to the source on the frequency spectrum of the mask. In the Hopkins imaging model [44], [46], the position of the pupil is shifted relative to the valid pixel position of the source to generate the aerial image in the imaging simulation (the pupil and source are respectively represented by the blue and red circles in Fig. 1). The transmission cross-coefficient (TCC) matrix is represented by the green shaded area formed by the overlap of the three circles: the source matrix, pupil matrix, and pupil-conjugate matrix. Along the f axis, the maximum movement length of the pupil matrix can be assumed to be R, 0 ≤ R < r s + r p . Therefore, the movement region of P is the circle with radius R.
The diffraction spectrum, which is utilized to generate the lithographic aerial image, is only a part of the complete spectrum. Fig. 2 illustrates the process of extracting the valid diffraction spectrum. The complete diffraction spectrum can be obtained by applying 2D FFF to the mask. Thus, the valid diffraction spectrum matrix can be acquired using the following rule: where M ext represents the valid diffraction spectrum matrix of the mask. The region of the valid diffraction spectrum is represented by the circle with radius R f . Assuming that the radius of the pixelated source matrix is smaller than that of the pupil matrix, then In a partially coherent system, the PCI process can be expressed as the sum of the aerial images produced by a series of coherent imaging processes. Additionally, the coherent imaging process can be represented as a spectral integral, as follows: Here, C ext represents the extracted illumination cross coefficient (ICC), and (4) confirms the generation of a coherent image by a unit source point in the pixelated source. The discrete coherent imaging process can be explained by (5), wherex i ,ŷ i , j, k = 1, 2, 3, . . . , N ext . Further, N ext denotes the sampling number of the extracted valid diffraction spectral in the lateral dimension. Therefore, for the discrete extended illumination source in lithography, the lithographic aerial image can be expressed as follows: In (6), N s represents the sampling number of the pixelated source in the lateral dimension. To guarantee the same dimension between the results of (1) and (6), the interpolation operation must be implemented. Resultantly, (6) can also be expressed as (7) via the multiplication of low-dimensional matrices: where S ext is the 1D matrix that consists of the effective source points employed to generate the aerial image. Let I ext , C ext , and S ext be represented by the real matrices, denoted by s,ext , and S ext ∈ R N 2 s,ext ×1 , respectively. Fig. 3 demonstrates the process of manipulating the matrices in (7).
Assuming that the illumination source is annular, as shown in Fig. 3(e), the number of valid source points becomes N s,ext . To expediently implement the matrix operation, the 2D source matrix, S, can be transformed into the 1D matrix, S ext , by respectively carrying out Extr[ ] and Reshape[ ]. In ICC ext , each column represents the 1D ICC matrix, which can be generated by converting the 2D C ext of (5) into the 1D vector matrix. According to (7), the multiplication operation of C ext and S ext can generate the 1D I ext matrix. With the reconstruction  To calculate the RP on the wafer, it is critical to ensure that the output aerial image has the same matrix size as that of the input mask. Upon executing the interpolation operation using Interp[ ], I ∈ R N ×N can be generated from I ext , as shown in Fig. 3 To achieve the optimal intensity distribution of the lithographic source in the SO model, the layout of RP can be generated using the resist effect, which is approximated using the sigmoid function. Therefore, the layout of RP obtained by the resist effect can be expressed as follows: Here, I RP represents the layout of RP. The function, Γ{ }, is a regular S-type function, which approximately replaces the threshold value; α is the steepness index, and t r is the threshold value of the photoresist.

B. Flow of the PSO-GA Approach
The flow of the PSO-GA approach is derived in this section. In this paper, the proposed hybrid PSO-GA, as a heuristic algorithm, only requires the merit function with the original format that has the free-gradient operation. To visibly estimate the optimization results in the SO process, the PEs and EPEs are utilized as the merit functions. The PE formula representations are as follows: The function, Π{x s }, in (9), which can help achieve a value close to the global minimum in the iteration procedure, is incorporated in the accumulation of the absolute value of the difference between the output RP and the input mask pattern, M * . Further, x s represents the variable matrix of the extracted valid source points, which can be updated in each loop, and x s ∈ [0, 1]. For the EPEs, the monitored regions of M * marked in the assigned length range are illustrated in Fig. 4, which are 2-pixel points inside and outside the margins of M * .
In Fig. 4(a), the green region represents the monitored pixels, which are utilized to calculate the change in the EPEs. The matrix, ICC Edge , shown in Fig. 4(b), can be generated by reconstructing the extracted pixel points in the green region into the 1D vector matrix. Thus, according to the different positions of the valid source points, the 1D vector matrix can be arranged into the N edge,ext × N s,ext matrix. Therefore, the EPEs can be formulated as In (10), the matrix, M ext , can be formed by extracting these points with the marked position in Fig. 4. The size of M ext is the same as the result of ICC Edge S ext {x s }, and they are both N edge × 1 matrices, where N edge represents the number of extracted points from the green region.
To ensure high-performance lithographic imaging, highefficiency optimization approaches as a part of ILTs are indispensable. Thus, a hybrid PSO-GA approach is proposed to improve the lithographic-imaging performance in the PCI process. Although PSO exhibits the good convergence performance in the early stage of the iteration process, the relatively low convergence aptitude causes the value of the merit function to fall to the local optimum. To enhance the optimization performance of the PSO algorithm, the mutation operation and crossover operation of GA are embedded into the PSO procedure. This approach has been used in the previous studies. Piotr et al. proposed to utilize the neuro-fuzzy system to improve the performance of a hybrid method combining PSO with GA [40]. Shang et al. applied the combination of GA and PSO into the management of workshop production scheduling to improve the production efficiency and product quality [47]. For the constrained engineering optimization problems, Zhu et al. used a dynamic adaptive inertia factor to balance the convergence rate and global search capacity of PSO, and employed the operators of GA, such as a selection operator, crossover operator, and n-point random mutation operator, to enhance the convergence ability [48]. Based on the random search mechanism, GA is capable of global optimization, and its solution is considered robust [49], [50]. Moreover, the superior scalability of GA enables it to be easily combined with other algorithms.
In the proposed PSO-GA approach, the mutation and crossover operations are utilized to update the initial variates of the pixelated source. Subsequently, these updated variates are employed as the inputs for the PSO to accomplish the entire iteration process. The optimization flow of the proposed approach is shown in Fig. 5. Assuming that the initial variate matrix named as the population is P, and P is represented by a real N p × N s,q matrix denoted by P ∈ R N p ×N s,q , where N p and N s,q respectively represent the individual number and the variate number of the valid source points in the first quadrant, the complete source can be achieved by executing the mirror operation according to the optical symmetry of the lithographic source. In the GA process, the following formulas can decide the result of the mutation operation: ς mum ∈ (0, 1) ; P (k, m) = P (k, m) * (1 + Δε) , otherwise.
Here, the increment, Δε, can be randomly updated with an increasing number of iterations, where η 1 , t, and τ represent the random number, current iteration number, and total iterations, respectively. In (12), the variates of the individuals in P can be indexed by the position (k, m), and the values of η 1 and ς mum determine the variable tendency of these variates in each iteration.
To maintain the diversity in population P, the crossover operation of GA is necessary to improve the search performance of the proposed approach. The multipoint crossover strategy is employed to improve the randomness of the variate-update process and expand the search scope. The selected part of this crossover operation in every individual can be determined by the index position (χ obj , χ node ), which is illustrated in Fig. 6. Assuming that the individual in the k th row needs to be updated in the current loop for the crossover process, row χ obj of the individual can be utilized as the target to perform the crossover operation. In addition, these variates indexed by column χ node in row χ obj of the individual are selected to swap with row k th .
Thus, population P, generated by the mutation and crossover operations of GA, is inputted into the PSO model. A nonlinear adaptive control strategy is employed to improve the search performance of PSO in the optimization process. The probability of maintaining the local optimum can be reduced using this strategy. Meanwhile, the search scope can be expanded by updating the weight coefficient. This strategy can be expressed by the following formula: where w(t) represents the weight coefficient in the t th iteration, which is utilized as the coefficient of calculating the velocity of searching variates in population P; w max and w min represent the maximum and minimum weight coefficients, respectively; Ψ[ ] is the hyperbolic tangent function, which is used to control the gradient process of the weight coefficient according to the Generate the initial search velocity matrix υ) and the source population (P) using the random function.
While t ≤ τ Do, execute the mutation operation, For i = 1 to N P If η mum < ς mum , η mum = rand, then Execute (11) value of the constant, ζ. Therefore, the variation denoted as the particle-search velocity in every individual can be represented as In (14), υ k (t) represents the variation of the k th individual, P k , in the t th iteration, where c 1 , c 2 represent the learning factor, and η 4 , η 5 are the random numbers. The current best value, Q best,k , and global best value, G best , can be calculated with the merit function in the lithographic-imaging process. Therefore, in the loop for updating population P, the current individual can be calculated using (15). The flow of the proposed PSO-GA algorithm in this work is illustrated in Algorithm 1.

III. SIMULATIONS
In this section, a set of simulation results is provided to verify the superior optimization performance of the proposed method to improve the performance of the lithographic-imaging process. The pixelated SO model is based on the 193 nm immersion lithography system in the 45-nm technology node. The numerical aperture (NA) of the imaging system is 1.35. The effective spectrum-extraction approach is utilized to achieve the PCI step of the lithography. In these simulations, the annular source shape is used as the initial illumination source. The inner partial coherent factor (σ inner ) and outer partial coherent factor (σ outer ) are 0.68 and 0.95, respectively. The pixelated source pattern in this model is a N s × N s matrix, where N s = 41. To reduce the degree of discretization in the intensity distribution and achieve a continuous gray gradient, the source is blurred via gaussian filtering. The sigmoid function is employed to approximately simulate the process of forming the RP, where α = 85 and t r = 0.21.
To evaluate the performance of the proposed method, four mask patterns are employed to accomplish the SO simulations (Patterns 1-4), as shown in Fig. 7. The first pattern is the horizontal block pattern, the middle two are both vertical array patterns with duty ratios of 1:1 and 1:2, and the last is a typical logical-circuit pattern. They are all N × N matrices, where N = 521. In this SO model, the size of a single pixel is 5.625 nm × 5.625 nm. For convenience in executing the SO procedure, only valid pixels in this source are extracted to form a N s,ext × 1 matrix, where N s,ext = 576. As a typical 4-fold symmetry structure, 144 pixels are utilized as the optimized variates in the iterative process. Using the mirror operation, the complete source shape can be directly recovered. Furthermore, two merit functions are employed as the iteration objectives to achieve the optimum source shape: PEs and EPEs. Three algorithms, including GA, PSO, and the proposed algorithm, are utilized to execute the SO simulations with different merit functions and mask patterns. For a reasonable comparison of the simulation results, the valid pixel values of the initial source, which are generated by the random function, were kept constant for the same mask pattern Fig. 8 illustrates the SO simulation results using different optimization algorithms with different mask patterns and the PE merit function as the optimization objective. The three rows from top to bottom show the optimization results of the source and the resist patterns obtained using three different optimization algorithms: GA, PSO, and PSO-GA. Based on the type of mask pattern, the simulation results can be divided into three columns. From left to right, they are Patterns 1-3, respectively. Further, each column exhibits the intensity distribution of the optimized source and resist patterns. In the optimization results of the second row, the grayscale changes among all the optimized sources are inconspicuous in the intensity distribution, as shown in Fig. 8(g), (i), and (k). On the contrary, the grayscale gradients show outstanding performance for the optimized source in the first and third rows. This is because the excellent performance of the PSO algorithm enables the easy achievement of the local optimum for the complex PCI model. The intensity distributions of the sources optimized by GA and the proposed method both show similar tendencies. In the SO model based on GA, the source shape with Pattern 1 is composed of four arcs, which are shown in Fig. 8(a). The vertical dipole illumination and four arcs in the inclined top, shown in Fig. 8(c), constitute the final source shape for Pattern 2. Additionally, the source shape for Pattern 3 has the approximate quadrupole illumination mode, where the patterns represented in Fig. 8(e) in four directions are closer to the circles. However, for the proposed SO method, the results of the optimizing sources, as shown in Fig. 8(m) and (q), show more pronounced changes corresponding to Patterns 1 and 2, respectively. In the layout of the resist patterns, there are no discernible differences. For the PEs with different algorithms, the proposed SO method can achieve large declines. Table I lists the PEs at the end of the iteration process.
Meanwhile, the optimization results for the simulations with different SO methods executed using the EPE merit function are shown in Fig. 9, including the optimized sources and the resist  patterns. These illustrations are arranged in the same manner as in Fig. 8. Similarly, it is evident that PSO still keeps the local optimum in the SO model, which is consistent with the results obtained with the PEs. The intensity distribution of the optimized sources for GA is evidently different from that in Fig. 8. However, for Patterns 1 and 2, there are similar intensity distributions between Figs. 8 and 9. Although there is a difference in intensity, the source shape in Fig. 8(q) is similar to that in Fig. 9(q). Comparing the optimization performance of the RPs based on the merit functions, PEs and EPEs, it is evident that there are two similar sets of results that are achieved using Patterns 1 and 3, respectively. However, the optimization results using Pattern 2 have a more noticeable difference. The EPEs simulated by different optimization algorithms are listed in Table II.  To ensure the feasibility of the proposed method in practical applications, a typical logical-circuit pattern was employed to complete the SO simulations. The optimized sources and the intensity distribution of the aerial image via the PE merit function are shown in Fig. 10. The simulations using different optimization algorithms were all performed five times. From top to bottom, the displayed simulation results were achieved using GA, PSO, and PSO-GA, chronologically. The first five columns are the optimized source, and the last column shows the intensity distribution of the aerial image. The distribution rules of the simulation results via the EPE merit function in Fig. 11 are consistent with those described above. Comparing the simulation results obtained with the two merit functions, the optimized sources obtained by the PSO algorithm have indistinct intensity distributions. Nevertheless, the SO results obtained using GA and PSO-GA are outstanding. In the simulation  Fig. 12 shows that the simulation results via PE and EPE merit functions. From left to right, the simulation results with Patterns 1 to 4 respectively are shown. The first two rows and last two rows represent the simulation results obtained the PE and EPE merit functions, respectively. The intensity distributions of optimized sources with different patterns are approximately the quadrupole illumination. During these simulations, 150 iterations were executed. Under different conditions, these simulations with GA, PSO, and PSO-GA were all executed five times to assess the convergence efficiency. And the average execution time of these simulations are listed in Table III. The simulations using GD only were run once because there is no randomness in this method. Their running time is shorter than the previous methods. For the different Fig. 11. Simulation results obtained with the EPE merit function for the logical-circuit pattern. From the first row to the third row, the simulation results obtained using GA, PSO, and PSO-GA, respectively, are shown. The first five columns are the optimized sources, and the last column is the intensity distribution of the aerial image.  patterns, the runtime of simulations via the PE merit function are 248.57, 231.76, 318.59, and 254.31, respectively. And for these simulations via the EPE merit function, the running time are respectively 286.21, 275.82, 304.38, and 217.29. The initial intensity distribution of the pixelated source was initialized to be the same matrix in the simulation process with the same input mask pattern and merit function. The convergence results are illustrated in Fig. 13. The convergence curves obtained with the merit functions, PEs and EPEs, are shown in Fig. 13(a)-(d) and (e)-(h), respectively. In Columns 1-4, the iterative results using different input mask patterns are shown, corresponding to Patterns 1-4, respectively. In all convergence curves, the PSO and GA algorithms stop the search from falling into the local optimum after iteration 20. This is because excess optimization variables complicate the optimization model, hampering the search speed of the method and causing the iteration process to stop prematurely. Conversely, for these SO results obtained by GA, the performance when searching for the global optimum is better than that using PSO. Around iteration 60, the convergence curves show a tendency to retain stability. Owing to the good generalization capability of GA, the optimization performance for the complex variate matrix can be significantly improved by combining PSO with GA in the SO model. The overall search scope can be clearly expanded near the local optimal value, which brings the variate matrix closer to the global optimum. It is, thus, confirmed that the optimization ability applied in the SO model can be enhanced by combining PSO and GA.

IV. CONCLUSION
In this paper, a hybrid PSO-GA algorithm was proposed to inversely obtain the optimal intensity distribution of the pixelated source in the lithographic-imaging process. To reduce the dispersion degree of the source, Gaussian filtering was implemented to render the pixelated source grayscale. In the proposed SO model, a special imaging model, which is based on the effective spectrum of the mask pattern, was employed for the PCI process. Considering the limitations of PSO and GA in the optimization of the complex variate matrix, they were combined to enhance the convergence performance in the global optimization process. To verify the improvement in the optimization performance, two merit functions: PEs and EPEs, were employed as the optimization objectives. Meanwhile, four mask patterns (including a horizontal block pattern, two different vertical array patterns, and a logical-circuit pattern) were employed as inputs for the optimization model. Upon comparing the simulation results, the global-search scope of the proposed method was found to be significantly improved around the local optimum, bringing it closer to the global optimum. The simulation results demonstrate the superior performance of the proposed SO method for inversely optimizing the intensity distribution of the lithographic-imaging source. Moreover, the proposed hybrid algorithm exhibits a higher convergence capacity than the other traditional algorithms individually.