## Pixel-based OPC optimization based on conjugate gradients |

Optics Express, Vol. 19, Issue 3, pp. 2165-2180 (2011)

http://dx.doi.org/10.1364/OE.19.002165

Acrobat PDF (928 KB)

### Abstract

Optical proximity correction (OPC) methods are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. In pixel-based OPC (PBOPC), the mask is divided into small pixels, each of which is modified during the optimization process. Two critical issues in PBOPC are the required computational complexity of the optimization process, and the manufacturability of the optimized mask. Most current OPC optimization methods apply the steepest descent (SD) algorithm to improve image fidelity augmented by regularization penalties to reduce the complexity of the mask. Although simple to implement, the SD algorithm converges slowly. The existing regularization penalties, however, fall short in meeting the mask rule check (MRC) requirements often used in semiconductor manufacturing. This paper focuses on developing OPC optimization algorithms based on the conjugate gradient (CG) method which exhibits much faster convergence than the SD algorithm. The imaging formation process is represented by the Fourier series expansion model which approximates the partially coherent system as a sum of coherent systems. In order to obtain more desirable manufacturability properties of the mask pattern, a MRC penalty is proposed to enlarge the linear size of the sub-resolution assistant features (SRAFs), as well as the distances between the SRAFs and the main body of the mask. Finally, a projection method is developed to further reduce the complexity of the optimized mask pattern.

© 2011 Optical Society of America

## 1. Introduction

1. A. K. Wong, *Resolution Enhancement Techniques* (SPIE Press, 2001), Vol. 1. [CrossRef]

2. X. Ma and G. R. Arce, *Computational Lithography*, Wiley Series in Pure and Applied Optics, 1st ed. (John Wiley and Sons, 2010). [CrossRef]

*λ*is the wavelength,

*NA*is the numerical aperture, and

*k*is the process constant which can be minimized through RET methods [3–6

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

1. A. K. Wong, *Resolution Enhancement Techniques* (SPIE Press, 2001), Vol. 1. [CrossRef]

2. X. Ma and G. R. Arce, *Computational Lithography*, Wiley Series in Pure and Applied Optics, 1st ed. (John Wiley and Sons, 2010). [CrossRef]

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

9. Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. **5**(2), 138–152 (1992). [CrossRef]

10. A. Erdmann, R. Farkas, T. Fühner, B. Tollküuhn, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE **5377**, 646–657 (2004). [CrossRef]

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

12. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlithogr., Microfabr., Microsyst. **5**, 043002 (2006). [CrossRef]

13. A. Poonawala and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. **16**(3), 774–788 (2007). [CrossRef] [PubMed]

14. X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A **25**(12), 2960–2970 (2008). [CrossRef]

15. X. Ma and G. R. Arce, “Binary mask optimization for forward lithography based on boundary layer model in coherent systems,” J. Opt. Soc. Am. A **26**(7), 1687–1695 (2009). [CrossRef]

16. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express **17**(7), 5783–5793 (2009). [CrossRef] [PubMed]

14. X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A **25**(12), 2960–2970 (2008). [CrossRef]

14. X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A **25**(12), 2960–2970 (2008). [CrossRef]

18. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. **21**(15), 2770–2777 (1982). [CrossRef] [PubMed]

13. A. Poonawala and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. **16**(3), 774–788 (2007). [CrossRef] [PubMed]

19. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express **15**, 15066–15079 (2007). [CrossRef] [PubMed]

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

21. K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE **6283**, 62830O (2006). [CrossRef]

22. K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE **6730**, 67304F (2007). [CrossRef]

## 2. Partially Coherent Imaging Model

### 2.1. Partially Coherent Illumination System

*λ*is placed at the focal plane of the first condenser, illuminating the mask. Common illuminations sources include dipole, quadrupole and annular shapes, all introducing partial coherence. The image of the photomask is formed by the projection optics onto the wafer [1

1. A. K. Wong, *Resolution Enhancement Techniques* (SPIE Press, 2001), Vol. 1. [CrossRef]

18. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. **21**(15), 2770–2777 (1982). [CrossRef] [PubMed]

2. X. Ma and G. R. Arce, *Computational Lithography*, Wiley Series in Pure and Applied Optics, 1st ed. (John Wiley and Sons, 2010). [CrossRef]

**r**= (

*x*,

*y*) is the coordinate on the image plane located on the wafer.

**r**= (

_{1}*x*

_{1},

*y*

_{1}) and

**r**= (

_{2}*x*

_{2},

*y*

_{2}) are the coordinates on the object plane located on the mask.

*M*(

**r**) is the mask pattern,

*γ*(

**r**−

_{1}**r**) is the complex degree of coherence, and

_{2}*h*(

**r**) represents the amplitude impulse response of the optical system. The complex degree of coherence

*γ*(

**r**−

_{1}**r**) is generally a complex number, whose magnitude represents the extent of optical interaction between two spatial locations

_{2}**r**= (

_{1}*x*

_{1},

*y*

_{1}) and

**r**= (

_{2}*x*

_{2},

*y*

_{2}) of the light source [1

1. A. K. Wong, *Resolution Enhancement Techniques* (SPIE Press, 2001), Vol. 1. [CrossRef]

23. B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A **13**(10, 2086–2090) (1996). [CrossRef]

18. B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. **21**(15), 2770–2777 (1982). [CrossRef] [PubMed]

### 2.2. Fourier Series Expansion Model

*A*defined by

*γ*(

**r**) needed are those inside the square area

*A*defined by

_{γ}*x*,

*y ∈*[−

*D*,

*D*]. Applying the 2-D Fourier series expansion,

*γ*(

**r**) can be rewritten as and where

*ω*

_{0}=

*π*/

*D*,

**m**= (

*m*,

_{x}*m*),

_{y}*m*and

_{x}*m*are integers, and · is the inner-product operation. Substituting Eqn. (2) into Eqn. (1), the light intensity on the wafer is given by where where It is observed from Eqn. (4) that PCI leads to the superposition of coherent systems. Since the Fourier series expansion model is based on direct discretization of the Hopkins imaging model, they have the same accuracy. For the annular illumination, the complex degree of coherence is where

_{y}*D*and

_{cl}*D*are the coherent lengths of the inner and outer circles respectively.

_{cu}*h*(

**r**) is defined as the Fourier transform of the circular lens aperture with cutoff frequency

*NA*/

*λ*[24, 25]; therefore,

## 3. Inverse Lithography Optimization

*M*(

*x*,

*y*) be the input binary mask to an optical lithography system

*T*{·}, with a partially coherent illumination. The PCI optical system is approximated by a Hopkins imaging model. The effect of the photoresist is modeled by a hard threshold operation. The output pattern is denoted as

*Z*(

*x*,

*y*) =

*T*{

*M*(

*x*,

*y*)}. Given a

*N*×

*N*desired output pattern

*Z̃*(

*x*,

*y*), the goal of OPC design is to find the optimized

*M*(

*x*,

*y*) called

*M̂*(

*x*,

*y*) such that the distance is minimized, where

*d*(·,·) is the square of the

*L*

_{2}norm. The OPC optimization problem can thus be formulated as the search of

*M̂*(

*x*,

*y*) over the

*N*×

*N*real space ℜ

^{N×N}such that

*t*is the process threshold, and

_{r}*a*dictates the steepness of the sigmoid function.

- The
*M*_{N×N}matrix represents the mask pattern with a*N*^{2}× 1 equivalent raster scanned vector representation, denoted as*m*. - A convolution matrix
*H*is a^{m}*N*^{2}×*N*^{2}matrix with an equivalent two-dimensional filter*h*.^{m} - The desired
*N*×*N*binary output pattern is denoted as*Z̃*. It is the desired light distribution sought on the wafer. Its vector representation is denoted as*Z̃*. - The hard threshold version of
*Z*is the binary output pattern denoted as*Z*. Its equivalent vector is denoted as_{b}*z*, with all entries constrained to 0 or 1._{b} - The binary optimized mask
*M̂*is the quantization of_{b}*M̂*. Its equivalent vector is denoted as*M̂*, with all entries constrained to 0 or 1._{b}

*ith*entry in this vector can be represented as where

*i*,

*jth*entry of the filter. In the optimization process,

*M̂*is searched to minimize the square of the

*L*

_{2}norm of the difference between

*z*and

*Z̃*. Therefore, where the cost function

*F*(·) is defined as: where ‖ · ‖

_{2}is the

*L*

_{2}norm and

*z*

*is represented in Eqn. (14). In order to reduce the above bound-constrained optimization problem to an unconstrained optimization problem, we adopt the parametric transformation [13*

_{i}13. A. Poonawala and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. **16**(3), 774–788 (2007). [CrossRef] [PubMed]

*θ*

*∈ (−∞, ∞) and*

_{j}*m*

*∈ [0,1]. Defining the vector*

_{j}*θ*= [

*θ*

_{1},...,

*θ*

_{N2}]

*, the optimization problem is formulated as*

^{T}27. R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. **7**, 149–154 (1964). [CrossRef]

*P*

_{0},

*P*

_{1},...,

*P*} with some properties, along which the cost function is successively minimized. Each vector

_{l}*P*is updated from

_{i}*P*

_{i}_{−1}, and is the linear combination of the past gradients. Thus, each new minimizing direction

*P*inherits the useful information from its ancestor. In the past, a variety of CG methods have been proposed, such as Fletcher-Reeves method, Polak-Ribière method, and so on [17]. In this paper, we adopt the Fletcher-Reeves method. The comparison among these CG algorithms and the method to select the proper algorithm are out of the scope of this paper.

_{i}^{N×N}is the equivalent matrix of

*θ*. The gradients Δ

*F*(Θ) can be calculated as follows [14

**25**(12), 2960–2970 (2008). [CrossRef]

*F*(Θ) ∈ ℜ

^{N×N}, ⊙ is the element-by-element multiplication operator, (

*h*

**)° rotates**

^{m}*h*

**by 180° degrees in both row and column directions, * is the conjugate operation, and**

^{m}**1**

_{N×N}∈ ℜ

^{N×N}has all entries equal to 1. Assuming Θ

*is the*

^{k}*k*iteration result, then the CG method is summarized as following:

^{th}**Iteration 0:**Given the desired binary mask pattern

*Z̃*, calculate According to Eqn. (19), assigning Θ

^{0}(

*i*,

*j*) =

*π*or Θ

^{0}(

*i*,

*j*) = 0 would degrade the gradient to 0 and therefore the iteration would be terminated. Thus, we select the angle values

^{0},

*P*

^{0}∈ ℜ

^{N×N}.

**Iteration k:**

**Stop:**

*t*is a global threshold. We define the pattern error

_{m}*E*as the square of the

*L*

_{2}norm of the distance between the desired output image

*Z̃*and the actual binary output pattern

*Z*: In general, lower pattern errors result in lower critical dimension (CD) errors. There are some other metrics to evaluate the performance of the optical lithography, such as the process window and so on. However this topic is out of the scope of this paper, and will be studied in the future work. In the proposed algorithm, when the pattern error is reduced to a tolerable level, or the iteration number reaches the prescribed upper limit, the CG iteration is terminated. In the following simulations, the tolerable error level is set to 0. That means we would like to seek a printed image on the wafer that is the same as the desired pattern. Therefore, the CG algorithm is terminated when the printed image is the same as the desired pattern, or the iteration number reaches the prescribed upper limit. The CG method above is tailored for the quadratic problem [17]. However, the inverse lithography (ILT) optimization formulated in Eqn. (18) is a non-quadratic problem, and the cost function possesses numerous local minima. Thus, the CG method is easily trapped in these local minima. For the SD method used in prior work, the step length is a constant. Thus, the SD method also may get stuck in a local minimum, but a different one. Our extensive simulations show that setting the step length as a constant makes the CG method have much faster convergence. It also leads to lower pattern errors than the results obtained by the SD algorithm. Intuitively, the constant step length helps the CG method skip the local minimum. Thus, in the following simulations, we replace the

_{b}**Step 1**above with a constant step length. In practice, the proposed CG method often leads to linear or superlinear convergence speed [17, 28

28. H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Dev. **16**, 431–433 (1972). [CrossRef]

30. K. Ritter, “On the rate of superlinear convergence of a class of variable metric methods,” Numerische Mathematik **35**, 293–313 (1980). [CrossRef]

*σ*= 0.3 and

_{inner}*σ*= 0.4. For both methods, the convolution kernel is shown in Eqn. (8) with

_{outer}*NA*= 1.25 and

*λ*= 193

*nm*. We assume

*h*(

**r**) vanishes outside the area

*A*defined by

_{h}*x*,

*y*∈ [−56.25

*nm*, 56.25

*nm*]. In the sigmoid function, we assign the parameters

*a*= 25 and

*t*= 0.19. The global threshold is

_{r}*t*= 0.5, the pixel size is 5.625

_{m}*nm*× 5.625

*nm*. Generally, smaller step lengths lead to stable but slower convergence, while larger step lengths lead to faster convergence or divergence. Based on this guideline, we set the step length equal to 1, which is moderate for the underlying optimization problem.

31. C. Vogel, *Computational Methods for Inverse Problems* (SIAM Press, 2002). [CrossRef]

*F*(

*m*) is the data-fidelity term and

*R*(

*m*) is the regularization term which is used to reduce the solution space and constrain the optimized results.

*γ*is the user-defined parameter to reveal the weight of the regularization. In the above simulation, the quadratic penalty [13

**16**(3), 774–788 (2007). [CrossRef] [PubMed]

19. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express **15**, 15066–15079 (2007). [CrossRef] [PubMed]

*γ*and

_{Q}*γ*too large can result in divergence of the optimization process. In the following simulations, the regularization weights of the quadratic penalty and wavelet penalty are

_{W}*γ*= 0.01 and

_{Q}*γ*= 0.025, respectively.

_{W}*Z̃*with dimension of 1035

*nm*× 1035

*nm*, and Fig. 3(d) is its output pattern on the wafer with pattern error of 5052. Figure 3(b) is the optimized OPC pattern from the SD method, and Fig. 3(e) is its output pattern with pattern error of 1368. Figure 3(c) is the optimized OPC pattern from the CG method, and Fig. 3(f) is its output pattern with pattern error of 1288. Compared to the SD method, the CG method reduces the pattern error by 6%

*x*-axis is the iteration number and

*y*-axis is the pattern error. In Fig. 4, the convergence of SD and CG methods are shown in black solid line and blue dashed line, respectively. The SD method converges to pattern error of 1368 after 150 iterations using 996 seconds, while the CG method converges to a pattern error of 1288 after 33 iterations using 220 seconds. The performance and convergence comparison between the two approaches are summarized in the 2

^{nd}to 3

^{rd}rows of Table 1.

## 4. MRC Penalty

*γ*= 0.025, 0.035 and 0.045, respectively. Figures 5(d), 5(e) and 5(f) show the output patterns corresponding to their above OPC patterns. It is shown that increasing the wavelet penalty weight

_{W}*γ*cannot enlarge the dimensions of the SRAFs or the distances between the SRAFs and the main body of the mask. In addition, larger wavelet penalty weight will increase the pattern error. Particularly, Fig. 5(f) shows that when

_{W}*γ*= 0.045, some bridges appear in the output pattern. Therefore, the wavelet penalty falls short to generate MRC-favorable optimized masks.

_{W}**t**

_{mN×N}∈ ℜ

^{N×N}has all entries equal to

*t*,

_{m}*g*is referred to as the MRC filter,

*g*⊗

*M*is a weighted sum of each pixel value and its neighbors, and

*M*−

**t**

_{m}_{N×N}judges whether the pixel value is above

*t*and carved on the binary mask or not.

_{m}*t*is a constant typically defined as 0.5. The meaning of the MRC penalty is explained as follows. Consider the value of a pixel located at (

_{m}*x*

_{0},

*y*

_{0}) that is larger than

*t*, i.e.,

_{m}*p*(

*x*

_{0},

*y*

_{0}) >

*t*. Then, (

_{m}*M*−

**t**

_{mN×N})(

*x*

_{0},

*y*

_{0}) > 0. If we want to obtain a low penalty value, we need to keep (

*g*⊗

*M*)(

*x*

_{0},

*y*

_{0}) as large as possible, which means that we want all pixel values around

*p*(

*x*

_{0},

*y*

_{0}) to be larger than

*t*. The similar analysis applies for the case of

_{m}*p*(

*x*

_{0},

*y*

_{0}) <

*t*. In other word, it is preferable to have the value of

_{m}*p*(

*x*,

*y*) −

*t*around the location of (

_{m}*x*

_{0},

*y*

_{0}) have the same sign of

*p*(

*x*

_{0},

*y*

_{0}) −

*t*, thus both of the mask opening and blocking areas are concentrated and enlarged. In our simulations,

_{m}*t*= 0.5, and

_{m}*g*=

**1**

_{3×3}, which means that we equally sum each pixel value with its eight neighboring pixel values. Through this way, the linear sizes of the SRAFs, and the distances from the SRAFs to the main body of the mask will be simultaneously increased. Eqn. (30) can be reformulated as As derived in Appendix B, the gradient of

*R*(Θ) is thus, the cost function in Eqn. (29) is adjusted as Figure 6(a) shows the resulting OPC pattern by optimizing Eqn. (33), where the desired pattern is the same as Fig. 3(a) and

_{M}*γ*= 0.005. Figure 6(b) shows the output pattern corresponding to Fig. 6(a). The pattern error, iteration number and the runtime are summarized in the 4

_{M}^{th}row of Table 1. The convergence of this simulation is shown by the green dotted line in Fig. 4. It is observed that the MRC penalty effectively enlarges the linear sizes of SRAFs and the distances from the SRAFs to the main body of the mask. As a tradeoff, the pattern error is increased by 21%.

## 5. Projection Method

*g*in Eqn. (30), the pattern error will be dramatically increased, while the number of the small SRAFs cannot be further reduced. This phenomenon is illustrated in Fig. 7. Figures 7(a), 7(b) and 7(c) show the optimized OPCs with

*g*=

**1**

_{3×3},

**1**

_{5×5}and

**1**

_{7×7}, respectively. Figures 7(d), 7(e) and 7(f) show the output patterns corresponding to their above OPC patterns. It is shown that increasing the MRC filter dimension leads to more details on the optimized OPC. In addition, larger MRC filter dimension will increase the pattern error. One approach to ameliorate the pattern distortions as the window size of the MRC filter is increased is to replace the linear filter by a median-type edge-preserving filter [32

32. M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust., Speech, Signal Process. **35**, 98–106 (1987). [CrossRef]

34. K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process **42**, 782–798 (1994). [CrossRef]

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

_{0}as described in Eqn. (20). In stage I, the CG method with the MRC penalty is applied to obtain a raw optimized gray-level mask pattern

*M̂*. If

*M̂*satisfies the prescribed termination conditions, we threshold

*M̂*by

*t*to obtain the final optimized binary mask

_{m}*M̂*. Otherwise, we move to stage II. In stage II, we cut off the high frequency components of

_{b}*M̂*in the 2D-DCT domain, resulting in

*M̂′*. Subsequently, in stage III we threshold

*M̂′*to obtain

*M̂′*. In addition, we update Θ according to

_{b}*M̂′*using method described in Eqn. (20). The updated Θ is used as the input of the stage I to continue the loop. Let

_{b}*S*represent the subspace of the optimal solutions of Eqn. (10).

_{F}*S*represents the subspace supported by the low frequency components in the 2D-DCT domain. The proposed method iteratively projects the pixel values on the mask between

_{DCT}*S*and

_{F}*S*, and drives the final solution into the intersection between the two subspaces.

_{DCT}## 6. Conclusion

## A. Appendix A

*R*(

_{Q}*m*) = 4

*m*

*(*

^{T}**1**−

*m*). For each pixel value, the corresponding penalty is the quadratic function

*r*(

*m*

*)=1 − (2*

_{i}*m*

*− 1)*

_{i}^{2},

*i*= 1,...,

*N*

^{2}. The gradient of

*R*(

_{q}*m*) is

*or*2,

*p*

_{1}= (

*p*+ 1)

*mod*2 and

*q*

_{1}=

*q*+ 1)

*mod*2.

## B. Appendix B

*g*=

**1**

_{3×3},

*g*= 1 and

_{kk}*g*=

_{pk}*g*. Therefore,

_{kp}*t*= 0.5 and

_{m}## References and links

1. | A. K. Wong, |

2. | X. Ma and G. R. Arce, |

3. | S. A. Campbell, |

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

5. | F. Schellenberg, |

6. | L. Liebmann, S. Mansfield, A. Wong, M. Lavin, W. Leipold, and T. Dunham, “TCAD development for lithography resolution enhancement,” IBM J. Res. Dev. |

7. | U. Okoroanyanwu, |

8. | S. Sherif, B. Saleh, and R. Leone, “Binary image synthesis using mixed integer programming,” IEEE Trans. Image Proc. |

9. | Y. Liu and A. Zakhor, “Binary and phase shifting mask design for optical lithography,” IEEE Trans. Semicond. Manuf. |

10. | A. Erdmann, R. Farkas, T. Fühner, B. Tollküuhn, and G. Kokai, “Towards automatic mask and source optimization for optical lithography,” Proc. SPIE |

11. | Y. Granik, “Solving inverse problems of optical microlithography,” Proc. SPIE |

12. | Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlithogr., Microfabr., Microsyst. |

13. | A. Poonawala and P. Milanfar, “Fast and low-complexity mask design in optical microlithography–an inverse imaging problem,” IEEE Trans. Image Process. |

14. | X. Ma and G. R. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A |

15. | X. Ma and G. R. Arce, “Binary mask optimization for forward lithography based on boundary layer model in coherent systems,” J. Opt. Soc. Am. A |

16. | X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express |

17. | J. Nocedal and S. J. Wright, |

18. | B. E. A. Saleh and M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains and by modal expansion,” Appl. Opt. |

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

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

21. | K. Kato, K. Nishizawa, and T. Inoue, “Advanced mask rule check (MRC) tool,” Proc. SPIE |

22. | K. Kato, Y. Taniguchi, K. Nishizawa, and M. Endo, “Mask rule check using priority information of mask patterns,” Proc. SPIE |

23. | B. Salik, J. Rosen, and A. Yariv, “Average coherent approximation for partially cohernet optical systems,” J. Opt. Soc. Am. A |

24. | M. Born and E. Wolfe, |

25. | R. Wilson, |

26. | M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Natl. Bur. Stand. |

27. | R. Fletcher and C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. |

28. | H. P. Crowder and P. Wolfe, “Linear convergence of the conjugate gradient method,” IBM J. Res. Dev. |

29. | A. Cohen, “Rate of convergence of several conjugate gredient algorithms,” SIAM J. Numer. Anal. |

30. | K. Ritter, “On the rate of superlinear convergence of a class of variable metric methods,” Numerische Mathematik |

31. | C. Vogel, |

32. | M. P. McLoughlin and G. R. Arce, “Deterministic properties of the recursive separable median filter,” IEEE Trans. Acoust., Speech, Signal Process. |

33. | S. Kalluri and G. R. Arce, “Fast algorithms for weighted myriad computation by fixed-point search,” IEEE Trans. Signal Process |

34. | K. Barner and G. R. Arce, “Permutation filters - a class of nonlinear filters based on set permutations,” IEEE Trans. Signal Process |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(110.4980) Imaging systems : Partial coherence in imaging

(110.5220) Imaging systems : Photolithography

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 10, 2010

Revised Manuscript: December 25, 2010

Manuscript Accepted: December 26, 2010

Published: January 20, 2011

**Citation**

Xu Ma and Gonzalo R. Arce, "Pixel-based OPC optimization based on conjugate gradients," Opt. Express **19**, 2165-2180 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2165

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