In modern optical element manufacturing, center artifacts are a common problem. A center artifact is a shape error that is rotationally symmetrical, steep, and localized at the center. These properties cause characteristic image defects different from those caused by ordinary irregularities. However, tolerancing center artifacts has not been fully discussed or properly carried out. We propose a simple mathematical model for center artifacts using normal distribution function as a figure model and showing that this function can be represented by a polynomial including odd-order terms. Our method enables appropriate optical simulation and tolerancing for center artifacts using general optical design software.
We theoretically represent the effectiveness of the odd-order surface for optical designs via an aberration theory. The theory employs an expression of the odd-order surface with aberration coefficients, which are derived by power-series expansion of wavefront aberration with respect to the coordinates of optical surfaces. This expression allows us to understand the aberration characteristics of odd-order surfaces. By applying this aberration theory to the design of extreme ultraviolet lithography optics, we show that odd-order surfaces are effective in reducing higher-order aberrations with fewer coefficients than even-order surfaces do.
Odd-order surfaces have begun to be used in optics. In order to investigate the aberration characteristics of such surfaces, Zernike expansion is widely used since it directly and explicitly corresponds to wavefront aberrations. Since the Zernike expansion of an odd-order surface contains an infinite number of terms, the convergence of the expanded sum and the possibility of termwise derivatives are not explicitly guaranteed mathematically. We give a complete proof for these problems. For an application of this result, we analyze the aberration characteristics of odd-order surfaces and present their effectiveness in optical design.