# The Inverse Optimization of an Optical Lithographic Source with a Hybrid Genetic Algorithm
## 2 Methodology
### Flow of Hybrid-GA-Based SO
#### 2.2.2 The Optimization Strategy for the Illumination Source  
In imaging systems for optical lithography, an extended source is employed as the illumination source, and is deemed to be partially coherent illumination. According to their different partial coherence factors and different intensity distributions, the illumination modes can be divided into circular illumination, annular illumination, dipolar illumination,quadrupole illumination, etc. In this study, the annular illumination mode is utilized as the initial source in the proposed SO model, as shown in Figure 2. The partially coherent factor (\(\sigma\)) of this annular source is determined by the source's effective radius (\(r_{s}\)) and the pupil's radius (\(r_{p}\)), \(\sigma=r_{s}/r_{p}\). For annular illumination in the optimization model, the maximum and minimum partially coherent factors are, respectively, \(\sigma_{out}=r_{s,max}/r_{p}\) and \(\sigma_{in}=r_{s,min}/r_{p}\), where \(r_{s,max}\) and \(r_{s,min}\) are the maximum and minimum radiuses of the illumination source, respectively. Because of a special property of symmetry with respect to the optical axis, an optical lithographic source can be decomposed into four identical parts according to the axis quadrants, which are \(J_{1}\), \(J_{2}\), \(J_{3}\), and \(J_{4}\) in Figure 2a. It is essential to solely optimize one of the four regions of an annular source to reduce the computational complexity of the optimization model. The region \(J_{1}\) shown in Figure 2b is hypothetically employed as the part to be optimized. Assuming that the size of the pixelated source is \(N_{J}\times N_{J}\), the coordinate range of the X-Y axis in the first quadrant is 0 to \((N_{J}-1)/2\). An indexing operation that is implemented with Equation (6) can be utilized to record the positions of valid pixels in \(J_{1}\). \[ \begin{array}{l}\mathbf{\mathcal{P}}~{}=\Psi\{J_{1}(x,y)==1\},\\ x=0,1,2,\cdots,(N_{J}-1)/2,\\ y=0,1,2,\cdots,(N_{J}-1)/2.\end{array}  \tag{6}\]  
In Equation (6), \(\mathbf{\mathcal{P}}\) is an index result that contains the position information of the valid pixel points in \(J_{1}\). Assuming that the number of valid pixel points is hypothetically \(m\), \(\mathbf{\mathcal{P}}\) can be represented by a real matrix, which is denoted by \(\mathbf{\mathcal{P}}\in\mathbb{R}^{m\times 2}\). \(\Psi\{\cdot\}\) is a query function that can achieve the indexing operation with the criterion of \(J_{1}(x,y)==1\). Then, the 1-D vector matrix \(\mathbf{J}_{1}\) shown in Figure 2c is generated as the objective to be optimized, and it is explained by \(\mathbf{J}_{1}\in\mathbb{R}^{1\times m}\). In the proposed hybrid algorithm, a GA is employed as the main body of the optimization framework to update the intensity value of a pixel point in \(J_{1}\). The initial population is necessarily created as the initial solution group, and it can be defined as a real matrix denoted by \(\mathbf{f}^{\prime\prime}\in\mathbb{R}^{n\times m}\). After several iterations, the  
Figure 2: Schematic of the optimization of a lithographic source by using the proposed SO method. In (**a**), the dotted circle represents a circle with a radius of 1. The annular source is divided into four identical parts: \(J_{1}\), \(J_{2}\),\(J_{3}\), and \(J_{4}\). \(J_{1}\) is utilized as the optimized region, which is shown in the form of a shadow. (**b**) The optimized region of the source; (**c**) the flow of matrix manipulation; (**d**) the optimized shape of \(J_{1}\); (**e**) the intensity distribution of the optimal source. initial population matrix can be continuously updated to select the optimal condition \(J\prime_{1}\) according to the value of the merit function \(f_{PE}\), where \(I_{1}^{\prime\prime}\in\mathbb{R}^{1\times m}\). Then, the intensity distribution of one quarter of the source can be attained by realigning the matrix \(I_{1}^{\prime\prime}\) via the recorded position of \(J_{1}\).