Networks of local minima for extreme ultraviolet lithographic objectives
Oana Marinescu and Florian Bociort
Optics Research Group, Delft University of Technology, The Netherlands Phone: (+31) 15 278 8109 Fax: (+31) 15 278 8105
E-mail: O.Marinescu@tnw.tudelft.nl
Introduction
Local minima situated in a multidimensional merit function space are connected via links that contain saddle points and form a network1.
Network detection
1) start from a local minimum
2) detect all saddle points connected with the local minimum
3) local optimization downwards on both sides of each saddle point -> new local minima
4) perform 2) and 3) for all new local minima 5) select the best solutions
m6.72525 m14.0456 m30.5273 m55.847 m203.541 m1592.24 m2973.37 m3435.5 m3913.88 s34.5787 s36.0449 s1052.75 s1731.37 s2802.48 s5951.91 s8139.24 s8416.16 s10921.3 s13786.4 s14544.9 s15719.5
We gratefully acknowledge the financial support of this research by ASML and TNO Science and Industry.
Reference
1. F. Bociort, E.van Driel, A. Serebriakov, “Networks of local minima in optical system optimization”, Optics Letters 29, 189-191 (2004)
Fig.4. Six-mirror microlithographic projection system with object heights
between 114 and 118 mm, a numerical aperture of 0.25, a magnification of 0.25, distortion below 1 nm and all incidence angles on the surfaces below 25û.The Strehl ratio is 0.995 and the merit function is 0.016 λ.
Conditions: variables: 6 curvatures
constraints: paraxial telecentricity
magnification
Fig.3 Network structure of a six-mirror system search, situated in a
four-dimensional merit function space.
The best local minimum is then reoptimized with all varia-bles and all constraints (Fig. 4).
The network method provides insight in the topography of the merit function space.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 -0.001 -0.0009 -0.0008 -0.0007 -0.0006 -0.0005 -0.0004 -0.0003 -0.0002 -0.0001 0 b • d • a c e
Fig.1 Network structure of a six-mirror system search, situated in a
five-dimensional merit function space. S represents the saddle points. M represents the minima. The value of the merit function is also shown.
The evolution of the merit function along a link is shown in Fig. 2.
Fig.2. Evolution of the merit function along the link. On the x-axis the
curvature of surface number three is represented. Systems a, b, c, d and e are shown in Fig. 1.
m0.0033845 m0.0666438 m0.732361 m2.45529 m9.65382 m22.4941 m131.427 m347.216 m374.199 m671.953 m6527.06 s0.621315 s1.20861 s11.9133 s16.8282 s49.3075 s132.329 s346.774 s809.94 s2406.97 s7094.32 d a b c e
Application
The best systems (represented in blue in Fig.1 and
Fig.3) remain in the network even when the number of
constraints is changed.
Conditions: variables: 6 curvatures
