## Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography

Optics Express, Vol. 16, Issue 19, pp. 14746-14760 (2008)

http://dx.doi.org/10.1364/OE.16.014746

Acrobat PDF (871 KB)

### Abstract

The continuous shrinkage of minimum feature size in integrated circuit (IC) fabrication incurs more and more serious distortion in the optical projection lithography process, generating circuit patterns that deviate significantly from the desired ones. Conventional resolution enhancement techniques (RETs) are facing critical challenges in compensating such increasingly severe distortion. In this paper, we adopt the approach of inverse lithography in the mask design, which is a branch of design methodology to treat it as an inverse mathematical problem. We focus on using pixel-based algorithms to design alternating phase-shifting masks with minimally distorted output, with the goal that the patterns generated should have high contrast and low dose sensitivity. This is achieved with a dynamic-programming-based initialization scheme to pre-assign phases to the layout when alternating phase-shifting masks are used. Pattern fidelity and worst case slopes are shown to improve with this initialization scheme, which are important for robustness considerations.

© 2008 Optical Society of America

## 1. Introduction

2. C. A. Mack, “30 years of lithography simulation,” Proc. SPIE **5754**, 1–12 (2004). [CrossRef]

3. F. Schellenberg, “Resolution enhancement technology: The past, the present, and extensions for the future,” Proc. SPIE **5377**, 1–20 (2004). [CrossRef]

4. A. K.-K. Wong, *Resolution enhancement techniques in optical lithography* (SPIE Press, Bellingham, Washington, 2001). [CrossRef]

5. K. Nashold and B. Saleh, “Image construction through diffraction-limited high-contrast imaging systems: An iterative approach,” J. Opt. Soc. Am. A **2**, 635–643 (1985). [CrossRef]

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

8. Y. Liu and A. Zakhor, “Binary and phase-shifting image design for optical lithography,” IEEE Trans. Semicond. Manuf. **5**, 138–151 (1992). [CrossRef]

10. N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” Proc. SPIE **2621**, 534–545 (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**, 2438–2452 (1994). [CrossRef]

13. C. Hung, B. Zhang, E. Guo, L. Pang, Y. Liu, K. Wang, and G. Dai, “Pushing the lithography limit: Applying inverse lithography technology (ILT) at the 65nm generation,” Proc. SPIE **6154**, 61541M (2006). [CrossRef]

14. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE **6283**, 62830X (2006). [CrossRef]

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

16. A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A nonlinear optimization approach,” Proc. SPIE textbf**6154**, 61543H (2006). [CrossRef]

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

16. A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A nonlinear optimization approach,” Proc. SPIE textbf**6154**, 61543H (2006). [CrossRef]

## 2. Optical lithography

### 2.1. Lithography systems modeling

*O*(

*x*,

*y*) be the mask pattern, and its spectrum

*Õ*(

*f*,

*g*). Similarly,

*H*(

*x*,

*y*) is the transfer function describing the projection optics, while its Fourier transform

*Ĥ*(

*f*,

*g*) is the pupil function. For unity magnification systems,

*Ĥ*(

*f*,

*g*) = 1 for

*J*

_{1}(

*x*) is the Bessel function of the first kind, order 1 [20]. To maintain a unity gain, often we need to normalize

*H*(

*x*,

*y*). The total E-field at the image plane is

*E*(

*x*,

*y*)=∫∞-∞∫∞-∞

*Ĥ*(

*f*,

*g*)

*Õ*(

*f*,

*g*)exp{-

*j*2

*π*(

*fx*+

*gy*)} d

*f*dg, and the intensity of the aerial image equals magnitude squared of the field, i.e.

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

*H*(

*x*,

*y*)|

^{2}and |

*O*(

*x*,

*y*)|

^{2}[20].

*tr*, any aerial image intensity higher than

*tr*will cause the development. It has been suggested that one can reasonably model the resist action by a sigmoid function sig{·} as [15

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

*a*> 0 is the parameter determining the contrast of the sigmoid function.

## 2.2. Optimization process in inverse lithography

*I*(

*x*,

*y*) with a certain mask pattern

*O*(

*x*,

*y*). For conventional chromium on glass (COG) masks, we can assume that

*O*(

*x*,

*y*) takes on binary values {0,1}. Apart from such masks, the most commonly used ones nowadays are phase-shifting masks (PSM) [4

4. A. K.-K. Wong, *Resolution enhancement techniques in optical lithography* (SPIE Press, Bellingham, Washington, 2001). [CrossRef]

*O*(

*x*,

*y*) takes on ternary values {-1,0,1}.

*F*between a desired circuit pattern

*Î*(

*x*,

*y*) and the output image due to an input mask

*O*(

*x*,

*y*) is given by

*Î*(

*x*,

*y*) can lead to large changes in

*O*(

*x*,

*y*) that can minimize

*F*. There can also be multiple solutions, and hence the optimal mask pattern is not necessarily unique.

14. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE **6283**, 62830X (2006). [CrossRef]

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

14. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE **6283**, 62830X (2006). [CrossRef]

*O*(

*x*,

*y*)≤1, add a penalty term

*R*= ϣ

*(-4.5*

_{x, y}*O*(

*x*,

*y*)

^{4}+

*O*(

*x*,

*y*)

^{2}+3.5), and minimize

*F*+

*where*

_{γ}R*γ*is a weighting constant for the penalty

*R*. The optimization is then solved iteratively by steepest descent [18

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

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

## 2.2.1. Active set method

*O*(

*x*,

*y*) ≤ 1 is a simple bound on the variables. It was suggested to use the trigonometric substitution

*O*(

*x*,

*y*)=cos

*θ*(

*x*,

*y*) to transform the optimization problem into an unconstrained one [15

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

16. A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A nonlinear optimization approach,” Proc. SPIE textbf**6154**, 61543H (2006). [CrossRef]

*θ*has a period of 2

*π*so for every 2

*π*we have a repeated solution. Another problem is the increment of nonlinearity. The objective function itself is nonlinear. Using a trigonometric substitution, we may bring additional local minima and stationary points. Nonlinearity also makes the gradient algorithm difficult to search, because often it affects stability and convergence negatively.

*O*(

*x*,

*y*) = 1 if

*O*(

*x*,

*y*) > 1, and

*O*(

*x*,

*y*)=-1, if

*O*(

*x*,

*y*)<-1. Active set method guarantees convergence, because the constraints form a

*k*-dimensional interval (or

*k*-cell [25]), which is convex. Hence the restriction to this

*k*-cell is a projection onto convex set (POCS). Another advantage of using active set method is that we can stop searching at (x i,yi) if

*O*(

*x*,

_{i}*y*) is a boundary point. Thus the number of variables will decrease monotonically.

_{i}## 2.2.2. Conjugate gradient

*O*(

^{k}*]*,

*y*) be the variable at

*k*-th iteration, and let

*G*=

*F*+

*. The following pseudo-code for conjugate gradient is then applicable:*

_{γ}R**Stage 0**Choose a starting point

*O*

^{0}(

*x*,

*y*), and set

*d*

_{0}= -

**∇**(

*G**O*

^{0}(

*x*,

*y*)).

**Stage**

*k*

*λ*= argmin

_{k}*G*(

*O*(

^{k}*x*,

*y*)+

*λd*).

_{k}*O*

^{k+1}(

*x*,

*y*)=

*O*(

^{k}*x*,

*y*)+

*λ*.

_{k}d_{k}*d*

_{k+1}= -∇

*G*(

*O*

^{k+1}(

*x*,

*y*))+

*β*.

_{k}d_{k}**Stop**If stopping criteria is satisfied, then END. Otherwise,

*k*←

*k*+1 and proceed Stage

*k*+1.

*G*requires ∇

*F*and ∇

*R*, which equal to:

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

## 3. Initialization

*π*phase alternation among the objects if the initial guess is all 0-phase.

*π*phase. Therefore, given a complicated mask pattern, if we can assign phases to objects as alternating as possible, we can improve the slope significantly. This is the motivation of our methodology in this paper. However, it is very common that phase conflicts would exist if we enforce this rule rigidly. For example, in Fig. 3(b), if we assign Object A with phase 0, and Object B with phase

*π*, then we will have trouble in assigning phase to Object C. This type of phase conflict problem is well known in the industry, and there are plenty of researches done [27

27. P. Berman, A. Kahng, D. Vidhani, H. Wang, and A. Zelikovsky, “Optimal phase conflict removal for layout of dark field alternating phase shifting masks,” IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems **19**, 175–187 (2000). [CrossRef]

28. A. Moniwa and T. Terasawa, “Heuristic method for phase-conflict minimization in automatic phase-shift mask design,” Jpn. J. Appl. Phys. **34**, 6584–6589 (1995). [CrossRef]

## 3.1. Extracting vertices

## 3.2. Distance matrix

*d*(

*i*,

*j*) to measure the degree of interaction between Object

*i*and Object

*j*. There are various distance functions that can be use for such degree measurement, and we discuss three reasonable designs here. Let

*W*be the width of overlapping region of two edges, and

*G*be the gap between two objects, as shown in Fig. 4(a). The first possible distance metric is

*k*refers to the

*k*-th edge pair under comparison. In the figure,

*k*= 2, as there are two pairs of edges being interacting with each other. Equation (8) measures the sum of ratios between

*G*and

*W*. Intuitively, if the gap

*G*is small, then image slope of

*I*

_{aerial}(

*x*,

*y*) will be low. If the width

*W*is large, then there are more parts of the edge suffering from poor slope. Clearly, Eq. (8) satisfies these two criteria.

*G*, because optical illumination is electromagnetic wave propagation. The E-field at a point is inversely proportional to the square of distance from the source. A summation is taken to represent superposition, and a reciprocal is taken because we want to reverse the orderings of the magnitude.

*l*, and charge distribution

*λ*, then the E-field at a point y from the mid-point of the rod is [30]

*d*

_{1}(

*i*,

*j*),

*d*

_{2}(

*i*,

*j*) and

*d*

_{3}(

*i*,

*j*). Our results show that if we arrange them in a descending order, the order will be very similar with each other despite the differences in the specific formulas used. Since the dynamic programming in the next Subsection depends on

*d*(

*i*,

*j*), this implies relative insensitivity to the choice among the formulas above.

## 3.3. Dynamic programming

*π*to the one with the largest interaction with A. We observe that this is equivalent to arranging the objects in an order such that the overall interaction is maximized. Using our terminology discussed above, this is equivalent to minimizing the cost.

*X*to denote the node at stage

_{k}*k*. Suppose we start from node

*X*

_{0}=

*D*in stage 1 (this is arbitrary). Then at stage 2, the possible destinations are

*A*,

*B*, and

*E*. So we can compute the cost to reach these nodes:

*X*

_{1}

*X*

_{2}=

*DA*= 38.3795

*X*

_{1}

*X*

_{2}=

*DB*= 64.3758

*X*

_{1}

*X*

_{2}=

*DE*= 109.0517.

*A*, we only have one choice (through

*DE*), and the cost is

*X*

_{1}

*X*

_{2}

*X*

_{3}=

*DEA*= 109.0517+64.3758 = 173.4275.

*B*, we have two choices:

*X*

_{1}

*X*

_{2}

*X*

_{3}=

*DAB*= 38.3795+53.3539 = 91.7334

*X*

_{1}

*X*

_{2}

*X*

_{3}=

*DEB*= 109.0517+54.7127 = 163.7644,

*B*at stage 3, we shall choose path

*DAB*.

*C*, we have two choices:

*X*

_{1}

*X*

_{2}

*X*

_{3}=

*DAC*= 38.3795+162.8364 = 201.2059

*X*

*1*

*X*

_{2}

*X*

_{3}=

*DBC*= 64.3758+12.7830 = 77.1588,

*DBC*.

*E*, we have

*X*

_{1}

*X*

_{2}

*X*

_{3}=

*DAE*= 38.3795+64.3758 = 93.0922

*X*

_{1}

*X*

_{2}

*X*

_{3}=

*DBE*= 64.3758+54.7127 = 119.0885,

*DAE*.

*B*), then we list out all possible ways to go to

*B*, and choose the one with lowest

*accumulated*cost. Repeating for other destinations, we can complete stage 4. See circled nodes at stage 4 in Fig. 6. Finally at stage 5, we can see that there are three feasible paths:

*X*

_{1}

*X*

_{2}

*X*

_{3}

*X*

_{4}

*X*

_{5}=

*DAEBC*= 160.5879

*X*

_{1}

*X*

_{2}

*X*

_{3}

*X*

_{4}

*X*

_{5}=

*DBCAE*= 304.3610

*X*

_{1}

*X*

_{2}

*X*

_{3}

*X*

_{4}

*X*

_{5}=

*DEACB*= 349.0369.

*DAEBC*for the whole graph.

*k*-1 to stage

*k*, we only need to consider the

*survivors*. For example from stage 2 to stage 3, we had two paths to go to

*B*(

*DAB*and

*DEB*). By comparing the accumulated cost we chose path

*DAB*, and discarded

*DEB*. So starting from stage 3 onwards we only considered the path

*DAB*(as circled in Fig. 6).

*D*. However, in order to search for the best possible path, we should start the dynamic programming at all possible nodes. Thus, we loop over all starting nodes. With each starting node, we run through the above mentioned algorithm, and give the best path. Among all these paths (best with respect to different starting points) we choose the one with the lowest cost.

*π*,0,

*π*,…) to the objects according to the optimal sequence. Figure 7 summarizes the above in a flow chart for the phase initialization algorithm.

## 4. Results

*DAEBC*. We assign phases according to this order:

*D*(0),

*A*(

*π*),

*E*(0),

*B*(

*π*),

*C*(0). The resulting pattern is shown in Fig. 8(a). Several other examples can be found in (b) to (f). In these figures, we display white = amplitude 1 phase 0, black = amplitude 1 phase

*π*, gray = amplitude 0. We can see that for complicated patterns, the dynamic program indeed finds reasonable solutions as well. If there is no phase conflict, such as in Fig. 8(b), the algorithm assigns phases from left to right, and up to down. If there are phase conflicts, the algorithm can assign phases so as to minimize the conflict, as demonstrated in Fig. 8(c) to (f).

*λ*= 193nm, numerical apertureNA = 0.85. As in numerical implementation, we represented the point spread function

*H*(

*x*,

*y*) as an array of size 121 × 121, which is adequate to cover 98% energy. For resist development, we set the sigmoid function cut-off threshold as

*t*= 0.3, and the sharpness of the function

_{r}*a*= 25. The weighting constant in the algorithm is

*γ*= 0.01. The initial guess follows from Section 3.

*H*(

*x*,

*y*) (large

*λ*). Therefore, the contrast is very low. On the other hand, the contrast of the aerial image using optimal PSM generated by inverse imaging is

4. A. K.-K. Wong, *Resolution enhancement techniques in optical lithography* (SPIE Press, Bellingham, Washington, 2001). [CrossRef]

*I*

_{threshold}is the threshold intensity (i.e.

*t*), CD is the critical dimension. NILS measures the tolerance of the aerial image when facing fluctuation in dose concentration. Therefore, a higher value is preferred.

_{r}*t*= 0.3, CD = 50nm, and use forward difference with Δ

_{r}*x*= 10nm/pixel. On the other hand, if we use the optimal PSM, the NILS is

## 5. Conclusion

## Acknowledgment

## References and links

1. | J. Plummer, M. Deal, and P. Griffin, |

2. | C. A. Mack, “30 years of lithography simulation,” Proc. SPIE |

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

4. | A. K.-K. Wong, |

5. | K. Nashold and B. Saleh, “Image construction through diffraction-limited high-contrast imaging systems: An iterative approach,” J. Opt. Soc. Am. A |

6. | B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. |

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

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

9. | Y. Liu and A. Zakhor, “Optimal binary image design for optical lithography,” Proc. SPIE |

10. | N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” Proc. SPIE |

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

12. | D. S. Abrams and L. Pang, “Fast inverse lithography technology,” Proc. SPIE |

13. | C. Hung, B. Zhang, E. Guo, L. Pang, Y. Liu, K. Wang, and G. Dai, “Pushing the lithography limit: Applying inverse lithography technology (ILT) at the 65nm generation,” Proc. SPIE |

14. | L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” Proc. SPIE |

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

16. | A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A nonlinear optimization approach,” Proc. SPIE textbf |

17. | S. H. Chan, A. K. Wong, and E. Y. Lam, “Inverse synthesis of phase-shifting mask for optical lithography,” in “OSA Topical Meeting in Signal Recovery and Synthesis,” (2007), p. SMD3. |

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

19. | J. W. Goodman, |

20. | J. W. Goodman, |

21. | Y. Granik, “Solving inverse problems of optimal microlithography,” Proc. SPIE |

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

23. | S. H. Chan and E. Y. Lam, “Inverse image problem of designing phase shifting masks in optical lithography,” in “IEEE International Conference on Image Processing,” (2008). |

24. | P. E. Gill, W. Murray, and M. H. Wright, |

25. | W. Rudin, |

26. | M. Minoux, |

27. | P. Berman, A. Kahng, D. Vidhani, H. Wang, and A. Zelikovsky, “Optimal phase conflict removal for layout of dark field alternating phase shifting masks,” IEEE Trans. Computer-Aided Design of Integrated Circuits and Systems |

28. | A. Moniwa and T. Terasawa, “Heuristic method for phase-conflict minimization in automatic phase-shift mask design,” Jpn. J. Appl. Phys. |

29. | L. G. Shapiro and G. C. Stockman, |

30. | D. Halliday, R. Resnick, and K. S. Krane, |

**OCIS Codes**

(100.2980) Image processing : Image enhancement

(100.3190) Image processing : Inverse problems

(110.5220) Imaging systems : Photolithography

**ToC Category:**

Image Processing

**History**

Original Manuscript: June 3, 2008

Revised Manuscript: August 7, 2008

Manuscript Accepted: August 30, 2008

Published: September 4, 2008

**Citation**

Stanley H. Chan, Alfred K. Wong, and Edmund Y. Lam, "Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography," Opt. Express **16**, 14746-14760 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-14746

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### References

- J. Plummer, M. Deal, and P. Griffin, Silicon VLSI Technology ??? Fundamentals, Practice and Modeling (Prentice Hall, 2000).
- C. A. Mack, "30 years of lithography simulation," Proc. SPIE 5754, 1-12 (2004). [CrossRef]
- F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future," Proc. SPIE 5377, 1-20 (2004). [CrossRef]
- A. K.-K. Wong, Resolution enhancement techniques in optical lithography (SPIE Press, Bellingham, Washington, 2001). [CrossRef]
- K. Nashold and B. Saleh, "Image construction through diffraction-limited high-contrast imaging systems: An iterative approach," J. Opt. Soc. Am. A 2, 635-643 (1985). [CrossRef]
- B. Saleh and S. Sayegh, "Reductions of errors of microphotographic reproductions by optical corrections of original masks," Opt. Eng. 20, 781-784 (1981).
- S. Sherif, B. Saleh, and R. Leone, "Binary image synthesis using mixed linear integer programming," IEEE Trans. Image Process. 4, 1252-1257 (1995). [CrossRef] [PubMed]
- Y. Liu and A. Zakhor, "Binary and phase-shifting image design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-151 (1992). [CrossRef]
- Y. Liu and A. Zakhor, "Optimal binary image design for optical lithography," Proc. SPIE 1264, 410-412 (1990).
- N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," Proc. SPIE 2621, 534-545 (1995). [CrossRef]
- Y. C. Pati and T. Kailath, "Phase-shifting masks for microlithography automated design and mask requirements," J. Opt. Soc. Am. A 11, 2438-2452 (1994). [CrossRef]
- D. S. Abrams and L. Pang, "Fast inverse lithography technology," Proc. SPIE 6154, 534-542 (2006).
- C. Hung, B. Zhang, E. Guo, L. Pang, Y. Liu, K. Wang, and G. Dai, "Pushing the lithography limit: Applying inverse lithography technology (ILT) at the 65nm generation," Proc. SPIE 6154, 61541M (2006). [CrossRef]
- L. Pang, Y. Liu, and D. Abrams, "Inverse lithography technology (ILT): What is the impact to the photomask industry?" Proc. SPIE 6283, 62830X (2006). [CrossRef]
- A. Poonawala and P. Milanfar, "Prewarping techniques in imaging: Applications in nanotechnology and biotechnology," Proc. SPIE 5674, 114-127 (2005). [CrossRef]
- A. Poonawala and P. Milanfar, "OPC and PSM design using inverse lithography: A nonlinear optimization approach," Proc. SPIE 6154, 61543H (2006). [CrossRef]
- S. H. Chan, A. K. Wong, and E. Y. Lam, "Inverse synthesis of phase-shifting mask for optical lithography," in "OSA Topical Meeting in Signal Recovery and Synthesis," (2007), p. SMD3.
- X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15066-15079 (2007). [CrossRef] [PubMed]
- J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
- J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publisher, Englewood, Colo, 2005), 3rd ed.
- Y. Granik, "Solving inverse problems of optimal microlithography," Proc. SPIE 5754, 506-526 (2004). [CrossRef]
- A. Poonawala and P. Milanfar, "Mask design for optical microlithography ???an inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007). [CrossRef] [PubMed]
- S. H. Chan and E. Y. Lam, "Inverse image problem of designing phase shifting masks in optical lithography," in "IEEE International Conference on Image Processing," (2008).
- P. E. Gill, W. Murray, and M. H. Wright, Practical optimization (Academic Press, London, 1986).
- W. Rudin, Principles of Mathematical Analysis (McGraw-Hill, 1976).
- M. Minoux, Mathematical programming theory and algorithms (John Wiley and Sons, Chichester, 1986).
- P. Berman, A. Kahng, D. Vidhani, H. Wang, and A. Zelikovsky, "Optimal phase conflict removal for layout of dark field alternating phase shifting masks," IEEE Trans.Comput.-Aided Des. 19, 175-187 (2000). [CrossRef]
- A. Moniwa and T. Terasawa, "Heuristic method for phase-conflict minimization in automatic phase-shift mask design," Jpn. J. Appl. Phys. 34, 6584-6589 (1995). [CrossRef]
- L. G. Shapiro and G. C. Stockman, Computer Vision (Prentice Hall, 2001).
- D. Halliday, R. Resnick, and K. S. Krane, Physics, (John Wiley and Sons, New York, 2002), 2nd ed., Vol. 2.

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