Semi-implicit level set formulation for lithographic source and mask optimization

Yijiang Shen; Fei Peng; Zhenrong Zhang

doi:10.1364/OE.27.029659

1. Introduction

With advances in microlithography pushing towards sub-22nm technology node, the engineering of printing circuits layout on wafers has become increasingly complicated and challenging, where the inherent distortion due to optical diffraction has to be compensated by resolution enhancement techniques (RET) [1] and optical proximity correction (OPC) [2]. To further improve the imaging performance, source and mask optimization (SMO) [3] leverages the sub-component interactions of the source and mask patterns to design lithography solutions increasing the resolution capacity of the lithography systems.

The development of freeform diffractive optical elements (DOEs) [4,5] enabling continuous-intensity pixelated sources, paves the way for the evolution of early non-pixel based SMO which optimize source and mask diffraction orders [6,7] to pixel-based SMO. The latter effectively improves lithographic imaging performance by prewarping the mask pattern and inserting the subresolution assist features (SRAFs) around the layout features [8]. Generally implemented iteratively, SMO approaches have to process excessive calculation of aerial images and gradients of cost functions in each iteration [9]. Moreover, with much higher degrees of optimization freedom than traditional rule-based [1] or edge-based OPC [10], the performance comes with intensive computational price, especially for sophiscated large-scale patterns. Consequently, gradient-based methods, including steepest descent [11–13], conjugate gradient [14,15], augmented Lagrangian [16], depth learning methods [17], compressive sensing [18,19], model-driven convolution neural network (CNN) methods [20] and alike have been extensively investigated for accelerated convergence to improve process robustness [21–23], compensate pupil and mask topology [15,24] and so on. Meanwhile, level-set formulation as a mathematical technique [25] with multiple applications in image processing [26], meriting autonomous topology change, easy implementation with numerical schemes and readily geometric representations, have offered significant algorithmic insights and pragmatic applications to inverse lithography problems [27,28]. Shen et al. present detail level-set formulation with appropriate finite-difference schemes for OPC in [29–31] and vector imaging systems [32,33]. A variational formulation with distance regularized level-set (DRLS) term enabling localized level-set evolution is further proposed to improve convergence with reduced optimization dimensionality [34].

While enriching the weaponry for inverse lithography problems, the convergence efficiency of the aforementioned gradient-based algorithms is seriously encumbered for stability by the prohibitive iterative stepsize, when the simple and popular explicit discretization schemes are applied. Likewise, Courant-Friedrichs-Lewy (CFL) condition asserting the numerical wave speed must be at least as fast as the physical wave, requests necessary small stepsizes for level-set evolutions [26]. In this paper, we propose to discretize the diffusion terms in the variational DRLS regularized formulation of SMO [34] implicitly and the non-diffusion terms explicitly, hence “semi-implicitly” circumvent the stability constraints enabling sufficiently large stepsize to improve convergence with much less iteration numbers. We further split the iterative process with respect to coordinate axes into one-dimensional problems, thereby source and mask patterns are updated by solving tridiagonal linear systems of equations most efficiently by the Thomas algorithm [35] with linear complexity. We apply additive operator splitting (AOS) [36,37] instead of locally one-dimension (LOD), alternative directional implicit (ADI) [38] and additive-multiplicative operator splitting (AMOS) [39] because AOS is accurate, easy to implement and does not discriminate the order of coordinate operators. We present the semi-impilict level-set formulation of lithographic SMO with sufficient details so others can follow, analyze and improve, with the goal of fostering more open discussion on ways to apply implicit discretization to inverse lithography.

2. OPtimization framework

2.1 Vector image formation

According to Fourier optics [40] and Abbe method [41], the vector image formation $\mathcal {T}\{\cdot , \cdot \}$ of lithography systems can be calculated in integrative and analytic matrix format to explicitly incorporate pixelated source and mask variables as

(1)$$\small \textbf{I}=\mathcal{T}\{\textbf{J }, \textbf{M}\}=sig(\textbf{I}_{a})=sig\left(\frac{1}{J_{sum}} \sum_{\alpha_s}\sum_{\beta_s}\textbf{J}(\alpha_s, \beta_s)\sum_{p=x,y,z}\|\textbf{H}_p(\alpha_s, \beta_s)\otimes\textbf{B}(\alpha_s, \beta_s)\odot \textbf{M}\|^2\right),$$

where $\textbf {J} \in \mathcal {R}^{N_s \times N_s}$ and $\textbf {M} \in \mathcal {R}^{N \times N}$ are scalar matrices representing the source pattern and mask pattern distributions, $\textbf {I}_{a}$ is the aerial image, $\textbf {J}(\alpha _s, \beta _s)$ is the intensity of the source point at $(\alpha _s, \beta _s)$, $J_{sum}= \sum _{(\alpha _s, \beta _s)}\textbf {J}(\alpha _s, \beta _s)$ is the sum of source intensities as a normalization factor. $\textbf {B}(\alpha _s, \beta _s)\in \mathcal {R}^{N \times N}$ is the matrix representing the oblique incidence effect of the light rays [42]. $\textbf {H}_p(\alpha _s, \beta _s)$ with $p = x, y, z$ are referred to as the equivalent filters of the $x, y, z$ components. $\odot$ and $\otimes$ are entry-by-entry multiplication operation and convolution operation, respectively. The resist effect is approximated using a logarithmic sigmoid function $sig(x)=\frac {1}{1+\mathrm {e}^{-a(x-t_r)}}$ because of its derivability, with $a$ being the steepness of the sigmoid function and $t_r$ being the threshold. In what follows, we will drop the argument $(\alpha _s, \beta _s)$ when there is no ambiguity.

2.2 DRLS regularized level-set formulation

The level-set formulation takes the perspective of viewing the contour of the source or mask distribution as the zero level set of function $\phi _{\textbf {l}}(\textbf {r}),~\textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ and treats the SMO as an inverse problem by tracking the evolution of $\phi _{\textbf {l}}(\textbf {r})$, hence, we give $\textbf {J}$ or $\textbf {M}$ a level-set description by relating $\phi _{\textbf {l}}(\textbf {r})$ to $\textbf {J}$ or $\textbf {M}$ as

(2)$$\textbf{l} = \left\{ \begin{array}{ll} l_{\textrm {int}} & {\textrm{for} \{\textbf{r}:\phi_\textbf{l}(\textbf{r})\,<\,0\}}\\ l_{\textrm {ext}} & {\textrm{for} \{\textbf{r}:\phi_\textbf{l}(\textbf{r})\,>\,0\}}\end{array} \right.,~\textbf{l}=\textbf{J}~{\textrm{or}}~\textbf{M},$$

where $\textbf {r}$ denotes spatial coordinate $(x,y)$, $l_{\textrm {int}}$ and $l_{\textrm {ext}}$ are predefined negative and positive numbers. Consequently, the inverse lithography problem of SMO is reformulated to handle the level-set functions (LSFs) $\phi _{\textbf {l}}$ instead of the source and mask patterns $\textbf {J}$ and $\textbf {M}$. With the synthesized source and mask patterns embedded as the evolution of zero level set of $\phi _{\textbf {J}}$ and $\phi _{\textbf {M}}$, it is necessary to maintain $\phi _{\textbf {J}}$ and $\phi _{\textbf {M}}$ in good conditions, so that the level set evolution is stable and the numerical computation is accurate, which is well satisfied by the signed distance property $|\nabla \phi _{\textbf {l}}|=1|, \textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ [43], where $\nabla$ is the gradient operation.

To keep the evolving $\phi _{\textbf {l}}$ as signed distance functions especially in the vicinity of the zero level set, we define an energy function $\mathcal {E}(\phi )$ of the LSF $\phi$ by

(3)$$\mathcal{E}(\phi)=\mu\mathcal{R}_{\mathcal{P}}(\phi)+\mathcal{E}_{ext}(\phi),$$

where $\mathcal {R}_{\mathcal {P}}(\phi )$ is the distance regularized level set (DRLS) term [43], $\mu$ is a constant and $\mathcal {E}_{ext}(\phi )$ is the external energy term that minimizes the the designed distance between the given target pattern $\textbf {I}_0\in \mathbb {R}^{N\times N}$ and the printed wafer image $\textbf {I}=\mathcal {T}\{\textbf {J }, \textbf {M}\}$. Hereafter, we will drop the subscript $\textbf {l}={\textbf {J}}~{\textrm {or}}~\textbf {M}$ of the LSF $\phi$ without ambiguity to denote the same actions of $\phi _{\textbf {J}}$ and $\phi _{\textbf {M}}$, unless specified otherwise. The DRLS term is formulated as

(4)$$\mathcal{R}_p(\phi)=\frac{1}{2}\int_{\Omega\in\mathbb{R}^{N\times N}}\left(|\nabla\phi|-1\right)^2 {d}\textbf{r}.$$

$\mathcal {E}_{ext}(\phi )$ is designed in this work to minimize the sum of the mismatches between $\textbf {I}_0$ and $\textbf {I}$ over all locations, or namely, the pattern error (PE), which with a least squares leads to

(5)$$\mathcal{E}_{ext}(\phi)=\frac{1}{2}\int_{\Omega\in\mathbb{R}^{N\times N}}\left(\mathcal{T}\{\phi_\textbf{J}, \phi_\textbf{M}\}-\textbf{I}_0\right)^2 {d}\textbf{r}.$$

Besides, a parametric transformation

\textbf{J}=\frac{1+\cos\theta_{\textbf{J}}}{2}~~{\textrm{and}}~~\textbf{M}=\frac{1+\cos\theta_{\textbf{M}}}{2}

where $\theta _{\textbf {J}}\in \mathbb {R}^{N_s\times N_s}$ and $\theta _{\textbf {M}}\in \mathbb {R}^{N\times N}$, is applied to reduce the binary-constrained optimization problem to an unconstrained one in the optimization procedure. Consequently, the level-set flow of the energy $\mathcal {E}(\phi )$ is computed as

(6)$$\begin{aligned} \frac{\partial\phi}{\partial t}&={-}\mu\frac{\partial\mathcal{R}_p}{\partial\phi}|\nabla\phi|-\frac{\partial\mathcal{E}_{ext}}{\partial\phi}|\nabla\phi|\\ &={-}|\nabla\phi|v_{\textbf{l}}(\textbf{r},t)-\mu|\nabla\phi|\left[\triangle\phi-\nabla\cdot\left(\frac{\nabla\phi}{|\nabla\phi|}\right)\right], ~\textbf{l}=\textbf{J}~{\textrm{or}}~\textbf{M},\end{aligned}$$

where $\triangle$ is the Laplacian operator, $t$ is the artificial time and $v_{\textbf {l}}(\textbf {r},t)$ is defined as

(7)$$\begin{aligned} v_{\textbf{J}}(\textbf{r},t)&={-}\mathcal{J}\{\phi_{\textbf{J}}\}^T(\mathcal{T}\{\phi_{\textbf{J}}, \phi_{\textbf{M}}\}-\textbf{I}_0)=\frac{1}{2}\frac{\partial}{\theta_{\textbf{J}}}(\textbf{I}-\textbf{I}_0)^2\\ &={-}a\sin\theta_{\textbf{J}}\sum_{\alpha_s, \beta_s}\frac{\sum_{p=x,y,z}\|\textbf{E}_p^{\alpha_s\beta_s}\|^2-\textbf{I}_{a}}{\sum_{\alpha_s, \beta_s}\textbf{J}}\odot(\textbf{I}_0-\textbf{I})\odot \textbf{I} \odot(\textbf{1}-\textbf{I}),\end{aligned}$$

and

(8)$$\begin{aligned} v_{\textbf{M}}(\textbf{r},t)=&-\mathcal{J}\{\phi_{\textbf{M}}\}^T(\mathcal{T}\{\phi_{\textbf{J}}, \phi_{\textbf{M}}\}-\textbf{I}_0)=\frac{1}{2}\frac{\partial}{\theta_{\textbf{M}}}(\textbf{I}-\textbf{I}_0)^2=\frac{a\sin\theta_{\textbf{M}}}{J_{sum}}\sum_{\alpha_s, \beta_s}\sum_{p=x,y,z}\textbf{J}(\alpha_s, \beta_s)\\ &{\textrm{Real}}\left[\left(\textbf{B}\right)^*\odot \left(\left(\textbf{H}_p\right)^{*^\circ}\otimes \left\{\textbf{E}_p^{\alpha_s\beta_s}\odot(\textbf{I}_0-\textbf{I})\odot \textbf{I} \odot(\textbf{1}-\textbf{I})\right\}\right)\right], \end{aligned}$$

with $\mathcal {J}\{\cdot \}$ being the Jacobian of $\mathcal {T}$, $*$ being the conjugate operation, $\circ$ flipping the matrix in the argument in both up-down and right-left directions, $\textbf {1}\in \mathbb {R}^{N\times N}$ being the all-ones matrix and $\textbf {E}_p^{\alpha _s\beta _s}=\textbf {H}_p(\alpha _s, \beta _s)\otimes (\textbf {B}(\alpha _s, \beta _s)\odot \textbf {M})$.

3. Semi-implicit discretization scheme

We employ discrete times $t_k=k\tau , k=0,1,2,\dots$ with $\tau$ being the stepsize and divide $\theta _{\textbf {l}}, \textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ by a uniform mesh of spacing $h=1$. We further represent the parametrically transformed source and mask patterns $\theta _{\textbf {l}}, \textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ by vectors $\omega _{\textbf {J}}\in \mathbb {R}^{N_s^2\times 1}, \omega _{\textbf {M}}\in \mathbb {R}^{N^2\times 1}$ by concatenating the rows of $\theta _{\textbf {l}}$, whose components of $\omega _{\textbf {l}}, \textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ contains the pixel values. Solving the partial differential equation in Eq. (6) for $\textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ in same formulation with different $v_{\textbf {l}}(\textbf {r},t)$, the evolution of level-set functions $\theta _{\textbf {l}}, \textbf {l}=\textbf {J}~{\textrm {or}}~\textbf {M}$ perform source and mask patterns similarly in one iteration. Likewise, we will drop the subscript $\textbf {l}$ hereafter to denote same operations for $\omega _{\textbf {l}}$, unless specified otherwise.

The time-dependent model in Eq. (6) is a partial differential equation, which can be readily solved by finite difference schemes [29]. However, most computational approaches are based on explicit discretization schemes which require very small stepsize. To overcome the stability constraints of prohibitive stepsize, we rewrite Eq. (6) as a “unified model” [26] to give

(9)$$\frac{\partial\omega}{\partial t}=\mu|\nabla\omega|\nabla\cdot\left(\frac{\nabla\omega}{|\nabla\omega|}\right)\ +|\nabla\omega|g\left(\textbf{r}, t \right),$$

where $g\left (\textbf {r}, t \right )=-v(\textbf {r},t)-\mu \triangle \omega$ is referred as the balloon force. Denoting $\omega _i^k$ as the approximation of $\omega (\textbf {r}_i, t_k)$ where pixel $i$ corresponds to some grid node $\textbf {r}_i$, the semi-implicit formulation of Eq. (9) is defined by discretizing the diffusivity $\frac {1}{|\nabla \omega |}$ implicitly and the balloon force explicitly to give

(10)$$\omega_i^{k+1}=\omega_i^k + \tau\sum_{j \in \mathcal{N}(i)} \frac{2\mu}{\left( \left| \nabla \omega \right| \right)_i^k + \left( \left| \nabla \omega \right| \right)_j^k} \frac{\omega_j^{k+1} - \omega_i^{k+1}}{h^2}+\tau\left| \nabla \omega \right|_i^k g(i,k),$$

with $\mathcal {N}(i)$ being the $4$-neighbors of the pixel $i$ and $g(i,k)$ being the discretization of $g\left (\textbf {r}, t \right )$. A harmonic averaging scheme is applied to set $\omega _{i}^{k+1}\colon = \omega _{i}^k$ for pattern positions with $\left | \nabla \omega \right |_{i}=0$ to avoid the vanishing denominator. In matrix-vector notation, Eq. (10) becomes

(11)$$\omega^{k+1} = \omega^k +\tau\left| \nabla \omega \right|^k g^k+ \tau\sum_{l \in \left\{x,y\right\}}A_l\left(\omega^k\right) \omega^{k+1},$$

where $A_l$ represents the interaction in the $l$ direction, with $A_l(\omega ^k)=\left [a_{ijl}(\omega ^k)\right ]$ given by

(12)$$a_{ijl}(\omega^k) = \left\{ \begin{array}{rcl} \frac{2\mu}{\left( \left| \nabla \omega \right| \right)_i^k + \left( \left| \nabla \omega \right| \right)_j^k} & & [j \in \mathcal{N}_l(i)] \\ -\sum\limits_{m \in \mathcal{N}_l(i)}\frac{2\mu}{\left( \left| \nabla \omega \right| \right)_i^k + \left( \left| \nabla \omega \right| \right)_m^k} & & (j = i) \\ 0\hspace{0.8cm} & & (else) \\ \end{array} \right.,$$

with $\mathcal {N}_l(i)$ being the $2$-neighbors of the pixel $i$ with respect to the $l$ coordinate. However, the solution $\omega ^{k+1}$ cannot be determined directly from this scheme, but requires to solve the linear system of equations

(13)$$\left(I - \tau\sum_{l \in \left\{x,y\right\}}A_l\left(\omega^k\right)\right) \omega^{k+1} =\omega^k+\tau \left| \nabla \omega \right|^k g^k,$$

where $I$ denotes the unit matrix. Although the structure of the system matrix $\left (I - \tau \sum\limits _{l \in \left \{x,y\right \}}A_l\left (\omega ^k\right )\right )$ in Eq. (13) depends on pixel numbering with at most $5$ nonvanishing entries per row, it is not possible to order the pixels in the $i_{th}$ row to bound the nonvanishing entries diagonally within the positions $\left [i, i-2\right ]$ and $\left [i, i+2\right ]$. As a result, Eq. (13) is a very large sparse system (of size $N_s^2\times N_s^2$ for $\omega _{\textbf {J}}$ and $N^2\times N^2$ for $\omega _{\textbf {M}}$) with much larger bandwidth.

Applying direct algorithms such as Gaussian elimination or iterative methods to 2-D problems in Eq. (13) would lead to an immense storage and computation effort. Instead, we consider its additive operator splitting (AOS) variant

(14)$$\omega^{k+1} = \frac{1}{2}\sum_{l \in \left\{x,y\right\}} \left(I-2\tau A_l\left(\omega^k\right)\right)^{{-}1}\left(\omega^k +\tau \left| \nabla \omega \right|^k g^k\right).$$

The AOS scheme in Eq. (14) offers one important advantage: The operators $\left (I-2\tau A_l\left (\omega ^k\right )\right )$ lead to strictly diagonally dominate tridiagonal linear systems which can be solved very efficiently by the Thomas algorithm with linear complexity and easy implementation [35]. It should also be noted that Eq. (14) is stable only for $|\tau g|\leq 0.5$ [26], however, in our simulations for $\tau \leq 1.5$ the constraint is not severe.

4. Simulation results

Simulations are performed on a partially coherent imaging system with an annular illumination source $\textbf {J}_0$ represented as a $29\times 29$ matrix whose outer radius is $\sigma _{out}={0.9}$ and inner radius is $\sigma _{in}=0.6$. The illuminating wavelength $\lambda$ is set at $193nm$ and the numerical aperture $NA$ is $1.35$. The resolution of the mask pattern is $\delta x=\delta y=4nm$. The resist effect is approximated by a sigmoid function with steepness $a=85$ and the threshold $t_r=0.3$.

In order to demonstrate the validity of the proposed SMO approach, we illustrate the simulations in Fig. 1 and compare the performance with conventional level-set SMO method. We also define pattern error (PE) as the square of the $L_2$ norm of the difference between the target pattern $\textbf {I}_0$ and the resist image $\textbf {I}$. In the columns of Fig. 1 from left to right lie the illuminating source $\textbf {J}$, the illuminated input mask pattern $\textbf {M}$ and the printed wafer image $\textbf {I}$. In row (a) without OPC, severe distortion of pattern error (PE) $4139$ in the printed wafer image is incurred by the low pass nature of the lithography system when a dense poly pattern (target pattern $\textbf {I}_{01}$ represented as a $257\times 257$ matrix) is illuminated by the original source $\textbf {J}_0$. In row (b) and (c), pattern fidelities of the printed wafer images are significantly improved to PE $591$ and $586$ when synthesized masks are illuminated by the synthesized sources derived from the conventional level-set based SMO and the proposed approach. CFL condition [26] is enforced to guarantee the stabililty when conventional level-set SMO is applied and stepsize $\tau =1.3$ is used in the proposed approach. The linearity of the runtimes of the simulations in Fig. 1 with respect to iteration number are illustrated in Fig. 2, indicating convergence in $0.2843$hrs with $15$ iterations for the proposed approach as compared to $1.0212$hrs with 60 iterations for the conventional level-set SMO, achieving a $3.6$-fold speedup.

Fig. 1. Simulation of lithographic imaging with a poly dense pattern $\textbf {I}_{01}$. Columns from left to right: illuminating source $\textbf {J}$, illuminated mask pattern $\textbf {M}$ and printed wafer image $\textbf {I}$. Row (a), (b) and (c): simulation results with desired target pattern $\textbf {I}_{01}$ as $\textbf {M}$ illuminated by the original illuminating source $\textbf {J}_0$, synthsized mask illuminated by the synthesized source by conventional level-set SMO method and by the proposed semi-implicit SMO approach with $\tau =1.3$, respectively.

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Fig. 2. Runtimes for conventional level-set SMO method and the proposed SMO approach with $\tau =1.3$ in Fig. 1 with respect to iteration numbers.

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Figure 3 compares the PE convergence for the simulations by conventional level-set SMO and the proposed SMO approach applying stepsizes $\tau =0.55, 0.7, 0.85, 1, 1.15, 1.3$ with respect to iteration number. Considerably stable convergence is observed in all the simulations, which is especially important for level set methods to maintain regularity of the LSFs. Also, the proposed SMO approach greatly accelerates the convergence while achieving similar PE improvements. A more quantitative comparison of the simulations in Fig. 2 is given in Table 1. Same average iteration times are duely noted from Table 1 for different $\tau$ justified by solving linear systems of equations with same size, $N_s^2\times N_s^2$ for updating source and $N^2\times N^2$ for updating mask in Eq. (14) in each iteration, therefore gradually increasing stepsize $\tau$ from $0.55$ to $1.3$ in the proposed approach requires less iterations to converge, reducing the overall convergence time. Moreover, solving linear systems of equations with the Thomas Algorithm, although very efficient, is comparatively slower than that with the explict discretization scheme, however, the overall convergence performance with the proposed approach is greatly improved with much larger stepsize $\tau$ thereby much less number of iterations. Table 1 quantitatively demonstrates average iteration time 0.017hrs for the conventional level-set SMO, faster than the 0.019hrs for the proposed approach, yet bounded average stepsizes $0.4429$ for $\omega _{\textbf {J}}$ and $0.2796$ for $\omega _{\textbf {M}}$ by CFL condition severely encumber the convergence.

Fig. 3. Convergence performance for the simulations in Fig. 1 using the conventional level-set SMO and the proposed approach with stepsizes $\tau =0.55, 0.7, 0.85, 1, 1.15, 1.3$ improving PE from $4139$ to $580$, $574$, $568$, $568$, $574$, $586$ in 40, 35, 30, 25, 20, 15 iterations, respectively.

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Table 1. Convergence Stepsize and Runtime (in hours) of Simulations in Fig. 1.

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Simulations are also performed on another target pattern $\textbf {I}_{02}$ illuminated by the same original source $\textbf {J}_0$ with the lithographic imaging performance in Fig. 4. In row (a) with $\textbf {I}_{02}$ as input where no OPC is involved, degraded pattern fidelity of PE 4631 is recorded in printed wafer image $\textbf {I}$. Row (b) and (c) show the OPC results when synthesized mask patterns are illuminated by the synthesized source patterns derived by the conventional level-set SMO and the proposed SMO approach with improved PE 444 and 435. Convergence for the simulations by conventional level-set SMO and the proposed SMO approach applying stepsizes $\tau =0.55, 0.7, 0.85, 1, 1.15, 1.3$ with respect to iteration number is presented in Fig. 5 showing stable level-set function evolution, computation linearity with iteration number and accelerated convergence with less interation number for the proposed method.

Fig. 4. Simulation of lithographic imaging with $\textbf {I}_{02}$. Columns from left to right: illuminating source $\textbf {J}$, illuminated mask pattern $\textbf {M}$ and printed wafer image $\textbf {I}$. Row (a), (b) and (c): simulation results with desired target pattern $\textbf {I}_{02}$ as $\textbf {M}$ illuminated by the original illuminating source $\textbf {J}_0$, synthsized mask illuminated by the synthesized source by conventional level-set SMO method and by the proposed semi-implicit SMO approach with $\tau =1.3$, respectively.

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Fig. 5. Convergence performance for simulations in Fig. 4 using the conventional level-set SMO and the proposed approach with stepsizes $\tau =0.55, 0.7, 0.85, 1, 1.15, 1.3$ improving PE from $4631$ to $424$, $465$, $428$, $418$, $436$, $435$, respectively.

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5. Conclusion

We have presented an efficient semi-implicit computation strategy for source and photomask design in optical microlithography, surmounting the stabilility constraint of prohibitive stepsize by explicit discretization schemes. The distance regularized level-set (DRLS) incorporation in the variational level-set formulation, besides maintaining signed distance properties of LSFs, provides the diffusivity which is discretized implicitly and the laplacian operator as a part of the “balloon force” discretized explicitly enabling computational stability for source and mask syntheses with sufficient large stepsizes. In addition, computation efficiency is rooted in splitting the two dimensional problem with respect to coordinate axes into one dimensional tridiagonal linear systems of equations solved by the Thomas algorithm. Simulation results merit the significant convergence improvement by the proposed approach combining stablilty and efficiency. The systematic incorporation of semi-implicit discretization and operator splitting, therefore, is a potential substitute of the popular explicit discretization schemes in pragmatic OPC computations.

Funding

National Natural Science Foundation of China (61875041); Natural Science Foundation of Guangdong Province (2015A030310290, 2016A030313709); Guangzhou Municipal Science and Technology Project (201607010180); Natural Science Foundation of Guangxi Province (2013GXNSFCA019019, 2017GXNSFAA198227).

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Semi-implicit level set formulation for lithographic source and mask optimization

Abstract

1. Introduction

2. OPtimization framework

2.1 Vector image formation

2.2 DRLS regularized level-set formulation

3. Semi-implicit discretization scheme

4. Simulation results

5. Conclusion

Funding

References

Cited By

Figures (5)

Tables (1)

Equations (15)

Optics Express

	stepsize $τ$		Avg. itr. time	Total time	Itr. num.	PE
Level-set	Avg. $τ$ for $ϕ_{J}$	Avg. $τ$ for $ϕ_{M}$	0.017	1.0212	60	591
	$0.4429$	$0.2796$
Proposed	$0.55$		0.019	0.76	40	580
Proposed	$0.7$		0.019	0.665	35	574
Proposed	$0.85$		0.019	0.57	30	568
Proposed	$1$		0.019	0.475	25	568
Proposed	$1.15$		0.019	0.38	20	574
Proposed	$1.3$		0.019	0.285	15	586