A set of functions is presented which is orthonormal over the surface of a cylinder. The functions are useful for describing surface errors on the types of near cylindrical optics found in X-ray and some extreme ultra violet systems. In addition, the functions provide a convenient means of using surface deformation data to separate rigid body motions (misalignments) from surface errors. The functions are harmonic in the azimuthal direction, and polynomial in the axial direction. In this paper, the functions are detailed in terms of the form of their corresponding surface errors. Also, some general relationships are given, which relate the surface errors defined by these functions to the resulting wavefront errors. These relationships depend explicitly on the nature of the wavefront incident on the optic. This paves the way for describing the wavefront in a metrology configuration in terms of the errors on the optic and the misalignments in the system. Finally, some parallels are drawn between the use of these functions for describing near cylindrical optics, and the use of Zernike polynomials for describing conventional optics.
"A Set Of Orthonormal Surface Error Descriptors For Near Cylindrical Optics", Proc. SPIE 0429, Precision Surface Metrology, (15 November 1983); doi: 10.1117/12.936358; https://doi.org/10.1117/12.936358