David Hilbig

• Gradient based measurement techniques for transmission testing of optical components represent a relatively small group in optical metrology. Nevertheless, some of them are already in wide spread use and others have the potential to be an all-around tool for the extensive characterization of optical components. This thesis will provide a thorough introduction into the theoretical background connected with these techniques. Furthermore, it will introduce several new methods to correctly determine two of the most relevant parameters of optical systems from gradient measurement with high accuracy. One of these is the effective focal length, whose correct determination still pose a problem for available measurement techniques, which deviate from its definition and provide the user with a result that suffers from aberrations effects. Especially for fast lenses with small diameter, these influences may generate errors that fairly excel common specified tolerances. Three numerical analysis methods are discussed and compared that evaluate the effective focal length from gradient measurement. This is done by simulations of a very strong spherical lens with an f/#‑number of 1. The advantage of using ray slopes over wavefront is not having to determine the exact position of the principle plane, since the slopes are invariant along the ray propagation in homogeneous media. Results from experiments demonstrated that these methods combined with a certain gradient method are able to retrieve a focal value with an error of only 0.063 %. The second parameter of interest covered in this work is the modulation transfer function (MTF), which is a quantitative measure of image quality, describing the ability of an optical system to transfer different levels of detail from an object to an image. Its value is of high practical relevance and traditionally measured from imaging appropriate test target units. Several methods will be discussed that allow to generate the MTF from gradient measurement. One of these is also suitable for highly corrected optical systems, which are beyond the limits of other methods. From the gradient techniques, experimental ray tracing was demonstrated to be capable of retrieving the shape of an aspherical lens from transmission test, provided that certain assumptions apply. The performance of the retrieval is bound to the utilized model function of the aspherical surface. In this case, traditional surface descriptions of aspheres are inefficient and numerical unstable when it comes to modeling surface feature in the mid-spatial regime. With Forbes' Q-polynomials, two sets of orthogonal polynomials were found, that are superior to the standard equation and promise to be a suitable replacement. Their positive properties solely result from their orthogonality. Recurrence relations will be demonstrated that enable the evaluation of the polynomials to arbitrary high orders on the base of lower order terms. In gradient techniques, the lateral resolution is commonly limited. In these cases, the Q-polynomials, defined in the continuous sense, will lose their positive properties. Within this work, a process will be proposed that retains the properties of this polynomials in case of discrete data sets by discrete orthonormalization. Orthogonal polynomials play a vital role in various parts of this work and therefore, will be discussed in more detail.