Lithography simulation employing rigorous solutions to Maxwell's equations

Proceedings of SPIE - The International Society for Optical Engineering

Volumn 3334, Issue , 1998, Pages 176-196

  Gordon, Ronald L ^a   Mack, Chris A ^a  

^a KLA Tencor Austin   (United States)

Author keywords

Absorbing boundary conditions; Kirchhoff approximation; Lithography simulation; Maxwell's equations

Indexed keywords

BOUNDARY CONDITIONS; COMPUTER SOFTWARE; LIGHT TRANSMISSION; LITHOGRAPHY;

ABSORBING BOUNDARY CONDITIONS; KIRCHHOFF APPROXIMATION; LITHOGRAPHY SIMULATION; MATE RIAL PROPERTIES; MAXWELL'S EQUATIONS;

MAXWELL EQUATIONS;

EID: 0011257412     PISSN: 0277786X     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1117/12.310747     Document Type: Conference Paper

References (29)

1. C. Mack, Inside PROLITH, Austin: FINLE Technologies, Inc., 1997, Ch.2 B & C.
2. Interesting discussions of diffraction problems for which the Kirchhoff approximation represents an exact solution can be found in F. Kottler, "Diffraction at a black screen", Progress in Optics, vol. IV, pp. 281-314 (1966),
3. and E.W. Marchand & E. Wolf, "Consistent Formulation of Kirchhoff's Diffraction Theory", J. Opt. Soc. Am. 56, 1712-1722 (1966).
4. G.W. Forbes, D.J. Butler, R.L. Gordon, A.A. Asatryan, "Algebraic corrections for paraxial wave fields", J. Opt. Soc. A,. A 14, 3300-3315 (1997).
5. C. Yuan, "Calculation of One-Dimensional Lithographic Aerial Images Using the Vector Theory", IEEE Trans. Electron Devices 40, 1604-1613 (1993).
6. A.K. Wong & A.R. Neureuther, "Mask Topography Effects in Projection Printing of Phase-Shifting Masks", IEEE Trans. Electron Devices 41, 895-902 (1994).
7. C. Pierrat, A. Wong, S. Vaidya, M. Vernon, "Phase-Shifting Mask Topography Effects on Lithographic Image Quality", SPIE Vol. 1927, 28-41 (1993).
8. G. Wojcik, J. Mould, Jr., R. Ferguson, R. Martino, K.K. Low, "Some image modeling issues for I-line, 5x phase shifting masks", SPIE Vol. 1927, 455-465 (1994).
9. R.A. Ferguson, A.K. Wong, T.A. Brunner, L.W. Liebmann, :Pattern-Dependent Correction of Mask Topography Effects for Alternating Phase-Shifting Masks", SPIE Vol. 1927, 349-360 (1995).
10. V.D. Barger & M.G. Olsson, Classical Electricity and Magnetism, Boston: Allyn and Bacon, Inc., 1987, p. 381.
11. M. Born & E. Wolf, Principles of Optics, Oxford: Pergamon Press, 1980, p. 52.
12. See for example, T.B.A. Senior & J.L. Volakis, Approximate boundary conditions in electromagnetics, London: The Institution of Electrical Engineers, 1995.
13. B.B. Baker & E.T. Copson, The Mathematical Theory of Huygen's Principle, Oxford: Clarendon Pres, 1950, Ch. 1, Secs. 5-6.
14. S.N. Karp, "A convergent far-field expansion for the two-dimensional radiation functions", Commun. Pure Appl. Math. 14, 427-434 (1961).
15. K.S. Yee, "Numerical Solution of initial boundary value problems involving Maxwell's equations in isotropic media", IEEE Trans. Ant. Prop., 14, 302-307 (1966).
16. W.H. Press, S.A. Teukolsky, W.T. Vetterling, & B.P. Flannery, Numerical Recipes in C, Cambridge: Cambridge University Press, 1992, Sec. 19.1. They go on to show that, with this so-called "staggered leapfrog" type of scheme, for the ID wave equation, there is also no amplitude dissipation suffered. The proof for the 2D case here is a simple extension.
17. A.K. Wong, A.R. Neurether, "Rigorous Three-Dimensional Time-Domain Finite-Difference Electromagnetic Simulation for Photolithographic Applications", IEEE Trans. Semiconductor Manufacturing 8, 419-431 (1995).
18. There are many works in the literature dealing with this topic. The first to suggest a pseudo-differential operator approach [and an approximation with a rational function] appears to be E.L. Lindman, "'Free-Space' Boundary Conditions for the Time-Dependent Wave Equation"" J. Comp. Phys. 18, 66-78 (1975).
19. The most cited work on these boundary conditions is B. Engquist and A. Madja, "Absorbing boundary conditions for the numerical simulation of waves", Math. Comp. 31, 629-651 (1977).
20. The first paper geared toward the solution of Maxwell's equation rather than the wave equation was G. Mur, "Absorbing Boundary Conditions for the Finite-Difference Approximation of the Time-Domain Electromagnetic Field Equations", IEE Trans. Elec. Comp. EMC-23, 377-382 (1981). Eq. (2.25) can be derived using analysis from this latter paper.
21. P.A. Tirkas, C.A. Balanis, R.A. Renaut, "Higher Order Absorbing Boundary Conditions for the Finite-Difference Time-Domain Method", IEEE Trans. Ant. Prop. 40, 1215-1222 (1992).
22. L.N. Trefethen, L. Halpern, "Well-Posedness of One-Way Wave Equations and Absorbing Boundary Conditions", Math. Comp. 47, 421-435 (1986).
23. π/2dθcosθ. For the zeroth-order boundary condition, δ=,π-3≈142 , while for that in Eq. (2.28), δ=19/3-2π≈0.050.
24. For an implementation of such a method, see Ref. 18. The analysis performed in this work is greatly simplified using a result from K. McInturff and P.S. Simon, "Closed-Form Expressions for Coefficients Used in FD-TD High-Order Boundary Conditions", IEEE Microwave and Guided Letters 3, 222-223 (1993).
25. C.A. Mack, "Analytical expression for the standing wave intensity in photoresist", Appl. Opt. 25, 1958-1961 (1986).
26. This form of Sommerfeld's solution can be found in Ref. 10, pp. 569-570.
27. See H.H. Hopkins, "The concept of partial coherence in optics", Proc. Royal. Soc., A208, 263-277 (1951). Note that this approximation here takes the form of the assumption that the point spread function of an optical system only depends on the difference of the coordinates in the object and image planes. This is also presented in great detail in Born & Wolf, pp. 526-532.
28. Alfred K. Wong, "Rigorous Three-Dimensional Time-Domain Finite-Difference Electromagnetic Simulation", Ph.D. Thesis, Memorandum No. UCB/ERL M94/69, U. Cal., Berkeley (1994). The question of how many samples over σ-space are actually needed to produce a certain accuracy is addressed in detail in Robert J. Socha, "Propagation Effects of Partially Coherent Light in Optical Lithography and Inspection", Ph.D. Thesis, Memorandum No. UCB/ERL M97/55, U. Cal., Berkeley (1997).
29. Inside PROLITH, p. 33.