微影光學在半導體製程中扮演一個重要的角色：將電路的元件以及線路圖快
速的投射到矽晶圓表面，以增加製成的產量。但是在元件的尺寸接近甚至低於光
源波長時，光學鄰近效應開始發生，於是電路的平面圖無法忠實的反應在矽晶圓
表面。因此，今天的微影光學需要有鄰近效應修正的步驟，而其中最重要的步驟
之一就是光學影像的模擬。本論文的中心即是討論增加微影光學中模擬速度的可
能性。
微影光學成像模擬是將光源以及光罩圖以二維卷積疊加而成。而在進行此過程之
前，光罩通常會被分割成網格。網格的大小對於模擬的時間和精確度有很大的影
響：大的網格需要較低的模擬時間但是精確度低，反之小網格有高精確度但是需
要很長的時間和電腦資源。我們嘗試利用多重網格以增加精確度同時提高模擬速
度：光罩中沒有或較少幾何圖形的區域使用大網格，而需要高精確度的區域使用
小網格，因而大量提高了模擬的效率；而此方法所以可行源於二維卷積疊加的線
型特性。在此論文中我們進一步使用一靜態記憶體的基本線路展示了利用多重網
格進行微影光學鄰近效應修正的方法。

Photolithography is a key step in semiconductor process. The desired layout patterns are transferred to silicon surface repeatedly to increase the manufacturing throughputs. However, as the feature sizes of the circuit layout decreases beyond the light source wavelength, severe optical proximity problems occur. To alleviate these problems, optical proximity correction (OPC) method is essential to today’s IC manufacturing. The core of OPC is efficient and accurate simulation of the photolithography imaging. It is the aim of this thesis to explore the possibility to increase the simulation efficiency.

The key step in photolithography simulation is the 2-D convolution of layout patterns and light sources. The layout patterns are usually discretized into grids. Accurate simulations need small grid size while using a large amount of computer resources, time and memory. We explore multi-grid method to improve simulation time while keeping memory space to minimum. This approach is found to be feasible due to the linearity of the convolution operation. Good speed up is obtained. A workable OPC methodology using this multi-grid approach was also demonstrated in this thesis.

Acknowledgement 4
Contents 5
1 Introduction 7
1.1 Motivation 7
1.2 Related works 8
1.3 Convolution 9
1.4 The organization of this thesis 9
2 One-slit convolution in 1-D 10
2.1 Convolution in 1-D 10
2.2 Slit width and intensity 11
2.3 Point number and intensity 13
2.4 Point number and total power 15
2.5 Slit width and line width 16
2.6 OPC (Optical proximity Correction) in 1-D 17
3 Double-Slit convolution in 1-D 20
3.1 Double-Slit convolution 20
3.2 Double-Slit interval with convolution 21
3.3 Double-Slit interval and line width 21
4 Inverse mask 24
4.1 Convolution Schematic 24
4.2 LU decomposition 24
4.3 Modify transmission of mask 28
4.4 Discrete Fourier transform method 33
5 Convolution in 2-D 38
5.1 Convolution in 2-D 38
5.2 Compensation for corner rounding 42
5.3 Aberration 46
6 Spline interpolation in 2-D 53
6.1 Spline interpolation in 1-D concept 53
6.2 Apply spline interpolation to find smaller gird size pattern 56
6.3 Apply spline interpolation to real mask 60
6.4 OPC (optical proximity correction in 2-D) 82
6.5 Multiple gird size Convolution 94
7 Conclusion and future work 100
Reference 102

[1] B. J. Lin and B. J. Lin, "Optical lithography: here is why," 2010: SPIE Bellingham.
[2] J. F. Chen and J. A. Matthews, "Mask for photolithography," ed: Google Patents, 1993.
[3] C. A. Mack, "Corner rounding and line-end shortening in optical lithography," in Microlithographic Techniques in Integrated Circuit Fabrication II, 2000, vol. 4226: International Society for Optics and Photonics, pp. 83-92.
[4] C. A. Mack, "Optical proximity effects," Microlithography World, vol. 5, no. 2, pp. 22-23, 1996.
[5] K. Lai, "Review of computational lithography modeling: focusing on extending optical lithography and design-technology co-optimization," Advanced Optical Technologies, vol. 1, no. 4, pp. 249-267, 2012.
[6] X. Ma and G. R. Arce, Computational lithography. John Wiley & Sons, 2011.
[7] H. Yang, S. Li, Y. Ma, B. Yu, and E. F. Young, "GAN-OPC: Mask optimization with lithography-guided generative adversarial nets," in 2018 55th ACM/ESDA/IEEE Design Automation Conference (DAC), 2018: IEEE, pp. 1-6.
[8] M. Garza, J. V. Jensen, N. K. Eib, and K. K. Chao, "Optical proximity correction method and apparatus," ed: Google Patents, 2001.
[9] J. Ralston, "Gaussian beams and the propagation of singularities," Studies in partial differential equations, vol. 23, no. 206, p. C248, 1982.
[10] D. K. Lam, E. D. Liu, M. C. Smayling, and T. Prescop, "E-beam to complement optical lithography for 1D layouts," in Alternative Lithographic Technologies III, 2011, vol. 7970: International Society for Optics and Photonics, p. 797011.
[11] M. Endo and M. Sasago, "Pattern formation method and exposure system," ed: Google Patents, 2006.
[12] C. A. Mack, Field guide to optical lithography. SPIE Press Bellingham, WA, 2006.
[13] W. T. Rhodes, "Acousto-optic signal processing: convolution and correlation," Proceedings of the IEEE, vol. 69, no. 1, pp. 65-79, 1981.
[14] S. Liu, Q. Wang, and G. Liu, "A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses," Wear, vol. 243, no. 1-2, pp. 101-111, 2000.
[15] W. Qingwang, L. Huan, and G. Yanfeng, "Discriminating multiple kernel learning for joint classification of optical and LiDAR data in urban area," in 2015 7th Workshop on Hyperspectral Image and Signal Processing: Evolution in Remote Sensing (WHISPERS), 2015: IEEE, pp. 1-4.
[16] A. N. K. Reddy and D. K. Sagar, "Half-width at half-maximum, full-width at half-maximum analysis for resolution of asymmetrically apodized optical systems with slit apertures," Pramana, vol. 84, no. 1, pp. 117-126, 2015.
[17] A. Jameson and E. Turkel, "Implicit schemes and ùêøùëà decompositions," Mathematics of Computation, vol. 37, no. 156, pp. 385-397, 1981.
[18] J.-B. Martens, "Recursive cyclotomic factorization--A new algorithm for calculating the discrete Fourier transform," IEEE transactions on acoustics, speech, and signal processing, vol. 32, no. 4, pp. 750-761, 1984.
[19] S. McKinley and M. Levine, "Cubic spline interpolation," College of the Redwoods, vol. 45, no. 1, pp. 1049-1060, 1998.
[20] K. Nii et al., "2RW dual-port SRAM design challenges in advanced technology nodes," in 2015 IEEE International Electron Devices Meeting (IEDM), 2015: IEEE, pp. 11.1. 1-11.1. 4.