Gradient based transmission testing is a small area of optical metrology in which especially the technique of experimental ray tracing recently developed to an extraordinary technique able to characterize the performance of almost any optical component with high resolution. Measuring the gradient instead of the actual parameter itself, will result in a more efficient sensing. Irrelevant constant factors are rejected by the inherent differentiation. In case of transmission testing of optical components, the gradient is connected to geometrical light rays, which are the orthogonal trajectories to surfaces of equivalent optical path length, denoted as geometrical wavefronts. Both parameters are therefore closely related and one of them may be reconstructed from measurement of the other. Techniques for detecting the gradient in transmission focus on detecting the slope of the rays and derive all other relevant parameters from it. Modal integration proves to be the correct choice for retrieving the wavefront. However, as was shown for the simple case of a spherical wavefronts, integration of the slopes will only retrieve an approximation of the sphere valid for very limited cases. A problem that arises when trying to detect a curved wavefront with respect to a flat detection plane. A method was demonstrated, that was able to successfully reconstruct the actual spherical wavefront connected to the detected slopes. Furthermore, expressions were derived for the exact determination of the radius of curvature of a Gaussian reference sphere based on the radial terms of the Zernike polynomials. Under certain conditions, this radius can be identified with the focal length of the optical system under test, probably its most import parameter. Aside from this, three more method were introduced, where the emphasis was set on retrieving the effective focal length instead of a more general focal length measured over the complete aperture of the lens and therefore, suffering the effects of spherical aberrations. In case of lenses with small focal length to diameter ratios, these aberrations will lead to a significant deviation from the design focal length.
The three methods retrieve the effective focal length from ray slope measurement by interpolation or extrapolation on the existing data based on polynomial model functions. Detecting focal length from ray slopes offers the enormous advantage of being freed from the necessity to determine the position of the principle plane, a task many wavefront related techniques struggle to fulfill. While the wavefront changes its shape over propagation, the slope of the rays stays the same in homogeneous media. The methods were compared using ray tracing simulations where they were able to retrieve the design focal length with an error in the range of 10-2 – 10-4 %. Introducing tilt as
Summary
a misalignment to lens under test, showed only a minor impact on the result with an increase of 0.005% for a tilt of 0.5°. Experimental ray tracing was suggested as a specially suited technique that can easily incorporate these methods for testing of optical components. Monte-Carlo simulations were applied to demonstrate how an uncertainty related to the poisoning system, as used in an experiment ray tracing setup, will propagate to the final focal length value. From all methods, the local curvature analysis (LCA) showed the best overall performance with lowest resulting uncertainty connected to the focal length and a lower sensitivity to tilt in the lens.
Furthermore, three different methods were discussed to obtain the modulation transfer function from a gradient measurement. This transfer function 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. Two of the demonstrated methods, which are almost identical, are based on the evaluation of a spot diagram in the focal plane of the optical system, where the spot diagram represents the intersection points of the test rays with that plane. Another more elaborate method utilizes the retrieved wavefront aberration function to create a virtual representation of a focused light spot in the focal plane by propagation using Fourier transform. Though the aberrations are a result from geometrical wavefronts, the method allows to introduce diffraction effects resulting from the limited extend of the lens pupil to the calculation. Results from these methods were referenced against the outcome of a widely used commercial ray tracing package. The comparison showed outstanding resemblance to the given references, which is surprising, since the evaluation of the modulation transfer function by the reference differs significantly in certain parts.
Aside from purely functional testing, experimental ray tracing showed in the past its capability to identify smallest deviations in an aspherical surface from its design values with sub-micrometer precision. This surface retrieval involves an elaborate minimization process that strongly relies on its model of the aspherical surface. Using the standard equation for aspherical surfaces as a model proved to have its limitations. The simple power series expansion as part of the description is inefficient in a minimization situation due to strong cancellation between the individual terms and has a tendency to become numerical unstable at higher orders. With Forbes' Q-polynomials, two sets of orthogonal polynomials for the description of aspherical surfaces were found, that are superior to the standard equation in terms of numerical stability and efficiency. The demonstrated recurrence relations enable the evaluation of these polynomials to arbitrary high orders on the base of lower order terms. This is especially useful when a polynomial expansion is not only used to
Summary deviations in testing of aspherical lenses. Furthermore, orthogonal polynomial sets offer a simplified solution to the least-squares problem as discussed within this work. This solution is much simpler to realize programmatically, significantly less computationally intensive and therefore, especially suited for embedded systems or real-time applications. However, the superiority of the Q-polynomials is strongly linked to their orthogonality defined over a continuous range. But a measurement always results in discretely distributed sample points. In case of techniques for gradient transmission testing, the sample points are distributed over an even grid with fixed spacing.
To retain the numerical advantages of the Q-polynomials, it was proposed to perform a Gram-Schmidt process to orthonormalize realizations of the polynomials to the sample grid of the measurements. It was proved that using this, orthogonality could be successfully retained in case of discrete data. Mid-spatial frequency components could be successfully modeled with high details close to the nano-meter region. The associated fit contained a polynomial expansion using more than 150 terms from the Q-polynomials performed with high efficiency.
Gradient based techniques will continue to evolve into more precise and fast measurement techniques for all kind of relevant parameters of optical components . Especially, experimental ray tracing shows the capability to be a general all-around tool in optical metrology. Though the presented work is focusing on testing components in transmission, it was also demonstrated how the principles discussed here can readily be applied to characterization of reflective optical elements.
Further extending the application of ray tracing to surface measurement in reflection forms a promising topic for future research.
Acknowledgements
Acknowledgments
At this point I want to thank everyone who has contributed to the success of this work. Without the support of these people, the work presented here would not have been possible.
I am deeply grateful to my academical advisers, Prof. Dr. rer. nat. Thomas Henning and Prof. Dr.-Ing. Friedrich Fleischmann from the City University of Applied Sciences Bremen for their encouragement, advise and interest in my work. They successfully managed to mentor me through the period of my studies from the beginning as an undergraduate student.
I would like to express my deepest appreciation to my supervisor Prof. Dr.-Ing. Dietmar Knipp, who supported me to the end and believed in the success of my work.
I would also like to express my appreciation to my fellow colleagues Dr. Ufuk Ceyhan for all the interesting conversations on and off of the topic and for laying the foundation of this work as well as Ralf Lüning for sharing his expertise in experimental implementation. Furthermore, I want to thank the students Tobias Binkele for his richness of ideas, Jan Schulze for his keen eye for details and Gustavo Barreto for his patience and persistence in finding the accurate solution. Working with them added a tremendous factor of inspiration and amusement.
I would like to thank my parents, Birgit and Hans-Joachim Hilbig, whom I owe my existence in the first place. They made every effort to ensure I was raised in a sheltered and carefree childhood.
They always believed in me and mentioned me with pride.
Finally and most importantly, I want to thank my lovely wife Nina, who with great patience and kindness endured whatever happened over the course of my PhD-study period. She always supported me with all her heart and energy, granting me all the space and time needed to successfully finish this thesis. This work at all positive things that might result from it are first and foremost dedicated to her.
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