Multi-objective lithographic source mask optimization to reduce the uneven impact of polarization aberration at full exposure field

Tie Li; Yiyu Sun; Enze Li; Naiyuan Sheng; Yanqiu Li; Pengzhi Wei; Yang Liu

doi:10.1364/OE.27.015604

1. Introduction

The advanced source and mask optimization (SMO) technology has being developed by means of extensive simulation before it is adopted in integrated circuits (IC) manufacture in order to improve the lithography process and yield of product [1,2]. In 2001, the SMO is first proposed by Rosenbluth et al. to increase the optimization degree of freedom and lithography image fidelity [3]. Until 2010, the experimental study of SMO was developed based on novel and advanced freeform illumination sources [4]. In 2011, the SMO technology has just been adopted in advanced lithography process with extremely expensive facility IC manufacture factory [5]. Then many excellent SMO methods have been proposed to improve lithography imaging robustness, mask manufacturability, and computational efficiency [6–12]. However, the most of conventional SMO methods don’t take into account the polarization aberration (PA), which is defined as the spatial variations of light’s polarization state cross the pupil of immersion projection optics (PO) [13,14]. PA innately exists in hyper numerical aperture (NA) immersion PO, and usually relates to the position of exposure field [15,16], which means the PA are uneven at full exposure field of PO.

Several solid work systematically and amply analyze the influence of PA on hyper-NA lithography imaging [17–20]. However, there are very few studies on the impact of uneven PA at full exposure field on SMO. Our early work proposed a robust mask optimization method accounting the assigned PA of single field point [21]. But even if the above robust mask optimization can be developed into an assigned PA aware SMO, it still aims to improve the lithography performance at only one assigned field point. For large-field and immersion PO, the value of PA is field dependent, so the optimized source and mask attained by the SMO accounting the assigned PA at one field point would not apply to other field points at all, which are severely against for the uniformity of image fidelity and the overlap of process window (PW) at full exposure field. Furthermore, using this assigned PA aware SMO method, every field point would correspond to a pair of different optimized source and mask. It is absolutely unacceptable in full exposure field of real lithography system to use more than one pair of source pattern and mask pattern. Thus, this assigned PA aware SMO is not applicable in real lithography process. It is necessary to propose a novel SMO method to compensate and balance the uneven impact of different PA on lithography imaging at full field.

In this paper, we firstly build a new SMO model accounting the assigned PA at single field point and investigate the serious impact of different PA on SMO. The simulation results show the PA of marginal field point aware SMO is not suitable for center field point and the PA of center field point aware SMO is not suitable for marginal field point, where the pattern error (PAE) increases by 84.8% and 121.3%, respectively. Then, we innovate a multi-objective SMO (MOSMO) to reduce the uneven impact of field dependent PA on lithography imaging. To the best of our knowledge, this is the first paper focusing on balancing lithography imaging of full exposure field in SMO. Different from aforementioned assigned PA aware SMO method, the proposed MOSMO designs a novel multi-objective cost function including the PA information of almost all field points. So the optimized source and mask by MOSMO can apply to full exposure field. Aiming at the PO designed by our group, the cost function of MOSMO is designed as the weighted sum of lithography imaging PAE with nine representative field points. Compared to the assigned PA aware SMO, the simulation results demonstrate the proposed MOSMO approach reduces the average of PAE distribution at full field from 503 to 243, narrows the range of PAE distribution at full field from 822 to 687, and improves the maximum exposure latitude of global field from 4% to 6.7%. Thus, the proposed MOSMO can balance imaging quality and improve process robustness at full exposure field.

The remainder of this paper is organized as follows. The assigned PA aware SMO and the impact of different PA on SMO are illustrated in Section 2. The model, algorithm, and simulations of MOSMO are provided in Section 3. Conclusions are presented in Section 4.

2. The impact of PA on SMO

In this section, we propose a novel SMO method accounting the assigned PA at single field point and investigate the serious impact of different PA on SMO results for the first time. The simulations demonstrate different PA at different field point would result in different SMO results, which can’t be accepted in real lithography process.

As the aforementioned statement, the value of PA depends on the position of exposure field. Figure 1 shows the off-axis rectangle exposure field and the position of field point for the PO designed by our group with NA 1.2 [22]. In our early work, we proposed the robust mask optimization method accounting assigned PA at one field point. The vectorial imaging model described in [21] is accurate to describe the impact of PA on hyper-NA lithography imaging. In this model, the aerial image I can be formulated as:

I = \frac{1}{J_{s u m}} \sum_{x_{s}} \sum_{y_{s}} (J (x_{s}, y_{s}) \sum_{p = x, y, z} {| H_{p}^{x_{s} y_{s}} \otimes (B^{x_{s} y_{s}} ⊙ M) |}_{2}^{2}),

where

J (x_{s}, y_{s})

is the intensity of source point at

(x_{s}, y_{s})

, and M denotes the distribution of mask transmissivity. The notation

\otimes

represents matrix convolution. The

H_{p}^{x_{s} y_{s}}

is complex spatial filter of p-component and the

B^{x_{s} y_{s}}

is mask diffraction matrix. The

H_{p}^{x_{s} y_{s}}

involves the impact of PA on electric vector, which can be formulated as:

H_{p}^{^{x_{s} y_{s}}} = F^{- 1} {\frac{2 π}{n_{w}} \times C \times V^{x_{s} y_{s}} ⊙ U ⊙ P A ⊙ E_{i}^{^{x_{s} y_{s}}}},

where notation

F^{- 1}

represents inverse Fourier transform.

Fig. 1 The off-axis rectangle exposure field and field point position in image plane.

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Here we firstly build a novel SMO model including the assigned PA_i at ith field point. The Euler distance $D_{i}$ between resist image $Z_{i} (P A_{i})$ and target pattern $\tilde{Z}$ is designed as a new cost function accounting the PA_i, which is different from that of previous SMO:

D_{i} = d (Z_{i} (P A_{i}), \tilde{Z}) = {‖ Z_{i} (P A_{i}) - \tilde{Z} ‖}_{2}^{2} .

Thus, the assigned PA_i aware SMO problem can be expressed as:

(\hat{J}, \hat{M}) = \underset{J, M}{\arg \min} D_{i} (J, M) .

In the following, we mainly focus on the SMO accounting the assigned PA at center field point F3 and the assigned PA at marginal field point F11. For simplicity, the above two assigned PA aware SMO are named as F3-SMO and F11-SMO, respectively.

The sigmoid function [23] is adopted to approximate the photoresist effect for the continuity of cost function. Thus, the resist image on the wafer can be formulated as:

Z = s i g (I) = \frac{1}{1 + \exp [- a (I - t_{r})]},

where t_r is the photoresist threshold, and a indicates the steepness of the sigmoid function. It is worth noting that the photoresist threshold t_r can approximately represent the exposure dose in threshold model. In a sense, the double t_r is approximately equivalent to the half exposure dose for the final resist image. Thus the t_r is used to represent exposure dose in following Fig. 6.

The steepest-descent method is used to solve the assigned PA aware SMO problem. In future work, we would introduce more advanced algorithms to solve this problem effectively and efficiently, however, which is out of the scope of this paper. According to our previous work about gradient-based SMO [24], the solving process of SMO problem is provided in Table 1 in the form of pseudo-code. The derivation of key gradient formula for cost function is shown in the Appendix. The original source pattern is annular illumination with inner coherent factor 0.82 and outer coherent factor 0.97. The original mask pattern is the same as target pattern.

Table 1. Pseudo-code of the assigned PA aware SMO algorithm.

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Next, we would compare simulation results and lithography imaging evaluations between F3-SMO and F11-SMO to illustrate the serious impact of PA on SMO. In simulation, the parameters of lithography system are as follows: the illumination wavelength is 193nm, the illumination polarization is TE-polarization, the reduction ratio is 1:4, and the NA on the wafer side is 1.2. The t_r in Eq. (5) is 0.12 and the t_m in Table 1 is 0.5. Figure 2 shows two target patterns used in simulations. Both two test patterns are represented by a 201 × 201 matrix with pixel size of 5.625nm × 5.625nm on wafer scale, which means the mask dimension is finite 4522.5 nm × 4522.5 nm in simulation. Thus, the size of one period pattern on mask is very small in rectangle field. The following simulations are based on the hypothesis: different regions in the same period pattern correspond to the same field point whereas different period patterns correspond to different field points.

Fig. 2 Two target patterns used in the simulation. The red lines mark the locations for PW calculation.

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The Kirchhoff approximation is adopted to calculate the diffraction of mask which does not decrease the significant of our method in this paper, because thick mask effect can be compensated via active pupil wavefront optimization as reported in our earlier work [25].

Figures 3(a)-3(h) respectively display the optimized source pattern, the optimized mask pattern, the printed image at center field-of-view (FOV) F3 and the printed image at marginal FOV F11, which are obtained by F3-SMO and F11-SMO for pattern #1. We use PAE as the criterion to evaluate the pattern fidelity of lithography imaging. The PAE is defined as:

P A E = {‖ \tilde{Z} - Ξ {I - t_{r}} ‖}_{2}^{2},

where

Ξ {\cdot} = 0

if the argument is smaller than 0; otherwise,

Ξ {\cdot} = 1

.

Fig. 3 Optimization results and evaluations of the F3-SMO and F11-SMO for pattern #1. Left to right: optimized source pattern, optimized mask pattern, printed image at center FOV F3 and marginal FOV F11.

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In Fig. 3, the optimized source patterns of F3-SMO and F11-SMO are similar, but the optimized mask patterns of F3-SMO and F11-SMO are very different. That may be because the PA is mainly compensated by optimized mask in SMO. But there is no doubt that considering different PA in SMO would result in different optimized source and mask, which can’t be accepted in real lithography process. Moreover, for F3-SMO, the PAE at F11 field point is much larger than the PAE at F3 field point, increasing by 121.3%. Such uneven lithography imaging at different field points is unable to satisfy the requirement of lithography process. Similarly, the F11-SMO is not suitable for center field point F3, where the PAE increases to 1096 from 592. The simulations show the assigned PA aware SMO can’t simultaneously apply to all field points.

3. MOSMO model, algorithm, and simulation

To overcome the drawback of the assigned PA aware SMO and balance the lithography imaging at full exposure field, we innovate a MOSMO method against the uneven impact of field dependent PA on lithography imaging. The common solutions of multi-objective optimization problem mainly include weighted sum method [26] and non-dominated sorting genetic algorithm (NSGA)-II method [27]. But several works [28,29] indicated the genetic algorithm requires lengthy calculation times, which is not suitable for pixelated SMO with high degree-of-freedom. Thus, in this paper we adopt weighted sum method to transform the MOSMO problem to a single objective optimization. According to Section 2, there are different PA at different field points and the data of PA at all field points are known. We design the novel cost function of MOSMO as the weighted sum of pattern fidelity with each field point, which includes the PA information at almost all field points. The multi-objective cost function D can be formulated as:

D = \sum_{i = 1}^{n} ω_{i} D_{i} = \sum_{i = 1}^{n} ω_{i} {‖ Z_{i} (P A_{i}) - \tilde{Z} ‖}_{2}^{2},

where

ω_{i}

is the weighting factor of the ith field point and constrained by

\sum_{i = 1}^{n} ω_{i} = 1

. Particularly, in this paper we adopt nine representative field points shown in Fig. 1 to establish the multi-objective cost function. Thus, the specific cost function D is:

D = ω_{1} D_{1} + ω_{2} D_{2} + ω_{3} D_{3} + ω_{6} D_{6} + ω_{7} D_{7} + ω_{8} D_{8} + ω_{11} D_{11} + ω_{12} D_{12} + ω_{13} D_{13} .

The MOSMO problem can be formulated as:

(\hat{J}, \hat{M}) = \underset{J, M}{\arg \min} D (J, M) .

Because the cost function D is the weighted sum of D_i, we can also use the steepest-descent method to solve MOSMO problem. According to [30], the approximate Pareto-front can be attained by solving the MOSMO problem under all kinds of weighting vectors. Considering the length of this paper, we introduce two representative weighting strategy. In the first strategy, all weighting factors are equal. This simple strategy, which is called mean-MOSMO, is very acceptable because all field points are equal in the lithography imaging process. Table 2 provides the pseudo-code of solving process for mean-MOSMO problem with the similar form as Table 1. In simulation, the original source pattern, the original mask pattern, and the parameters of lithography system are the same as described in Section 2.

Table 2. Pseudo-code of the mean-MOSMO algorithm.

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The second weighting strategy is considering the different impact for different field points. So in this strategy, we adaptively update the weighting factor in every iteration based on the PAE distribution of last iteration. We call this weighting strategy adaptive-MOSMO. Table 3 provides the pseudo-code of solving process about adaptive-MOSMO. Except weighting factor, other parameters are the same as Table 2.

Table 3. Pseudo-code of the adaptive-MOSMO algorithm.

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In order to demonstrate the superiority of the proposed MOSMO, we compare the lithography performance at full exposure field between the F11-SMO and two MOSMO methods. Figures 4(a)-4(l) respectively display the optimized source pattern, the optimized mask pattern, the printed image at center field point F3 and the printed image at marginal field point F11, which are obtained by different SMO methods for pattern #1.

Fig. 4 Optimization results and evaluations of the F11-SMO, mean-MOSMO, and adaptive-MOSMO for pattern #1. Left to right: optimized source pattern, optimized mask pattern, printed image at center FOV F3 and marginal FOV F11.

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Generally, the center field point corresponds to better imaging quality than marginal field point owing to lighter PA. However, for F11-SMO method, Figs. 4(c) and 4(d) show the PAE at marginal field point F11 is very little and the PAE at center field point F3 is very large. It is because the optimized source and mask by F11-SMO method are only suitable for field point F11. In addition, for both MOSMO methods, the difference between PAE of two field points is less than that of F11-SMO method, which shows the MOSMO effectively balance the lithography imaging for different FOVs. Particularly, compared to mean-MOSMO, the adaptive-MOSMO attains better and closer imaging quality for both FOVs.

To further analyze above phenomena, Fig. 5 demonstrates the comparison of PAE between F11-SMO, mean-MOSMO, and adaptive-MOSMO at each field point for pattern #1. Although the F11-SMO method has the better performance at some field points, such as F1, F6, and F11, the overall data still clearly shows that the proposed two MOSMO methods have more balanced and consistent PAE distribution at full field. Table 4 provides a quantitative comparison about the statistics of PAE distribution with different SMO methods. The sharp decline of PAE mean-value shows both two MOSMO methods integrally improve pattern fidelity at full field compared to F11-SMO. The less standard deviation and range mean the PAE distribution with MOSMO is more stable and consistent than that of F11-SMO method at full field, which is because all PAEs of nine field points are controlled simultaneously in the MOSMO flow. In addition, the adaptive-MOSMO method attains the less standard deviation and range than mean-MOSMO, which demonstrates the adaptive weighting strategy sufficiently reduces extreme PAE and ensures the consistence of lithography imaging. The comparison of PAE distribution proves the adaptive-MOSMO has superior lithography performance to other two SMO methods in imaging quality and imaging uniformity.

Fig. 5 The PAE with different methods at each field point for pattern #1.

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Table 4. The statistics of PAE distribution with different SMO methods for pattern #1

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Besides PAE, PW is the other important criterion to evaluate the lithography performance. The PW consists of all pairs of depth of focus (DOF) and exposure latitude (EL) that satisfy the linewidth and position measurement specification, typically ± 10% to ideal target pattern. In this paper, the global-field PW (GPW) for all field points shown in Fig. 1 and all calculated locations marked by the red lines shown in Fig. 2 is used to evaluate the lithography process stability and robustness in the following part. As shown in Fig. 6, the GPW is defined as the overlap of all PWs at each field point, which means the GPW is typically determined by the worst performance at one field. Thus, the proposed MOSMO should result in a better GPW than the F11-SMO in theory. Figure 6 illustrates the GPW obtained by F11-SMO method, mean-MOSMO method, and adaptive-MOSMO method. In Fig. 6(a), every general color curve represents the PW of one field point, and the bold blue curve represents overlap of all PWs of each field point for F11-SMO method. We can find the overlap area of multiple color curves in Fig. 6(a) is very small. That is why the F11-SMO method can’t attain large GPW. Figure 6(b) and 6(c) provide the relationship between PW and field point for mean-MOSMO and adaptive-MOSMO in a similar form with Fig. 6(a), respectively. In both Fig. 6(b) and 6(c), the area of bold blue curve is much larger than that of Fig. 6(a), which confirms the proposed MOSMO can enlarge lithography GPW at full field.

Fig. 6 The overlap of PW in full field with different SMO methods. (a) F11-SMO method; (b) mean-MOSMO method; (c) adaptive-MOSMO method.

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Based on Fig. 6, Fig. 7 compares the GPW of F11-SMO and MOSMO in the form of EL-DOF curve for pattern #1. The blue curve, red curve, green curve in Fig. 7 corresponds to Fig. 6(a), 6(b), 6(c), respectively. The size of blue curve is the smallest in three curves, which is tally with the result of Fig. 6. Particularly, the F11-SMO method can only attain quite low EL with 4%, which is unable to satisfy the industrial requirement with 5% EL. It is noted that the green curve corresponds the largest size, which shows adaptive-MOSMO attains larger GPW. Thus, according to simulation results of imaging uniform and GPW, adaptive-MOSMO is the better weighting strategy the mean-MOSMO.

Fig. 7 The comparison of EL-DOF curve with F11-SMO method (blue curve), mean-MOSMO method (red curve), and adaptive MOSMO method (green curve) for pattern #1.

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Runtime may be a main challenge for the proposed MOSMO. In theory, the calculation complexity of MOSMO method would increase nine times compared with that of F11-SMO. But we can reduce the runtime via parallel computing. In this paper, the simulation code is implemented in MATLAB, and the computations are carried out on an Intel(R) Xeon(R) E5-2650 v2 CPU, 2.30 GHz, 64GB of RAM. If only one computing core is used, the runtime of F11-SMO is 6064 seconds, and the runtime of mean-MOSMO is 20358s. In the case of parallel computing with 4 computing cores, the runtime of mean-MOSMO decreases to 13847s. The runtime of the proposed MOSMO would be further reduced via using more computing cores and more advanced algorithms.

To demonstrate the universality of the proposed MOSMO method, Fig. 8 provides the optimization results and evaluations of F11-SMO, mean-MOSMO, and adaptive-MOSMO for pattern #2 in a similar form with Fig. 4. Due to the lower complexity of pattern #2, the PAE of pattern #2 are smaller than the PAE of pattern #1 for all SMO methods. But the distribution feature between PAE and field point is still related to SMO method and independent of target pattern. The PAE with F11-SMO method at marginal FOV F11 is much larger than center FOV F3, and the MOSMO methods still result in relatively uniform PAE distribution for both two field points. It is noted that adaptive-MOSMO attains much closer imaging quality than mean-MOSMO for two field points, which reflects the superior of adaptive weighting strategy in balance lithography imaging.

Fig. 8 Optimization results and evaluations of the F11-SMO, mean-MOSMO, and adaptive MOSMO for pattern #2. Left to right: optimized source pattern, optimized mask pattern, printed image at center FOV F3 and marginal FOV F11.

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Figure 9 demonstrates the comparison of PAE between F11-SMO, mean-MOSMO, and adaptive-MOSMO at each field point for pattern #2 in a similar form as Fig. 5. Table 5 provides the quantitative comparison statistics of PAE distribution with different SMO methods. The comparison of PAE distribution still demonstrates the proposed MOSMO methods has superior imaging quality and imaging uniformity to the F11-SMO at full field for pattern #2, which shows the universality of the proposed MOSMO. Particularly, compared to mean-MOSMO method, the standard deviation of adaptive-MOSMO decrease 38.2% and the range of adaptive-MOSMO decrease 47.4%, which again reflects the superior of adaptive weighting strategy in balance lithography imaging.

Fig. 9 The PAE with different methods at each field point for pattern #2.

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Table 5. The statistics of PAE distribution with different SMO methods for pattern #2

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Figure 10 compares the GPW in the form of EL-DOF curve attained by F11-SMO method, mean-MOSMO method, and adaptive-MOSMO method for pattern #2. Due to the lower complexity of pattern #2, the pattern #2 has larger size of EL-DOF curve for all SMO methods than those of pattern #1. However, we can still find the proposed MOSMO methods result in larger GPW than F11-SMO method in Fig. 10. In addition, adaptive-MOSMO attains a little larger GPW compared to mean-MOSMO. Figure 9 and 10 demonstrate that one pair of mask and source optimized by proposed MOSMO methods, particularly adaptive-MOSMO, can completely realize the more consistent lithography image and larger process window even if the different PAs exist at full field.

Fig. 10 The comparison of EL-DOF curve with F11-SMO method (blue curve), and mean-MOSMO method (red curve), and adaptive-MOSMO (green curve) for pattern #2.

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4. Conclusion

This paper firstly develops an assigned PA aware SMO method and investigates the serious impact of PA on SMO. Because the lithography imaging and SMO results dependence of PA is associated with the field point, the assigned PA aware SMO is not applicable to other field points. To overcome this drawback, we innovate a MOSMO against uneven PA to balance lithography imaging at full exposure field. The weighted sum of PAE at each field point is designed as a novel multi-objective cost function in MOSMO method. Two weighting strategy, mean-MOSMO and adaptive-MOSMO are introduced, implemented, and discussed in this paper. Compared to the assigned PA aware SMO, extensive simulations demonstrate the proposed MOSMO can improve the consistence of PAE distribution and enlarge the GPW at full exposure field for different test patterns. Moreover, the simulation results demonstrate and prove the adaptive-MOSMO is the better weighting strategy. In conclusion, the proposed MOSMO can balance lithography performance and contribute to lithography process robustness improvement at full exposure field.

Appendix

For simplicity, some formulas derived in our previous papers [21,24] are directly used in the following derivation without detailed explanation.

The derivation of gradient for cost function D is as follows:

\nabla D = \sum_{i} ω_{i} \nabla D_{i} .

According to [21], the aerial image I could be rewrite as:

I_{i} (P A_{i}) = \frac{1}{J_{s u m}} \sum_{x_{s}} \sum_{y_{s}} (J (x_{s}, y_{s}) \sum_{p = x, y, z} {| H_{p}^{x_{s}, y_{s}} (P A_{i}) \otimes (B ⊙ M) |}^{2}) .

In this case, our previous work [24] has proven the gradient of D_i respect to the source pattern and mask pattern can be calculated as:

\nabla_{J} D_{i} = \frac{a \sin Ω_{J}}{J_{s u m}} ⊙ 1_{N \times 1}^{T} [\sum_{p = x, y, z} {| E_{p}^{w a f e r} |}^{2} ⊙ (\tilde{Z} - Z) ⊙ Z ⊙ (1 - Z)] 1_{N \times 1};

\nabla_{M} D_{i} = \frac{2 a \sin Ω_{M}}{J_{s u m}} ⊙ \sum_{x_{s}} \sum_{y_{s}} [J^{x_{s} y_{s}} \sum_{p = x, y, z} Re {B^{x_{s} y_{s}}^{*} ⊙ \begin{matrix} [H_{p}^{x_{s} y_{s}}^{* ο} \otimes Λ_{p} \end{matrix}]}] .

Thus, the gradient of D respect to the source pattern and mask pattern can be calculated as:

\nabla_{J} D = \sum_{i} ω_{i} \nabla_{J} D_{i} (P A_{i});

\nabla_{M} D = \sum_{i} ω_{i} \nabla_{M} D_{i} (P A_{i}) .

Funding

National Natural Science Foundation of China (NSFC) (61675026, 11627808); National Science and Technology Major Project (2017ZX02101006-001).

Acknowledgments

We gratefully acknowledge KLA-Tencor Corporation for providing academic use of PROLITH.

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1. Input: Set the original source parameter $Ω_{J}$ , the original mask parameter $Ω_{M}$ , the source step size $s_{J}$ , the mask step size $s_{M}$ , the maximum of iteration number $l_{S M O}$ , and $k = 0$ .
2. Update the source and mask parameters: *while* $k \leq l_{S M O}$
	$k \leftarrow k + 1$ ;
		Calculate the gradients $\nabla_{J} D_{i}^{(k - 1)}$ and $\nabla_{M} D_{i}^{(k - 1)}$ using Eq. (12) and Eq. (13), respectively; Update the source and mask parameters: $Ω_{J}^{(k)} = Ω_{J}^{(k - 1)} - s_{J} \nabla_{J} D_{i}^{(k - 1)},$ $Ω_{M}^{(k)} = Ω_{M}^{(k - 1)} - s_{M} \nabla_{M} D_{i}^{(k - 1)}$ ; Keep source pattern quadrant symmetry, and constrain mask pattern to 0 or 1 via a threshold t_m.
	*end*
3. Output the optimized source and mask parameters.

1. Input: Set the original source parameter $Ω_{J}$ , the original mask parameter $Ω_{M}$ , the source step size $s_{J}$ , the mask step size $s_{M}$ , the maximum of iteration number $l_{S M O}$ , the weighting factor $ω_{i}$ = 1/9, and $k = 0$ .
2. Update the source and mask parameters:
	while $k \leq l_{S M O}$
			$k \leftarrow k + 1$ ;
				Calculate the gradients $\nabla_{J} D^{(k - 1)}$ and $\nabla_{M} D^{(k - 1)}$ using Eq. (14) and Eq. (15), respectively; Update the source and mask parameters: $Ω_{J}^{(k)} = Ω_{J}^{(k - 1)} - s_{J} \nabla_{J} D^{(k - 1)},$ $Ω_{M}^{(k)} = Ω_{M}^{(k - 1)} - s_{M} \nabla_{M} D^{(k - 1)}$ ; Keep source pattern quadrant symmetry, and constrain mask pattern to 0 or 1 via a threshold t_m.
		*end*
3. Output the optimized source and mask parameters.

1. Input: Set the original source parameter $Ω_{J}$ , the original mask parameter $Ω_{M}$ , the source step size $s_{J}$ , the mask step size $s_{M}$ , the maximum of iteration number $l_{S M O}$ , and $k = 0$ .
2. Calculate the every PAE _i of ith field point using Eq. (3), set the original weighting factor $ω_{i} = \frac{P A E_{i}}{\sum_{i} P A E_{i}} .$
3. Update the source and mask parameters:
	while $k \leq l_{S M O}$
		$k \leftarrow k + 1$ ;
			Calculate the gradients $\nabla_{J} D^{(k - 1)}$ and $\nabla_{M} D^{(k - 1)}$ using Eq. (14) and Eq. (15), respectively; Update the source and mask parameters: $Ω_{J}^{(k)} = Ω_{J}^{(k - 1)} - s_{J} \nabla_{J} D^{(k - 1)},$ $Ω_{M}^{(k)} = Ω_{M}^{(k - 1)} - s_{M} \nabla_{M} D^{(k - 1)}$ ; Keep source pattern quadrant symmetry, and constrain mask pattern to 0 or 1 via a threshold t_m; Calculate the every $P A E_{i}^{(k)}$ of ith field point using Eq. (3); Update the weighting factor $ω_{i}^{(k + 1)} = \frac{P A E_{i}^{(k)}}{\sum_{i} P A E_{i}^{(k)}} .$
	*end*
4. Output the optimized source and mask parameters.

Statistics of PAE distribution	Mean-value	Standard deviation	Range
F11-SMO method	822	195	503
Mean-MOSMO method	690	103	263
Adaptive-MOSMO method	687	91	243

Statistics of PAE distribution	Mean-value	Standard deviation	Range
F11-SMO method	338	141	392
Mean-MOSMO method	243	68	206
Adaptive-MOSMO method	229	42	126

Multi-objective lithographic source mask optimization to reduce the uneven impact of polarization aberration at full exposure field

Abstract

1. Introduction

2. The impact of PA on SMO

3. MOSMO model, algorithm, and simulation

4. Conclusion

Appendix

Funding

Acknowledgments

References

Cited By

Figures (10)

Tables (5)

Equations (15)

Optics Express