Abstract
The coherence function offers new possibilities for optical metrology that are not available with conventional wave field sensing. Its measurement involves a spatio-temporal sampling of the wave fields modulated by the object under investigation. Temporal sampling is well known e. g. by means of White Light Interferometry (WLI) and spatial sampling can e. g. performed by Computational Shear Interferometry (CoSI). The present paper describes an approach that combines both temporal and spatial sampling using a robust common-path setup. While the evaluation of the coherence function is more elaborate than approaches that either sample the temporal or the spatial domain, an information theoretical treatment shows that it also delivers more information about the object under investigation. Our approach is based on the mutual information that represents the reduction of uncertainty about the object as a consequence of the measurements performed. Using a simplified measurement case, we calculate the mutual information for different measurement situations and demonstrate that spatio-temporal sampling of the coherence function results in a higher mutual information as compared to classical approaches. Based on the proposed approach, we identify further open research tasks for an efficient information extraction from the coherence function to surpass current limitations of optical metrology.
Zusammenfassung
Die Kohärenzfunktion bietet neue Möglichkeiten für die optische Messtechnik, die mit konventioneller Wellenfeldsensorik nicht verfügbar sind. Die Messung der Kohärenzfunktion beinhaltet eine raum-zeitliche Abtastung der vom Messobjekt veränderten Wellenfelder. Die zeitliche Abtastung z. B. mittels Weißlichtinterferometrie (WLI) ist wohlbekannt, eine räumliche Abtastung kann z. B. mittels Computational Shear Interferometry (CoSI) erfolgen. Die vorliegende Veröffentlichung beschreibt einen Ansatz, der sowohl eine zeitliche als auch eine räumliche Abtastung mit einem robusten Common-Path-Aufbau kombiniert. Während die Auswertung der Kohärenzfunktion aufwändiger ist als Ansätze, die entweder eine zeitliche oder eine räumliche Domäne abtasten, zeigt eine informationstheoretische Betrachtung, dass sie auch mehr Informationen über das Messobjekt liefert. Unser Ansatz basiert auf der Transinformation, die die Verringerung der Unsicherheit über das Messobjekt als Folge der durchgeführten Messungen darstellt. Anhand einer vereinfachten Messsituation berechnen wir die Transinformation für verschiedene Messsituationen und zeigen, dass die raum-zeitliche Abtastung der Kohärenzfunktion im Vergleich zu klassischen Ansätzen zu einer höheren Transinformation führt. Basierend auf dem vorgeschlagenen Ansatz identifizieren wir weitere offene Forschungsfragen für eine effiziente Informationsextraktion aus der Kohärenzfunktion, um derzeitige Beschränkungen der optischen Messtechnik zu überwinden.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: 265388903
Funding statement: Gamma-Profilometry, Grant No. 265388903, at BIAS and SFB/TRR 136 project C06 at BIMAQ both funded by the Deutsche Forschungsgemeinschaft (DFG).
About the authors
Ralf B. Bergmann studied physics in Heidelberg and Freiburg, received his doctorate with his work at the Max Planck Institute for Solid State Research (MPI-FKF) from the University of Stuttgart, worked as a postdoc at of the University of New South Wales and habilitated at the University of Freiburg. After leading a research group at the University of Stuttgart he headed the department of Applied Physics at the Central Research and Advance Engineering facility of the Robert Bosch GmbH and later the Physical Analyses Laboratory in the Automotive Electronics division. Since 2008 he is a professor at the University of Bremen in the Faculty of Physics and Electrical Engineering and head of the Bremen Institute for Applied Beam Technology (BIAS) with the field of “Optical Metrology and Optoelectronic Systems”.Foto: Marcus Windus / BIAS
Andreas Fischer studied electrical engineering, completed his PhD at the Technische Universität Dresden in 2009 and his habilitation in 2014. Since 2016, he is a professor at the University of Bremen in the Faculty of Production Engineering and head of the Bremen Institute for Measurement, Automation and Quality Science (BIMAQ). His research areas cover optical measurement principles for flow and production processes, in-process applications of model-based measurement systems, and the investigation of fundamental limits of measurability.
Carsten Bockelmann received the Dipl.-Ing. and Ph. D. degrees in electrical engineering from the University of Bremen, Germany, in 2006 and 2012, respectively. Since 2012, he has been a Senior Research Group Leader at the University of Bremen within the Faculty of Physics and Electrical Engineering coordinating research activities regarding the application of compressive sensing and machine learning to communication problems. His research interests include massive machine-type communication, ultra-reliable low latency communications and industry 4.0, compressive sensing and channel coding.
Armin Dekorsy has more than ten years of industrial experience in leading research positions, such as an DMTS at Bell Labs Europe and the Head of Research Europe Qualcomm, Nuremberg. Since 2010 he is Professor for Communications Engineering and Head of the Department of Communications Engineering at the University of Bremen in the Faculty of Physics and Electrical Engineering. His current research focuses on distributed signal processing, compressed sampling, information bottleneck method, and machine learning leading to further development of communication technologies for 5G/6G, industrial wireless communications, and NewSpace satellite communications. He investigates wireless communication and signal processing for the baseband of transceivers of future communication systems, the results of which are transferred to the pre-development of industry through political and strategic activities.
Alberto Garcia-Ortiz obtained the diploma degree in Telecommunication Systems from the Polytechnic University of Valencia (Spain) in 1998. After working for two years at Newlogic in Austria, he started the Ph. D. at the Institute of Microelectronic Systems, Darmstadt University of Technology, Germany and received his PhD in 2003. From 2003 to 2005, he worked as a Senior Hardware Design Engineer at IBM Deutschland Development and Research in Böblingen. He then joined the start-up AnaFocus (Spain), where he was responsible for the design and integration of AnaFocus’ next generation Vision Systems-on-Chip. Since 2010 he is a full professor for the chair of integrated digital systems at the University of Bremen in the Faculty of Physics and Electrical Engineering. His interests include low-power design and estimation, communication-centric design, SoC integration, and hardware accelerators for machine learning.
Claas Falldorf studied physics at the University of Bremen, where he received his doctorate at the Faculty of Physics and Electrical Engineering in 2009. Since then he heads the group “Coherent Optics and Nano-Photonics” at BIAS – Bremen Institute of Applied Beam Technology. His research focusses on optical metrology, coherence theory, signal processing and optimization theory.Foto: Marcus Windus / BIAS
References
1. J. W. Goodman, Statistical Optics (John Wiley & Sons, Inc., Hoboken, USA, 2000). p. 63.Search in Google Scholar
2. C. Falldorf, M. Agour, A. F. Müller, and R. B. Bergmann, “Gamma-Profilometry: a new paradigm for precise optical metrology,” Opt. Exp. 29(22), 36100–36110 (2021).10.1364/OE.434510Search in Google Scholar PubMed
3. R. Bergmann, M. Kalms, and C. Falldorf, “Optical In-Process Measurement: Concepts for Precise, Fast and Robust Optical Metrology for Complex Measurement Situations,” Appl. Sci. 11, 10533 (2021). https://doi.org/10.3390/app112210533.10.3390/app112210533Search in Google Scholar
4. A. F. Müller, C. Falldorf, M. Lotzgeselle, G. Ehret, and R. B. Bergmann, “Multiple Aperture Shear-Interferometry (MArS): a solution to the aperture problem for the form measurement of aspheric surfaces,” Opt. Exp. 28(23), 34677–34691 (2020).10.1364/OE.408979Search in Google Scholar PubMed
5. C. Falldorf, M. Agour, and R. B. Bergmann, “Digital holography and quantitative phase contrast imaging using computational shear interferometry,” Opt. Engin. 54, 024110 (2015).10.1117/1.OE.54.2.024110Search in Google Scholar
6. C. Falldorf, J.-H. Hagemann, G. Ehret, and R. B. Bergmann, “Sparse light fields in coherent optical metrology,” Appl. Opt. 56, F14–F19 (2017).10.1364/AO.56.000F14Search in Google Scholar PubMed
7. D. Malacara, Interferogram Analysis For Optical Testing (CRC Press, Hoboken, NJ, 2005).10.1201/9781420027273Search in Google Scholar
8. C. Krause, R. B. Bergmann, and C. Falldorf, “Statistical analysis of phase values for the determination of step heights in multi-wavelength interferometry,” TM-Technisches Messen 89(6), 430–437 (2022). https://doi.org/10.1515/teme-2021-0139.10.1515/teme-2021-0139Search in Google Scholar
9. B. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).Search in Google Scholar
10. T. M. Cover and J. A. Thomas, Entropy, Relative Entropy, and Mutual Information, chap. 2, pp. 13–55 (John Wiley & Sons, Ltd, 2005). https://onlinelibrary.wiley.com/doi/pdf/10.1002/047174882X.ch2, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/047174882X.ch2.Search in Google Scholar
11. T. M. Cover and J. A. Thomas, Differential Entropy, chap. 8, pp. 243–259 (John Wiley & Sons, Ltd, 2005). https://onlinelibrary.wiley.com/doi/pdf/10.1002/047174882X.ch8, URL https://onlinelibrary.wiley.com/doi/abs/10.1002/047174882X.ch8.Search in Google Scholar
12. Y. C. Eldar and G. Kutyniok, eds., Compressed Sensing - Theory and Applications (Cambridge, 2012).10.1364/FIO.2012.FM4C.1Search in Google Scholar
13. Y. Wang, Q. Yao, J. T. Kwok, and L. M. Ni, “Generalizing from a Few Examples: A Survey on Few-Shot Learning,” ACM Comput. Surv. 53(3) (2020). URL https://doi.org/10.1145/3386252.10.1145/3386252Search in Google Scholar
14. C. Bishop, Pattern Recognition and Machine Learning (Springer, 2006). URL https://www.microsoft.com/en-us/research/publication/pattern-recognition-machine-learning/.Search in Google Scholar
15. R. Shwartz-Ziv and N. Tishby, “Opening the Black Box of Deep Neural Networks via Information,” CoRR abs/1703.00810 (2017). 1703.00810, URL http://arxiv.org/abs/1703.00810.Search in Google Scholar
16. M. Lefebvre, L. Moreau, R. Dekimpe, and D. Bol, “7.7 A 0.2-to-3.6TOPS/W Programmable Convolutional Imager SoC with In-Sensor Current-Domain Ternary-Weighted MAC Operations for Feature Extraction and Region-of-Interest Detection”, in 2021 IEEE Internat. Solid- State Circuits Conference (ISSCC), vol. 64, pp. 118–120 (2021).10.1109/ISSCC42613.2021.9365839Search in Google Scholar
17. L. Alba, R. Domínguez Castro, F. Jiménez-Garrido, S. Espejo, S. Morillas, J. Listán, C. Utrera, A. García-Ortiz, M. D. Pardo, R. Romay, C. Mendoza, A. Jiménez, and Á. Rodríguez-Vázquez, “New Visual Sensors and Processors”, in Spatial Temporal Patterns for Action-Oriented Perception in Roving Robots, P. Arena and L. Patanè, eds., pp. 351–369 (Springer Berlin Heidelberg, Berlin, Heidelberg, 2009). URL https://doi.org/10.1007/978-3-540-88464-4_8.10.1007/978-3-540-88464-4_8Search in Google Scholar
18. M. Kalms and R. B. Bergmann, “Structure function analysis of powder beds in additive manufacturing by laser beam melting,” Addit. Manuf. 36, 101396 (2020). https://Doi.org/10.1016/j.addma.2020.101396.10.1016/j.addma.2020.101396Search in Google Scholar
19. E. C. Teague, T. V. Vorburger, D. Maystre, and R. D. Young, “Light Scattering from Manufactured Surfaces,” CIRP Annals 30(2), 563–569 (1981).10.1016/S0007-8506(07)60168-1Search in Google Scholar
20. A. Fischer and D. Stöbener, “In-process roughness quality inspection for metal sheet rolling,” CIRP Annals 68, 523–526 (2019).10.1016/j.cirp.2019.04.069Search in Google Scholar
21. J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technology 8, 1379 (13 pp.) (1997).10.1088/0957-0233/8/12/002Search in Google Scholar
22. Y. Barranger, P. Doumalin, J. C. Dupré, and A. Germaneau, “Strain measurement by digital image correlation: influence of two types of speckle patterns made from rigid or deformable marks,” strain 48, 357–365 (2012).Search in Google Scholar
23. A. Tausendfreund, D. Stöbener, and A. Fischer, “In-process measurement of three-dimensional deformations based on speckle photography,” Applied Sciences 11, 4981 (11 pp.) (2021).10.3390/app11114981Search in Google Scholar
24. A. Fischer, “Limiting uncertainty relations in laser-based measurements of position and velocity due to quantum shot noise,” Entropy 21, 264 (19 pp.) (2019).10.3390/e21030264Search in Google Scholar PubMed PubMed Central
25. A. Fischer, “Fundamental flow measurement capabilities of optical Doppler and time-of-flight principles,” Experiments in Fluids 62(2), 37 (19 pp.) (2021).10.1007/s00348-020-03127-xSearch in Google Scholar
26. A. Fischer, “Fundamental uncertainty limit for speckle displacement measurements,” Appl. Opt. 56, 7013–7019 (2017).10.1364/AO.56.007013Search in Google Scholar PubMed
© 2022 Walter de Gruyter GmbH, Berlin/Boston
