This manuscript provides a compendium for the application of the non- uniform wave model to standard problems of two beam interference and addresses common educational issues associated with wave propagation at attenuating interfaces. It also reports scientific advances in interferometric applications focusing on an interferometric point sensor with optical path length modulation and an areal measuring RGB-interferometer for fast topography measurements in the sub millisecond regime. Schake Markus C. Advanced theory of optical wave propagation and interferometric sensors for topography measurement Markus C. Schake ISBN 978-3-7376-0840-4 kassel university 9 783737 608404 press Markus C. Schake Advanced theory of optical wave propagation and interferometric sensors for topography measurement kassel university press This work has been accepted by Faculty of Electrical Engineering / Computer Science of the University of Kassel as a thesis for acquiring the academic degree of Doktor der Ingenieur- wissenschaften (Dr.-Ing.). Supervisor: Prof. Dr.-Ing. Peter Lehmann Co-Supervisor: Prof. Dr.-Ing. Tino Hausotte Defense day: 18. Oktober 2019 This document – excluding quotations and otherwise identified parts – is licensed under the Creative Commons Attribution-Share Alike 4.0 International License (CC BY-SA 4.0) https://orcid.org/0000-0002-9883-3494 (Markus C. Schake) Bibliographic information published by Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.dnb.de. Zugl.: Kassel, Univ., Diss. 2019 ISBN 978-3-7376-0840-4 (print) ISBN 978-3-7376-0841-1 (e-book) DOI: https://dx.doi.org/doi:10.17170/kobra-202007031403 © 2020, kassel university press, Kassel http://kup.uni-kassel.de Printed in Germany i Acknowledgments I want to thank everybody who contributed to the successful completion of this manuscript by ideas, discussions, distracting exhilaration or any other possible means including the mere bearing of my company, throughout the years I have been obsessively occupied with my dissertation. Special thanks, to my supervisor Prof. Lehmann and my colleagues for supporting my work by sharing their scientific expertise, proofreading of many contributions, sitting through hours of presentations and discussions, and making our department a place I liked to work at and will keep in good memory. ii iii Abstract The industrial demand in non-contact, perturbation resistant, fast and dynamic topog- raphy measurement systems for quality control catalyzes proliferating research in optical metrology. Interferometry enables axial resolution in the nano meter and even sub nano meter range in combination with an adjustable lateral resolution depending on the em- ployed microscope objectives. Therefore, research and development of interferometric devices and the corresponding phase retrieval algorithms are evermore evolving. This manuscript is organized in three major chapters, considering advances in different fields of interferometry related research. First, presenting advances in teaching and inter- pretation of non-uniform wave propagation in dissipative media. This manuscript provides a compendium for the application of the non-uniform wave model to standard problems of two beam interference. Especially considering wave propagation at attenuating interfaces and inside dissipative media. Second, presenting advances in development and applicability of a fiber coupled interfer- ometric system with periodical optical path length modulation employing a long coherent laser illumination. This system employs a miniaturized probe head, which is feasible to perform measurements inside hollow structures or at measuring objects with complex geometry. The present manuscript introduces novel details considering the plane of ref- erence inside the miniaturized fiber probe, the design procedure of a similar probe with low aperture for measurements in an expanded depth of field and correction procedures for systematic phase retrieval deviations related to time discrete Hilbert filters. Also the capability of a 3D-topography measuring system based on the fiber coupled sensor for sur- face measurements with dynamic focus tracking is presented. An analytic approach for extension of the unambiguity range in multi wavelength interferometry is implemented in a real time digital signal processor and applied for surface measurements and the feasibil- ity of the interferometric point sensor for measurements of rough surfaces is demonstrated with experimental results. Third, a novel measurement procedure for fast, perturbation resistant, areal surface mea- surement employing pulsed RGB-illumination and optical path length modulation in the reference arm is introduced. The presented interferometric system employs two interfer- ence images in quadrature and a microscopic image in a separate color channel of the recording device to retrieve the phase of the wavefront reflected from the surface. By synchronization of the pulse illumination with the short camera exposure gap between two consecutive images, the total data acquisition time of the interferometer is < 100 µs. This causes a high robustness to environmental perturbations and enables topography measurements of objects in lateral motion. The characteristics of the employed interfer- ometers and recording devices, like influence of dispersion effects or color crosstalk are discussed alongside a presentation of topography results illustrating the capabilities of the measuring procedure. iv CONTENTS 1 Contents Abstract iii Contents 1 1 State of the art in optical surface metrology 5 1.1 Laser interferometers with optical path length modulation and correspond- ingphaseretrievalmethods.......................... 12 1.2 Low coherence-, polarizing- and RGB-interferometrysystemsfortopographymeasurement......... 16 2 Theory of wave propagation and interference 21 2.1 Uniform plane wave equation ......................... 23 2.2 Non-uniform plane wave equation ....................... 25 2.3Wavepolarization................................ 27 2.4 Maxwell’s equations ............................... 31 2.5 Boundary conditions of the electric and magnetic field components parallel totheinterface................................. 32 2.6 Boundary conditions of the electric and magnetic flux density components orthogonaltotheinterface........................... 35 2.7 Maxwell’s equations and the dispersion relation of the non-uniform plane wave....................................... 38 2.8 Snell’s Law for the non-uniform wave model ................. 40 2 2.9Fresnel’sequationinnon-attenuatingmedia................. 42 2.9.1 Reflection and transmission of TE-light polarized perpendicular to theplaneofincidence.......................... 43 2.9.2 Reflection and transmission of TM-light polarized parallel to the planeofincidence............................ 44 2.10 Fresnel equations for lossy media in complex phasor expression ....... 46 2.11 Interference equations employing the non-uniform wave model ....... 51 2.11.1 Time averaged poynting vector in real field expression ........ 53 2.11.2 Time averaged poynting vector in complex phasor expression .... 56 2.11.3 Interference conditions of the common path interferometer ..... 58 2.12 Complex electric permittivity ......................... 65 3 Fiber coupled dual wavelength common path laser interferometer with periodical optical path length modulation 69 3.1 Identification of the reference plane location in a common path interferometer 71 3.2 Low aperture, fiber coupled, interferometric probe for focus tracking .... 74 3.3 Phase retrieval algorithms for real time implementation ........... 79 3.3.1 Lock-indetectionalgorithm...................... 80 3.3.2 Time discrete Hilbert filters ...................... 81 3.3.3 Characteristic deviations of discrete Hilbert filters .......... 84 3.3.4 Signal resampling to reduce systematic deviations in discrete Hilbert transformation............................. 86 3.3.5 Frequencyselectivequadraturefilters................. 89 3.4Focus-trackingsystem............................. 90 3.5 Analytic extension of the unambiguity range in dual wavelength interfer- ometry ..................................... 96 3.6 Applicability of the fiber coupled laser interferometer for roughness mea- surements....................................101 3 4 RGB-interferometry with pulsed LED illumination 105 4.1 RGB-Linnik-Interferometer for parameter identification based phase retrieval106 4.2RGB-Michelson-Interferometerforquadraturebasedphaseretrieval....110 4.3 Dispersion in the Michelson interferometer with pulsed RGB illumination . 115 4.4 Offset intensity distribution retrieval from the microscopic image ......121 4.5Crosstalkcorrection...............................131 4.6 Measuring objects in lateral motion ......................133 5 Conclusion and future work 139 Appendix 145 List of Publications 148 Nomenclature 148 Bibliography 161 4 5 Chapter 1 State of the art in optical surface metrology Optical surface metrology is a fast developing field of research. The industrial needs for fast and non destructive, inline quality control require precise optical instruments offering high lateral and height resolution, which yield reliable results in the presence of vibrations or other perturbations. Tactile measurement systems like the atomic force microscope and the stylus instrument, are limited by their slow measurement speed. Also the tactile probe may cause