Abstract

Inverse lithography technology (ILT) treats photomask design for microlithography as an inverse mathematical problem. We show how the inverse lithography problem can be addressed as an obstacle reconstruction problem or an extended nonlinear image restoration problem, and then solved by a level set time-dependent model with finite difference schemes. We present explicit detailed formulation of the problem together with the first-order temporal and second-order spatial accurate discretization scheme. Experimental results show the superiority of the proposed level set-based ILT over the mainstream gradient methods.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, Bellingham, WA, 2001).
  2. F. Schellenberg, "Resolution enhancement technology: the past, the present, and extensions for the future," Proc. SPIE 5377, 1-20 (2004).
    [CrossRef]

2004 (1)

F. Schellenberg, "Resolution enhancement technology: the past, the present, and extensions for the future," Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

Proc. SPIE (1)

F. Schellenberg, "Resolution enhancement technology: the past, the present, and extensions for the future," Proc. SPIE 5377, 1-20 (2004).
[CrossRef]

Other (1)

A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, Bellingham, WA, 2001).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Simulation of lithographic imaging with different mask patterns computed using different spatial schemes. The first column denotes the input U(x), the second column I aerial(x), and the third column I(x). Row (a) uses the target circuit pattern I 0 as input. Row (b) uses the pattern derived by our level set algorithm, with first-order temporal accuracy and ENO1 spatial accuracy. Row (c), (d) and (e) are similar to (b), but with ENO2, ENO3 and WENO spatial accuracy, respectively.

Fig. 2.
Fig. 2.

Simulation of lithographic imaging with different mask patterns computed using different temporal schemes. The first column denotes the input U(x), the second column I aerial(x), and the third column I(x). Row (a) uses the target circuit pattern I 0 as input. Row (b) uses the pattern derived by our level set algorithm, with ENO2 spatial and first-order temporal accuracy. Row (c) and (d) are similar to (b), but with second-order and third-order temporal accuracy, respectively.

Fig. 3.
Fig. 3.

Simulation of lithographic imaging with different mask patterns computed using gradient method and level set approach. The first column denotes the input U(x), the second column I aerial(x), and the third column I(x). Row (a) uses the target circuit pattern I0 as input. Row (b) uses the pattern derived by gradient method. Row (c) uses the target pattern derived by our approach with ENO2 spatial and first-order temporal accuracy. Row (d) uses the target pattern derived by gradient method with target circuit pattern in Fig 1.

Tables (3)

Tables Icon

Table 1. Normalized Computation Time (against that in Fig. 1(c) using ENO2) and Pattern Error (pixel difference) in Fig. 1

Tables Icon

Table 2. Normalized Computation Time (against that in Fig. 2(b) using first-order temporal accuracy) and Pattern Error (pixel difference) in Fig. 2

Tables Icon

Table 3. Table 3. Normalized Computation Time (computation time in Fig. 3(b) and Fig. 3(d) using gradient method normalized against that in Fig. 3(c) and Fig. 1(c) using the proposed method, respectively) and Pattern Error (pixel difference) in Fig. 3

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

I ( x ) = { U ( x ) } ,
Û ( x ) = arg min U ( x ) d ( I 0 ( x ) , { U ( x ) } ) ,
I aerial ( x ) = H ( x ) * U ( x ) 2 ,
sig ( U ( x ) ) = 1 1 + e a ( U ( x ) t r )
I ( x ) = { U ( x ) } = sig ( I aerial ( x ) ) = sig ( H ( x ) * U ( x ) 2 ) .
𝓡 1 ( U ) = Δ U L 2 = Ω ( U x ) 2 + ( U Y ) 2 d x ,
󑓡 2 ( U ) = U L 2 = Ω ( 2 U x 2 + 2 U Y 2 ) 2 d x ,
𝓡 3 ( U ) = Ω U d x = Ω ( U x ) 2 + ( U y ) 2 d x .
minimize Ω U d x
subject to Ω ( sig ( H * U 2 ) I 0 ) 2 d x = ε ,
λ · ( U U ) + α ( x ) = 0 ,
1 2 ( Ω ( sig ( H * U 2 ) I 0 ) 2 d x Ω ε ) = 0 ,
α ( x ) = 1 2 U ( sig ( H * U 2 ) I 0 ) 2
= α { H * [ ( I 0 sig ( H * U ) ) sig ( H * U ) ( 1 ( H * U ) ) ( H * U ) ] } ,
min U Ω ( λ U + 1 2 ( sig ( H * U 2 ) I 0 ) 2 ) d x ,
U t = α ( x , t ) + λ · ( U U ) ,
U t = U α ( x , t ) + λ U · ( U U ) .
U ( x ) = { U int for { x : ϕ ( x ) < 0 } U int for { x : ϕ ( x ) > 0 }
F ( U ) = 1 2 𝒯 ( U ) I 0 2 .
δ ϕ + ∇ϕ · δ x = 0 ,
δ x = α ( x , t ) ϕ ϕ .
α ( x , t ) = J ( U ) T ( 𝓣 ( U ) I 0 ) ,
ϕ t = ϕ α ( x , t ) .

Metrics