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Abstract
Source mask optimization (SMO) is one of the indispensable resolution enhancement techniques to guarantee the image fidelity and process robustness for the 2Xnm technology node and beyond. The optimization capacity and convergence efficiency of SMO are important, especially for full-chip SMO. An SMO method using the covariance matrix adaptation evolution strategy (CMA-ES), together with a new source representation method, is proposed in this paper. Based on the forward vector imaging formulation, the encoding and decoding methods of the source and the mask, and the constructed merit function, the source and the mask are optimized using the CMA-ES algorithm. The solution search space and the search step size are adaptively updated during the optimization procedure. Considering the sparsity of the optimal source, the source is represented by a set of ideal point sources with unit intensity and adjustable positions. The advantageous spatial frequency components of the source for imaging performance improvement are identified through the aggregation of the point sources. Simulations and comparisons verify the superior optimization capacity and convergence efficiency of the proposed method.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Lithography is one of the key technologies for the very-large-scale integrated circuits (VLSI) manufacturing. Resolution is one of the three key performance metrics for lithography. As the technology node continues to shrink, resolution enhancement techniques (RETs) [1] are indispensable for overcoming the significant optical proximity effects (OPEs) and improving the lithographic imaging quality, such as resolution, image fidelity and process robustness. Source mask optimization (SMO) proves to be an effective resolution enhancement technique for 22nm technology node and beyond [2].
Rosenbluth et al. introduced the concept of SMO in 2001 [3]. Since then, many researches related to SMO have been carried out, involving optimization algorithms and strategies to accelerate the convergence speed and improve the searching ability for global optimum solution, representation methods of the source/mask to balance the freedom degree and efficiency of optimization, and intensity calculation formulations convenient for different optimization objects. Extended SMO methods improve the imaging quality and process robustness by establishing various evaluation metrics, such as the process window (PW), the normalized image log slope (NILS), the edge placement error (EPE), the pattern error (PE), the mask error enhancement factor (MEEF) and so on.
Early SMO was realized by joint optimization of the parametric source and the mask, with the mask modified via optical proximity correction (OPC) or sub-resolution assist feature (SRAF) insertion. For better image fidelity and larger process window, pixelated SMO methods [4–18] are successively proposed to exploit the higher degree of optimization freedom provided by the freeform source and pixelated mask [19–21].
Pixelated SMO methods include gradient-based methods such as gradient descent (GD) [4], steepest descent (SD) [5,6], conjugate gradient (CG) [7,8], augmented Lagrange [9], depth learning method [10], level-set based method [11,12], compressive sensing based method [13,14], and SMO methods using heuristic algorithms like genetic algorithm (GA) [15,16], particle swarm optimization (PSO) [17], and self-adaptive differential evolution (JADE) [18].
In gradient-based SMO methods, using a sigmoid function to approximate the resist effect [4–9] enables a differentiable merit function, which is convenient for calculating the derivate for the source and mask variables and adjusting the variables along the descent direction of the gradient. However, it is difficult and time-consuming to calculate the gradients of a more complex merit function for a lithographic imaging model considering the complicated resist effect, limiting the applicability of the gradient-based SMO methods. Heuristic algorithms are popular, because they are free of gradient calculations, free of the prior knowledge for the specific problem, and search for a suboptimal solution as an alternative global optimal solution. They are compatible with various lithographic imaging models. The existing SMO methods using heuristic algorithms [15–18] pick out the optimal solution by continuously comparing the fitness values of different encoded individuals and generating new populations. Hence, these methods are easy to implement. Nevertheless, these methods lack control of the solution search space and the search step size during the optimization stage, and make little attempt to find out the best direction for the variables’ adjustment, causing an inefficient convergence.
Using a large number of variables to encode the source pixels’ intensity and the mask pixels’ transmittance permits a rather large solution space for SMO, but suffers from an intensive computation load conversely. Several researchers proposed alternative source representation methods to speed up the source optimization process. Wu et al. introduced the Zernike polynomials as basis functions and characterized the source as a weighted superposition of well-chosen Zernike polynomial functions [22]. The source variables corresponds to the coefficients of chosen Zernike polynomial functions, so the number of source variables reduces significantly. However, the continuity of Zernike polynomials forces the optimized source to be concentrated in certain regions without highlighting the important spatial frequency components of the source [23]. Yang et al. proposed a multipole source representation under the prerequisite of small pupil filling ratio [16]. The source consists of a group of circular poles with finite-size. The source is modulated by tuning the poles’ locations and intensities. Artificial restriction on the number of circular poles according to a given pupil filling ratio exactly improves the convergence efficiency, but the small number of poles may limit the global optimization capacity because of insufficient coverage of the solution space composed of locations and intensities. Moreover, it is noted that the optimized sources using the multipole representation and the full pixel representation are quite different.
While the pixelated source provides many degrees of freedom, only part of the source pixels on behalf of specific illumination directions contribute to improving the imaging quality. Most source pixels usually have zero intensity after source optimization. It is inefficient to optimize all pixels’ intensity without taking advantage of the sparse property of the optimal source.
In this paper, we propose a pixelated SMO method using the covariance matrix adaptation evolution strategy (CMA-ES). CMA-ES is an evolutionary strategy based on Gaussian mutation and deterministic selection [24]. It is one of the best choices against ill-conditioned, non-convex and non-separable black-box optimization problems in the continuous domain. CMA-ES adaptively adjusts the solution search space by adapting the covariance matrix using rank-µ-update and rank-1-update mechanism, and controls the global step size related to the scale of the distribution. Good solutions in former generations have higher probabilities to reappear in later generations. To further identify the most advantageous spatial frequency components of the source, we propose a new source representation method. The source comprises a set of ideal point sources with unit intensity and adjustable positions. The number of the ideal point sources is predetermined based on the total number of source pixels and partial coherence factor of the illumination system. According to the position coordinates, the point sources fall into different source pixels. A source pixel with more point sources inside has a higher relative intensity. The positions of the point sources determine the intensity distribution of the pixelated source and affect the imaging performance. Consequently, the source optimization is converted to the adjustment of the point sources’ positions. The source representation method proposed in this paper reduces the number of source variables and helps to highlight the spatial frequency components contributing to better imaging quality. Lithographic imaging systems with complicated resist models limit the application of the gradient based SMO methods, because it is difficult to calculate the analytical derivate of the merit function. By contrast, the CMA-ES based SMO is compatible with any type of resist model and more applicable than gradient based SMO methods.
Simulation results show that the CMA-ES based SMO together with the new source representation improve the optimization capacity and the convergence efficiency.
2. Methodology
2.1 Forward vector imaging formulation
Figure 1 illustrates the schematic of immersion lithography system. An extended light source is placed at the focal plane of the condenser to generate Köhler illumination [25]. The light rays emitted from the source are diffracted when passing through the mask. Only low-frequency diffracted light rays pass the projection lens due to the low-pass filtering effect of the pupil. When the light rays reach the film stack on the wafer, they expose the resist and change the solubility of the resist. After post exposure bake (PEB) and development, a feature is printed on the wafer. The aim of SMO as well as other types of RETs is to make the feature printed on the wafer as close to the target pattern (TP) as possible.
The imaging process in the lithography system belongs to partial coherence imaging. According to Abbe imaging theory [25], the aerial image is calculated as
Alternatively, by exchanging integral order in Eq. (1), the Hopkins’ formulation for lithographic imaging is obtained [25]. Transmission cross coefficient (TCC) has been extensively exploited in fast aerial image calculation, because it has integrated all the information of the imaging system except that of the mask. TCC is decomposed into multiple kernels via the singular value decomposition (SVD) as
In this paper, the resist effect is approximated with the sigmoid model. If necessary, the sigmoid model can be replaced with a more accurate and physical resist model. The resist image (RI) is given by
The parameter $\alpha $ and $tr$ are the steepness and threshold of the resist, respectively. Besides, we use the constant threshold resist model to generate the printed image (PI) on the wafer directly from the comparison between the aerial image and the threshold,
$\Gamma ({\cdot} )$ is the hard threshold function, which says $\Gamma (x) = 1,\;if\;x > 0$. The parameter $tr$ is the same as above. Unlike the continuously valued RI, PI is a binary wafer image depending on whether the resist material is removed after development. These two models are essentially the same if we use a threshold of 0.5 to truncate the resist image for the printed wafer image. A threshold of 0.5 for development process is a reasonable assumption, because the resist image intensities are as close to 0 or 1 as possible if SMO is well done, far away from the hard threshold used in the constant threshold resist model.
2.2 Encoding and decoding of the source and mask
Encoding of the source and mask is to use a set of variables to characterize the constructive elements for the source and mask, while decoding of the source and mask is to perform some transformations on the variables to recover the source and mask. For pixelated SMO, both the source and the mask are gridded into pixels in the Cartesian system. Let positive odd numbers ${N_S}$ and ${N_M}$ be the lateral dimensions of the source and the mask, respectively. The pixelated source is an ${N_S} \times {N_S}$ matrix, and the pixelated mask is an ${N_M} \times {N_M}$ matrix.
We use $(\hat{f},\hat{g})$ to denote the spatial frequency coordinates of the source pixels. Every source pixel represents a unique illumination direction, and the pixels’ values reveal the illumination directional intensity distribution. Owing to the restriction of cut-off frequency from the pupil, only those source pixels in the unit circle participate in imaging. The discrete values for $\hat{f}$ are from -1 to 1 with an increment of ${2 / {({{N_S} - 1} )}}$, and $\hat{g}$ shares the same discrete values with $\hat{f}$. The spatial frequency coordinates $(\hat{f},\hat{g})$ match with the position indexes $(m,n)$ for each source pixel with the relationship of $m = {{(\hat{f} + 1)({N_S} - 1)} / \textrm{2}} + 1$, $n = {{(\hat{g} + 1)({N_S} - 1)} / \textrm{2}} + 1,\;\textrm{1} \le m \le {N_S},\;1 \le n \le {N_S},\;m \in {\mathbb Z},\;n \in {\mathbb Z}$. To prevent pattern shift caused by a mono-pole source, we use a 4-fold source which is symmetric with respect to the horizontal and vertical axes. In this paper, we use ${N_{pre}}$ point sources with unit intensity and adjustable positions to represent the source. ${N_{pre}}$ is the number of point sources in the first quadrant. We assume that the point sources are ideal and infinitesimal. The point sources in other quadrants are obtained by mirror operation. Figure 2 shows the source representation with a predetermined number of point sources. The circles with gradient colors indicate the blur effect of the source that will be mentioned later.
The point sources fall into different source pixels. A source pixel with more point sources inside has a higher relative intensity. The positions of the point sources determine the intensity distribution of the pixelated source and affect the imaging performance.
The source is encoded with all the point sources’ polar coordinates as
To avoid duplicated count of point sources falling in the middle row or column that act as the symmetry axes, we divide the count number at those locations by 2. In mathematic form, it is written as
There exists a practical limit to the sharpness of defined illumination directions. R. Socha suggested to roughly account for the finite illuminator resolution by convolving the pixel-based source with a blur function [24]. The source with blur effect is defined as
${\otimes} $ is the convolution operator, PSF is a Gaussian function with a support related to the source pixel size, ${S_{blurred}}$ is the source with blur effect. The source blur effect is applied to all the subsequent SMO implementations in this paper. The source is normalized with all pixels’ intensities scaled by the maximum intensity, $\hat{S}$ is the decoded source. Keeping the pixels’ maximum intensity unchanged as one helps to avoid sharp spikes that may damage lenses [28]. From the perspective of efficient imaging with ICC method, the normalized source $\hat{S}$ is converted to a NS2×1 vector SV column-wise.For source representation with full pixels, the source is encoded with intensities of all the pixels in the concentric ring region with inner and outer radii of ${\sigma _{in}}$ and ${\sigma _{out}}$, respectively. The number of source variables depends on the total number of source pixels in that concentric ring region.
Figure 3 illustrates the mask representation with opaque and transparent pixels. The parameter P denotes the period of the mask pattern.
Encoding and decoding of the mask depends on the symmetry of the mask pattern. We take the asymmetric mask and 4-fold symmetric mask for examples to demonstrate the encoding and decoding of the mask.
The asymmetric mask is encoded with all the mask pixels’ transmittance extracted column by column to form a row vector,
For a 4-fold symmetric mask, the encoding and decoding operations are similar to those of a 4-fold symmetric source. We encode the mask with the mask pixels in the first quadrant, and update the pixels in other quadrants using mirror operation and halve the pixels’ transmittance in the middle row and column. The number of variables for an asymmetric mask encoding is ${D_M} = {{{{({1 + {N_M}} )}^2}} / 4}$.
The resulting transmittance matrix M is the recovered mask with continuous transmittance. In this paper, the continuous transmittance mask is just the intermediate object for variable updates. We always use a binary mask for imaging evaluation. Therefore, the last step of mask decoding is to binarize a mask with threshold of 0.5,
$\Gamma ({\cdot} )$ is the same hard threshold function as mentioned above. If the transmittance of a mask pixel is larger than 0.5, reset the pixel’s transmittance to one. Otherwise, reset the pixel’s transmittance to zero. The binary mask MB is the decoded mask.2.3 Flow of the CMA-ES based SMO
In this paper, the merit function for the SMO process is defined as the square of the Euclidean distance between the resist image (RI) and the target pattern (TP),
The resist image and the target pattern have the same size as the mask pattern. They are both ${N_M} \times {N_M}$ matrices. Given the encoded ${x_S}$ and ${x_M}$, the source and mask are recovered to calculate RI according to the flow introduced in section 2.1. Other kinds of metrics for imaging quality evaluation are available for different motivations [23,29,30].The aim of SMO is to minimize the merit function by optimizing the source and mask variables under certain constraints as follows,
We use CMA-ES as the optimization algorithm to implement the sequential optimization of the source and mask. The flow of CMA-ES based SMO is shown in Fig. 4. The source and the mask are encoded as ${x_S}$ and ${x_M}$, respectively. The source variables are initialized randomly according to their ranges to generate the initial source. The mask variables are initialized corresponding to the pixel-based target pattern, which is directly sampled from the mask pattern. Therefore, the initial mask is the same as the target pattern. Then, set other parameters related to the lithographic system such as the illumination wavelength, numerical aperture, refraction index of the immersion liquid, reduction ratio, aberration and so on. When the initialization is finished, we start the one-loop of sequential optimization of the source and the mask, followed by an additional refinement of the source to match the optimized mask. For the sequential SMO, each stage of the optimization is a greedy search for optimal solutions based on current conditions. After many tests, we find that it is amazing to obtain good enough solutions with only one loop of sequential source and mask optimization owing to the excellent performance of CMA-ES, instead of the need for iterative efforts when using other optimization algorithms.
CMA-ES suits ill-conditioned, non-convex and non-separable black-box optimization problems in the continuous domain due to its invariance properties. CMA-ES use a multivariate normal distribution to characterize the distribution of solutions. By updating the covariance matrix with good solutions found in the populations along with the generations, together with a control of the search step size, CMA-ES helps to find an optimal solution more efficiently. For an optimization problem, CMA-ES only needs the encoding and decoding methods of the solution vector, constraints of the solution variables and the merit function. Among them, the dimension of the solution vector is the most important factor for CMA-ES, because many other parameters inherent to CMA-ES depend on the dimension. As a result, the general optimization flow of CMA-ES usually starts with the construction of the merit function according to a specific problem. The solution vectors are the interfaces to call the merit function. ${x_S}$ and ${x_M}$ are the solution encoding for source optimization and mask optimization via CMA-ES, respectively. The corresponding dimensions of the solution vector are ${D_S}$ and ${D_M}$.
We take the source optimization via CMA-ES for example to describe the optimization flow of CMA-ES, and the same flow goes for the mask optimization by changing the dimension, encoding and decoding of the solution variables.
The dimension N equals ${D_S}$ in the source optimization, and the source is encoded as ${x_S}$. Given N, we can set the internal parameters of CMA-ES. The population size λ is given by
The function $floor(x)$ rounds a real number x towards negative infinity. The parent population size µ, i.e. the number of selected individuals to update the mean of the distribution, is given byDecreasing weights for these µ individuals are given by
To make the sum of all the weights be one, the weights are normalized with the summand as with$\sum\limits_{i = 1}^\mu {{W_i}} = 1,\;{W_1} > {W_2} > \cdots > {W_\mu } > 0$. Initialize the covariance matrix as ${C^{(0)}} = {I_{N \times N}}$, the evolution path for the covariance matrix and step-size as$p_c^{(0)} = {\bf 0}$, $p_\sigma ^{(0)} = {\bf 0}$, and choose distribution mean ${m^{(0)}} \in {{\mathbb R}^N}$ and step-size ${\sigma ^{(0)}}$ depending on the problem. The superscript (0) refers to the parameters at the generation 0.The CMA-ES generates a population with λ individuals as the next generation by sampling from a multivariate normal distribution ${\cal N}({{m^{(g)}},\;{\sigma^{{{(g)}^2}}}{C^{(g)}}} )$ of current generation,
All the individuals are ranked in ascending order according to their merit function values,
$F(x_{1:\lambda }^{(g + 1)}) \le F(x_{2:\lambda }^{(g + 1)}) \le \cdots \le F(x_{\lambda :\lambda }^{(g + 1)})$, $x_{i:\lambda }^{(g + 1)}$ is the individual with the ${i_{th}}$ smallest merit function values within the population at generation g+1. For each generation, the call times of merit function will increase by λ. This cumulative count number of merit function call is used for adjustment of the covariance matrix and termination judgement. The mean of the distribution is updated with the first µ individuals with the smallest merit function values,
The evolution path for the step-size isThe accumulation means a learning strategy taken over a number of generations, using the evolution path to reintroduce the sign information, which is related to the search direction and vital for searching the optimal solution. The evolution path for the covariance matrix is
The CMA-ES based SO flow will terminate according to several criteria. The parameter maxEval gives the maximum call times of the merit function. The parameter maxCond denotes the maximum condition number of the covariance matrix. The parameter minF denotes the threshold to stop the optimization if an individual’s merit function value is less than the threshold. The parameter nLimit denotes the maximum generations with the best merit function values unchanged. The parameters for termination judgement are selected depending on the specific problem, considering the time consumption and the memory occupation. The optimization process with CMA-ES continues until any of the termination criteria is satisfied.
The above optimization flow using CMA-ES is also applied to the mask optimization. The dimension N equals DM in the mask optimization, and the mask is encoded as ${x_M}$. The parameters for termination judgement are adjusted accordingly. When the whole SMO flow terminates, the optimal solution $x_S^\ast $ and $x_M^\ast $ are output, and the optimized source $\hat{S}$ and mask MB are synthesized according to the decoding operations introduced in section 2.2.
3. Simulations and analysis
This Section presents several simulations to verify the optimization capacity and convergence efficiency of the proposed method. In this paper, all of the simulations are implemented on a computer with Intel Core i5-6300HQ CPU, 2.3GHz, and 8GB of RAM.
The simulations exploit the vector imaging theory for the 193nm immersion lithography system. The source of the partially coherent imaging system is pixelated as a 51×51 matrix with the inner partial coherence factor ${\sigma _{in}} = 0.4$ and the outer partial coherence factor ${\sigma _{out}} = 0.95$. The polarization mode of the illumination source is set as tPol, i.e. linear polarization with tangential orientation. The numerical aperture NA is 1.35, the reduction of the projection optics is 4, and the refractive index of the immersion liquid is 1.44. We use a sigmoid function to approximate the resist effect, with the resist steepness $\alpha \textrm{ = }\textrm{85}$ and the resist threshold $tr = 0.24$.
Figure 5 shows the initial mask pattern used in the simulations, which is also the target pattern for SMO. The size of the mask pattern is 1215nm×1215nm. The mask is pixelated as a 81×81 matrix with the pixel size being 15nm×15nm, which is big enough to ensure the manufacturability of the mask. Pixels in the black region are opaque pixels with transmittance of zero, and pixels in the white region are transparent pixels with transmittance of one. Two red dashed lines intersect the boundary of the transparent region at four points T1, T2, M1 and M2. These two cut-lines are located in the relatively sparse region where breaking or shrinking of the line are more likely to emerge. The NILS at the edges is a good evidence for the validity of SMO, because the NILS reveals the slope of the aerial image on the image boundary. The NILS is defined as
The first simulation is to compare the optimization performances between SMO using CMA-ES and several popular heuristic algorithms such as JADE, PSO, and GA. In this simulation, the source is represented with full pixels limited in the range of ${\sigma _{in}}$ and ${\sigma _{out}}$. Taking account of the 4-fold symmetry of the source, there are 377 source pixels in total in a quadrant. Similarly, there are 1681 mask pixels in a quadrant. The SMO methods are used to optimize the intensity for each source pixel and the transmittance for each mask pixel. SMO is run by iterative source optimization (SO) and mask optimization (MO).
Table 1 lists the parameters for different optimization algorithms for SMO in the simulation. The definitions for the parameters of JADE, PSO and GA can be found in [30], [17] and [16], respectively. The procedures of SO and MO share the parameters that are not distinguished by subscripts SO or MO. For fair comparison of the optimization performances between difference SMO methods, all of the SMO flows start with the same initial source and mask. Figure 6 demonstrates the results of SMO using different optimization algorithms with the full-pixel source representation. Four columns from left to right represent the source, the mask, the resist image (RI), and the comparison between the resist contour (RC) and the target pattern (TP), respectively. RC is the contour of the printed image (PI) defined in Eq. (6), and outlined with red solid lines. TP appears as the background pattern in dark red.
Fig. 6. The results of SMO using different optimization algorithms with the full-pixel source representation. From left to right: the source, the mask, the resist image (RI), the comparison of the resist contour (RC) and the target pattern (TP). From top to bottom: the simulation results with the initial source and mask, the simulation results after SMO using CMA-ES, JADE, PSO, and GA, respectively.
Table 1. The parameters for different optimization algorithms for SMO
From visual point of view, it is intuitive that the SMO using CMA-ES obtains a more uniformly distributed resist image in the third column, and outputs a smooth contour which is more consistent with the boundary of the target pattern in the fourth column, than other SMO methods. Looking at the last column, SMO using JADE provides a very close resist contour to that obtained from CMA-ES based SMO, while the results from the other two SMO methods are quite different. Because GA is not very suitable for high-dimensional optimization problems [31] like pixelated SMO. Therefore, even with generous adjustment of the GA algorithm parameters, GA based SMO shows very poor performance, especially in mask optimization with a relatively large optimization dimension. One notable point is that the SMO using CMA-ES offers a much more compact source than other SMO methods. Comparing the four optimized sources in the first column, we find that the pixels with larger intensities are mainly located at the two separated poles in the vertical direction. Even the GA-based SMO with poor optimization performance shows a slight but similar behavior. Unlike the dispersed distribution of source generated from other SMO methods, the source produced via CMA-ES based SMO is more concentrated. The aggregation of some point sources in the same source pixels decreases the source fill ratio. It is reasonable to conclude that CMA-ES based SMO has the ability to capture the most advantageous spatial frequency components of the source for better imaging quality, and suppress the spatial frequency components degrading the imaging quality at the same time. On the contrary, the other three SMO methods fail to distinguish different spatial frequency components sufficiently according to their importance. In other words, CMA-ES based SMO has superior optimization capacity.
We further analyze the simulation results in Fig. 6 quantitatively. Table 2 lists several evaluation metrics about the simulation results corresponding to Fig. 6, including the final merit function values, pattern error (PE), pattern error ratio (PER), normalized image log slope (NILS) after SMO, the runtime of individual SO and MO stages and the whole SMO procedure, and the times for forward imaging evaluations. PE is defined as the Euclidean distance between PI and TP. Because both PI and TP are binary matrixes, PE refers to the number of pixels with errors. PER is the ratio of PE to the total number of pixels for the pixelated mask. The runtime for SMO is slightly longer than the sum of the runtime for individual SO and MO stages, owing to extra calculations for encoding, decoding and assignment operations.
Table 2. Comparison between the results of SMO using different optimization algorithms
The merit function values given by SMO using CMA-ES and JADE are rather close, much smaller than those given by SMO using PSO and GA. Similar data trends are found in PE, confirming the effectiveness of the merit function. The obtained PER with SMO using CMA-ES is less than 1%, which is a common level in other advanced SMO methods. SMO using CMA-ES improves the NILS along the cut-lines, which reveals a better imaging contrast and greater potential for a larger process window. There is no significant difference in runtime of the whole SMO procedure using different optimization algorithms. However, it takes the SMO using CMA-ES much less times of forward imaging evaluation to reach the optimal solution. The eigenvalue decomposition of the covariance matrix is a time-consuming operation in the CMA-ES algorithm, and occupies most of the runtime in SMO using CMA-ES. Nevertheless, the adaptive updates of the covariance matrix and the search step size indeed improve the ability of searching the optimal solution. Compared with the SMO using JADE that has a comparable merit function value, SMO using CMA-ES behaves better in convergence efficiency because of a shorter runtime.
To further identify the most advantageous spatial frequency components of the source, we propose a new source representation method. The source is comprised of a set of ideal point sources with unit intensity and adjustable positions. The source pixels that are more influential to improving the imaging quality tend to have more point sources falling inside and relatively larger intensities. The number of the point sources affects the relative intensity distribution of the source, as well as the convergence efficiency. The variable number for the source represented with full pixels is 377. Using the source representation of point sources, each point source corresponds to two variables as the polar coordinates. Considering the sparsity of the optimal source, it is a natural dimension reduction operation to set the number of point sources less than 188. Through the aggregation of these point sources, the vital spatial frequency components are highlighted. Figure 7 illustrates the results of SMO using CMA-ES with the source represented by three different numbers of point sources, 50, 100, and 150 respectively. From the uniformly distributed resist image and well-matched resist contour with the target pattern, all of the three cases seem to achieve an optimal combination of the source and the mask. Although the initial sources for three cases are quite different, the output optimal sources are similar to each other. They are sparser than the optimized source in Fig. 6(e) obtained with full-pixel source representation.
Fig. 7. Results of SMO using CMA-ES with the source represented by three different numbers of point sources. From left to right: the source, the mask, RI, the comparison of RC and TP. Row #1, #3, #5: the simulation results with the initial source and mask for Npre=50, 100 and 150, respectively. Row #2, #4, #6: the simulation results with the optimized source and mask for Npre=50, 100 and 150, respectively.
To investigate the effect of the number of point sources and the source representation methods on the optimization performance, we further analyze the simulation results in Fig. 7 quantitatively. Table 3 lists the same evaluation metrics as those in Table 2 about the simulation results from Fig. 7. Besides, the evaluation metrics for SMO based on the full-pixel source representation are copied from Table 2, as a comparison of the source representation methods. Source representations with point sources lead to smaller merit function values than the full-pixel source representation. This new source representation method further reduces the pattern error and increases the NILS values. The numbers of source variables differ among the listed four cases of source representation and affect the runtime for SO significantly. With a smaller number of point sources to represent the source, the reduced dimension of the source variables leads to a relatively faster convergence but limited optimization capacity. The limited number of source points are only enough for figuring out the important spatial frequency components, but fail to produce the required relative intensity distribution across the source plane. Because the relative intensities have something to do with the number of point sources collected in different source pixels. By contrast, a larger number of point sources balances the ability of searching the advantageous spatial frequency components and the production capacity of desired relative intensity distribution. This is supported by the comparison of the PE values after SMO with ${N_{pre}}$ being 50 and 150.
Table 3. Comparison of results with CMA-ES based SMO between different source representations
It is noted that with the increment of the number of point sources, the runtime of SMO did not increase significantly. This phenomenon can be explained for three reasons. First, the incremental number of point sources increases the time consumption in SO stage, but the time increment is slight relative to the total time consumption for SMO, because the SO stage with linear operations is not so time-consuming. Second, the excellent optimization performance of CMA-ES, together with the proposed source representation method, enables similar optimized sources in three cases. The similar optimized sources affect the MO stage slightly, presented as the small difference of runtime for MO stage among three cases. Third, the time consumptions for extra calculations such as encoding, decoding and assignment operations in three cases are almost the same.
To validate the ability of finding out the key spatial frequency components with the proposed source representation method, we use a metric called cosine similarity [32] to quantify the correlation degree between any pair of two sources after SMO. The cosine similarity is defined as
The parameters${S_1}$ and ${S_2}$ refer to two individual sources. The subscript u points to the ${u_{th}}$ element of the source matrix. ${C_{1,2}}$ is the cosine similarity between the two sources. The closer ${C_{1,2}}$ is to one, the more similar source ${S_1}$ is to source ${S_2}$. The closer ${C_{1,2}}$ is to zero, the greater the difference between source ${S_1}$ and source ${S_2}$ is. The cosine similarity among the three optimized sources corresponding to ${N_{pre}}$ = 50, 100 and 150 are calculated and listed in Table 4. In the 3×3 symmetric matric, each entry except the ones on the diagonal has a value close to one, indicating the three optimized sources are highly correlated. In other words, the new source representation method helps to find the critical spatial frequency components of the source and improve the imaging quality.
4. Conclusion
In this paper, we develop a pixelated SMO method based on CMA-ES. The forward vector imaging formulations, the encoding and decoding of the source and the mask, and the flows of SMO using CMA-ES are studied. On the one hand, benefiting from the adaptive adjustment of the solution search space and search step size, CMA-ES outperforms the heuristic algorithms such as JADE, PSO and GA in optimization performance. On the other hand, considering the sparsity of the optimal source, a new source representation with point sources is proposed to further identify the advantageous spatial frequency components of the source for imaging performance improvement. Moreover, the CMA-ES based SMO is more applicable than gradient based SMO methods in lithographic imaging systems with complicated resist models which are not analytically differentiable. Simulation results confirm that the SMO using CMA-ES together with the new source representation improve the optimization capacity and the convergence efficiency of pixelated SMO effectively.
Table 4. The cosine similarity among the optimized sources obtained from CMA-ES based SMO using the new source representation method with different values of Npre
Funding
National Major Science and Technology Projects of China (2017ZX02101004-002, 2017ZX02101004); Natural Science Foundation of Shanghai (17ZR1434100).
Disclosures
The authors declare no conflicts of interest.
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