## PSM design for inverse lithography with partially coherent illumination

Optics Express, Vol. 16, Issue 24, pp. 20126-20141 (2008)

http://dx.doi.org/10.1364/OE.16.020126

Acrobat PDF (691 KB)

### Abstract

Phase-shifting masks (PSM) are resolution enhancement techniques (RET) used extensively in the semiconductor industry to improve the resolution and pattern fidelity of optical lithography. Recently, a set of gradient-based PSM optimization methods have been developed to solve for the inverse lithography problem under coherent illumination. Most practical lithography systems, however, use partially coherent illumination due to non-zero width and off-axis light sources, which introduce partial coherence factors that must be accounted for in the optimization of PSMs. This paper thus focuses on developing a framework for gradient-based PSM optimization methods which account for the inherent nonlinearities of partially coherent illumination. In particular, the singular value decomposition (SVD) is used to expand the partially coherent imaging equation by eigenfunctions into a sum of coherent systems (SOCS). The first order coherent approximation corresponding to the largest eigenvalue is used in the PSM optimization. In order to influence the solution patterns to have more desirable manufacturability properties and higher fidelity, a post-processing of the mask pattern based on the 2D discrete cosine transformation (DCT) is introduced. Furthermore, a photoresist tone reversing technique is exploited in the design of PSMs to project extremely sparse patterns.

© 2008 Optical Society of America

## 1. Introduction

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

*λ*is the wavelength,

*NA*is the numerical aperture, and

*k*is the process constant which can be minimized through RET methods [2, 3

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

5. 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]

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

7. M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices **ED-29**, 1828–1836 (1982). [CrossRef]

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

9. 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]

12. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express **15**, 15,066–15,079 (2007). [CrossRef]

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

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

*λ*is placed at the focal plane of the first condenser (

*L*

_{1}), illuminating the mask. The image of the photomask is formed by the projection optics onto the wafer [1

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

14. E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38 (1981). [CrossRef]

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

15. P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” in *Proc. SPIE* (San Jose, CA, 2008). [CrossRef]

16. X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in *Proc. SPIE* (Taiwan, 2008). [CrossRef]

17. X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008). [CrossRef]

16. X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in *Proc. SPIE* (Taiwan, 2008). [CrossRef]

17. X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008). [CrossRef]

9. 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]

19. J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in *Simulation of Semiconductor Processes and Devices*, 12 (2007). [CrossRef]

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

## 2. Partially coherent imaging models

## 2.1. Hopkins diffraction model

20. 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 (1982). [CrossRef] [PubMed]

**r**=(

*x*,

*y*),

**r**

_{1}=(

*x*

_{1},

*y*

_{1}) and

**r**

_{2}=(

*x*

_{2},

*y*

_{2}).

*M*(

*r*) is the mask pattern,

*γ*(

**r**

_{1}-

**r**

_{2}) is the complex degree of coherence, and

*h*(

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

*γ*(

**r**

_{1}-

**r**

_{2}) is generally a complex number, whose magnitude represents the extent of optical interaction between two spatial locations

**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*, 1 (SPIE Press, 2001). [CrossRef]

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

*γ*(

**r**)=1. In this case, the intensity distribution in Eq. (1) is separable on

**r**

_{1}and

**r**

_{2}, and thus

*I*(

**r**)=|

*M*(

**r**)⊗

*h*(

**r**)|

^{2}, where ⊗ is the convolution operation. For the completely incoherent case, the illumination source is of infinite extent and thus,

*γ*(

**r**)=

*δ*(

**r**). In this case, the intensity distribution reduces to

*I*(

**r**)=|

*M*(

**r**)|

^{2}⊗|

*h*(

**r**)|

^{2}. In the frequency domain, Eq. (1) is translated as

*M*̃(

*f*

_{1},

*g*

_{1}) and

*M*̃(

*f*

_{2},

*g*

_{2}) are the Fourier transforms of

*M*(

*x*

_{1},

*x*

_{2}) and

*M*(

*x*

_{2},

*y*

_{2}), respectively.

*TCC*(

*f*

_{1},

*g*

_{1};

*f*

_{2},

*g*

_{2}) is the transmission cross-coefficient, which indicates the interaction between

*M*̃(

*f*

_{1},

*g*

_{1}) and

*M*̃(

*f*

_{2},

*g*

_{2}). Specifically,

*γ*(

*f*,

*g*), referred to as the effective source, is the Fourier transform of

*γ*(

*x*,

*y*).

*h*̃(

*f*,

*g*) is the Fourier transform of

*h*(

*x*,

*y*).

## 2.2. SVD model as the sum of coherent systems

*M*(

*x*,

*y*), referred to as

*M*(

*m*,

*n*),

*m*,

*n*=1,2, …,

*N*, the intensity distribution on the wafer shown in Eq. (2) can be reformulated as a function of matrices

*H*is the conjugate transposition operator. ŝ is an

*N*

^{2}×1 vector, and the

*ith*entry of s̃ is

*M*̃(

*p*,

*q*)=

*FFT*{

*M*(

*m*,

*n*)}, and

*FFT*{·} is the FFT operator.

*p*=

*i*

*mod*

*N*,

*A*is an

*N*

^{2}×

*N*

^{2}matrix including the information of the transmission cross-coefficient

*TCC*. Specifically, the

*ith*row and

*jth*column entry of

*A*is

*Ai*

*=*

_{j}*TCC*(

*p*,

*q*;

*r*,

*u*), where

*p*=

*i*

*mod*

*N*,

*r*=

*j*

*mod*

*N*, and

*p*,

*q*) and (

*r*,

*u*) in the argument of

*TCC*should be separated by the singular value decomposition. The result of the singular value decomposition of

*A*is

*α*

*is the*

_{k}*kth*eigenvalue, and

*α*

_{1}>

*α*

_{2}>…>

*α*

_{N}^{2}. The

*N*

^{2}×1 vector

*V*

*is the eigenfunction corresponding to*

_{k}*α*

*. Thus Eq. (4) becomes*

_{k}*S*

^{-1}(·) be the inverse column stacking operation which converts the

*N*

^{2}×1 column vector

*V*

*into a*

_{k}*N*×

*N*square matrix

*S*

^{-1}(

*V*

*). In particular,*

_{k}*V*

*,*

_{k}*is the*

_{i}*ith*entry of

*V*

*. Taking the inverse FFT of*

_{k}*h*̃

*(*

_{k}*p*,

*q*) leads to the

*kth*equivalent kernel of the SOCS model,

*N*

^{2}coherent systems. The scheme of the SOCS decomposition by SVD is depicted in Fig. 2. The

*ith*order coherent approximation to the partially coherent system is defined as

*N*=51. The pixel size is 11

*nm*×11

*nm*. Thus, the effective source is

*NA*=1.35 and λ=193

*nm*. The amplitude impulse response is defined as the Fourier transform of the circular lens aperture with cutoff frequency

*NA*/

*λ*[21, 22]; therefore,

*h*(

*x*,

*y*) is

9. 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]

## 3. PSM Optimization Using Inverse Lithography

### 3.1. Optimization using the SOCS model

*M*(

*m*,

*n*) be the input phase-shifting mask to an optical lithography system

*T*{·}, with a partially coherent illumination. The PCI optical system is approximated by a Hopkins diffraction

*Z*(

*m*,

*n*)=

*T*{

*M*(

*m*,

*n*)}. Given a

*N*×

*N*desired output pattern

*Z**(

*m*,

*n*), the goal of PSM design is to find the optimized

*M*(

*m*,

*n*) called

*M*̂(

*m*,

*n*) such that the distance

*d*(·, ·) is the mean square error criterion. The PSM inverse lithography optimization problem can thus be formulated as the search of

*M*̂(

*m*,

*n*) over the

*N*×

*N*real space ℜ

*N*×

*N*such that

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

*U*(

*x*-

*t*

*), which is approximated by the sigmoid function*

_{r}*t*

*is the process threshold, and a dictates the steepness of the sigmoid function.*

_{r}*M*

_{N×N}a real-valued matrix represents the mask pattern with a

*N*

^{2}×1 equivalent raster scanned vector representation, denoted as

*m*̱.

*H*

_{1}is a

*N*

^{2}×

*N*

^{2}matrix. Its equivalent two-dimensional filter is the first equivalent kernel

*h*

_{1}(

*m*,

*n*) of the SOCS model.

*N*×

*N*binary output pattern is denoted as

*Z**. It is the desired light distribution sought on the wafer. Its vector representation is denoted as

*z*̱*.

*N*×

*N*image denoted as:

*Z*=sig{|

*H*

_{1}{

*m*̱}|

^{2}}. The equivalent vector is denoted as

*z*̱.

*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}*N*×

*N*real-valued mask denoted as

*M*̂ minimizes the distance between

*Z*and

*Z**, ie,

*m*̱̂∈[-1,1].

*M*̂

*is the quantization of*

_{tri}*M*̂. Its equivalent vector is denoted as

*m*̱̂

*, with all entries constrained to -1, 0 or 1.*

_{tri}*z*̱=sig{|

*H*

_{1}{

*m*̱}|

^{2}}, the ith entry in this vector can be represented as

*h*

_{1}

_{,ij}is the

*i*,

*jth*entry of the first equivalent kernel

*h*

_{1}(

*m*,

*n*). In the optimization process,

*m*̱̂ is searched to minimize the

*L*

_{2}norm of the difference between

*z*̱ and

*z*̱*. Therefore,

*F*(·) is defined as:

*z*̱

*in Eq. (20) is represented in Eq. (18). In order to reduce the above bound-constrained optimization problem to an unconstrained optimization problem, we adopt the parametric transformation [10]. Let*

_{i}*m*̱

*=cos(*

_{j}*θ*̱

*),*

_{j}*j*=1,…,

*N*

^{2}, where

*θ*̱

*∈(-∞,∞) and*

_{j}*m*̱

*∈[-1,1]. Defining the vector*

_{j}*F*(

*θ*̱)

*̱ can be calculated as follows:*

_{θ}*θ*̱

*is the*

^{k}*k*

*iteration result, then at the*

^{th}*k*+1

*iteration:*

^{th}*s*

*θ is the step-size.*

_{̱}*m*̱̂

_{tri}_{,i}=sgn(

*m*̱̂

*)*

_{i}*U*(|

*m*̱̂

*|-*

_{i}*t*

*),*

_{m}*i*=1, …,

*N*

^{2}, where

*t*

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

_{m}*E*as the distance between the desired output image

*Z** and the actual binary output pattern

*Z*̄

*evaluated by the equation Eq. (2) and a hard threshold operator,*

_{b}## 3.2. Discretization regularization

24. 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 following, the discretization penalty is discussed.

*J*(

*m*̱)=

*F*(

*m*̱)+

*γ*

_{D}*R*

*(*

_{D}*m*̱). In our simulations for σ=0.3, discretization regularization attains near-trinary optimized mask and reduce 30% output pattern error.

## 3.3. Simulations

*nm*×561

*nm*. The matrices representing all of the patterns have dimension of

*N*×

*N*, where

*N*=51. The pixel size is 11

*nm*×11

*nm*. The partially coherent illumination is a circular illumination with small partial coherence factor σ=0.3. In Figure 6, the top row illustrates the input masks of (left) the desired pattern

*Z**, (center) the optimized real-valued mask

*M*̂, and (right) the optimized trinary mask

*M*̂

*. The optimized trinary mask, referred to as the alternating phase-shifting mask, includes clear areas and shifting areas, which introduce 180° phase difference with each other. In particular, the transmission coefficients of the clear area and shifting area are assigned to 1 and -1, respectively. The binary output patterns are shown in the bottom row. White, grey and black represent 1, 0 and -1, respectively. The effective source and the amplitude impulse response are shown in Eq. (11) and Eq. (12) with*

_{tri}*NA*=1.35 and

*λ*=193

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

*a*=200 and

*t*

*=0.003. The binary output patterns in the bottom row are evaluated by the equation Eq. (2) followed by a hard threshold operator with threshold*

_{r}*t*

*=0.33. The step length and the regularization weights are*

_{m}*s*

*̱=0.2 and*

_{θ}*γ*

_{D}=0.1. The initial mask pattern is the same as the desired binary output pattern

*Z**. For

*θ*̱, we assign the phase of

## 4. Post-processing of Mask Pattern Based on 2D DCT

19. J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in *Simulation of Semiconductor Processes and Devices*, 12 (2007). [CrossRef]

*M*̂′ is the inverse 2D DCT of the maintained low frequency components. The post-processed trinary mask

*M*̂′

*is the discretization of*

_{tri}*M*̂′.

## 5. PSM Optimization with Photoresist Tone Reversing

1. A. K. Wong, *Resolution enhancement techniques*, 1 (SPIE Press, 2001). [CrossRef]

*k*=0.29,

*λ*=193

*nm*and

*NA*=1.35. The resolution limit is

*nm*×561

*nm*contains two pairs of vertical bars, each with pitch width of 22

*nm*<

*R*. In this simulation, all of the parameters are the same as the simulation shown in Fig. 6, except for

*t*

*=0.0003. In Fig. 10, from left to right, the first figure shows the desired pattern. The second figure shows the binary output pattern of the desired pattern. The third figure shows the optimized trinary PSM. The last figure shows the binary output pattern of the optimized PSM. White, grey and black represent 1, 0 and -1, respectively. Note that the binary output pattern of the optimized PSM is totally different from the desired pattern, indicating that the PSM optimization approach cannot attain the desired output pattern on the wafer. The reason is that the dimension of the features in the desired pattern is smaller than the resolution limit without application of the photoresist tone reversing.*

_{r}*t*

*=0.01, and the optimized trinary mask is shown in the middle figure. The binary output pattern of the optimized trinary mask is shown in the right figure. White, grey and black represent 1, 0 and -1, respectively. It is obvious that photoresist tone reversing method is effective to expose a sparse feature, whose resolution limit is much higher than the traditional case without application of photoresist tone reversing.*

_{r}## 6. Conclusions

## Acknowledgments

## References and links

1. | A. K. Wong, |

2. | S. A. Campbell, |

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

4. | F. Schellenberg, |

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

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

7. | M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, “Improving resolution in photolithography with a phase-shifting mask,” IEEE Trans. Electron Devices |

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

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

10. | A. Poonawala and P. Milanfar, “OPC and PSM design using inverse lithography: A non-linear optimization approach,” in |

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

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

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

14. | E. Wolf, “New spectral representation of random sources and of the partially coherent fields that they generate,” Opt. Commun.38 (1981). [CrossRef] |

15. | P. S. Davids and S. B. Bollepalli, “Generalized inverse problem for partially coherent projection lithography,” in |

16. | X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” in |

17. | X. Ma and G. R. Arce, “Binary mask opitimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A25 (2008). [CrossRef] |

18. | N. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. thesis, University of California at Berkeley (1998). |

19. | J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, “Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT),” in |

20. | 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 (1982). [CrossRef] [PubMed] |

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

22. | R. Wilson, |

23. | N. Cobb and A. Zakhor, “Fast sparse aerial image calculation for OPC,” in |

24. | C. Vogel, |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(100.3190) Image processing : Inverse problems

(110.4980) Imaging systems : Partial coherence in imaging

(110.5220) Imaging systems : Photolithography

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: October 2, 2008

Revised Manuscript: November 10, 2008

Manuscript Accepted: November 18, 2008

Published: November 21, 2008

**Citation**

Xu Ma and Gonzalo R. Arce, "PSM design for inverse lithography with partially coherent illumination," Opt. Express **16**, 20126-20141 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-20126

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### References

- A. K. Wong, Resolution enhancement techniques, 1 (SPIE Press, 2001). [CrossRef]
- S. A. Campbell, The science and engineering of microelectronic fabrication, 2nd ed. (Publishing House of Electronics Industry, Beijing, China, 2003).
- F. Schellenberg, "Resolution enhancement technology: The past, the present, and extensions for the future, Optical Microlithography," in Proc. SPIE 5377, 1-20 (2004). [CrossRef]
- F. Schellenberg, Resolution enhancement techniques in optical lithography (SPIE Press, 2004).
- 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]
- A. Poonawala and P. Milanfar, "Fast and low-complexity mask design in optical microlithography - An inverse imaging problem," IEEE Trans. Image Process. 16, 774-788 (2007). [CrossRef] [PubMed]
- M. D. Levenson, N. S. Viswanathan, and R. A. Simpson, "Improving resolution in photolithography with a phase-shifting mask," IEEE Trans. Electron Devices ED-29, 1828-1836 (1982). [CrossRef]
- Y. Liu and A. Zakhor, "Binary and phase shifting mask design for optical lithography," IEEE Trans. Semicond. Manuf. 5, 138-152 (1992). [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]
- A. Poonawala and P. Milanfar, "OPC and PSM design using inverse lithography: A non-linear optimization approach," in Proc. SPIE, 6154, 1159-1172 (San Jose, CA, 2006).
- X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," in Proc. SPIE (San Jose, CA, 2007).
- X. Ma and G. R. Arce, "Generalized inverse lithography methods for phase-shifting mask design," Opt. Express 15, 15,066-15,079 (2007). [CrossRef]
- B. Salik, J. Rosen, and A. Yariv, "Average coherent approximation for partially cohernet optical systems," J. Opt. Soc. Am. A 13 (1996). [CrossRef]
- E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38 (1981). [CrossRef]
- P. S. Davids and S. B. Bollepalli, "Generalized inverse problem for partially coherent projection lithography," in Proc. SPIE (San Jose, CA, 2008). [CrossRef]
- X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," in Proc. SPIE (Taiwan, 2008). [CrossRef]
- X. Ma and G. R. Arce, "Binary mask opitimization for inverse lithography with partially coherent illumination," J. Opt. Soc. Am. A 25 (2008). [CrossRef]
- N. Cobb, "Fast optical and process proximity correction algorithms for integrated circuit manufacturing," Ph.D. thesis, University of California at Berkeley (1998).
- J. Zhang, W. Xiong, M. Tsai, Y. Wang, and Z. Yu, "Efficient mask design for inverse lithography technology based on 2D discrete cosine transformation (DCT)," in Simulation of Semiconductor Processes and Devices, 12 (2007). [CrossRef]
- 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 (1982). [CrossRef] [PubMed]
- M. Born and E. Wolfe, Principles of optics (Cambridge University Press, United Kingdom, 1999).
- R. Wilson, Fourier Series and Optical Transform Techniques in Contemporary Optics (John Wiley and Sons, New York, 1995).
- N. Cobb and A. Zakhor, "Fast sparse aerial image calculation for OPC," in BACUS Symposium on Photomask Technology, Proc. SPIE, 2440, 313-327 (1995).
- C. Vogel, Computational methods for inverse problems (SIAM Press, 2002). [CrossRef]

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