Xiaofei Wu

Level-set-based inverse lithography for mask synthesis using the conjugate gradient and an optimal time step

Inverse mask synthesis is achieved by minimizing a cost function on the difference between the output and desired patterns. Such a minimization problem can be solved by a level-set method where the boundary of the pattern is iteratively evolved. However, this evolution is time-consuming in practice and usually converges to a local minimum. The velocity of the boundary evolution and the size of the evolution step, also known as the descent direction and the step size in optimization theory, have a dramatic influence on the convergence properties. This paper focuses on developing a more efficient algorithm with faster convergence and improved performance such as smaller pattern error, lower mean edge placement error, wider defocus band, and higher normalized image log slope. These improvements are accomplished by employing the conjugate gradient of the cost function as the evolution velocity, and by introducing an optimal time step for each iteration of the boundary evolution. The latter is obtained from an...

Efficient source mask optimization with Zernike polynomial functions for source representation

In 22nm optical lithography and beyond, source mask optimization (SMO) becomes vital for the continuation of advanced ArF technology node development. The pixel-based method permits a large solution space, but involves a time-consuming optimization procedure because of the large number of pixel variables. In this paper, we introduce the Zernike polynomials as basis functions to represent the source patterns, and propose an improved SMO algorithm with this representation. The source patterns are decomposed into the weighted superposition of some well-chosen Zernike polynomial functions, and the number of variables decreases significantly. We compare the computation efficiency and optimization performance between the proposed method and the conventional pixel-based algorithm. Simulation results demonstrate that the former can obtain substantial speedup of source optimization while improving the pattern fidelity at the same time.

Robust and efficient inverse mask synthesis with basis function representation

Mask optimization is essential in the resolution scaling of optical lithography due to its strong ability to overcome the optical proximity effect. However, it often demands extensive computation in solving the nonlinear optimization problem with a large number of variables. In this paper, we use a set of basis functions to represent the mask patterns, and incorporate this representation into the mask optimization at both the nominal plane and various defocus conditions. The representation coefficients are updated according to the gradient to the coefficients, which can be easily obtained from the gradient to the pixel variables. To ease the computation of the gradient, we use an adaptive method that divides the optimization into two steps, in which a small number of kernels is used as the first step, and more kernels are used for fine optimization. Simulations performed on two test patterns demonstrate that this method can improve the optimization efficiency by several times, and the optimized patterns have better manufacturability compared with regular pixel-based representation.

Fast aerial image simulations using one basis mask pattern for optical proximity correction

Aerial image simulation is one of the key parts in the model-based optical proximity correction (OPC) technique, which has become a must have process to improve lithography performance with ever-decreasing feature sizes. In this paper, a fast aerial image simulation approach is proposed by using one basis mask pattern to generate a lookup table, where the convolutions of the basis pattern with the partially coherent kernels are precalculated and stored. A rectilinear polygon mask pattern used in integrated circuit layouts can be decomposed into several shifted basis patterns. Its convolutions with kernels for use in aerial image calculation can then be quickly obtained from the precalculated lookup table by applying the translation-invariant property of two-dimensional convolution. Simulations conducted by using the proposed approach have demonstrated that this approach yields a superior quality in the fields of aerial image calculation and OPC optimization, due to the advantage of dramatically decreasing...

Illumination source optimization in optical lithography via derivative-free optimization

Illumination source optimization (SO) in optical lithography is generally performed under a simulation model that does not consider critical effects such as the vectorial nature of light and mask topography. When a numerical aperture becomes large and the critical dimension reaches subwavelength, the prediction of this model generally fails; therefore, the previous works based on this model become inaccurate. In order to correctly compute SO, we first propose a new source pattern representation method that has moderate parameter variations but remains complete in solution space. Then we develop a derivative-free optimization (DFO) method to optimize these parameters under a rigorous simulation model. Unlike gradient-based techniques, DFO methods do not require a closed-form formulation of the model and are independent of the form of cost function.

Iterative method for in situ measurement of lens aberrations in lithographic tools using CTC-based quadratic aberration model

This paper proposes an iterative method for in situ lens aberration measurement in lithographic tools based on a quadratic aberration model (QAM) that is a natural extension of the linear model formed by taking into account interactions among individual Zernike coefficients. By introducing a generalized operator named cross triple correlation (CTC), the quadratic model can be calculated very quickly and accurately with the help of fast Fourier transform (FFT). The Zernike coefficients up to the 37th order or even higher are determined by solving an inverse problem through an iterative procedure from several through-focus aerial images of a specially designed mask pattern. The simulation work has validated the theoretical derivation and confirms that such a method is simple to implement and yields a superior quality of wavefront estimate, particularly for the case when the aberrations are relatively large. It is fully expected that this method will provide a useful practical means for the in-line monitoring of the imaging quality of lithographic tools.

Incorporating photomask shape uncertainty in computational lithography

The lithographic performance of a photomask is sensitive to shape uncertainty caused by manufacturing and measurement errors. This work proposes incorporating the photomask shape uncertainty in computational lithography such as inverse lithography. The shape uncertainty of the photomask is quantitatively modeled as a random field in a level-set method framework. With this, the shape uncertainty can be characterized by several parameters, making it computationally tractable to be incorporated in inverse lithography technique (ILT). Simulations are conducted to show the effectiveness of using this method to represent various kinds of shape variations. It is also demonstrated that incorporating the shape variation in ILT can reduce the mask error enhancement factor (MEEF) values of the optimized patterns, and improve the robustness of imaging performance against mask shape fluctuation.

Sparse nonlinear inverse imaging for shot count reduction in inverse lithography.

Inverse lithography technique (ILT) is significant to reduce the feature size of ArF optical lithography due to its strong ability to overcome the optical proximity effect. A critical issue for inverse lithography is the complex curvilinear patterns produced, which are very costly to write due to the large number of shots needed with the current variable shape beam (VSB) writers. In this paper, we devise an inverse lithography method to reduce the shot count by incorporating a model-based fracturing (MBF) in the optimization. The MBF is formulated as a sparse nonlinear inverse imaging problem based on representing the mask as a linear combination of shots followed by a threshold function. The problem is approached with a Gauss-Newton algorithm, which is adapted to promote sparsity of the solution, corresponding to the reduction of the shot count. Simulations of inverse lithography are performed on several test cases, and results demonstrate reduced shot count of the resulting mask.

Comparison of three TCC calculation algorithms for partially coherent imaging simulation

Three kinds of TCC (transmission cross coefficient) calculation algorithms used for partially coherent imaging simulation, including the integration algorithm, the analytical algorithm, and the matrix-based fast algorithm, are reviewed for their rigorous formulations and numerical implementations. The accuracy and speed achievable using these algorithms are compared by simulations conducted on several mainstream illumination sources commonly used in current lithographic tools. Simulation results demonstrate that the integration algorithm is quite accurate but time consuming, while the matrix-based fast algorithm is efficient but its accuracy is heavily dependent on simulation resolution. The analytical algorithm is both efficient and accurate but not suitable for arbitrary optical systems. It is therefore concluded that each TCC calculation algorithm has its pros and cons with a compromise necessary to achieve a balance between accuracy and speed. The observations are useful in fast lithographic simulation for aerial image modeling, optical proximity correction (OPC), source mask optimization (SMO), and critical dimension (CD) prediction.

Impact of photomask shape uncertainties on computational lithography

We devise an algorithm that can incorporate the photomask shape uncertainties in computational lithography such as inverse lithography technique. The photomask patterns are expressed by a random field under a level-set framework to represent the uncertain shape variation. The Karhunen-Loève expansion is introduced so that only several parameters are used to delineate the random field, and thus it can be incorporated into the optimization algorithm in inverse lithography. Simulations show that this method is effective to improve the lithographic imaging performance.