MITIGATION OF AMPLITUDE AND PHASE DISTORTION OF SIGNALS UNDER MODIFIED VON KARMAN TURBULENCE USING ENCRYPTED CHAOS WAVES Dissertation Submitted to The School of Engineering of the UNIVERSITY OF DAYTON In Partial Fulfillment of the Requirements for The Degree of Doctor of Philosophy in Engineering By Fathi H.A. Mohamed UNIVERSITY OF DAYTON Dayton, Ohio August, 2016 MITIGATION OF AMPLITUDE AND PHASE DISTORTION OF SIGNALS UNDER MODIFIED VON KARMAN TURBULENCE USING ENCRYPTED CHAOS WAVES Name: Mohamed, Fathi H.A. APPROVED BY: _____________________ ____________________ Monish R. Chatterjee, Ph.D. Partha P. Banerjee, Ph.D. Advisory Committee Chairman Committee Member Professor Professor Department of Electrical and Department of Electrical and Computer Engineering Computer Engineering Director Electro-Optics Graduate Program _____________________ ____________________ Eric J. Balster, Ph.D. Muhammad N. Islam, Ph.D. Committee Member Committee Member Associate Professor Professor Department of Electrical and Department of Mathematics Computer Engineering ____________________________ ___________________________ Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean Professor School of Engineering School of Engineering ii ABSTRACT MITIGATION OF AMPLITUDE AND PHASE DISTORTION OF SIGNALS UNDER MODIFIED VON KARMAN TURBULENCE USING ENCRYPTED CHAOS WAVES Name: Mohamed, Fathi H.A. University of Dayton Advisor: Dr. Monish R. Chatterjee Atmospheric turbulence as an agency affecting the propagation of electromagnetic (EM) waves in different regions of the earth relative to the ground plane has been studied extensively over the past several decades. Mathematical models describing turbulence itself relative to EM waves have been developed by a variety of investigators in the last 50 or more years. It turns out that the majority of these models are essentially in the spatial domain, involving transverse spatial coordinates and their spatial frequency counterparts in the spectral domain. Most turbulence models start out by assuming a random dependence of the medium permittivity on the turbulence. This leads to a random model describing what is commonly referred to as the refractive index power density spectrum. It is well known that propagation through standard atmospheric turbulence creates ripples, random distortions, phase variations and also for monochromatic cases scintillations in the recovered signals. One idea that was proposed to the investigators of this research was that perhaps pre-packaging the EM signal inside a trackable chaos waveform might offer some measure of shielding for the signal even as iii the overall EM wave passes through turbulence. With this objective in mind, this work began by first establishing standard numerical simulations of EM propagation through homogeneous regions upon passage through a variety of apertures. This standard application involved the use of the Fresnel-Kirchhoff diffraction integral implemented in two ways: (a) as a direct propagation from an object to an image plane, and (b) segmented propagation over uniform incremental layers of the medium in the longitudinal direction. The latter approach was put into place in anticipation of the later introduction of a turbulent layer in the system. Following successful implementation of this technique, turbulence was inserted once again in two different ways: (a) assuming a relatively narrow region of turbulence, modeled as a planar random phase screen derived from the use of the well-known modified von Karman spectrum (MVKS) for refractive index; and (b) the case of an extended random region which is modeled by inserting multiple planar random screens along the propagation path. These initial approaches led to the determination of the resulting scalar output fields numerically derived as complex entities. In the first half of this work, time statistics of the scalar fields were obtained by repeating the simulation multiple times on the basis of an assumed (relatively low) frequency of variation of the turbulence phenomenon (of the order of, say, 20-100 Hz). These time statistics were then incorporated into a transfer function model involving two random processes: (a) the MVKS phase turbulence for which the time statistics were derived as mentioned; and (b) a purely time-dependent chaos wave generated via an acousto-optic (A-O) Bragg cell under feedback, whose first-order optical output is thereby encrypted by an input signal waveform. Use the transfer function approach, cross spectral densities and corresponding cross-correlation functions between the two iv random phenomena were numerically derived with the final cross correlation product containing the vital message information. Retrieving the message signal from the turbulence-chaos cross correlation product became a prohibitive task, and therefore, even though further investigations are needed, a new approach was developed to complete the intended work. In this approach, a modulated carrier wave which is both time and space- dependent, is propagated through a region of homogeneous space, and upon diffraction through the region, is picked up at the receiver and the embedded message is recovered using appropriate electronics. Thereafter, the same process is repeated in the presence of spatial turbulence, and the recovered signal waveforms are averaged over multiple runs of the simulation representing the time statistics of the turbulence. It is demonstrated that signals recovered under varying degrees of turbulence indeed suffer moderate to severe phase and amplitude distortion, as expected. It must be noted that all numerical simulations reported here are based on strictly near-isoplanatic and paraxial or low 2 propagation angle basis, such that the essential turbulence parameter, 퐶푛 is h-independent for all practical purposes. In the final application of this strategy, an encrypted chaos wave riding on an optical carrier is propagated through narrow turbulence of varying strengths, and recovered using a chaos-based heterodyne detection technique. It is shown that indeed encapsulation of the message inside the chaos reduces the distortions in the recovered signal which occur when chaos is not used. v Dedicated to my parents for their unconditional love and support AND To my wife and children for giving me the drive to succeed. vi ACKNOWLEDGMENTS I would like to express my appreciation to my advisor, Professor Monish Chatterjee, for his vision, wisdom, continuous guidance, friendship, and enthusiasm through the completion of this research effort. Without his constant feedback, encouragement, and guidance I would still be struggling to compile and document this research. Also, I would like to thank Professor Partha Banerjee for his dedication, depth of knowledge and positive criticism throughout this work. His insight, advice, and valued comments have been academically valuable. Thank you also to my committee members, Dr. Eric Balster and Dr. Muhammad Islam. I am deeply grateful to Professor Arun Majumdar for his guides and valued comments at the early stage of this work. Additionally, I would like to thank the head of the department, Dr. Guru Subramanyam, for continuous support, making it possible for me to travel to several conferences. Also, I would like to extend my appreciation to the ECE department staff, especially Nancy Striebich, for her cheerful help in administrative issues. I want to thank my family. I cannot thank enough my parents, for their continuous love, guidance, and support throughout my entire educational path. Finally, I want to thank my wife, for initiating my drive to pursue my Ph.D. and providing me with all of the necessary emotional support. I also thank my children, for being a continuous inspiration for me to succeed. vii TABLE OF CONTENTS ABSTRACT………………………………………………………………….…………..iii DEDICATION……………………………………………………………………………vi ACKNOWLEDGMENTS……………………………………………………………….vii LIST OF FIGURES…………………………………………………………………...…xii LIST OF TABLES……………………………………………………………………..xxxi CHAPTER 1 BACKGROUND……………...…………………………………………...1 1.1 Introduction………………………………………………………………………….1 1.2 Turbulence- a brief overview ……………………………………………………….3 1.3 Outline of this dissertation…………………………………………………………..5 1.3.1 Acousto-optic chaos……………………………………………………………..5 1.3.2 Aperture diffraction properties with and without turbulence…………………...7 1.3.3 Propagation of Gaussian beams through narrow and extended turbulence……..8 1.3.4 Auto and cross correlations, power spectral densities and the transfer function formalism ……………………………………………………………………..10 1.3.5 Propagation of non-chaotic and chaotic EM waves through turbulence using modulation theory……………………………………………………………...12 CHAPTER 2 ACOUSTO-OPTIC (A-O) CHAOS AND CHAOS GENERATION…...14 2.1 Introduction…………………………..…………………………………………….14 viii 2.2 Nonlinear dynamics of the A-O Bragg cell under feedback…………………….…17 2.3 Chaos time, frequency, and power spectral density representations………………20 CHAPTER 3 DIFFRACTION PROPERTIES OF THE PROFILED BEAM TRANSMISSION THROUGH BINARY APERTURES AND RANDOM PHASE SCREEN …………………………………….………………………………...28 3.1 Introduction………………………………………………………………………...28 3.2 Split-beam propagation method……………………………………………………30 3.3 Fresnel-Kirchhoff diffraction integral…………………………………………...…34 3.4 Numerical simulations, results and interpretations...………………………………36 3.4.1 Split - step method versus direct Fresnel diffraction integral approach…....36 3.4.2 Far-field (Fraunhofer) diffraction pattern with binary aperture………..…..39