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## 4.2.3 Zernike Polynomials for General Aberration Terms

General aberration terms are characterized most conveniently by the circle polynomials bc(,) of Zernike [11, pp. 464-468]. The aberration function (n, m) is expanded as follows:

 (4.66)

As before, refers to the maximal optical path difference of the lens measured in units of the wavelength, while the factor Aac is determined by the object location (rocos, rosin),

 (4.67)

The orientation of the incident wavevector is given by the polar coordinates (,),

 (4.68)

The Zernike polynomials in real-valued form are defined as

 (4.69)

whereby bc() are the so-called orthogonal radial polynomials given by [11, p. 465]

 (4.70)

A thorough discussion can be found in, e.g., [11, Appendix VII]. In Table 4.2 the explicit forms of (4.79) for the first few values of the indices b and c are listed. The normalization has been chosen so that bc(1) = 1 for all permissible values of b and c.

Table 4.2: Radial polynomials bc() for b7, c7 (after [11, p. 465]).

 c b 0 1 2 3 4 5 6 7 0 1 2 - 1 6 - 6 + 1 20 - 30 + 12 - 1 1 3 - 2 10 - 12 + 3 35 - 60 + 30 - 4 2 4 - 3 15 - 20 + 6 3 5 - 4 21 - 30 + 10 4 6 - 5 5 7 - 6 6 7

The lens aberrations have a variety of impacts on the lithographic performance [117,119]. The most severe problems are shifts in the image position, image asymmetry, reduction of the process window, and the appearance of undesirable imaging artifacts. Aerial image simulation provides a powerful tool to investigate most of them. In Table 4.3 the imaging consequences of the first 11 Zernike polynomials are listed.

Table 4.3: Imaging consequences of first 11 Zernike polynomials (after [120]).

 (b, c) Name Imaging consequence (0, 0) Piston None (1, 1) Lateral translation Shift of image, independent of pattern (2, 0) Defocus Image degradation (2, 2) Astigmatism (hor./vert. or 45o) Orientation dependent shift of focus (3, 1) Lateral coma Image asymmetry and pattern dependent shift of image (3, 3) Three-leaf clover (rotated 30o) Imaging anomalies with threefold symmetry (4, 0) Third-order spherical Pattern dependent focus shift

Next: 4.2.4 In-Lens Filters Up: 4.2 Lens Aberrations and Previous: 4.2.2 Power Series Representation
Heinrich Kirchauer, Institute for Microelectronics, TU Vienna
1998-04-17