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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 22 — Oct. 25, 2010
  • pp: 23331–23342
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Impacts of cost functions on inverse lithography patterning

Jue-Chin Yu and Peichen Yu  »View Author Affiliations


Optics Express, Vol. 18, Issue 22, pp. 23331-23342 (2010)
http://dx.doi.org/10.1364/OE.18.023331


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Abstract

For advanced CMOS processes, inverse lithography promises better patterning fidelity than conventional mask correction techniques due to a more complete exploration of the solution space. However, the success of inverse lithography relies highly on customized cost functions whose design and know-how have rarely been discussed. In this paper, we investigate the impacts of various objective functions and their superposition for inverse lithography patterning using a generic gradient descent approach. We investigate the most commonly used objective functions, which are the resist and aerial images, and also present a derivation for the aerial image contrast. We then discuss the resulting pattern fidelity and final mask characteristics for simple layouts with a single isolated contact and two nested contacts. We show that a cost function composed of a dominant resist-image component and a minor aerial-image or image-contrast component can achieve a good mask correction and contour targets when using inverse lithography patterning.

© 2010 OSA

1. Introduction

2. Methodology

Optical microlithography simulations are made possible by the Köhler’s illumination model [15

15. M. Born, and E. Wolf, Principles of Optics, 7th(expanded) ed. (Cambridge University Press, 1999).

17

17. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).

]. Currently, the industry standard illumination source still employs an ArF excimer laser with a 193 nm wavelength. The quasi-monochromatism of the ArF excimer laser results in partially coherent images on the wafer, which are also referred as the aerial images. The optical intensity of an aerial image can be formulated by a Sum of Coherent Systems (SOCS) model [17

17. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).

19

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

], where the illumination system including the source and projectors is decomposed into eight kernels using Singular Value Decomposition (SVD) [20

20. J. S. Leon, Linear Algebra with applications, 6th ed. (Prentice-Hall, 2002).

,21

21. 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]

]. Therefore, the electric fields are constructed by a convolution of different coherent kernels as indicated in Eq. (1). The total intensity at a given position (x, y) on the image plane is then the superposition of those kernels’ intensities, which can be expressed as Eq. (2).
Eq(x,y)=φq(x,y)o(x,y),
(1)
I(x,y)=q=1nλq|Eq(x,y)|2,
(2)
where E and I represent the electrical field and image intensity respectively. φq is the q th optical kernel, ⊗ denotes the convolution calculation, and o the mask function. λq is the eigenvalue of the q th kernel with n kernels in total.

The aerial image represents the distribution of optical intensity on the wafer, which corresponds to the exposure condition of a photoresist. Some photoresist models employ a Constant Threshold Resist (CTR) [22

22. W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE 5377, 1536–1543 (2001). [CrossRef]

], where the developed resist profile can then be described by a sigmoid transformation of the aerial image [8

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

,9

9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]

], that is,

T(I)=11+ea(Itr).
(3)

In Eq. (3), the parameter a describes the sensitivity of the photoresist reacting with the light, which depicts the slope of sidewall profiles, and tr is the parameter of the constant threshold level. Here, the value is set and normalized to 0.5.

D=[11O11O1111]N×N,
(6)

In Eq. (5), DI performs the operation that calculates the difference in the row direction, and ID T computes the column difference.

In order to apply these objective functions in an optimization algorithm, we also need to define the target aerial image based on the input patterns. The target aerial image is parameterized as
It(r)={α1+eb(rr0),GeometricCenterrBoundary,ec(rr0),rBoundary,
(7)
where It(r) is the target image as a function of the pixel index r. r is arranged in the radial direction from the geometric center of a drawn pattern, where r0 is the index of the boundary pixels. The generation of the pixel index is adopted from a wavefront expansion technique previously developed by us [11

11. J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE 7140, 714014 (2008). [CrossRef]

]. For any particular layout, we construct a wavefront from the feature’s edges both inwards and outwards. This way, we can construct a complete wavefront from a small region (or even a single point) inside each feature all the way to the edge of the image. Each wavefront, starting from the innermost, receives an index, and all the pixels on the layout are processed in order according to this index. A few examples of wavefront templates can be found in Ref [11

11. J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE 7140, 714014 (2008). [CrossRef]

]. As a result, the indices of boundary pixels end up being the same for individual features. It is also worth noting that this indexing technique will clearly identify pixels that are equidistant from two or more features. Moreover, in Eq. (7), α, b and c are constants to adjust for the target aerial image profile. Here, we choose a target image that is unity within the drawn and zero outside, corresponding to α = 1, b = -∞ and c = ∞. Since the designed target is normalized to unity at the maximal, the optical model used for simulations must be calibrated with experimental values. Once the target aerial image is defined, the target resist image is obtained by the corresponding sigmoid transformation and the target image contrast is the differential operation of the target aerial image. As a result, the three objective functions that evaluate the differences between the desired and the calculated profiles can be expressed as in Eq. (8), Eq. (9) and Eq. (10), respectively:

FI=ItI2,
(8)
FR=T(It)T(I)2,
(9)
FC=x(ItIa)2+y(ItIa)2,
(10)

Here, FI, FR and FC represent the costs for the aerial image, resist image, and aerial image contrast, respectively. The norm 2 denotes the square of Euclidean distance which denotes the inner product of the same vector. Moreover, we can further combine the components into a total cost function, F by assigning three weighting coefficients, γI, γR, and γC to the aerial image, resist image, and aerial image contrast, respectively. The final expression is then as follows:

F=γIFI+γRFR+γCFC.
(11)

It is important to note that the numeric cost values of these three objective functions, i.e. the variations between their minimal and maximal values, are not always in the same range. The individual costs not only depend on the drawn layouts, but also the nature of different objective functions. For example, the cost of the aerial-image component is often larger than that of the resist-image for simple layouts, since the cost of the former is contributed by all of the pixels on mask, while that of the latter mainly arises from the contour pixels. The cost variation for the resist-image component is therefore very limited compared to that for the aerial image. The normalization approach is essential to remove such a dependency on different objective functions. Therefore, for each case studied, we normalize the costs of the individual functions to the same interval of [0,10] by performing a linear transformation for costs at the initial (maximal) and at the final (minimal) iterations. The normalization process is not absolutely stringent, but necessary to compare the characteristics of different objective functions on the same basis for different drawn layouts. Therefore FI, FR, and FC in Eq. (11), represent normalized objective functions.

o=1+cos(θ)2.
(13)

We note that θ,oRN×N,0θπ,0o1. The converted mask then allows a continuous optical transmission value between zero and unity with θ varying between zero and π. Moreover, Eq. (13) can also be extended to a phase-shift mask (PSM) intuitively by multiplying a complex phase term [9

9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]

].

The cost function gradientFcan be derived as shown in Eq. (14)
F=γIFI+γRFR+γCFC.
(14)
The explicit expressions of the objective functions for the aerial image [7

7. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5, 043002 (2006). [CrossRef]

], resist image [8

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

,9

9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]

] and image contrast are listed in Eq. (15), Eq. (16) and Eq. (17), respectively:
FI(θ)=(q=1Q[2(ItI)2Eq]φqflip)(sin(θ)2),
(15)
FR(θ)=(qQ[2(T(It)T(I))a(1T(I))T(I)2Eq]φqflip)(sin(θ)2),
(16)
FC(θ)=(q=1Q[(2DT(D(ItI))2((ItI)DT)D)2Eq]φqflip)(sin(θ)2),
(17)
where · is the element-by-element multiplication operator and ⊗ is the convolution operator. Furthermore, φqflip is the up-down and left-right flip of φq, i.e. φqflip(i,j)= φq(Ni+1,Nj+1).

Finally, we employ a steepest-descent approach [8

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

,23

23. M. Minoux, Mathematical programming theory and algorithms (John Wiley and Sons, Chichester, 1986).

] to find a solution to the inverse problem. The step length is chosen to be 2 for the tradeoff between speed and convergence. The diagram in Fig. 1
Fig. 1 Steepest descent algorithm. Three parameters γR, γI and γC are used to bias the relative cost of aerial image, resist image and image contrast.
shows the general procedure of the iterative calculation.

The iterative algorithm can be summarized by the following pseudo-code:

  • .Initialization:
  • Assign an initial guess θ 0 and calculate thed0=F(θ0)F(θ0).
  • Initialize a constant step length λ.
  • .Iterative Step:
  • dk=F(θk)F(θk).
  • θk+1=θk+λdk.
  • .Stop condition:
  • If the stop condition, F< Fσ is satisfied, the algorithm terminates.

3. Results and discussion

In this section, we employ the gradient descent algorithm described previously to analyze two examples: a 190nm × 190nm isolated contact and two horizontally-aligned contacts each with an area of 100nm × 100nm and a spacing of 100nm. The test patterns are chosen such that the single contact is above the Rayleigh criterion: R = 0.61λ/NA ~170nm, while for the pair, both the spacing and dimensions are below the limit. Both layouts are represented in 256 × 256 pixel tiles with a pixel dimension of 10 nm. An annular illumination source is employed with σin = 0.4 and σout = 0.7. The partially coherent illumination model contains eight kernels, where λ = 193nm, NA = 0.7. Moreover, the constant threshold for the aerial image intensity is set and normalized to 0.5. Therefore the threshold parameter tr, required for the resist image transformation in Eq. (3) is set to 0.5, while a is chosen to be 90 to represent a conventional resist profile. We discuss the impact that the previously defined cost function components in various superposition configurations have on the resulting characteristics of the corrected grey-level masks, the contours and the aerial image intensities.

Figure 2(a)
Fig. 2 The inverse results of a large isolated contact evaluated by every cost function component individually where (a) (γI, γR, γC) = (1, 0, 0), (b) (γI, γR, γC) = (0, 1, 0) and (c) (γI, γR, γC) = (0, 0, 1). The corresponding contours and aerial images of (a)-(c) are shown in (d)-(f) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.
2(c) shows the generated gray-level masks using cost functions with a single component, that is, coefficients, (γI, γR, γC) = (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively. The corresponding aerial images and contours of the single large contact are shown in Fig. 2(d)2(f). As shown in Fig. 2(a), the cost function based on only the aerial-image component results in mask features concentrated on the drawn pattern, while in Fig. 2(b) the correction shows very large serifs, and in Fig. 2(c) concentric rings. The mask features very much reveal the nature of each objective function as discussed below.

Second, as shown in Fig. 2(b), the mask features show very large serifs. Since the sigmoid transformation can convert a number of different aerial images into resist images with sharp corners, as expressed in Eq. (3), the cost of the resist-image cost function mainly arises from the difference on the edges defined in Eq. (9). Therefore the nature of this objective function tends to drive the mask correction towards the threshold contour, in this case, a square. It is known that large serifs are signature structures to obtain contours with sharp corners. The aggressive corner correction is also confirmed by the green contour shown in Fig. 2(e). However, since there is a tradeoff between sharp corner contours and aerial image contrast, the overall optical intensity is weak. Moreover, because of the low intensity, this cost function may have limited performance for small features as well. From a mask characteristics point of view, the gray-level features surrounding the main pattern are not very significant, implying minimal EPEs arising from filtering out such features. Therefore, the mask resulting from the resist image is also adequate for manufacturing.

Finally, as shown in Fig. 2(c), the mask features arising from the aerial-image-contrast cost function appear as rings with a high grey-level contrast, which are very similar to a Fresnel zone plate in optics. The Fresnel zone plate functions as a lens and is used for focusing and imaging in optical systems. In other words, in order to obtain a high aerial-image contrast as dictated by the cost function, the resulting mask features converge to a pattern that can focus light. Due to the partial coherence of the illumination source, the resulting mask pattern does not have an analytical expression as a Fresnel zone plate which is often designed for a coherent source with a single wavelength [24

24. E. Hecht, Optics, 4thed (Addison Wesley, San Francisco, 2002).

]. However, based on the mask shown in Fig. 2(c), the dominant kernel that represents one of the coherent sources in the illumination model dictates the mask characteristics, making the zone-plate type correction still evident. Moreover, due to the distributed mask features, the spreading of light distribution on the resist, as shown in the red color of Fig. 2(f), is also higher than that of the other two cost functions. However, as long as the light spreading does not trigger the exposure of photoresist, i.e. remains below the threshold, the features have no side effects on patterning. Still, such mask features are not favorable in mask manufacturing, relatively speaking. Furthermore, the contour shows relatively large EPEs, which means that convergence is difficult. Overall, mask making is relatively feasible for the aerial-image and the resist-image cost functions, but not so much for the image-contrast. Finally, it is worth noting that the aerial image intensity on wafer is largest with the aerial-image-contrast, then the aerial-image, and lastly, the resist-image.

The superposition of various objective functions is described next, as shown in Fig. 3(a)
Fig. 3 The inverse results of a large isolated contact as with different combination of cost functions components where (a) (γI, γR, γC) = (1, 1, 0), (b) (γI, γR, γC) = (1, 0, 1), (c) (γI, γR, γC) = (0, 1, 1) and (d) (γI, γR, γC) = (1, 1, 1). The corresponding contours and aerial images of (a)-(d) are shown in (e)-(h) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.
3(d) for (γI, γR, γC) = (1, 1, 0), (1, 0, 1), (0, 1, 1), and (1, 1, 1), respectively. When dealing with combined objective functions, we find that while the resulting mask features have the footprint from both components, one component always appears dominant to the others. So, in Fig. 3(a), 3(b), and 3(d) the aerial image dominates in the cost function, while in Fig. 3(c), the aerial image contrast dominates the resist image. The resulting contours and aerial image intensities are thus also dictated by the dominant objective function, as shown in Fig. 3(e)3(h). However, since all the linear combinations of the cost function components can result in the exposure of photoresist for the large isolated contact, it is not conclusive as to what combination is best for inverse lithography patterning.

The same three individual cost function components are applied to the correction of two nested square contacts, where both the edge length and spacing are equal to 100 nm. This input pattern is challenging to the illumination model used in this work, and that reflects on the limitations of the different cost functions. As shown in Fig. 4(a)
Fig. 4 The inverse results of two close contacts evaluated by every cost function component individually where (a) (γI, γR, γC) = (1, 0, 0), (b) (γI, γR, γC) = (0, 1, 0) and (c) (γI, γR, γC) = (0, 0, 1). The corresponding contours and aerial images of (a)-(c) are shown in (d)-(f) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.
, the corrected mask using the aerial-image component maintains similar mask characteristics as discussed previously, which only corrects for the main patterns. However, the corresponding aerial image shown in Fig. 4(d) clearly show that the cost function has hit a limitation for small features, as the photoresist is underexposed. The similarity of the corrected mask to a configuration achievable by segment-based OPC further highlights that such an approach is no longer sufficient for critical CMOS nodes.

Next, as shown in Fig. 4(b), the correction using the resist image component was trapped in a local minimum during optimization, and only the left contact was successfully exposed. Since the resist cost function allows a number of aerial images to be transformed into a similar resist image, as evidenced by Eq. (3), its solution space is the largest among these three cases. Therefore, the advantage of the resist image cost function is a more complete exploration of possible mask corrections, while the disadvantage is the higher probability of getting trapped in local minima. Still, this disadvantage may be mitigated with the assistance of other objective functions, as will be discussed later. Furthermore, the exposed contact has four large mask features at the corners, known as sub-resolution assist features (SRAFs). These features help the exposure of the drawn patterns by bring up the intensity level, but they themselves do not print. The corner SRAFs can be thought of as being evolved from the large serifs of a big contact, as previously shown in Fig. 2(b). Therefore, particular footprint of the resist image cost function is still present. While not complete, the contour of the exposed contact shown in Fig. 4(e) is also very round and symmetric in shape, distinguishing itself from those corrected by the aerial-image or image-contrast to be shown afterward.

Expectedly, the corrected mask in Fig. 4(c) using the image contrast results in zone-plate type patterns. The optical intensity from the image-contrast function shown in Fig. 4(f) over exposes the photoresist, in contrast to an insufficient exposure shown in Fig. 4(d). We found that the convergence of small mask features can rarely be obtained by single-component cost functions. The contours in Fig. 4 reflect the limitations of the individual objective functions. However, Fig. 4 also provides evidence that SRAFs are mandatory for pushing forward the resolution limit of this illumination model.

As shown in Fig. 6(a)
Fig. 6 The inverse results of two close contacts with different combinations of cost function components where (a) (γI, γR, γC) = (1, 0, 32.8947), (b) (γI, γR, γC) = (0.0041, 1, 0) and (c) (γI, γR, γC) = (0, 1, 0.0447). The corresponding contours and aerial images of (a)-(c) are shown in (d)-(f) where the cyan and green contours respectively label the drawn pattern edges and aerial image threshold contours.
, the weighting coefficient of the image-contrast component can be gradually increased until contours appear. However, as seen in Fig. 6(a), after many iterations, the contours are still slightly deformed and the EPEs relatively large. In general, the superposition of the aerial image and image contrast results in unstable convergence for small features, as both of them have the same effect of increasing the intensity on input patterns, and therefore, a small change in the surrounding features could result in large intensity variations. Finally, since the influence of the resist image is the weakest among the three, in order to achieve a solution with the pattern characteristics of the resist image component, the coefficients of the other two have to be reduced and adjusted to achieve optimized EPEs. The masks obtained with a minor addition of aerial image and image contrast components are shown in Fig. 6(b) and 6(c), respectively. As seen in Fig. 6(b), the mask pattern contains two SRAFs near the corners of the central space, while in Fig. 6(c), there are six corner SRAFs. In both cases, the contours are very round in shape, unlike those obtained by the dominant aerial-image component by itself. The mask features are relatively adequate since the SRAFs are large in size, compared to those obtained by just the aerial-image component. Moreover, the mask features in Fig. 6(b) and 6(c) also coincide with general guidelines for SRAF placement in small contacts, which are usually determined by either rule-based or design of experiments (DOE) techniques. Here, we show that the best mask patterns and contours are obtained by using the resist image component with the assistance of either the aerial image or the aerial image contrast to avoid convergence issues.

4. Conclusion

In conclusion, we developed a gradient descent approach to investigate three different objective functions and their combinations. All of them show very unique characteristics in the resulting mask patterns, aerial images, and contours. We demonstrate that a clever mix of the objective functions can push the resolution limits while maintaining manufacturing-friendly masks. In this work, a cost function composed of a dominant resist-image component and a minor aerial-image or image-contrast component achieves a good mask correction and contours close to the target for the resolution-challenging twin contacts. The intermediate results also validate the necessity of using sub-resolution assist features in advanced CMOS nodes. We believe that these findings can provide informative guidance for the optimization of inverse lithography patterning in specific cases.

Acknowledgments

This work is founded by National Science Council in Taiwan under grant number 96-2221-E-009-095-MY3 and partly by the TSMC-University joint development program.

References and links

1.

D. S. Abrams and L. Pang, “Fast inverse lithography technology,” Proc. SPIE 6154, 534–542 (2006).

2.

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]

3.

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]

4.

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] [PubMed]

5.

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

6.

B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. 20, 781–784 (1981).

7.

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

8.

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

9.

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]

10.

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), 14746–14760 (2008). [CrossRef] [PubMed]

11.

J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE 7140, 714014 (2008). [CrossRef]

12.

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

13.

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

14.

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]

15.

M. Born, and E. Wolf, Principles of Optics, 7th(expanded) ed. (Cambridge University Press, 1999).

16.

J. W. Goodman, Statistical Optics (John Wiley and Sons, 1985).

17.

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE Press, 2005).

18.

N. B. Cobb, Fast optical and process proximity correction algorithms for integrated circuit manufacturing (University of California at Berkeley, Berkely, California, 1998).

19.

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

20.

J. S. Leon, Linear Algebra with applications, 6th ed. (Prentice-Hall, 2002).

21.

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]

22.

W. Huang, C. Lin, C. Kuo, C. Huang, J. Lin, J. Chen, R. Liu, Y. Ku, and B. Lin, “Two threshold resist models for optical proximity correction,” Proc. SPIE 5377, 1536–1543 (2001). [CrossRef]

23.

M. Minoux, Mathematical programming theory and algorithms (John Wiley and Sons, Chichester, 1986).

24.

E. Hecht, Optics, 4thed (Addison Wesley, San Francisco, 2002).

OCIS Codes
(100.3190) Image processing : Inverse problems
(110.3960) Imaging systems : Microlithography
(110.1758) Imaging systems : Computational imaging
(110.3010) Imaging systems : Image reconstruction techniques

ToC Category:
Imaging Systems

History
Original Manuscript: August 23, 2010
Revised Manuscript: October 11, 2010
Manuscript Accepted: October 15, 2010
Published: October 20, 2010

Citation
Jue-Chin Yu and Peichen Yu, "Impacts of cost functions on inverse lithography patterning," Opt. Express 18, 23331-23342 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-22-23331


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References

  1. D. S. Abrams and L. Pang, “Fast inverse lithography technology,” Proc. SPIE 6154, 534–542 (2006).
  2. 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]
  3. 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]
  4. 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] [PubMed]
  5. K. Nashold and B. Saleh, “Image construction through diffraction-limited high-contrast imaging systems: An iterative approach,” J. Opt. Soc. Am. A 2(5), 635–643 (1985). [CrossRef]
  6. B. Saleh and S. Sayegh, “Reductions of errors of microphotographic reproductions by optical corrections of original masks,” Opt. Eng. 20, 781–784 (1981).
  7. Y. Granik, “Fast pixel-based mask optimization for inverse lithography,” J. Microlith. Microfab. Microsyst. 5, 043002 (2006). [CrossRef]
  8. A. Poonawala and P. Milanfar, “Mask design for optical microlithography--an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007). [CrossRef] [PubMed]
  9. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]
  10. 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), 14746–14760 (2008). [CrossRef] [PubMed]
  11. J.-C. Yu, P. Yu, and H.-Y. Chao, “Model-based sub-resolution assist features using an inverse lithography method,” Proc. SPIE 7140, 714014 (2008). [CrossRef]
  12. X. Ma and G. Arce, “Binary mask optimization for inverse lithography with partially coherent illumination,” J. Opt. Soc. Am. A 25(12), 2960–2970 (2008). [CrossRef]
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