## Level-set-based inverse lithography for photomask synthesis

Optics Express, Vol. 17, Issue 26, pp. 23690-23701 (2009)

http://dx.doi.org/10.1364/OE.17.023690

Acrobat PDF (522 KB)

### 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

## 1. Introduction

### 1.1. Inverse lithography

3. L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop **45(5)**, 651–665 (2001).
[CrossRef]

4. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE **6607**, 660739 (2007).
[CrossRef]

5. S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express **16(19)**, 14746–14761(2008).
[CrossRef]

6. E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express **17(15)**, 12259–12268 (2009).
[CrossRef]

7. A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE **7275**, 727509 (2009).
[CrossRef]

9. Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE **1463**, 382–399 (1991).
[CrossRef]

10. S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. **4(9)**, 1252–1257 (1995).
[CrossRef]

11. Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A **11(9)**, 2438–2452 (1994).
[CrossRef]

12. Y. Granik, “Solving inverse problems of optical microlithography,” *Proc. SPIE* **5754**, 506–526 (2004).
[CrossRef]

13. Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” *Proc. SPIE* **6349**, 63494R (2006).
[CrossRef]

14. A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE **5674**, 114–127 (2005).
[CrossRef]

15. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. **16(3)**, 774–788 (2007).
[CrossRef]

5. S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express **16(19)**, 14746–14761(2008).
[CrossRef]

18. X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express **16(24)**, 20126–20141 (2008).
[CrossRef]

20. A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” *Proc. SPIE* **6520**, 65202Q (2007).
[CrossRef]

22. N. Jia, A. K. Wong, and E. Y. Lam, “Robust photomask design with defocus variation using inverse synthesis,” *Proc. SPIE* **7140**, 71401W (2008).
[CrossRef]

4. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE **6607**, 660739 (2007).
[CrossRef]

23. L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” *Proc. SPIE* **6924**, 69240T (2008).
[CrossRef]

### 1.2. Level set method

24. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. **169(2)**, 463–502 (2001).
[CrossRef]

25. D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. **120(1)**, 128–144 (1995).
[CrossRef]

28. J. A. Sethian and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf. **10**, 167–184 (1997).
[CrossRef]

30. F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. **1**, 17–33 (1996).
[CrossRef]

31. S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. **171(1)**, 272–288 (2001).
[CrossRef]

32. A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. **22**, 387–405 (2000).
[CrossRef]

## 2. The Constrained level set time-dependent model formulation

### 2.1. Mathematical model

**x**denotes spatial coordinates (

*x,y*), and

*𝓣*{·} maps the input intensity function

*U*(

**x**) to the output intensity function

*I*(

**x**). Due to the lowpass nature of the optical lithography imaging system,

*I*(

**x**) is typically a blurred version of

*U*(

**x**). Suppose we denote the desired circuit pattern as

*I*

_{0}(

**x**). The objective of inverse lithography is to find a predistorted input intensity function

*U*̂(

**x**) which minimizes its distance with the desired output, i.e.,

*d*(·, ·) is an appropriately defined distance metric, such as the

*ℓ*

_{2}norm.

*H*(

**x**) is taken as the inverse Fourier transform of a disc function [33]. The resist effects can be approximated using a logarithmic sigmoid function [15

15. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. **16(3)**, 774–788 (2007).
[CrossRef]

*a*being the steepness of the sigmoid and

*t*being the threshold. Putting together, we can write the image formation equation for a coherent imaging system as

_{r}**x**whenever there is no ambiguity.

### 2.2. Optimization framework

*𝓡*. This functional takes as its argument an image

*U*and returns a scalar that quantifies the quality of the pattern. Smaller values of

*𝓡*correspond to more desirable images.

*𝓡*

_{1},

*𝓡*

_{2}, and

*𝓡*

_{3}below are common examples of the regularization (note that ∇ denotes the gradient, Δ denotes the Laplacian [32

32. A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. **22**, 387–405 (2000).
[CrossRef]

^{2}which contains

*U*):

*𝓡*

_{3}above is called the total variation (TV) regularization. Quadratic penalty terms and complexity penalty terms can also be used [15

15. A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. **16(3)**, 774–788 (2007).
[CrossRef]

*U*, thus allowing the recovery of its edges, we opt to use it in this work. The inverse lithography

*ε*is the parameter controlling the approximation of the aerial image and the hard thresholding to the desired circuit pattern, with a smaller

*ε*indicating better approximation. The Lagrangian of the above expression is

*U*, and its Euler-Lagrange equations, with homogeneous Neumann boundary conditions for

*U*, are

*α*(

**x**) is defined as

*α*(

**x**) in Eq. (12) requires a few steps and can be found in [5

5. S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express **16(19)**, 14746–14761(2008).
[CrossRef]

**16(3)**, 774–788 (2007).
[CrossRef]

*λ*to be known, the equivalent unconstrained problem of Eq. (9) can be written as

### 2.3. Time-dependent scheme

*t*denoting time and

*α*(

**x**) also updated to a time-dependent function. The solution procedure is a parabolic equation with time as an evolution parameter. Equation (14) moves each level curve of

*U*normal to itself, with velocity equal to the curvature of the level surface divided by the magnitude of the gradient of

*U*. The constraints are included in

*α*(

*x, t*) to prevent distortion and to obtain a nontrivial steady state.

*U*and regularizes the parabolic term in a nonlinear way, by multiplying the right hand side of Eq. (14) with the magnitude of the gradient [24

24. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. **169(2)**, 463–502 (2001).
[CrossRef]

32. A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. **22**, 387–405 (2000).
[CrossRef]

**22**, 387–405 (2000).
[CrossRef]

*U*normal to itself with velocity equal to -

*α*(

*x, t*), combined with a curvaturedriven flow characterized by velocity

**22**, 387–405 (2000).
[CrossRef]

### 2.4. Obstacle reconstruction

*U*

_{int}=1 and

*U*

_{ext}=0 if we are dealing with binary masks. Now we can define the inverse lithography problem as finding

*ϕ*(x) such that

*𝓣*(

*U*)≈

*I*

_{0}. Solving this with a least squares fit to the approximation is equivalent to seeking the minimizer of

*U*where

*U*=

*U*

_{int}is governed by the zero level set of

*ϕ*, namely,

*ϕ*(

**x**)=0. By taking the variation of the equation

*ϕ*(

**x**)=0, we find

*δϕ*and

*δ*

**x**represent the small variation of

*ϕ*and

**x**, respectively. If

*ϕ*depends on both

**x**and time

*t*, we associate the evolution of the subregions in

*U*with

*ϕ*(

**x**,

*t*). The minimum requirement for

*δϕ*(

**x**,

*t*) is that

*F*(

*U*) be a decreasing function of time, i.e., the resulting variation of

*F*(

*U*) or

*δF*(

*U*) is negative. Assuming that each point moves perpendicular to the surface with normal velocity

*α*(

**x**,

*t*),

*δ*

**x**satisfies

30. F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. **1**, 17–33 (1996).
[CrossRef]

*α*(

*x, t*) as

*J*(

*U*) is the Jacobian of

*𝓣*(

*U*) at

*U*, will produce a small variation

*δϕ*that reduces

*F*(

*U*), which leads to

*α*(

**x**,

*t*) is exactly the same as Eq. (15). The above equation is different from it only without the convection part, and Eq. (15) is easily derived from Eq. (20) by adding a normal speed which is λ times the curvature of the level surface.

*U*, which is undesirable in photomasks [34]. Therefore, we will use Eq. (20) to solve the inverse lithography problem in this paper. Besides, when enforcing mask manufacturability rules, the inverse lithography solution becomes a constrained optimization if we regularize the inverse problem, and using the above equation will be more convenient in terms of describing the geometrical constraints, for example, regularization terms including, but not limited to, mean curvature regularization (TV norm) and average mean curvature affect the geometry of the contour, the former minimizes the length of the contours while the latter minimizes the length of the boundaries keeping the total area of the enclosed regions fixed, giving preference to smoother contours and contours enclosing larger regions.

*α*(

**x**,

*t*) is governed by the convolution operation. Therefore, the algorithmic complexity is

*𝒪*(

*N*

^{2}log

*N*), assuming an

*N*×

*N*image.

### 2.5. Numerical schemes

*ϕ*and

*α*are defined at every grid point on the Cartesian grid, the first-order derivatives in space and time of Eq. (20) can be approximated using finite difference techniques. An appropriate difference scheme should be designed for our level set approach. It is natural to use a higher order accurate difference scheme, for example, the 2–4 implicit scheme in [35], if we want to achieve a better input mask. However, more accurate difference scheme requires more computation, and therefore a tradeoff should be made between accuracy and computation time.

*ϕ*using a first-order accurate forward difference and a backward difference, or a second-order accurate central difference, together with upwind differencing [36] which chooses an approximation to the spatial derivatives based on the sign of

*α*. However, we can improve the first-order accurate upwind scheme significantly by a more accurate approximation of the spatial derivative of

*ϕ*. Harten et al. develop the essentially nonoscillatory (ENO) polynomial interpolation method that uses the smoothest possible polynomial interpolation to find

*ϕ*, and then differentiates it to get the spatial derivatives [37

37. A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. **71**, 231–303 (1987).
[CrossRef]

38. C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. **77(2)**, 439–471 (1988).
[CrossRef]

39. C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. **83(1)**, 32–78 (1989).
[CrossRef]

24. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. **169(2)**, 463–502 (2001).
[CrossRef]

**22**, 387–405 (2000).
[CrossRef]

38. C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. **77(2)**, 439–471 (1988).
[CrossRef]

## 3. Experimental Results

*λ*=193nm,

*NA*=0.85, resolution=10nm/pixel, thresholding

*t*=0.3, and therefore the same PSF

_{r}*H*(

**x**) for various experiments. The optimization approach stops after 50 iterations.

*U*(

**x**), the second column is the corresponding aerial image

*I*

_{aerial}(

**x**), and the third column is the binary output circuit pattern

*I*(

**x**). The various rows denote different input patterns. In (a), we use the target circuit pattern as input, while in (b)–(e), we design the input mask using the target circuit pattern as input to our level set algorithms. All of them use a first-order temporal accuracy, but the difference is that (b) uses an ENO1 spatial accuracy, (c) uses ENO2, (d) uses ENO3 and (e) uses WENO. All images are of size 170×140 pixels, while

*H*(

**x**) is a jinc function of size 51×51.

**16(19)**, 14746–14761(2008).
[CrossRef]

**16(3)**, 774–788 (2007).
[CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | A. K.-K. Wong, |

2. | F. Schellenberg, “Resolution enhancement technology: the past, the present, and extensions for the future,” Proc. SPIE |

3. | L. W. Liebmann, S. M. Mansfield, A. K. Wong, M. A. Lavin, W. C. Leipold, and T. G. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Develop |

4. | L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE |

5. | S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express |

6. | E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express |

7. | A. K. Wong and E. Y. Lam, “The nebulous hotspot and algorithm variability,” Proc. SPIE |

8. | Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” in |

9. | Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” Proc. SPIE |

10. | S. Sherif, B. Saleh, and R. De Leone, “Binary image synthesis using mixed linear integer programming,” IEEE Trans. Image Process. |

11. | Y. C. Pati and T. Kailath, “Phase-shifting masks for microlithography: automated design and mask requirements,” J. Opt. Soc. Am. A |

12. | Y. Granik, “Solving inverse problems of optical microlithography,” |

13. | Y. Granik, K. Sakajiri, and S. Shang, “On objectives and algorithms of inverse methods in microlithography,” |

14. | A. Poonawala and P. Milanfar, “Prewarping techniques in imaging: applications in nanotechnology and biotechnology,” Proc. SPIE |

15. | A. Poonawala and P. Milanfar, “Mask design for optical microlithography: an inverse imaging problem,” IEEE Trans. Image Process. |

16. | S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in |

17. | X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express |

18. | X. Ma and G. R. Arce, “PSM design for inverse lithography with partially coherent illumination,” Opt. Express |

19. | V. Singh, B. Hu, K. Toh, S. Bollepalli, S. Wagner, and Y. Borodovsky, “Making a trillion pixels dance,” |

20. | A. Poonawala, Y. Borodovsky, and P. Milanfar, “ILT for double exposure lithography with conventional and novel materials,” |

21. | N. Jia, A. K. Wong, and E. Y. Lam, “Regularization of inverse photomask synthesis to enhance manufacturability,” |

22. | N. Jia, A. K. Wong, and E. Y. Lam, “Robust photomask design with defocus variation using inverse synthesis,” |

23. | L. Pang, G. Dai, T. Cecil, T. Dam, Y. Cui, P. Hu, D. Chen, K. Baik, and D. Peng, “Validation of inverse lithography technology (ILT) and its adaptive SRAF at advanced technology nodes,” |

24. | S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. |

25. | D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography I: algorithms and two-dimensional simulations,” J. Comput. Phys. |

26. | D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography II: three-dimensional simulations,” J. Comput. Phys. |

27. | D. Adalsteinsson and J. A. Sethian, “A unified level set approach to etching, deposition and lithography III: complex simulations and multiple effects,” J. Comput. Phys. |

28. | J. A. Sethian and D. Adalsteinsson, “An overview of level set methods for etching, deposition, and lithography development,” IEEE Trans. Semicond. Manuf. |

29. | S. Osher and N. Paragios, |

30. | F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Contröle Optim. Calc. Var. |

31. | S. Osher and F. Santosa, “Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum,” J. Comput. Phys. |

32. | A. Marquina and S. Osher, “Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal,” SIAM J. Sci. Comp. |

33. | A. K.-K. Wong, |

34. | T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” in |

35. | Y. Shen, N. Wong, and E. Y. Lam, “Interconnect thermal simulation with higher order spatial accuracy,” in |

36. | S. Osher and R. Fedkiw, |

37. | A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, “Uniformly high order accurate essentially non-oscillatory schemes III,” J. Comput. Phys. |

38. | C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes,” J. Comput. Phys. |

39. | C. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock-capturing schemes II,” J. Comput. Phys. |

40. | M. Minoux, |

**OCIS Codes**

(110.3960) Imaging systems : Microlithography

(110.5220) Imaging systems : Photolithography

(110.1758) Imaging systems : Computational imaging

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: October 30, 2009

Revised Manuscript: December 4, 2009

Manuscript Accepted: December 4, 2009

Published: December 10, 2009

**Citation**

Yijiang Shen, Ngai Wong, and Edmund Y. Lam, "Level-set-based inverse lithography for photomask synthesis," Opt. Express **17**, 23690-23701 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-23690

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### References

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

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