Inverse pupil wavefront optimization for immersion lithography

Chunying Han; Yanqiu Li; Lisong Dong; Xu Ma; Xuejia Guo

doi:10.1364/AO.53.006861

Applied Optics
Vol. 53,
Issue 29,
pp. 6861-6871
(2014)
•https://doi.org/10.1364/AO.53.006861

Inverse pupil wavefront optimization for immersion lithography

Chunying Han, Yanqiu Li, Lisong Dong, Xu Ma, and Xuejia Guo

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Chunying Han, Yanqiu Li, Lisong Dong, Xu Ma, and Xuejia Guo, "Inverse pupil wavefront optimization for immersion lithography," Appl. Opt. 53, 6861-6871 (2014)

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Abstract

As the critical dimension of integrated circuits is continuously shrunk, thick mask induced aberration (TMIA) cannot be ignored in the lithography image process. Recently, a set of pupil wavefront optimization (PWO) approaches has been proposed to compensate for TMIA, based on a wavefront manipulator in modern scanners. However, these prior PWO methods have two intrinsic drawbacks. First, the traditional methods fell short in building up the analytical relationship between the pupil wavefront and the cost function, and used time-consuming algorithms to solve for the PWO problem. Second, in traditional methods, only the spherical aberrations were optimized to compensate for the focus exposure matrix tilt and best focus shift induced by TMIA. Thus, the degrees of freedom were limited during the optimization procedure. To overcome these restrictions, we build the analytical relationship between the pupil wavefront and the cost function based on Abbe vector imaging theory. With this analytical model and the Fletcher–Reeves conjugate-gradient algorithm, an inverse PWO method is innovated to balance the TMIA including 37 Zernike terms. Simulation results illustrate that our approach significantly improves image fidelity within a larger process window. This demonstrates that TMIA is effectively compensated by our inverse PWO approach.

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1.	Initialize $k = 0$ , the starting Zernike coefficients $c_{i}^{(0)} = 0.000001$ , the pupil wavefront step size $s_{c}$ , the maximum iteration number $l_{pwo}$ , and $P^{(0)} = - \nabla F (c_{i}^{(0)})$ .
2.	Update the Zernike coefficients $c_{i}^{(k + 1)}$ :
	while $k \leq l_{pwo}$ & $d (Z^{(k)} - \tilde{Z}) > 0$
	$k \leftarrow k + 1$ ;
	Calculate the gradient, i.e., $\nabla F (c_{i}^{(k)})$ , using Eq. (10);
	$Calculate β^{(k)} : β^{(k)} = \frac{{‖ \nabla F (c_{i}^{(k)}) ‖}_{2}^{2}}{{‖ \nabla F (c_{i}^{(k - 1)}) ‖}_{2}^{2}};$
	$Update P^{(k)} : P^{(k)} = - \nabla F (c_{i}^{(k)}) + β^{(k)} P^{(k - 1)};$ ;
	$Update c_{i}^{(k)} : c_{i}^{(k + 1)} = c_{i}^{(k)} + s_{c} P^{(k)}$ .
	end
3.	Calculate the optimized wavefront manipulator $\hat{W}$ by the optimized Zernike coefficients ${\hat{c}}_{i}$ using Eq. (3).

			PWO Algorithm
Target	Criterion	Initial	SD	FR-CG	FR-CG+Reg
Line-Space	PAE	1191	601	553	561
	CDE (nm)	14.4	6.3	5.7	5.8
	PLE (nm)	2.7	2.5	2.5	2.5
	DOF (nm)	22	97	100	105
	$Δ BF$ (nm)	97	7	4	8
	Runtime (s)	—	5704	2519	2827
Complex	PAE	1533	1184	958	978
	CDE (nm)	6.6	5.8	4.9	4.9
	PLE (nm)	5.8	4.6	4.2	4.2
	DOF (nm)	19	74	116	119
	$Δ BF$ (nm)	71	32	21	12
	Runtime (s)	—	9508	3608	4023

1.	Initialize $k = 0$ , the starting Zernike coefficients $c_{i}^{(0)} = 0.000001$ , the pupil wavefront step size $s_{c}$ , the maximum iteration number $l_{pwo}$ , and $P^{(0)} = - \nabla F (c_{i}^{(0)})$ .
2.	Update the Zernike coefficients $c_{i}^{(k + 1)}$ :
	while $k \leq l_{pwo}$ & $d (Z^{(k)} - \tilde{Z}) > 0$
	$k \leftarrow k + 1$ ;
	Calculate the gradient, i.e., $\nabla F (c_{i}^{(k)})$ , using Eq. (10);
	$Calculate β^{(k)} : β^{(k)} = \frac{{‖ \nabla F (c_{i}^{(k)}) ‖}_{2}^{2}}{{‖ \nabla F (c_{i}^{(k - 1)}) ‖}_{2}^{2}};$
	$Update P^{(k)} : P^{(k)} = - \nabla F (c_{i}^{(k)}) + β^{(k)} P^{(k - 1)};$ ;
	$Update c_{i}^{(k)} : c_{i}^{(k + 1)} = c_{i}^{(k)} + s_{c} P^{(k)}$ .
	end
3.	Calculate the optimized wavefront manipulator $\hat{W}$ by the optimized Zernike coefficients ${\hat{c}}_{i}$ using Eq. (3).

			PWO Algorithm
Target	Criterion	Initial	SD	FR-CG	FR-CG+Reg
Line-Space	PAE	1191	601	553	561
	CDE (nm)	14.4	6.3	5.7	5.8
	PLE (nm)	2.7	2.5	2.5	2.5
	DOF (nm)	22	97	100	105
	$Δ BF$ (nm)	97	7	4	8
	Runtime (s)	—	5704	2519	2827
Complex	PAE	1533	1184	958	978
	CDE (nm)	6.6	5.8	4.9	4.9
	PLE (nm)	5.8	4.6	4.2	4.2
	DOF (nm)	19	74	116	119
	$Δ BF$ (nm)	71	32	21	12
	Runtime (s)	—	9508	3608	4023

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Equations (26)

Applied Optics