Compensation of higher order rotationally symmetric Zernike coefficients caused by the defocusing of spherical sections

The sphericity calibration of spherical sections at PTB involves a focus scan of the sample under test, during which the topography of the sample under test is retrieved at multiple points close to the ideal confocal position. By linear interpolation the form of the sample under test, represented by Zernike polynomials, in the confocal position is retrieved. This is necessary, since defocusing gives rise to spherical aberrations represented by higher order, rotationally symmetric Zernike coefficients. The reason for this is, that the Zernike coefficient for defocus describes a full hemisphere above the unit circle. If spherical wavefronts with slightly different radius of curvature are compared in the same limited aperture, their difference does contain higher order rotationally symmetric Zernike terms and may not be described by pure defocus. In this article the influence of this effect on sphericity measurements of spherical sections is considered and a method is introduced to retrieve the topography of the sample under test for the condition of full compensation of the spherical wavefront part in the confocal position.