# The Inverse Optimization of an Optical Lithographic Source with a Hybrid Genetic Algorithm
## 2 Methodology  
### Partially Coherent Imaging Model  
The forward imaging process in optical lithography can be described by a partially coherent imaging model (PCI), which is equivalent to the superposition of a series of coherent images. Assuming that a point source on the optical axis is utilized to complete completely coherent imaging (CCI), this imaging process can be expressed as follows [34]:  
\[I^{\prime}(x,y)=\left|\int\int_{+\infty}^{-\infty}H(f+f^{\prime},g+g^{\prime}) M(f^{\prime},g^{\prime})exp(-i2\pi f^{\prime}x+g^{\prime}y)df^{\prime}dg^{ \prime}\right|^{2} \tag{1}\]  
where \((x,y)\) are the spatial coordinates in the imaging plane, \((f,g)\) are the spectral coordinates of the pupil, and \((f^{\prime},g^{\prime})\) are the spectral coordinates of the mask spectrum. \(I^{\prime}(x,y)\) represents the intensity distribution of an image obtained via CCI. \(H(f,g)\), which is a pupil function, can be regarded as a low-pass filter, which severely restricts the higher-order diffraction information of the feature pattern from participating in the imaging process. In addition, it can be used to simulate the effects of defocus, aberration, and apodization by modifying the phase information of the pupil. \(M(f^{\prime},g^{\prime})\) denotes the frequency spectrum of the feature pattern in the mask, which can be acquired with a Fourier transform operation.  
To facilitate the imaging process of optical lithography, the illumination source can be depicted as discrete matrices with a pixelated representation, where a pixel represents a point source. For the off-axis point source, in the process of PCI, the imaging process can be implemented by shifting the pupil according to the position of this point source. The intensity distribution of an image obtained via PCI can be found by accumulating the images obtained by each point source. Hence, the intensity distribution of an aerial image for optical lithography can be calculated as follows:  
\[\begin{array}{ll}I(x,y)&=\int\int_{+\infty}^{-\infty}J(f,g)\cdot l^{\prime} dfdg\\ &=\int\int_{+\infty}^{-\infty}J(f,g)\cdot\left|\int\int_{+\infty}^{-\infty}H( f+f^{\prime},g+g^{\prime})M(f^{\prime},g^{\prime})exp(-i2\pi f^{\prime}x+g^{ \prime}y)df^{\prime}dg^{\prime}\right|^{2}dfdg\\ \\ \end{array} \tag{2}\]  
In Equation (2), \(I(x,y)\) represents the intensity distribution of an aerial image for optical lithography. \(J(f,g)\) is a discrete illumination source. In order to concisely express the imaging process of PCI, Equation (2) can be transformed into a format for matrix multiplication--a source matrix and mask matrix--and the result of CCI can be described by using the illumination cross-coefficient (_ICC_).  
\[\mathbf{I}=\mathbf{ICC}\cdot\mathbf{J} \tag{3}\]  
To evaluate the imaging performance in optical lithography, it is vital to describe the resistance patterns on the surface of a wafer. In this study, the formation process of a resistance pattern can be approximately explained by a sigmoid model, which is given by:  
\[R=sig(I)=\frac{1}{1+exp[-\alpha(I-t_{r})]} \tag{4}\]  
In Equation (4), \(R\) represents the contour layout of the resistance pattern formed due to the photo-resistance effect. \(sig(\cdot)\) is a sigmoid function that can approximately achieve a binary distribution. The parameters \(\alpha\) and \(t_{r}\) denote the steepness index and the threshold of photo-resistance, respectively.