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Zernike Piston Statistics in Turbulent Multi-Aperture Optical Systems

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Zernike Piston Statistics in Turbulent Multi-Aperture Optical Systems Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-2020 Zernike Piston Statistics in Turbulent Multi-Aperture Optical Systems Joshua J. Garretson Follow this and additional works at: https://scholar.afit.edu/etd Part of the Optics Commons Recommended Citation Garretson, Joshua J., "Zernike Piston Statistics in Turbulent Multi-Aperture Optical Systems" (2020). Theses and Dissertations. 3161. https://scholar.afit.edu/etd/3161 This Thesis is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. ZERNIKE PISTON STATISTICS IN TURBULENT MULTI-APERTURE OPTICAL SYSTEMS THESIS Joshua J. Garretson, Capt, USAF AFIT-ENG-MS-20-M-023 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. The views expressed in this document are those of the author and do not reflect the official policy or position of the United States Air Force, the United States Department of Defense or the United States Government. This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States. AFIT-ENG-MS-20-M-023 ZERNIKE PISTON STATISTICS IN TURBULENT MULTI-APERTURE OPTICAL SYSTEMS THESIS Presented to the Faculty Department of Electrical and Computer Engineering Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command in Partial Fulfillment of the Requirements for the Degree of Master of Science in Electrical Engineering Joshua J. Garretson, B.S.E.E. Capt, USAF March, 2020 DISTRIBUTION STATEMENT A APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED. AFIT-ENG-MS-20-M-023 ZERNIKE PISTON STATISTICS IN TURBULENT MULTI-APERTURE OPTICAL SYSTEMS THESIS Joshua J. Garretson, B.S.E.E. Capt, USAF Committee Membership: Stephen Cain, Ph.D Chair Maj David Becker, Ph.D Member Capt Joseph Tompkins, M.S.E.E Member AFIT-ENG-MS-20-M-023 Abstract There is an ever growing difficulty in acquiring an accurate and clear situational awareness of all objects in the orbit of Earth. The growing difficulty is a result of the shear number of objects being placed in orbit, advancing technology, and a growing threat from America's adversaries. There are currently several methods of detecting and tracking objects in orbit that include radar and optical telescopes. Optical telescopes have a better resolution and can yield better intelligence if they take advantage of the most up-to-date optical technology and astronomy practices. Some of these technologies include atmospheric monitors, adaptive optics, stellar interferometers, and fringe trackers. All of these technologies are an attempt to obtain a diffraction limited image buy canceling out the effects of the atmosphere. There is currently a lack of research into how the atmosphere effects Zernike piston. This Zernike piston is a coefficient related to the average phase delay of a wave. Usually Zernike piston can be ignored over a single aperture because it is merely a delay added to the entire wavefront. For multi-aperture interferometers though piston cannot be ignored. The statistics of Zernike piston could supplement and improve atmospheric monitoring, adaptive optics, stellar interferometers, and fringe tracking. This research will focus on developing a statistical model for Zernike piston introduced by atmospheric turbulence. iv Table of Contents Page Abstract . iv List of Figures . vii List of Tables . ix I. Introduction . .1 1.1 The Problem . .1 1.1.1 Problem Background . .1 1.1.2 Problem Statement . .5 1.2 This Research . .5 1.2.1 Basic Concepts . .5 1.2.2 Hypothesis . .5 1.2.3 Research Objectives . .6 1.2.4 Approach . .6 II. Background and Literature Review . .7 2.1 Zernike Decomposition . .7 2.2 Turbulence Models . 10 2.2.1 Von Karman Turbulence . 11 2.2.2 Other Outer Scale Models . 12 2.3 Covariances of Zernike Coefficients . 13 2.4 Current Atmospheric Profiling and Monitoring Techniques . 16 2.5 Atmospherically Induced Piston Differential . 18 2.6 Interferometry . 19 2.6.1 Coherence . 19 2.6.2 Temporal Delay in Spatial Interferometers . 23 2.6.3 Visibility and Fringe Tracking . 25 2.7 Summary of Background and Literature Review . 28 III. Methodology . 29 3.1 Variance of Piston . 29 3.1.1 Derivation . 29 3.1.2 Numerical Analysis . 31 3.2 Spatiotemporal Piston Autocovariance . 32 3.3 Tilt Variance and Spatiotemporal Autocovariance . 33 3.4 Piston Tracking with Fringe Tracking . 36 3.4.1 State Definition . 36 3.4.2 Conditional Mean and Covariance of the States . 37 v Page 3.4.3 Stochastic Model . 39 3.5 Data Collection . 41 3.5.1 Lab Set Up . 42 3.5.2 Estimation of Seeing and Outer Scale . 44 3.6 Summary of Methodology . 45 IV. Results and Analysis . 47 4.1 Results of Simulations . 47 4.1.1 Piston Monte Carlo Simulation with No Temporal Correlation . 47 4.1.2 Piston Monte Carlo Simulation with Temporal Correlation . 47 4.1.3 Fringe Tracking Performance . 49 4.2 Results of Data Collection . 51 4.2.1 Tilt Correlation and Atmospheric Parameters . 51 4.2.2 Piston Fringe Data . 52 4.3 Comparisons . 54 4.4 Summary of Results . 54 V. Conclusions . 55 5.1 Summary of Research . 55 5.2 Recommendations and Future Work . 55 5.3 Significance and Impact . 57 Appendix A. Math Appendix . 58 1.1 Covariance From Spatial to Wiener Spectrum . 58 1.1.1 Transformation of Zernike Coefficient Variance to Wiener Spectrum from Spatial Domain . 58 1.1.2 Transformation of Zernike Coefficient Spatial Autocovariance to Wiener Spectrum from Spatial Domain . 59 1.2 Multivariate Normal Conditional Mean and Covariance . 60 Bibliography . 62 vi List of Figures Figure Page 1. The Magdalena Ridge Optical Interferometer . .4 2. Zernike polynomials . .9 3. Kolmogorov and von Karman PSD Comparison . 12 4. Comparison of 4 PSDs . 13 5. Visual layout of a DIMM . 17 6. The effects of atmospheric distortion on path difference . 19 7. Young's Experiment . 21 8. The Michelson Interferometer . 23 9. Typical Stellar Interferometer . 24 10. Visibility of a Rectangular Spectrum . 27 11. Piston variance per L0 ........................................... 31 12. Autocovariance of piston . ..
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