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
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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
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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
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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.
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Dedicated to my parents for their unconditional love and support
AND
To my wife and children
for giving me the drive to succeed.
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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.
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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
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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
3.4.2.1 Uniform plane wave input………………………………...……………….39
3.4.2.2 Gaussian beam input….……………………………………………………44
3.4.2.3 Single-slit with Gaussian beam…………………………………...………..48
3.4.2.4 Thin sinusoidal amplitude grating with uniform plane wave………….…..49
3.4.2.5 Single-slit aperture with Bessel beam…….…….………………...………..50
3.4.3 Far-field (Fraunhofer) diffraction pattern of aperture with
random phase distribution in the aperture plan…………………….....………..51
3.4.3.1 Rectangular aperture……………………………………………..………..51
3.4.3.2 Circular aperture…………………………………………...……………...52
CHAPTER 4 INVESTIGATION OF PROFILED BEAM PROPAGATION
THROUGH NARROW AND EXTENDED TURBULENT MEDIUM………………..55
4.1 Introduction……………………………………………………………...…………55
4.2 Power spectrum models for refractive index fluctuations…………………………58
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4.3 Atmospheric turbulence……………………………………………...………….…61
4.3.1 Atmospheric turbulence models……………………………………………….62
4.3.2 Thin phase screen generation………………………………………………..…65
4.4 Fried parameter, scintillation index, and fringe visibility………………………….68
4.5 Propagation of profiled beam through narrow turbulence………….....…………...69
4.6 Temporal statistics…………………………………………………………………83
4.7 Propagation of profiled beam through extended turbulence…………….…..……..85
CHAPTER 5 A TRANSFER FUNCTION BASED FREQUENCY MODEL FOR
PROPAGATION OF A CHAOS WAVE THROUGH MODIFIED VON KARMAN
TURBULENCE UNDER VARIOUS CHAOS TURBULENCE CONDITIONS……..102
5.1 Introduction……………………………………………………………………….102
5.2 Turbulence power spectral density…………………………………...…………..103
5.2.1 Atmospheric turbulence strength……………………………………………..105
5.2.2 Phase screen position……………………………………………..……….….108
5.3 Turbulence time waveform ET(t) derived from spatial model………...………….111
5.4 Cross-correlation and cross-power spectral density functions……………………112
5.5 Results and discussion for specific cases of chaos and turbulence……………….113
CHAPTER 6 DIFFRACTIVE PROPAGATION AND RECOVERY OF
MODULATED (INCLUDING CHAOTIC) EM WAVES THROUGH
UNIFORM AND TURBULENT MEDIUM…………………………………...………123
6.1 Introduction……………………………………………………….………………123
6.2 Spectral approach to propagate the (non-chaotic) EM waves
through turbulence using SVEA and Fourier transforms………………….…..…125
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6.3 Numerical simulations, results and interpretations……………………………….127
6.3.1 A uniform (non-turbulent) propagation prototype………………...……….…128
6.3.2 Propagation through weak turbulence…………………………...……………129
6.3.3 Propagation through strong turbulence………………………...…………..…133
6.4 Spectral approach to encrypted chaotic wave propagation through
turbulence using SVEA and Fourier transforms ………..……………………..…138
6.5 Numerical simulations, results and interpretations………………...……………..140
6.5.1 A uniform (non-turbulent) propagation prototype………………...……….…140
6.5.2 Chaotic propagation through weak turbulence with mean
frequency fT = 50Hz……………………………………………………….….141
6.5.3 Chaotic propagation through strong turbulence with mean
frequency fT =50 Hz…………………………………………………………..143
CHAPTER 7 CONCLUDING REMARKS AND FUTURE WORK……….…………145
REFERENCES…………………………………………………………………………150
APPENDICES……………………………………………………………………….…157
A. ANALYTICAL PLOTS OF DIFFRACTION PATTERN…………………..…157
B. PUBLICATIONS RESULTING FROM THIS WORK………..………………163
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LIST OF FIGURES
Fig.2.1. Schematic of A-O system. (a) The Bragg regime (thick),
(b) The Raman-Nath regime (thin)………………………………...……………15
Fig.2.2. An A-O modulator with first-order feedback in the Bragg regime……………..17
Fig.2.3. A-O monostable and bistable regimes with hysteresis loop…………………….19
Fig.2.4. Monostable, bistable, multistable, and chaotic regimes……………….………..20
Fig.2.5. Chaotic wave data for (a) snapshot of time waveform,
and (b) Fourier transform……………..…………………...……...…….………22 ~ Fig.2.6. Chaotic wave PSD for = 4, ˆ0 = 2, Iinc =1, and TD = 1ms………………….22
Fig.2.7. Chaotic wave data for (a) snapshot of time waveform,
and (b) the Fourier transform………………………………...…………………23
Fig.2.8. Chaotic wave PSD for = 4, = 2, Iinc =1, and TD = 100µs……...……….23
Fig.2.9. Chaotic wave data for (a) snapshot of time waveform,
and (b) the Fourier transform……………………………………...……………24
Fig.2.10. Chaotic wave PSD for = 4, = 2, Iinc =1, and TD = 10µs………...……..24
Fig.2.11. Chaotic wave data for (a) snapshot of time waveform,
and (b) the Fourier transform…………..………………………………………25
Fig.2.12. Chaotic wave PSD for = 4, = 2, Iinc =1, and TD = 5µs…….…………..25
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Fig.2.13. Theoretical versus graphical data for average chaos
frequency for different TD values……………………..…………..…………..27
Fig.3.1. Schematic illustration and physical interpretation of the BPM…………...…….31
Fig.3.2. Geometry representation of the aperture and observation planes………….……35
Fig.3.3. Near-field diffraction pattern using Fresnel-integral diffraction
(direct method) for square aperture: (a) Input Gaussian beam
profile, (b) Gaussian beam profile at Z=L, (c) and (d) 2D section
along (xi,0) and (0,yi) respectively…………………………………………..….37
Fig.3.4. Near-field diffraction pattern using split-step method for square
aperture: (a) Input Gaussian beam profile, (b) Gaussian beam
profile at Z=L, (c) and (d) 2D section along (xi,0) and (0,yi) respectively……...37
Fig.3.5. Near-field diffraction pattern using Fresnel-integral diffraction
(direct method) for rectangular aperture: (a) Input Gaussian beam
profile. (b) Gaussian beam profile at Z=L, (c) and (d) 2D section
along (xi,0) and (0,yi) respectively……………………………………………...38
Fig.3.6. Near-field diffraction pattern using split-step method for rectangular
aperture. (a) Input Gaussian beam profile. (b) Gaussian beam profile
at Z=L. (c) and (d) 2D section along (xi,0) and (0,yi) respectively……….....…..38
Fig.3.7. Far-field diffraction pattern for binary double-slit aperture:
(a) Binary double-slit, (b) Far-field diffraction pattern,
(c) and (d) 2D section along (xi,0) and (0,yi) respectively…………….………..40
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Fig.3.8. Far-field diffraction pattern for very narrow horizontal double-slit:
(a) Very narrow horizontal double-slit aperture, (b) Far -field diffraction
pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively……………..41
Fig.3.9. Far-field diffraction pattern for circular aperture:
(a) Circular aperture, (b) Far -field diffraction pattern,
(c) and (d) 2D section along (xi,0) and (0,yi) respectively……………..……….42
Fig.3.10. Far-field diffraction pattern for annular aperture:
(a) Annular aperture, (b) Far -field diffraction pattern,
(c) and (d) 2D section along (xi,0) and (0,yi) respectively……….……………43
Fig. 3.11. Far-field diffraction pattern for infinitely narrow annual aperture:
(a) Infinitely narrow annual aperture, (b) Far -field diffraction
pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.………..…44
Fig.3.12. Far-field diffraction pattern for double slit aperture with wide
Gaussian beam: (a) Double slit aperture, (b) Far -field diffraction
pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively…………….45
Fig.3.13. Far-field diffraction pattern for double slit aperture with narrow
Gaussian beam: (a) Double-slit aperture, (b) Far -field diffraction
pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively………...…..46
Fig.3.14. Far-field diffraction pattern for very narrow double-slit aperture
with wide Gaussian beam: (a) Very narrow double-slit aperture,
(b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0)
and (0,yi) respectively…………………………………………………....……47
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Fig.3.15. Far-field diffraction pattern for very narrow double-slit aperture
with narrow Gaussian beam: (a) Very narrow double-slit aperture,
(b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0)
and (0,yi) respectively……...…………………………….……………………48
Fig. 3.16. Far-field diffraction pattern for single slit aperture with
Gaussian beam: (a) Single slit aperture, (b) Far -field diffraction
pattern, (c), and (d) 2D section along (xi,0) and (0,yi) respectively……….….49
Fig.3.17. Far-field diffraction pattern for thin sinusoidal amplitude grating
aperture: (a) Thin sinusoidal amplitude grating, (b) Far-field
diffraction pattern, (c) and (d) 2D section along (xi,0) and
(0,yi) respectively………………...…………………………………..……….50
Fig.3.18. Far-field diffraction pattern for single slit aperture with Bessel beam:
(a) Single-slit aperture, (b) Far-field diffraction pattern, (c) and
(d) 2D section along (xi,0) and (0,yi) respectively……………………………..51
Fig.3.19. Far-field diffraction pattern for rectangular aperture with
random phase distribution: (a) Rectangular aperture with
random phase distribution, (b) Far-field diffraction pattern,
(c) and (d) 2D section along (xi,0) and (0,yi) respectively……………………..52
Fig.3.20. Far-field diffraction pattern for circular aperture with
random phase distribution: (a) Circular aperture with
random phase distribution, (b) Far-field diffraction pattern,
(c) and (d) 2D section along (xi,0) and (0,yi) respectively…………………….53
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Fig. 4.1. Depiction of the process of turbulent decay, showing the energy
cascade and subsequent division of turbulent eddies in the atmosphere……....63
Fig. 4.2. 2D random phase screen distribution profile……………………….…………..67
Fig. 4.3. 3D random phase screen distribution profile………………….………………..67
Fig. 4.4. Schematic illustration of propagation through narrow turbulence……...……...69
Fig. 4.5. Gaussian beam propagation to distance z=L (phase screen at the
object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-
section, (c) Gaussian beam profile, (d) Random phase screen
profile, and (e) Gaussian beam phase distribution in transverse output plane.…72
Fig.4.6. Gaussian beam propagation to distance z = L (phase screen at 0.5L).
(a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian
beam profile, (d) Random phase screen profile, and (e) Gaussian
beam phase distribution in transverse output plane……………………..………72
Fig. 4.7. Gaussian beam propagation to distance z=L (phase screen at L
or image plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………………………...... …………….73
Fig. 4.8. Gaussian beam propagation to distance z=L (phase screen at the
object plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………………………………………...73
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Fig. 4.9. Gaussian beam propagation to distance z = L (phase screen at
0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section,
(c) Gaussian beam profile, (d) Random phase screen profile,
and (e) Gaussian beam phase distribution in transverse output plane…………74
Fig. 4.10. Gaussian beam propagation to distance z=L (phase screen at L or
image plane ). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………….…………………………….74
Fig.4.11. Gaussian beam propagation to distance z=L (phase screen at the
object plane ). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………………………….…………….76
Fig. 4.12. Gaussian beam propagation to distance z = L (phase screen at
0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section,
(c) Gaussian beam profile, (d) Random phase screen profile,
and (e) Gaussian beam phase distribution in transverse output plane………..77
Fig. 4.13. Gaussian beam propagation to distance z=L (phase screen at
L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………………………………………..77
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Fig. 4.14. Gaussian beam propagation to distance z=L (phase screen at
the object plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………..………………………………………78
Fig. 4.15. Gaussian beam propagation to distance z=L (phase screen at
0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section,
(c) Gaussian beam profile, (d) Random phase screen profile,
and (e) Gaussian beam phase distribution in transverse output plane………...78
Fig. 4.16. Gaussian beam propagation to distance z = L (phase screen at
L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane…………………...….………………………………..79
Fig.4.17. Gaussian beam propagation to distance z=L (phase screen at
object plane ). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane…………………….………………………………….80
Fig.4.18. Gaussian beam propagation to distance z = L ( phase screen at
0.5 L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section,
(c) Gaussian beam profile, (d) Random phase screen profile,
and (e) Gaussian beam phase distribution in transverse output plane………..81
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Fig.4.19. Gaussian beam propagation to distance z = L (phase screen at
L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………….…………………………….81
Fig.4.20. Gaussian beam propagation to distance z=L (phase screen at
object plane ). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane……………………………………..…………………82
Fig.4.21. Gaussian beam propagation to distance z = L ( phase screen at
0.5L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………………..………………………………82
Fig.4.22. Gaussian beam propagation to distance z = L (phase screen at
L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam
cross-section, (c) Gaussian beam profile, (d) Random phase
screen profile, and (e) Gaussian beam phase distribution in
transverse output plane………………….…………………………………….83
Fig.4.23. On-axis mean and variance of the amplitude and phase of diffracted
Gaussian beam at Z=L for different random phase screen positions……..…..85
Fig.4.24. Schematic illustration of propagation through extended turbulence………..…86
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Fig. 4.25. Gaussian beam propagation to incremental distance Δz.
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………………..87
Fig. 4.26. Gaussian beam propagation to distance Z = 0.5 L (250 Δz).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution……….……….88
Fig.4.27. Gaussian beam propagation to distance Z = L ( 500 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………….…….88
Fig.4.28. Gaussian beam propagation to incremental distance Δz.
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………………..89
Fig. 4.29. Gaussian beam propagation to distance Z = 0.5 L (250 Δz).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution…………….….89
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Fig. 4.30. Gaussian beam propagation to distance Z = L (500 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution……………..…90
Fig. 4.31. Gaussian beam propagation to incremental distance Δz.
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………………..91
Fig. 4.32. Gaussian beam propagation to distance Z = 0.5 L (50 Δz).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution……….……….92
Fig. 4.33. Gaussian beam propagation to distance Z = L ( 100 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution…….………….92
Fig. 4.34. Gaussian beam propagation to incremental distance Δz.
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………………..93
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Fig. 4.35. Gaussian beam propagation to distance Z = 0.5 L ( 500 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………………..93
Fig. 4.36. Gaussian beam propagation to distance Z = L ( 1000 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution……………..…94
Fig. 4.37. Gaussian beam propagation to distance Z = L ( 5000 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………………..95
Fig. 4.38. Gaussian beam propagation to distance Z = L ( 100 Δz ).
(a) 3D Gaussian beam, (b) its transverse plane intensity
distribution, (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution……………..…96
Fig. 4.39. The scintillation index plotted as a function of number of phase screens.
The (constant) parameters are: w0=30mm, r0=10mm, ℓ0=10mm, and
L0=1km. The propagation distance is given in the legend in Kilometers……..97
Fig. 4.40. The scintillation index plotted as a function of number of phase screens.
The (constant) parameters are: w0=30mm, r0=0.01mm, ℓ0=10mm, and
L0=1 km. The propagation distance is given in the legend in Kilometers….…98
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Fig. 4.41(a). Double Gaussian beam propagation to distance Z = L through
weak turbulence. (a) 3D Double Gaussian beam, (b) its transverse
plane intensity distribution, (c) 2D intensity profile……………….....……99
Fig. 4.41(b). Double Gaussian beam propagation to distance Z = L through
strong turbulence. (a) 3D Double Gaussian beam, (b) its transverse
plane intensity distribution, (c) 2D intensity profile…………………...…100
Fig. 4.42. The Fringe visibility plotted as a function of number of phase screens.
The parameters that held constant are: w 0=30mm, r0=10mm, beam
separation= 70 mm, ℓ0=10mm, and L0=1km. The propagation distance
is given in the legend in Kilometers………………………..…..………..…100
Fig. 4.43. The Fringe visibility plotted as a function of number of phase screens.
The parameters that held constant are: w0=30mm, r0=0.01mm, beam
separation= 70 mm, ℓ0=10mm, and L0=1km. The propagation distance
is given in the legend in Kilometers…………………………..………..……101
Fig.5.1. Point source propagation to distance z = L. (a) 3D point source;
(b) transverse plane intensity distribution at z = L, (c) 2D intensity
profile, (d) random phase screen distribution profile, and
(e) 3D field phase angle distribution in output plane………………………….105
Fig.5.2. (a) Time snapshot of the average spatial turbulence intensity,
and (b) PSD of turbulence derived from spectrum shown in (a)………….…..106
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Fig.5.3. Point source propagation to distance z = L. (a) 3D point source;
(b) transverse plane intensity distribution at z = L, (c) 2D intensity
profile, (d) random phase screen distribution profile, and
(e) 3D field phase angle distribution in output plane………………....……….106
Fig.5.4. (a) Time snapshot of the average spatial turbulence intensity,
and (b) PSD of turbulence derived from spectrum shown in (a)………………107
Fig.5.5. Point source propagation to distance z = L. (a) 3D point source;
(b) transverse plane intensity distribution at z = L, (c) 2D intensity
profile, (d) random phase screen distribution profile, and
(e) 3D field phase angle distribution in output plane……………………...…..107
Fig.5.6. (a) Time snapshot of the average spatial turbulence intensity,
and (b) PSD of turbulence derived from spectrum shown in (a)………………108
Fig.5.7. Point source propagation to distance z = L. (a) 3D point source;
(b) transverse plane intensity distribution at z = L, (c) 2D intensity
profile, (d) random phase screen distribution profile, and
(e) 3D field phase angle distribution in output plane…………….……………109
Fig.5.8. (a) Time snapshot of the average spatial turbulence intensity,
and (b) PSD of turbulence derived from spectrum shown in (a)…………..…..109
Fig.5.9. Point source propagation to distance z = L. (a) 3D point source;
(b) transverse plane intensity distribution at z = L, (c) 2D intensity
profile, (d) random phase screen distribution profile, and
(e) 3D field phase angle distribution in output plane…………….……………110
xxiv
Fig.5.10. (a) Time snapshot of the average spatial turbulence intensity,
and (b) PSD of turbulence derived from spectrum shown in (a)………….…110
Fig.5.11. Propagation of uniform plane wave through narrow turbulence region…...…112
Fig.5.12. Time waveform of the turbulence……………………………………………112
Fig.5.13. Uniform plane wave propagation to distance z=L (phase screen
at half way between aperture and image planes). (a) 3D uniform
plane wave; (b) its transverse plane intensity distribution at the
image plane; (c) 2D intensity profile, (d) random phase screen
distribution profile, and (e) 3D field phase angle distribution………...…….114
Fig.5.14. (a) chaos time waveform, (b) turbulence time waveform,
(c) the cross correlation function, (d) the cross-power spectral
density function, (e) chaos power spectral density, (f) transfer
function magnitude, (g) phase of the transfer function, and
(h) inverse Fourier transform of the transfer function…………....……….115
Fig.5.15. (a) modulated ( encrypted) chaos time waveform,
(b) its auto-correlation function, (c) power spectral
density of the modulated chaos, (d) the cross-power
spectral density function, (e) the cross correlation function……………..….116
Fig.5.16. (a) chaos time waveform, (b) turbulence time waveform,
(c) the cross correlation function, (d) the cross-power spectral
density function, (e) chaos power spectral density, (f) transfer
function magnitude, (g) phase of the transfer function, and
(h) inverse Fourier transform of the transfer function…………...……….117
xxv
Fig.5.17. (a) modulated ( encrypted ) chaos time waveform,
(b) its auto-correlation function, (c) power spectral
density of the modulated chaos, (d) the cross-power
spectral density function, (e) the cross correlation function………...……....118
Fig.5.18. (a) chaos time waveform, (b) turbulence time waveform,
(c) the cross correlation function, (d) the cross-power spectral
density function, (e) chaos power spectral density, (f) transfer
function magnitude, (g) phase of the transfer function, and
(h) inverse Fourier transform of the transfer function……………...…….119
Fig.5.19. (a) modulated ( encrypted ) chaos time waveform,
(b) its auto-correlation function, (c) power spectral
density of the modulated chaos, (d) the cross-power
spectral density function, (e) the cross correlation function…………....…...120
Fig.5.20. (a) chaos time waveform, (b) turbulence time waveform,
(c) the cross correlation function, (d) the cross-power spectral
density function, (e) chaos power spectral density, (f) transfer
function magnitude, (g) phase of the transfer function, and
(h) inverse Fourier transform of the transfer function……………....……121
Fig.5.21. (a) modulated ( encrypted ) chaos time waveform,
(b) its auto-correlation function, (c) power spectral
density of the modulated chaos, (d) the cross-power
spectral density function, (e) the cross correlation function...... …122
xxvi
Fig.6.1. Propagation of EM waves through uniform medium. (a) 3D Gaussian
beam input ( w0 = 30mm ) ; (b) modulating signal; (c) 3D Gaussian
beam output (w0 ≈ 80mm) ; (d) cross-section of the output Gaussian
intensity; (e) 2D output Gaussian profile; and (f) recovered signal……...……128
Fig.6.2. Propagation of EM waves through weak turbulence ( fT = 20Hz).
(a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal;
and (c) random phase screen distribution profile………………………….…..129
Fig.6.3. Propagation of EM waves through weak turbulence ( fT = 20Hz).
(a) 3D Gaussian beam output (w0=80mm ); (b) cross-section of
the output Gaussian intensity; (c) 2D output Gaussian profile;
and (d) recovered signal………………………….…………………………....130
Fig.6.4. Propagation of EM waves through weak turbulence ( fT = 50Hz ).
(a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal;
and (c) random phase screen distribution profile……………………………..131
Fig.6.5. Propagation of EM waves through weak turbulence (fT = 50Hz).
(a) 3D Gaussian beam output (w0 ≈ 80mm); (b) cross-section of
the output Gaussian intensity; (c) 2D output Gaussian profile;
and (d) recovered signal………….……………………………………………131
Fig.6.6. Propagation of EM waves through weak turbulence (fT = 100Hz).
(a) 3D Gaussian beam input (w0 = 30mm) ; (b) modulating signal;
and (c) random phase screen distribution profile…………………....…….…..132
xxvii
Fig.6.7. Propagation of EM waves through weak turbulence (fT = 100Hz).
(a) 3D Gaussian beam output (w0 ≈ 80mm); (b) cross-section of
the output Gaussian intensity; (c) 2D output Gaussian profile;
and (d) recovered signal…...……………….……………………………...…..132
Fig.6.8. Propagation of EM waves through strong turbulence ( fT = 20Hz ).
(a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D
field before the phase screen; and (d) 3D field after the phase screen………...134
Fig.6.9. Propagation of EM waves through strong turbulence ( fT = 20Hz ).
(a) 3D Gaussian beam output ( w0≈ 80mm ); (b) cross-section of
the output Gaussian intensity; (c) random phase screen distribution
profile; (d) 2D output Gaussian profile; and (e) recovered signal…………….134
Fig.6.10. Propagation of EM waves through strong turbulence ( fT = 50Hz ).
(a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D
field before the phase screen; and (d) 3D field after the phase screen………135
Fig.6.11. Propagation of EM waves through strong turbulence ( fT = 50Hz ).
(a)3D Gaussian beam output ( w0 ≈ 80mm) ; (b) cross-section of
the output Gaussian intensity; (c) random phase screen distribution
profile; (d) 2D output Gaussian profile; and (e) recovered signal…………..136
Fig.6.12. Propagation of EM waves through strong turbulence ( fT = 100Hz ).
(a)3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D
field before the phase screen; and (d) 3D field after the phase screen………137
xxviii
Fig.6.13. Propagation of EM waves through strong turbulence (fT = 100Hz).
(a) 3D Gaussian beam output; (b) cross-section of the output
Gaussian intensity; (c) random phase screen distribution profile;
(d) 2D output Gaussian profile; and (e) recovered signal………….………..137
Fig.6.14. Propagation of modulated chaotic waves through a uniform medium.
(a) 3D Gaussian beam input ( w0= 30mm ); (b) modulating signal;
(c) modulated signal; (d) 3D Gaussian beam output; (e) 2D output
Gaussian profile; and (f) recovered signal……………...……...……………141
Fig.6.15. Propagation of modulated chaotic waves through weak turbulence
( fT = 50Hz ). (a) 3D Gaussian beam input ( w0 = 30mm ); (b)
modulating signal; (c) modulated signal; (d) 3D Gaussian beam
output; (e) 2D output Gaussian profile; and (f) recovered signal……………142
Fig.6.16. Propagation of modulated chaotic waves through strong turbulence
( fT = 50Hz ). (a) 3D Gaussian beam input ( w0 = 30mm ); (b)
modulating signal; (c) modulated signal; (d) 3D Gaussian beam
output; (e) 2D output Gaussian profile; and (f) recovered signal………...….143
Fig. A1. Very narrow horizontal double-slit geometry…………………………………157 Fig.A2. Analytic plots of the far-field due to very narrow horizontal double slit…..…158
Fig.A3. Analytic plots of the far-field due to circular aperture………………..………159
Fig.A4. Annular aperture with radii a and b…………………..………………………159
Fig.A5. Analytic plots of the far-field due to annular aperture………...………………160
Fig.A6. Infinitely narrow annular aperture……………………………….……………160 Fig.A7. Analytic plots of the far-field due to infinitely narrow annular aperture…..…161
Fig.A8. Thin sinusoidal amplitude grating aperture……………………………………161
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Fig.A9. Analytic plots of the far-field due to thin sinusoidal amplitude grating….…...162
xxx
LIST OF TABLES
Table 2.1. Time delay (TD) and average chaos frequency (theoretical and graphical)…..26
Table 4.1. Structure parameter, Fried parameter, and turbulence strength………………71
xxxi
CHAPTER 1
BACKGROUND
1.1 Introduction
Atmospheric turbulence has a significant impact on the quality of free-space EM propagation over long distances. The Earth's atmosphere can be described as a locally homogeneous medium in which its properties vary with respect to temperature, pressure, wind velocities, humidity and other factors. Inhomogeneities in the temperature and pressure of the atmosphere lead to variations of the refractive index along the transmission path. These variations of the refractive index can cause fluctuations in both amplitude and phase of the received signal (in some instances, the random index fluctuations may also cause the phenomenon of scintillation), which increase the bit errors in a digital communication link; additionally, the above lead to distortion and other degradations of the transmitted signal or message. In order to quantify the performance limitations, a better understanding of the effect of turbulence-induced intensity fluctuations on the received signal under different turbulence categories is needed. Three major parameters characterizing atmospheric turbulence are: the refractive index
2 structure parameter C푛, and the turbulence inner (ℓ0) and outer (L0) scales. The refractive index structure parameter was first introduced in turbulence theory by Kolmogorov and
Obukhov [1-3]. This theory is commonly referred to as the Kolmogorov turbulence theory. The refractive index structure parameter describes the strength of the (spatial)
1
refractive index fluctuations and marks the first major principle on which the development of the classical Kolmogorov atmospheric turbulence theory depends. When a laser beam propagates through the atmosphere, the randomly varying spatial distribution of refractive index that it encounters causes a number of effects.
Propagation through the turbulent atmosphere often consists a laser beam propagating through the otherwise clear atmosphere with very small changes in the refractive index present. These small changes in refractive index, which are typically on the order of
∆푛 ∼ - 6 푛 10 are related primarily to the small variations in temperature (on the order of 0.1-
1oC), which are produced by the turbulent motion of the atmosphere, and also typical thermal gradients between the earth’s surface and the atmosphere [4]. More details of the refractive index fluctuations and the derivation of the phase turbulence models are presented in chapter 4. Since the turbulence is irregular and random, typically any information transmitted will suffer amplitude and phase distortions upon propagation through the turbulence. There may be multiple means of trying to mitigate the effects of turbulence in the wave communication. One such, which, as described in somewhat more detail later, is the potential use of an encrypted chaotic carrier wave which carries the information and is then transmitted through the turbulence. Since chaos itself has a certain degree of randomness in its characteristics, the latter problem therefore becomes more complex, involving two different random effects, of which the turbulence possess one or more spatial models; however, the chaos is not equally amenable to mathematical modeling, and in the work presented here, occurs only in time and not in space.
2
1.2 Turbulence-a brief overview
As mentioned, atmospheric turbulence-induced random distortions of the optical phase result in intensity fading at the receiver, giving rise to system bit error rates potentially orders of magnitude higher than those in the absence of turbulence. In 1935,
G. I. Taylor assumed that turbulence is a random phenomenon and proceeded to introduce statistical tools for the analysis of homogeneous, isotropic turbulence. As it turns out, the impact of this early work has persisted even to the present. In addition,
Taylor analyzed experimental data generated by wind tunnel flow through a mesh to show that such flows could be viewed as homogeneous and isotropic. The success of this provided even further incentive for future application of the analytical techniques he had introduced. A further contribution, especially valuable for analysis of experimental data, was introduction of the “Taylor hypothesis” which provides a means of converting temporal data to spatial data. In reality, however, most turbulence index models are typically in the spatial domain, and do not incorporate temporal variations. Actual turbulence effects will likely have some degree of both spatial and temporal randomness.
In 1941 the Russian statistician A. N. Kolmogorov published three papers (in Russian)
[5] that provide some of the most important and widely quoted results of turbulence theory. This theory provides a way to parameterize the eddy size scale, velocity and temperature profiles, and provide structure constants that describe the nature of the turbulence within that range of profiles. The boundary for this characterization is dependent on the scale size of the turbulent eddies as described above. Using these structure terms, optical parameters can be developed and used to predict the behavior of a wavefront propagating through the turbulence. This theory is very useful; however it is
3
only applicable to modeling turbulence effects on optical performance within the inertial sub-range of turbulence. Kolmogorov provided two definitions from which to make the determination that a turbulence distribution could be called homogeneous and isotropic.
These definitions set the boundaries of the inertial subrange. Kolmogorov also contributed two hypotheses on similarity which provides a way to develop parameters for analysis assuming a homogeneous, isotropic distribution.
During the 1940s the ideas of Landau and Hopf (see refs. [6,7]) on the transition to turbulence became popular. They (separately) proposed that as the Reynolds number (a non-dimensional measure useful in fluid flow) is increased, a typical fluid flow undergoes an (at least countable) infinity of transitions during each of which an additional incommensurate frequency (and/or wavenumber) arises due to flow instabilities, leading ultimately to very complicated, apparently random, flow behavior.
The Reynolds number is directly proportional to velocity and inversely to viscosity; if the
Reynolds number exceeds 105, the flow motion is transferred from laminar to turbulent.
This scenario based on laminar to turbulent transitions was favored by many theoreticians even into the 1970s when it was shown to be untenable in essentially all situations. In fact, such transition sequences were never observed in experimental measurements, and they were not predicted by more standard approaches to stability analysis.
One of the interesting facts about the history of work related to laser propagation through a turbulent environment is that even though the first working laser was not announced until 1960, much of the necessary theoretical work and some of the experimental work had been done prior to its invention. Laser propagation in the atmosphere is really a subtopic of a more general problem, i.e., the propagation of an EM wave (of arbitrary
4
frequency) in a turbulent medium. This general topic includes a number of other practical applications, such as propagation of starlight trough the atmosphere, propagation of sound waves through the atmosphere and ocean, propagation of microwaves through planetary atmospheres, and propagation of radio waves through the ionosphere and interplanetary space. In the above, we have outlined briefly the origins, scope and diversity of potential physical problems associated with atmospheric and other turbulence. The interested reader can find much of the theory of turbulence and its development throughout the last century in ref. [8].
1.3 Outline of this dissertation
This section presents the reader an outline briefly summarizing the contents of this dissertation.
1.3.1 Acousto-optic (A-O) chaos
A-O devices are used in laser equipment for electronic control of the intensity and position of a laser beam. In 1921, Brillouin first predicted that A-O consists of the diffraction of monochromatic light waves by ultrasonic waves [9]. When the acoustic medium is homogeneous and isotropic there is no deviation in the monochromatic light through the medium. But when the medium is traversed by a high frequency sound wave, the light wave is diffracted. The acoustic wave is generated by applying an RF signal to a piezo electric transducer that is suitably bonded to a crystal. The piezoelectric transducer is attached to a material such as glass. An oscillating electrical signal drives the transducer to vibrate, which creates sound waves in the medium. When a sound wave passes through the medium (either solid or liquid), it causes compressions and rarefactions in the medium. This compression and rarefaction of the medium is related to
5
the broader aspects of birefringence in materials manifested through the photo elastic effect. The photo elastic effect changes the refractive index of the medium. Periodically alternating layers with different refractive indices thus form in the medium, since the ultrasonic sound wave is sinusoidal in nature. These layers move at the velocity of sound and are separated by half the acoustic wavelength. When light is incident at a certain angle (called the Bragg angle (θB)) on the medium and propagates through the above mentioned periodic structure, it undergoes scattering as defined by Bragg diffraction by the grating or periodic structure [10]. If the grating is effectively thick, the diffraction occurs in the so-called Bragg regime, consisting of only two scattered orders. If the grating is effectively thin, multiple scattered orders are generated, and operation is described as the Ram-Nath regime. In the late 1980s, Chrostwoski, Delisle and co- workers discovered that the A-O device under first-order hybrid feedback begins to exhibit monostable, bistable and multistable behavior en route to chaos [11]. There are three primary parameters in the closed-loop system that control the transition from one
~ state to another. These parameters are: the effective feedback gain ( ), the feedback time delay (TD), and the amplitude (Iinc) of the incident light.
As mentioned, time delay introduces bistability in the hybrid A-O feedback (HAOF) device. Monostability, bistability, possible multistability and chaos in the hybrid A-O
device may be obtained via three tuning effects viz., , TD, andˆ0 . Generation of A-O chaos involves careful tuning of these parameters and amplification of the feedback signal. From previous research carried out on this subject, many have successfully demonstrated the behavior of optical bistability and chaos [12,13] and, more recently, how chaos behaves with time. Optical monostability, bistability and multistability leading
6
to chaos have been successfully demonstrated by several groups, achieved by controlling the nonlinear dynamics as parameters affecting the system are varied. In 2008, a specific application of chaos was reported in which the chaos in the photocurrent was treated as a carrier, which was then modulated by sending information signals through the acoustic driver circuit. The resulting modulated chaos is essentially an encrypted waveform because of the general random nature of the chaos; it offers inherent security in that the signal embedded in the chaos cannot be extracted without knowledge of the parameter keys. The multiplicity of parametric keys to be matched at the receiver makes the system robust and offers immunity from hacking or interception. One goal of the current research is to combine the use of chaos with its inherent encryption properties, and utilizing the latter advantageously in order to reduce the damaging effects of atmospheric turbulence on signal-bearing EM waves. Chapter 2 gives the reader a general overview of
A-O chaos, outlining the mechanism by which the feedback device enters the chaos regime via appropriate selection of chaos parameters. Further, the time, frequency, and power spectral density characteristics of A-O chaos are reported in this chapter.
1.3.2 Aperture diffraction properties with and without turbulence
In this work, propagation of uniform and profiled EM beams through apertures with binary amplitude transmission and random phase distributions is investigated in the far field (Fraunhofer limit). This study is based on two approaches: (a) using the Fresnel-
Kirchhoff diffraction integral directly (for the non-turbulent case), and (b) using a split- step propagation concept whereby the aperture or phase screen is placed at an arbitrary location along the propagation path, an approach appropriate for the turbulent case.
Results for uniform, Gaussian and Bessel profile beams propagating through a variety of
7
binary apertures are examined and compared with analytical predictions wherever feasible. A power spectrum density (PSD) of the MVKS model is also used to describe a planar aperture as a random phase distribution. Simulation results are limited to the diffraction intensity calculation of the intensity in the far-field or Fraunhofer regime evaluated in the in the transverse (image) plane. The simulations have consisted of applying the Fresnel-Kirchhoff diffraction integral and thereafter the far-field diffraction patterns of intensity have been obtained, along with the profiles along the horizontal and vertical axes. Additional examples, including diffraction through a thin sinusoidal amplitude grating and far-field diffraction following propagation through a random phase screen for profiled input beams are also presented. These results, derived serendipitously while examining turbulent propagation, provide insight into the mechanisms of diffraction through variable apertures, beam profiles and medium characteristics.
The work in this research consists of investigating the effect on Gaussian beam propagation for propagation through both narrow and extended MVKS-type phase turbulence, and eventually the propagation of encrypted chaotic waves through such turbulence. The work reported in chapter 3 is essentially a detailed study of aperture diffraction for a variety of aperture and medium conditions, and also of profiled beam transmission through random phase screens incorporating Fresnel-Kirchhoff diffraction theory.
1.3.3 Propagation of Gaussian beams through narrow and extended turbulence
Details for the study of the propagation of Gaussian beam through narrow and extended turbulent region with some additional measurements (scintillation index (SI) and fringe visibility (FV)) are reported in chapter 4. The impact of atmospheric phase
8
turbulence on Gaussian beam propagation along propagation paths of varying lengths is examined using signal and multiple random phase screens. The MVKS model makes a statement about large-scale eddies while preserving the behavior of the spectrum in the inertial range [14]. The reasons for choosing the MVKS in this work is that; (a) it contains inner (ℓ0) and outer (L0) scales; (b) it is relatively easy to simulate; and (c) it is defined when spatial wavenumber k 0, i.e. at low frequencies. In addition, the MVKS is simple to handle analytically and provides finite estimates for virtually all quantities of practical interest. In this investigations, a split step approach (SSA) (which is similar, though not identical to the split-step beam propagation method in EM theory) is used involving the Fresnel-Kirchhoff diffraction integral for propagating the profiled Gaussian beam whereby the EM wave alternately moves through a homogeneous medium under spatial diffraction and either a single or multiple random phase screen(s) to represent the propagation through narrow or extended turbulent region(s). Also, we examine the impact of the atmospheric turbulence on the propagating EM waves beam for different turbulence conditions and different medium conditions. Different criteria are typically used to examine the effect of atmospheric turbulence on laser beam propagation [15].
Usually, atmospheric turbulence is modeled under some assumptions, such as homogenous and isotopic media, for example. These models ( to be discussed in chapter 4) are expressed in the spatial or spatial frequency domain ((x,y) or (kx,ky)). On the other hand, the propagating laser beams modulated either directly by a message signal or by an encrypted chaotic carrier carrying a message signal is a function of both time and space, and the atmospheric effects on the transmitted message must therefore track
9
the time characteristics. Since turbulence itself is typically only a spatial effect, it becomes necessary to introduce time into the analysis in some manner.
In the research reported in this document, two distinctly divergent approaches are used to incorporate the temporal characteristics of both the propagating signals as well as the medium turbulence. In the first approach, cross power spectral densities and cross correlation functions involving both turbulence and chaos are developed using pre- determined time variations of the random processes, and recording the corresponding statistics via numerical simulations. This statistics are then incorporated into a transfer function formalism which relates the spectral densities for both turbulence and chaos.
Using this strategy, it was possible to numerically obtain the cross-correlation function between an encrypted chaos wave and a specific MVKS turbulence. Extraction of the encrypted chaos from this cross correlation turned out to be a bottleneck, and remains an ongoing investigative problem. An alternative approach was developed more recently in which a laser beam directly modulated by a time signal was propagated through narrow
MVKS turbulence, and its recovery in the image plane was made possible by extracting the time-dependent envelope of the modulated carrier. The signal was recovered from the envelope by using standard electronics. In the chapters that follow, both these methodologies are described along with numerical results.
1.3.4 Auto and cross correlations, power spectral densities and the transfer function
formalism
In this approach, the propagation process (including the input field, the homogenous medium modeled via Fresnel-Kirchhoff type paraxial diffraction, the random phase screen and the diffracted output field) is modeled as a linear system
10
defined in principle by a transfer function involving two random processes. Then the derived transfer function is used to examine the propagation of the encrypted (modulated) chaotic wave through atmospheric turbulence. First, in order to find the time waveform representation of the atmospheric turbulence, we propagate the uniform plane EM wave along the propagation path starting from the aperture plane through the homogenous medium; thereafter, the field will pass through MVKS-type random phase turbulence followed by further propagation until arrival at the destination. The on-axis field
(ET(0,0,z)) is then evaluated at the image plane. The turbulence time waveform is found by repeating the process over pre-selected time intervals. The time interval determines the effective frequency of the turbulence. Next, the cross-correlation RCT(τ) and the cross- power spectral density SCT(f) are calculated using a simple Fourier transform. The system transfer function may be calculated using SCT(f) and chaos power spectral density (SC(f)).
Thereafter, the cross-correlation RTCm(τ) between the turbulence and encrypted
(modulated) chaos wave is obtained from the cross-power spectral density STCm(f)
(calculated from the derived transfer function). In order to recover the message signal, we have to extract the modulated chaotic signal (ECm(t)) from RTCm(τ) and then apply the familiar heterodyne demodulation strategy to recover the original transmitted signal. A means of extracting the encrypted chaos from the cross-correlation is currently under investigation, and a workable solution is yet to be devised. Details of this methodology and results obtained are discussed in chapter 5.
11
1.3.5 Propagation of non-chaotic and chaotic EM waves through turbulence using
modulation theory
An alternative approach to the transfer function formalism discussed in chapter 5 is one that uses phasors and (spatial) Fourier transforms, and is based on simple modulation theory. The complex phasor wave is transmitted across a uniform or a turbulent medium using the Kirchhoff-Fresnel integral and incorporating the random phase screen. In this study, we use the slowly time-varying envelope approximation
(SVEA) with the (spatial) Fourier transform and Kirchhoff-Fresnel integral to propagate non-chaotic and encrypted chaotic waves through a uniform (homogenous) medium and through MVKS-type phase turbulence for different turbulence conditions including different (mean) temporal frequencies of turbulence. At the receiver port, the photodetector (PD) picks up the intensity of the received EM beam. Since, the output of the PD is a photocurrent proportional to the intensity of the received EM beam. This process eliminates the optical carrier at the PD output. Hence, for the case of a directly modulated optical carrier, heterodyne detection is not needed. On the other hand, in the case of an encrypted chaotic signal riding on the envelope of an optical carrier, propagation through a uniform or turbulent region would require heterodyning in order to filter the chaos frequency and its harmonics in order to recover the baseband (message) signal. Mathematical representations of these ideas and corresponding numerical results are discussed at length in chapter 6. Chapter 7 provides concluding remarks and some insights into future research. An appendix A showing graphical results comparing analytic versus numerical plots corresponding to uniform and profiled EM beam
12
propagation (discussed in chapter 3) through apertures with binary amplitude transmission appears at the end of this document.
13
CHAPTER 2
ACOUSTO-OPTIC (A-O) CHAOS AND CHAOS GENERATION
2.1 Introduction
A-O, where acousto- implies sound and optics implies light defines the diffraction of monochromatic light waves by ultrasonic waves whereby the monochromatic light passages through a rectangular structured cell and emerging them from the other side
[9,16]. In the case of homogeneous and isotropic medium, there is no deviation in the monochromatic light through the medium. But when the medium is traversed under high-frequency sound wave (also called a grating), the light wave is diffracted. Through the action of a piezoelectric transducer, electrical signals are converted into sound waves propagating in the medium. The ultrasonic wave propagating through a solid or liquid locally causes compression and rarefaction of the medium. These compression and rarefaction effects cause perturbations in the index of refraction of the medium, which in terms creates a phase grating in the material and splits the incident laser light into various diffracted orders. Because the sound wave is sinusoidal in nature, the compressions and rarefactions in the medium are periodic in nature. When the monochromatic incident light passes through the medium at an angle called the Bragg angle (훳B), and the grating is considered to be thick, the light is typically diffracted into two orders (called the Bragg regime), the diffracted light (first-order) and the undiffracted light (zeroth-order)
[10,17,18].
14
In the A-O modulator, there are two regimes, the Bragg regime (as mentioned above), and the Raman-Nath regime. These are shown schematically in the Fig.2.1 [19,20].
Einc.(ωi,ki) K First order E1(ω1,k1)
훳B 2훳B
Zeroth order E0(ω0,k0)
Ʌ (Ω,K)
(a)
K Einc.(ωi,ki) E 2 k 2훳B E1 훳inc
2훳B
E0
2훳B
E -1 Ʌ
L
(b)
Fig.2.1. Schematic of A-O system. (a) The Bragg regime (thick), (b) The Raman-Nath regime (thin).
15
A factor called Figure of merit Q similar to the quality factor in resonance was introduced by Klein and Cook [19] that explains the diffraction in A-O regime. Q is expressed by:
2 2 퐿퐾 2휋휆0퐿 2휋휆0훺 푄 = = 2 = 2 , (2.1) 푘 푛훬 푛푣푠 where,
Q - Klein-Cook Parameter.
L - Effective length of A-O Modulator. n - The index of refraction.
휆0- Wavelength of light in free-space.
Ω - Frequency of the sound wave. vs- Velocity of the sound wave. k - Wave number of light in medium.
K - Wave number of sound
Λ - Wavelength of sound inside the modulator.
Q is an important factor that determines of an A-O modulator operation regime. When Q is less than 1 (Q<1), the system operates in the Raman Nath Regime and when Q is greater than 1 (Q>1), the system operates in Bragg regime. In Bragg regime and when the light is incident at an angle (θB) called the Bragg angle, the light undergoes diffraction and generates first and zeroth orders as mentioned above. In 1996, Chatterjee and Chen showed that Q must be greater than 8π for the system to operate in strict Bragg regime
[21]. The Bragg angle θB is given by:
퐾 푄 = . (2.2) 퐵 2푘
In this research, we consider the Bragg regime and will deal with the first-order diffraction. In the Bragg regime, the incident light is diffracted into the zeroth- and first-
16
orders. The resulting first order is picked up by a linear photo detector at the output, amplified and then fed back to the adder in the feedback loop. The adder adds the feedback with the dc bias input and feeds it to the RF source that generates the RF input which is then incident on the Bragg cell via a transducer. The standard Bragg cell transmitter with feedback is shown in Fig.2.2.
1st order Diffracted Photo detector Incident Light beam(I1)
훳B 2훳B
Zeroth order Amplifier Bragg cell Undiffracted gain beam(I0) Piezo transducer Adder
RF source
α0 (bias)
Fig.2.2. An A-O modulator with first-order feedback in the Bragg regime.
2.2 Nonlinear dynamics of the A-O Bragg cell under feedback
(a) Mono- and bistability
The zeroth- and first-order beams in the Bragg cell are governed by two coupled differential equations for the phasor fields:
푑퐸1 αˆ = − j 0 E (2.3a) 푑휉 2 0
푑퐸0 ˆ = − 0 Ej , (2.3b) 푑휉 2 1
17
where αˆ 0 is the peak phase delay through the medium, is the normalized propagation
푧 distance (= ⁄퐿) , and L is the effective interaction length. The solution of these two equations is given by:
ˆ0 퐸1 = − jE sin( ) (2.4a) inc 2
ˆ 0 퐸0 = E cos( ) , (2.4b) inc 2
and the corresponding first-order detected intensity (I1(t)) follows the nonlinear dynamical equation [22]:
1 ~ 퐼 (푡) = 퐼 푠푖푛2 [ (ˆ (t) + 퐼 (푡 − 푇 ))] , (2.5) 1 푖푛푐 2 0 1 퐷 where,
- Effective feedback gain.
Iinc - Incident intensity.
TD- Feedback time delay including photo detector conversion delay.
- Bias input.
Under certain values of the three parameters mentioned above, the system can be driven through a series of dynamical regimes, namely, monostable, bistable, and multistable
[23]. When the feedback gain is less than 2, and with a finite time delay, the system exhibits monostability over a range of the bias input . When the feedback gain of the amplifier is greater than 2 and when the time delay in the loop is greater than 0, the system will transition from mono- to bistable for increasing values of the bias input .
We note here that for different choices of the incident light intensity and/or the initial
18
~ value of the first-order intensity, the threshold values of for system transition tends to vary. The monostable and bistable hysteresis characteristics are illustrated in Fig.2.3.
Hysteresis loop Hysteresis loop 1 1 =2 =2 0.9 I(0)=0 0.9 I(0)=0 T =5 ms D TD=5 ms I =1 0.8 inc. 0.8 Iinc.=1
0.7 0.7
) 0.6 ) 0.6 1 1
0.5 0.5
Intensity (I 0.4 Intensity (I 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 Bias voltage( ) Bias voltage( ) 0 0
Fig.2.3. A-O monostable and bistable regimes with hysteresis loop.
(b) Multistability and chaos
As mentioned, optical monostability, bistability, and possible multistability and chaos may be generated by driving the hybrid A-O system into chaos with a proper
choice of the parameters , αˆ 0 , and TD. By decreasing the time delay (TD ≈ 1ms) and increasing the value of (β ≈ 2.1) up to (β ≈ 2.25), it is found that multistable loops begin to appear transitioning into chaos. When is increased to 2.41 and higher (for the given values of Iinc and I1(0)), the system is driven into complete chaos as shown in Fig.2.4.
We note again that this threshold may be increased or decreased by adjusting the incident intensity and the initial value.
19
Hysteresis loop Hysteresis loop 1 1 =2.1 =2.2 I(0)=0 I(0)=0 T =1 ms 0.8 D 0.8 TD=1 ms I =1 inc. Iinc.=1 ) ) 1 0.6 1 0.6
0.4 0.4 Intensity (I Intensity (I
0.2 0.2
0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Bias voltage( 0 ) Bias voltage( 0 )
Hysteresis loop Hysteresis loop 1 1 =2.25 =2.41 I(0)=0 I(0)=0 0.8 TD=1 ms 0.8 TD=1 ms Iinc.=1 Iinc.=1 ) ) 1 0.6 1 0.6
0.4 0.4 Intensity (I Intensity (I
0.2 0.2
0 0 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 Bias voltage( ) Bias voltage( ) 0 0
Fig.2.4. Monostable, bistable, multistable and chaotic regimes.
2.3 Chaos time, frequency, and power spectral density representations
A study of the time and frequency characteristics of A-O chaos is necessary for the understanding of the oscillating behavior of chaos. As we mentioned above, chaotic
~ oscillations occur under certain thresholds of the parameters: , TD, and αˆ 0 . In this part, we examine the time, frequency, and PSD representations associated with a chaotic signal under different values of time delay (TD) with fixed values of ( = 4, and =2) and
Iinc =1 are kept constant as show in each figure.
20
The procedure for finding the chaos PSD adopted here is as follows:
1) Assign chosen values to the key chaos parameters as shown in each of the
following figures.
2) Carry out the statistical calculations for different (fixed) vales of TD.
3) Run the program multiple times (say 100) for given fixed parameters (thus we
obtain 100 snapshots of chaos).
4) For each time waveform we compute the Fourier transform.
5) For the all Fourier transforms (100 graphs), we find the power associated with
different values of frequencies (including the frequencies associated with the
highest intensity), assuming hypothetically that the field intensity is collected over
a fixed area over which the profile is approximately constant.
6) We compute the average transformed intensity over the 100 spectral waveforms,
and finally plot the derived average spectral densities over the chosen frequency
range. This should approximately result in an equivalent PSD for the chaos wave.
In the following figures (2.5-2.12), we show the snapshot of the one of the time waveforms, its Fourier transform and the PSD for different values of time delay
(TD= 1ms, 100µs, 10 µs, and 5 µs).
21
1 =4 I(0)=0 0.8 TD=1 ms
) =2 1 0 0.6
0.4 Intensity (I
0.2
0 100 105 110 115 120 125 130 time (ms) (a)
50
40
30
20 Amplitude
10
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 frequency (KHz) (b) Fig.2.5. Chaotic wave data for (a) snapshot of time waveform, and (b) the Fourier transform.
1
0.9
0.8
0.7
0.6
0.5
PSD( PSD( W / Hz) 0.4
0.3
0.2
0.1
0 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 frequency (KHz)
~ Fig.2.6. Chaotic wave PSD for = 4, ˆ0 = 2, Iinc =1, and TD = 1ms.
22
1 =4
) I(0)=0 1 TD=100 s 0.5 0=2 Intensity (I
0 40 40.2 40.4 40.6 40.8 41 41.2 41.4 41.6 41.8 42 time (ms) (a)
30
20
Amplitude 10
0 2 3 4 5 6 7 8 frequency (KHz) (b) Fig.2.7. Chaotic wave data for (a) snapshot of time waveform, and (b) the Fourier transform.
1
0.9
0.8
0.7
0.6
0.5
PSD( PSD( W / Hz) 0.4
0.3
0.2
0.1
0 4.9 4.95 5 5.05 5.1 5.15 5.2 5.25 5.3 frequency (KHz)
~ Fig.2.8. Chaotic wave PSD for = 4, ˆ0 = 2, Iinc =1, and TD = 100µs.
23
1 =4
) I(0)=0 1 TD=10 s 0.5 0=2 Intensity (I
0 4.5 4.55 4.6 4.65 4.7 4.75 4.8 time (ms) (a)
50 40 30 20 Amplitude 10 0 35 40 45 50 55 60 65 frequency (KHz) (b) Fig.2.9. Chaotic wave data for (a) snapshot of time waveform, and (b) the Fourier transform.
1
0.9
0.8
0.7
0.6
0.5
PSD( PSD( W / Hz) 0.4
0.3
0.2
0.1
0 49 49.5 50 50.5 51 51.5 52 52.5 53 frequency (KHz)
~ Fig.2.10. Chaotic wave PSD for = 4, ˆ0 = 2, Iinc =1, and TD = 10µs.
24
1 =4
) I(0)=0 1 TD=5 s 0.5 0=2 Intensity (I
0 2 2.05 2.1 2.15 time (ms) (a)
30
20
Amplitude 10
0 50 60 70 80 90 100 110 120 130 140 150 frequency (KHz) (b)
Fig.2.11. Chaotic wave data for (a) snapshot of time waveform, and (b) the Fourier transform. 1
0.9
0.8
0.7
0.6
0.5
PSD( PSD( W / Hz) 0.4
0.3
0.2
0.1
0 98 99 100 101 102 103 104 frequency (KHz)
~ Fig.2.12. Chaotic wave PSD for = 4, ˆ0 = 2, Iinc =1, and TD = 5µs.
25
From the previous figures, the average chaos frequencies in Table 2.1 were computed using the number of zero crossings, finding the mean, and taking the reciprocal of two cycles of oscillation (fch-graphically). Also, in this table we have calculated the theoretical chaos frequency using the formula fch=1/(2TD) (fch-theoretically).
Table 2.1. Time delay (TD) and average chaos frequency (theoretical and graphical).
TD fch(Theoretically)=1/(2*TD) fch(Graphically)
1 ms =1/(2*TD)=1/(2*1ms) = 500 Hz ≈ 1/(2*TD-av) ≈1/(2*1.05ms) = 476 Hz
100 µs =1/(2*TD)=1/(2*100µs) = 5 kHz ≈ 1/(2*TD-av) ≈1/(2*100.05µs) = 4.99 kHz
50 µs =1/(2*TD)=1/(2*50µs) = 10 kHz ≈ 1/(2*TD-av) ≈1/(2*50.05µs) = 9.99 kHz
20 µs =1/(2*TD)=1/(2*20µs) = 25 kHz ≈ 1/(2*TD-av) ≈1/(2*20.05µs) = 24.94 kHz 10 µs =1/(2*TD)=1/(2*10µs) = 50 kHz ≈ 1/(2*TD-av) ≈1/(2*10.3µs) = 48.544 kHz 5 µs =1/(2*TD)=1/(2*50µs) = 100kHz ≈ 1/(2*TD-av) ≈1/(2*50.05µs) = 99.9 kHz 2 µs =1/(2*TD)=1/(2*2µs) = 250 kHz ≈ 1/(2*TD-av) ≈1/(2*2.02µs) = 247.52 kHz
1.5 µs =1/(2*TD)=1/(2*1.5µs) = 333 kHz ≈ 1/(2*TD-av) ≈1/(2*1.48µs) = 337 kHz
1.2 µs =1/(2*TD)=1/(2*1.2µs) = 416 kHz ≈ 1/(2*TD-av) ≈1/(2*1..18µs) = 423 kHz
1 µs =1/(2*TD)=1/(2*1µs) = 500 kHz ≈ 1/(2*TD-av) ≈1/(2*1.01µs) =495.5 kHz
From Table 2.1, we note that the computed chaos frequency generally matches that predicted by the analytic formula.
26
500 Theortically 450 Graphically
400
350
300
250 (kHz) ch f 200
150
100
50
0 0 50 100 150 200 250 300 350 400 450 500 1/(2*T ) (kHz) D Fig.2.13. Theoretical versus graphical data for average chaos frequency for different TD values.
27
CHAPTER 3
DIFFRACTION PROPERTIES OF THE PROFILED BEAM TRANSMISSION
THROUGH BINARY APERTURES AND RANDOM PHASE SCREEN
3.1 Introduction
It is known that as an EM wave propagates through the atmosphere, turbulence induces amplitude and phase fluctuations. This work overview (which primarily deals with diffraction) originated with studying the propagation of chaotic EM waves through atmospheric turbulence. Atmospheric turbulence has been a subject of study for many years, and there is a wide interest in applications which consider effects of atmospheric turbulence on optical systems [24]. The foundations of the study of atmospheric turbulence were laid in the late 1960s and 1970s even though the field may be traced to many decades earlier. A vast amount of work regarding propagation through atmospheric turbulence is available in the literature. In several cases, turbulence is modeled via random phase screens that are either localized or extended over a physical space representing the turbulent medium [15,25,26]. Some of the commonly used models include the Kolmogorov, Tatarski, von Karman, and MVKS [15,27]. In our initial approach to representing atmospheric turbulence, we were motivated by recent work involving generation of A-O chaos and utilization of such chaos in encrypting the chaotic carrier with RF information signals which were then transmitted and
28
subsequently recovered using a set of matched keys (system parameters) [12,28,29,30].
It has been suggested [31] that propagation of a chaotic wave through turbulence may offer certain immunizing properties that would make the propagated signal relatively undisturbed by the turbulence in the medium. In pursuing this goal, the problem was set up by means of a thin, random phase screen representing turbulent phase fluctuations in a thin layer of the medium. Thereafter, profiled input EM waves would be transmitted over a fixed distance along the propagation path. The simulation and numerical analysis of this problem eventually led to the finding that this approach, which includes the use of a split-step algorithm whereby the EM wave alternately moves through a random-phase layer and then a thin layer of pure diffraction may be used conveniently as a good model simply to study both near- and far-field diffraction through arbitrary apertures and involving profiled EM beams, with and without atmospheric turbulence, and also with and without the use of the split-step algorithm. The focus in this chapter is therefore on the results obtained through applying the Fresnel-Kirchhoff integral to a variety of diffractive setups, including random phase screens. This work will may provide insight into not only regular far-field diffraction through common planar apertures, but also demonstrate a versatile means of numerically analyzing such problems by incorporating binary apertures, we also present two examples of propagation through thin random phase screens. The generation of such phase screens is also briefly described [15,27].
In this chapter, we have examined the propagation of uniform and profiled EM beams through apertures with binary amplitude transmission and random phase distributions at the far field (Fraunhofer limit). A verification of the split-step algorithm is conducted based on two approaches, (a) a straightforward application of the Fresnel-Kirchhoff
29
diffraction formalism from the aperture plane to the observation (or image) plane, and (b) by re-applying the split-step methodology whereby the intervening medium into infinitesimal segments, and carrying out the effective diffraction computation in multiple steps and compare the results. Results for uniform and Gaussian profile beams propagating through a variety of binary apertures are examined and compared with analytical predictions (appendix A) wherever feasible. In addition we also present two examples of propagation through aperture with thin random phase (distribution) screen.
3.2 Split-beam propagation method
Most atmospheric turbulence simulations use the split-step algorithm to mimic the turbulence and it is a common method for the analysis of laser beam propagation through inhomogeneous media. In this section, we discuss some of the basic concepts of an important numerical method, so-called split-step approach to EM wave propagation, also called simply (the beam propagation method (BPM)). BPM is a numerical technique commonly applied to solve nonlinear partial differential equations such as the nonlinear
Schrödinger equation (NLS). While this method is not necessarily a required component of the numerical solution of aperture diffraction problems, we present it here because our initial use of it in studying propagation through phase turbulence led to the re- examination of near- and far-field aperture diffraction via both direct integration and split-step integration. The method relies on computing the solution in small steps and on taking into account the linear and nonlinear (or non-deterministic) steps separately as illustrated in Fig.3.1 [32,33].
An adaptation of the above method permits combining the effects of propagational diffraction and the effects of an inhomogeneous (or non-deterministic) medium caused by
30
turbulence within the same algorithm. The key ideas behind the BPM are two-fold.
First, the expected propagation distance is sectioned into infinitesimal segments.
Secondly, the linear processes (such as diffractive integrals) are performed between the spatial frequency domain or k-space and the spatial coordinates, which are interrelated.
Correspondingly, the random phase function defined by the MVKS model (훷푛(푘)) is in the spatial frequency domain as such; it is then processed via a series of transformations so that we finally obtain a spatial phase distribution 휑푖푗 , where the subscripts (i,j) imply spatial coordinates of points on the chosen grid within which the phase distribution is applied. In the case of extended turbulence (with multiple random phase screens), the propagation distance, L, is divided into increments Δz=L/n , where n is the number of random phase screens [34].
phase screen 푒푗휑(푥,푦) Aperture plane Image plane Z=2Δz Z=3Δz Einc(r) Einc(r)
H(k , k ,Δz) H(kx, ky,Δz) x y H(kx, ky,Δz) … … … … … … .
Linear Linear Linear
Z=L Z=0 Δz Δz Δz L= n Δz
Fig.3.1. Schematic illustration and physical interpretation of the BPM.
31
In the BPM, we begin with a profiled EM beam at z = 0 which then becomes incident upon a planar aperture in the so-called aperture or “object” plane, characterized by the transverse coordinates(푥0, 푦0). The aperture function itself may be either binary, whereby the amplitude transmission is simply 1 within the boundary of the aperture, and 0 outside; alternatively, it may consist of a planar random phase function, such that the transmitted output across the aperture is simply phase-shifted by the random phase. Thus, if
푈푖푛(푥0, 푦0) is the incident profiled beam, and 푔(푥0, 푦0) is the so-called aperture function, then the output across the aperture may be expressed as:
+ 푈표푢푡(푥0, 푦0)(푎푡 푧 = 0 ) = 푈푖푛(푥0, 푦0)푔(푥0, 푦0) , (3.1) where, as mentioned, 푔(푥0, 푦0) may be either a binary function or a random phase distribution. In applying BPM, we next transmit the above scalar output field over an incremental distance Δz (from z to z + Δz) and find the diffracted field at this distance by using the familiar Fresnel-Kirchhoff diffraction integral, which is outlined in the next section. When the profiled EM wave reaches the first random phase screen, the second operator in the algorithm describes the effect of propagation in the absence of diffraction and in the presence of the medium inhomogeneities (random phase screen in our case); this is incorporated in the spatial domain. Hence, after a distance of Δz, the phase perturbations caused by refractive index fluctuation arising from turbulence effects are represented by multiplying the field by a phase function 푒푗휑(푥,푦) as:
푗휑(푥푖,푦푖) 푈표푢푡(푥푖, 푦푖) = 푈푖푛(푥푖, 푦푖) 푒 , (3.2) where 푈표푢푡(푥푖, 푦푖) is the field amplitude immediately after random phase screen, and
푈푖푛(푥푖, 푦푖) is the field before random phase screen. The above process is repeated until the field has traveled the desired distance. In the case of the extended random phase
32
turbulence, where this approach has been recently applied, the above field output is then passed through subsequent planar phase screens multiple times as needed, and finding the output for each screen by simply using the transmission product given by eq. (3.2).
We note at this stage that incremental propagation along the medium using the split-step approach allows us to study the problem in both near-field and far-field since a sufficiently extended propagation (from diffraction theory) automatically carries the diffraction integral from Fresnel to Fraunhofer. For the purpose of this research, however, we will present in a later section two examples involving random phase screens placed once only at z = 0, and then the field propagated by means of both the BPM and the direct integral strategies.
By applying the above approach to the single phase screen problem, it has been possible to verify that both strategies yield identical results. This attested to the validity of the split-step method in numerically studying random turbulence problems, including the propagation of chaotic EM waves through turbulence. In the process of arriving at this conclusion, it was found that generalized planar aperture diffraction problems might be readily studied by essentially using direct integrals (instead of the split-step approach) from the “object” to the “image” plane (with and without random phase screens), and at the same time incorporating arbitrary profiled input plane waves. Specifically, by using the direct integral approach, we limit ourselves in this work to looking at the far-field results only, even though as outlined above, the methodology described here may be just as well applied to studying the more complex near-field problems. Thus, in summary, this chapter addresses three different diffraction scenarios:
33
(1) Propagational diffraction through arbitrary binary aperture with uniform plane waves, which is the standard problem commonly studied;
(2) Diffraction through arbitrary binary apertures with non-uniform or profiled plane waves, including Gaussian and Bessel beams; and
(3) Diffraction through single or multiple random phase screens regarded as planar apertures with uniform plane waves.
It must be noted that case (3) above may also be extended to include non-uniform plane waves, and indeed such a problem is currently under detailed study as part of research on propagation through turbulence.
3.3 Fresnel-Kirchhoff diffraction integral
A general schematic for aperture diffraction of plane EM waves is shown in
Fig.3.2. Shown in the figure are the object or aperture plane (with coordinates x0, y0), the image plane (with coordinates xi, yi), and the radial distance r between points in the object and image planes [35].
Starting with the familiar Rayleigh-Sommerfeld and Huygens-Fresnel principles applied to generalized EM wave diffraction through a homogeneous medium, and after applying a series of assumptions including paraxial, binomial approximations, and zero obliquity
[36], one may write:
(푥표, 푦표), (푥푖, 푦푖) ≪ 푧푖 , and (3.3)
2 2 1 푥푖−푥표 푦푖−푦표 푟01 ≈ 푧푖 [1 + ( ) + ( ) ] , (3.4) 2 푧푖 푧푖 where 푟01 is the radial distance between object and image plane points, (푥0, 푦0) and
(푥푖, 푦푖) represent the object and image point coordinates, and z푖 is the longitudinal distance.
34
Fig.3.2. Geometry representation of the aperture and observation planes.
With the above approximations, one finally arrives at the well-known Fresnel-Kirchhoff diffraction formula:
푘 푗푘푧푖 {푗 [(푥 −푥 )2+(푦 −푦 )2]} 푒 ∞ 2푧 푖 표 푖 표 푈(푥푖, 푦푖) = . ∬ 푈(푥표, 푦표)푒 푖 푑푥표푑푦표, (3.5) 푗휆푧푖 −∞ where 푈(푥표, 푦표) is the output of the aperture as in eq. (3.1), k is the unbounded wave number and λ is the wavelength in the medium.
We note at this stage that the above integral works well for sufficiently paraxial problems while including moderately near-field conditions. However, since the work reported here involves principally far-field diffraction, the above equation is further approximated to the familiar Fraunhofer diffraction formula [37,38]:
푘 2휋 푗푘푧푖 푗푘푧푖 푗 (푥 2+푦 2) 푗 (푥 푥 + 푦 푦 ) 푒 푒 2푧 표 표 ∞ 휆푧 푖 표 푖 표 푈(푥푖, 푦푖) = 푒 푖 . ∬ 푈(푥표, 푦표) 푒 푖 푑푥표푑푦표, (3.6) 푗휆푧푖 푗휆푧푖 −∞ which is also expressible as the Fourier transform:
푈(푥푖, 푦푖) ∝ F 푈(푥표, 푦표) ,
where 푥푖 and 푦푖 are related to the spatial frequencies 푓푥푖 and 푓푦푖 as: 푥푖 = 푓푥푖휆푧푖 ,
푦푖 = 푓푦푖휆푧푖 .
35
3.4 Numerical simulations, results and interpretations
In this section, we present results from numerical simulations of the aperture diffraction problem extended to a variety of binary and random-phase apertures, including also separate cases involving uniform as well as profiled plane waves. Some of the results are the standard and well-known diffraction patterns, but the approach described in this work leads to much more generalized results that will shed some light upon the characteristics of aperture diffraction when combined with atmospheric turbulence and more realistic beam profiles. (Note: in all following cases λ=0.5µm).
3.4.1 Split - step method versus direct Fresnel diffraction integral approach
In this subsection, two approaches are used to examine the Fresnel diffraction pattern from an aperture with binary transmission function and an input Gaussian beam.
In the first approach, we apply the Fresnel diffraction integral to the input Gaussian beam
(with beam waist w0=50mm) through the aperture through the entire propagation distance as shown in Figs. 3.3 and 3.5, while in the second, we apply the BPM to eventually find the diffraction pattern at Z = L, where the propagation distance is divided into small distances (Δz = 10 m) and the split-step method is applied sequentially with an equivalent phase screen replaced simply by a constant phase (say 2mπ) in order to eliminate turbulence as illustrated in Figs. 3.4 and 3.6 . For both approaches, square and rectangular apertures (with dimensions, 400mmx400mm, 300mmx500mm, respectively) are used, primarily to demonstrate the diffractive spread in the near-field relative to aperture dimensions.
36
Fig.3.3. Near-field diffraction pattern using Fresnel-integral diffraction (direct method) for square aperture: (a) Input Gaussian beam profile, (b) Gaussian beam profile at Z=L, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
Fig.3.4. Near-field diffraction pattern using split-step method for square aperture: (a) Input Gaussian beam profile, (b) Gaussian beam profile at Z=L, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
37
Fig.3.5. Near-field diffraction pattern using Fresnel-integral diffraction (direct method) for rectangular aperture: (a) Input Gaussian beam profile. (b) Gaussian beam profile at Z=L, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
Fig.3.6. Near-field diffraction pattern using split-step method for rectangular aperture. (a) Input Gaussian beam profile. (b) Gaussian beam profile at Z=L. (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
38
From Figs.3.3-3.6, we observe that the two approaches yield identical results. Figs.3.3(b) and 3.4(b) show 3-D intensity broadening due to self-diffraction in the near-field (here chosen to be at L = 2 km) using direct integration and the split-step approach. The corresponding 2D sections along x and y (Figs.3.3(c,d) and 3.4(c,d), with y = 0 and x = 0 respectively) are the same because the aperture is symmetric and the diffraction will be the same at each edge. For the rectangular aperture (Figs.3.5 and 3.6), it is apparent that the diffracted beam spreads further along the X-direction since the input aperture is narrower along X.
3.4.2 Far-field (Fraunhofer) diffraction pattern with binary aperture
In what follows, we present a series of results pertinent to binary aperture diffraction with (a) uniform plane wave excitation, and (b) Gaussian plane waves with variable widths.
3.4.2.1 Uniform plane wave input i) Binary double-slit
Assuming unit-amplitude, normally-incident plane-wave illumination, the far- field diffraction pattern for the double slit (width= 50mm, height=5mm) is show in
Fig.3.7(a-d). The propagation of a plane wavefront is obstructed everywhere except within the aperture. Fig.3.7(a) shows the aperture with larger openings along X than Y.
Fig.3.7(b) shows the actual intensity pattern in the image plane. As expected, the pattern spreads farther vertically than horizontally due to the Fourier nature of the far-field.
Figs.3.7(c,d) show the 2D sections of the pattern along X and Y (corresponding to y = 0 and x = 0 respectively), and clearly demonstrate the nature of diffractive spreading as discussed above; moreover, since the double slits are laid out in the Y-direction, we see overlapping sinc2- type characteristics being more pronounced vertically in this case.
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Fig.3.7. Far-field diffraction pattern for binary double-slit aperture: (a) Binary double-slit, (b) Far-field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
ii) Very narrow horizontal double-slit aperture
As an interesting limiting case-study for the double slit, we next consider two slits
(width= 50mm, height=0.5mm) with arbitrarily narrow openings. This is shown in
Fig.3.8(a). Theoretically, in the limit that the slits become infinitely narrow, one obtains the cos2-type diffraction pattern, as discussed in Appendix A. Shown in Figs.3.8(a-d) are the aperture, the corresponding far-field diffraction pattern, and the 2D cross-section along X and Y, respectively. The results from the numerical diffraction solution are seen to match the analytic plot shown in Appendix A (Fig.A2).
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Fig.3.8. Far-field diffraction pattern for very narrow horizontal double-slit: (a) Very narrow horizontal double-slit aperture, (b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
iii) Circular aperture
We next consider a circular binary aperture (Radius = 15mm). For this case, we expect the usual Airy disk pattern via the Fourier-Bessel transform, as discussed in
Appendix A. The aperture, the corresponding numerical far-field diffraction pattern, and the 2D cross-sections are shown in Figs.3.9(a-d) respectively. Once again, we find that the results for the numerical simulation and the analytic plot (shown in Appendix A
(Fig.A3)) are identical.
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Fig.3.9. Far-field diffraction pattern for circular aperture: (a) Circular aperture, (b) Far - field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
iv) Annular aperture
A binary annular or ring aperture is formed by two concentric circles with different radii (b=15mm, a=7.5mm). The expected diffraction pattern in this case is the difference between the Airy patterns of the outer and inner circles, as discussed in
Appendix A. The annular aperture, its numerical far field diffraction pattern, and 2D cross-sections along X and Y are shown in Figs.3.10(a-d) respectively. The numerical plots are in agreement with the analytic plot shown in Appendix A (Fig.A5).
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Fig.3.10. Far-field diffraction pattern for annular aperture: (a) Annular aperture, (b) Far - field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
v) Infinitely narrow annular aperture
In the limit that the inner and outer radii (a =14.5mm, b =15mm) of the annular aperture become equal, the circle function representing the resulting binary aperture becomes a delta-function in the polar domain. This, after using the corresponding
Fourier-Bessel transform leads to a far-field pattern that looks like the zeroth-order
Bessel function, J0. This is analytically demonstrated in Appendix A. Upon passage of a uniform plane wave through this aperture, the diffraction pattern and the 2D sections shown in Figs.3.11(a-d) are obtained. These results are once again compatible with the analytic plot shown in Appendix A (Fig. A7). The zeroth-order Bessel beam characteristics are evident from the numerical plot.
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Fig. 3.11. Far-field diffraction pattern for infinitely narrow annual aperture: (a) Infinitely narrow annual aperture, (b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
3.4.2.2 Gaussian beam input i) Double-slit aperture with wide Gaussian beam
In this demonstration, we consider passage of a very wide Gaussian beam through a double-slit aperture (width= 50mm, height=5mm). The Gaussian beam waist (w0=
5000mm) is much larger than the aperture width (w0 >>> w). For this example, we assume w0 = 100 w. As seen in Figs.3.12(a-d), the far field diffraction pattern and the 2D cross sections along X and Y are almost to those for the uniform plane wave case, as in
Fig.3.7. The reason for that, the Gaussian beam with large beam waist compared with the aperture width can be treated as a uniform plane wave. Then the far field diffraction pattern will be proportional to the Fourier transform of the aperture.
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Fig.3.12. Far-field diffraction pattern for double slit aperture with wide Gaussian beam: (a) Double slit aperture, (b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
ii) Double-slit aperture with narrow Gaussian beam
If the (centered) Gaussian beam waist is close to the aperture width (w0 ≈ 2w in our case), along with the physical dimensions as given (i.e., w= 5 mm, w0=10 mm, slit separation=20 mm), we may no longer assume the Gaussian to be like a uniform plane wave. Most of the Gaussian beam will pass the aperture and diffract from the edges. In this case, as mentioned in eq. (3.6) the far field diffraction pattern is proportional to the
Fourier transform of the aperture function multiplied by Gaussian beam in spatial domain. In other words, the far field diffraction pattern will be the Fourier transform of the Gaussian beam convolved with the Fourier transform of the aperture. The simulation result is shown in Fig.3.13. Upon closer inspection, we observe that the diffraction due to the wider Gaussian (Fig.3.12) has finer sidelobes past the three main lobes, while that in the case of the narrower Gaussian does not exhibit any finer sidelobe. One may
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conjecture that if the Gaussian is made sufficiently narrow, the far-field will begin to approach a self-diffracted version of the Gaussian itself.
Fig.3.13. Far-field diffraction pattern for double slit aperture with narrow Gaussian beam: (a) Double-slit aperture, (b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
iii) Very narrow double-slit with wide Gaussian beam
As was mentioned in part (i), a very wide Gaussian beam can be considered as uniform plane wave. This explains why the simulation results in Fig.3.14 are well- matched to those in Fig.3.8. Note the cos2-type intensity distribution in the far-field, as happens with uniform plane waves.
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Fig.3.14. Far-field diffraction pattern for very narrow double-slit aperture with wide Gaussian beam: (a) Very narrow double-slit aperture, (b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
iv) Very narrow double-slit with narrow Gaussian beam
We next note that if the slits are arbitrarily narrow, then any finite-width Gaussian beam will appear similar to a uniform plane wave. However, a physically narrower
Gaussian beam in relation to the slit separation will have an impact in the diffraction outcome. Thus, we find from Figs.3.15 that while the cos2-type intensity pattern
(Fig.3.15(d)) along Y is reminiscent of a uniform plane wave diffraction, the actual intensity pattern (Fig.3.15(b)) and the cross-section along X (Fig.3.15(c)) are actually noticeably different from Fig.3.14 (b,c), where a wide Gaussian was considered. The diffraction pattern for the narrow Gaussian does not have finer fringe lines or sidelobes which are clearly present for the wider case.
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Fig.3.15. Far-field diffraction pattern for very narrow double-slit aperture with narrow Gaussian beam: (a) Very narrow double-slit aperture, (b) Far -field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
3.4.2.3 Single-slit with Gaussian beam
The far field diffraction pattern when the Gaussian beam waist w0 is wide enough in order to diffracted by single slit with width w (w0 ≈ 2w) is the Fourier transform of the
Gaussian beam convolve with the Fourier transform of the aperture. The single slit aperture, far field diffraction pattern, and 2D cross section are illustrated in Fig.3.16(a-d).
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(a) (b)
0.1 0.1 ) ) 2 0.08 2 0.08 0.06 0.06 0.04 0.04
Intensity (w Intensity m / 0.02 (w Intensity m / 0.02
-100 -50 0 50 100 -100 -50 0 50 100 x [m] y [m] (c) (d)
Fig. 3.16. Far-field diffraction pattern for single slit aperture with Gaussian beam: (a) Single slit aperture, (b) Far -field diffraction pattern, (c), and (d) 2D section along (xi,0) and (0,yi) respectively.
3.4.2.4 Thin sinusoidal amplitude grating with uniform plane wave
Thus far we have discussed apertures with binary transmittance functions (100% or 0% transmission). It is also possible to define an amplitude transmittance function inside the aperture that varies with (푥0, 푦0). One example is the thin sinusoidal amplitude grating (see Appendix A (Fig.A8)). The amplitude transmittance function inside the aperture, far field diffraction pattern, and 2D cross section are show in
Figs.3.17(a-d). Also, we find that the results for the numerical simulation and the analytic plot (shown in Appendix A (Fig.A9)) are identical.
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Fig.3.17. Far-field diffraction pattern for thin sinusoidal amplitude grating aperture: (a) Thin sinusoidal amplitude grating, (b) Far-field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
3.4.2.5 Single-slit aperture with Bessel beam
If the central lobe diameter of the Bessel beam is chosen to be close to the aperture width (ρ0 ≈ 2w in our case) as was done for the Gaussian beam, the Bessel beam will undergo edge diffraction and yet look different than a uniform plane wave. Thus, the far-field diffraction will be the Fourier transform of the Bessel beam convolved with that of the aperture. The corresponding single slit aperture, far-field diffraction pattern, and
2D cross sections are illustrated in Figs.3.18(a-d).
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Fig.3.18. Far-field diffraction pattern for single slit aperture with Bessel beam: (a) Single- slit aperture, (b) Far-field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
3.4.3 Far-field (Fraunhofer) diffraction pattern of aperture with random
phase distribution in the aperture plane
Until now, we have discussed apertures with binary transmission functions only.
We consider next an aperture with random phase distribution. The random phase distribution can be obtained by using the MVKS to model the turbulence within the aperture.
3.4.3.1 Rectangular aperture
We first consider the case of a rectangular aperture with a random phase distribution imposed across it in the manner of a planar, MVK-type phase turbulence.
The simulation results corresponding to uniform plane wave illumination can be seen in
Figs.3.19(a-d). We note here that the far-field pattern is theoretically the spectrum of the rectangular aperture with the random phase distribution, which is analytically non-
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tractable. The results shown in the figure indicate a substantial difference from the expected double-sinc2-pattern. Also, it is expected that the diffraction will vary for multiple applications of the MVKS phase screen.
Fig.3.19. Far-field diffraction pattern for rectangular aperture with random phase distribution: (a) Rectangular aperture with random phase distribution, (b) Far-field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
3.4.3.2 Circular aperture
Here we consider the case of a uniform plane wave passing through a circular aperture with a random, MVK-type phase distribution representing planar phase turbulence. Of course, the far-field diffraction pattern is now the Fourier transform of the
(random) aperture function. Due to the random nature of the phase screen, the spectrum corresponding to this aperture is not analytically tractable. The resulting numerical patterns and distributions are shown in Figs.3.20(a-d). The corresponding non-random, binary circular aperture results were shown earlier in Figs.3.9(a-d). The cross-section plots clearly show a pattern very different from the Airy disk characteristic of standard
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circular aperture diffraction. We conjecture that the diffraction results will be different for multiple runs of the phase screen, since the phase distributions are randomly generated.
Fig.3.20. Far-field diffraction pattern for circular aperture with random phase distribution: (a) Circular aperture with random phase distribution, (b) Far-field diffraction pattern, (c) and (d) 2D section along (xi,0) and (0,yi) respectively.
From the last figures, the far-field diffraction properties of uniform and profiled EM beams transmitted through binary and random-phase apertures using the Fresnel-
Kirchhoff diffraction integral heve been studied. A comparison between the split-step propagation method and direct Fresnel-Kirchhoff diffraction integral indicates identical results for both approaches. This affirms the validity of the split-step method applied to profiled beam propagation through random phase turbulence. In particular, diffraction patterns have been studied for uniform, Gaussian and Bessel beams incident upon the aperture. Wherever possible, the numerical plots have been compared with analytic
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plots, and have yielded essentially compatible results. The simulations have consisted of applying the Fresnel-Kirchhoff diffraction integral and thereafter the far field diffraction patterns of intensity have been obtained, along with the cross-sections along the horizontal and vertical axes. Additional plots have consisted of the case of a sinusoidal amplitude grating and of random phase distributions across the aperture plane corresponding to uniform plane wave propagation. The effect of phase turbulence upon
Gaussian beam propagation for both narrow and extended media, and eventually the propagation of modulated chaotic waves through such turbulence will be investigated extensively in the incoming chapters.
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CHAPTER 4
INVESTIGATION OF PROFILED BEAM PROPAGATION THROUGH NARROW
AND EXTENDED TURBULENT MEDIUM
4.1 Introduction
Atmospheric turbulence affects optical systems that operate in various atmospheric conditions. The characteristics of the optical wave transmitted through atmospheric turbulence can undergo dramatic changes resulting in degradation of system performance. Atmospheric turbulence causes index of refraction in-homogeneities such as different size eddies which affect optical wave propagation through the atmosphere.
These refractive index in-homogeneities cause fluctuations in both the intensity and the phase of the received signal [39,40]. Knowledge of atmospheric turbulence effects is helpful in the development of a wide class of atmospheric-optics systems including laser communication, energy transfer, remote sensing, and active and passive imaging systems
[24]. In recent research, propagation of plane EM waves through a turbulent medium with MVKS characteristics was modeled and numerically simulated using transverse planar apertures representing narrow and extended phase turbulence along the propagation path [25]. In the classical atmospheric turbulence theory, the refractive index structure parameter is the key parameter known to describe the strength of the atmospheric turbulence and accurate measurement of this parameter represents an
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important task. The optical property of the medium often referred to the index of refraction. The refractive index is typically dependent on wavelength of the incident radiation and the density of the medium [1-3]. Recently, atmospheric turbulence was modeled using the PSD of the refractive index. Included among these models are the
Kolmogorov, Tatarski, von Karman, and modified von Karman spectra. The refractive index structure parameter was first introduced in the turbulence theory developed by
Kolmogorov and Obukhov [1,2,24,25]. The key assumptions of classical (Kolmogorov) turbulence theory, such as statistical homogeneity and isotropy of the refractive index random field, are not always satisfied. This theory is now commonly referred to as the
Kolmogorov turbulence theory [41-43]. The phase fluctuations of the phase screen used to model the random phase distribution within the aperture are parameterized by the Fried parameter (ro), which describes the transverse coherence length, and the inner and outer scales that determine the amount of aberration seen by the propagating beam. The
2 refractive index structure parameter 퐶푛 describes the strength of the refractive index fluctuations and marks the first major principle on which the development of the classical
Kolmogorov atmospheric turbulence theory depends [15].
In this chapter, we have studied the effect of the atmospheric turbulence on the propagation of the Gaussian beam profile. This study was prompted by the eventual desire to propagate a modulated chaotic wave generated from an A-O cell with feedback through the turbulence [28,29]. A brief review of the MVKS is carried out relative to narrow and extended (wide) turbulence to mimic the random statistical behavior of the atmospheric turbulence. This is done by using the MVKS random phase screen(s). Two scenarios are followed to represent the atmospheric turbulence: (a) single random phase
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screen where the random phase screen is located at a particular distance from the aperture plane then the field intensity at the observation (image) plane is calculated, and (b) extended (multiple) random phase screens whereby a certain number of random phase screens are placed in equally infinitesimal distance (Δz) along the propagation path and then the field intensity at different positions and at the image plane is also calculated.
Modeling laser beam propagation through turbulence using successive phase screens provides an efficient tool for tracking the effect of atmospheric turbulence laser beam propagation. In this work, the BPM involving the Fresnel-Kirchhoff diffraction integral is used (either a single (narrow) random phase screen or extended (multiple) random phase screens are placed at arbitrary location(s) along the propagation path) to model the propagation of the EM wave through turbulent medium where the propagated EM beam alternatively passes through a purely diffractive region (absence of in-homogeneities) and propagation through a non-diffractive in-homogenous medium. In both scenarios, three parameters are chosen in this work, (a) strength of turbulence (weak and strong), (b) propagation distance, and (c) Δz for extended phase screen. More details about theses parameters will be discussed later in this chapter. Also, some work regarding time statistical in atmospheric turbulence is done where a single random phase screen is located at a certain distance and the amplitude and phase of the field intensity at the image plane are computed. The work reported here also includes the results of some additional investigations. Also, The SI of Gaussian beam propagation through extended turbulence as well as the FV due to interference between a twin set of Gaussian beams propagating through extended turbulence are also calculated.
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4.2 Power spectrum models for refractive index fluctuations
The index of refraction is one of the most significant parameters of the atmospheric for optical wave propagation. The index of refraction of the atmosphere,
푛(r⃗), is modeled as the sum of a mean index of refraction, 푛0, and a randomly fluctuating term, 푛1(r⃗) [39]:
푛⃗ (r) = 푛0 + 푛1(r⃗) , (4.1) where 푟⃗ is a three-dimensional vector position, 푛0 = 〈푛(푟⃗)〉 ≈ 1 is the mean value of the index of refraction of air, and 푛1(푟⃗) represents the random deviation of 푛(푟⃗) from its mean value (〈푛1(푟⃗)〉 = 0). Hence,
푛⃗ (r) = 1 + 푛1(r⃗) . (4.2) The fluctuations in the index of refraction are related to corresponding temperature and pressure fluctuations as follows;
푃(⃗r⃗) 푛(r⃗) = 1 + 79x10−6 , (4.3) 푇(⃗r⃗) where P is the pressure in millibars and T is the temperature in degree kelvin.
The statistical description of the random field of turbulence-induced fluctuations in the atmospheric refractive index is similar to that for the related random field of turbulent velocities. In particular, an inertial sub-range is bounded by outer and inner scales, 퐿0, and ℓ0, respectively. In this sub-range, statistical properties of the refractive index are homogenous and isotropic [44]. The covariance function of 푛(r⃗) may be expressed by:
퐵푛(푟⃗1, 푟⃗2) = 퐵푛(푟⃗1, 푟⃗1 + 푟⃗) = 〈푛1(푟⃗1)푛1(푟⃗1 + 푟⃗)〉 , (4.4) where 푟⃗1 and 푟⃗2 are two points in space, and 푟⃗= 푟⃗2 − 푟⃗1. Assuming a homogeneous and isotropic turbulent medium, the covariance function reduces to a function of only the
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scalar distance 푟 = |푟⃗2 − 푟⃗1|. The assumption of homogeneity implies that only the separation between two points and not the location of the two points in a random medium. This may also be viewed as relative to a spatial or translational invariance [41].
Locally homogeneous fields are usually not characterized by the covariance function, but by the structure function 퐷푛(푟) defined as the mean-squared difference of the refractive index at two points:
2 퐷푛(푟) = 〈[푛(푟1 + 푟) − 푛(푟1)] 〉 = 2[퐵푛(0) − 퐵푛(푟)]. (4.5)
By substituting eq. (4.4) into eq. (4.5) and applying the well-known two-thirds law
(Kolmogorov-Obukhov “two-thirds” power law), we get the structure function of refractive index fluctuations for separations in the intermediate range between 퐿0 and ℓ0:
2 2 ⁄3 퐷푛(푟) = 퐶푛 푟 , ℓ0 ≪ 푟 ≪ 퐿표 (4.6)
2 where 퐶푛 is the index of refraction structure parameter, also called the structure constant.
2 −17 -2/3 퐶푛 typically ranges from 10 m or less for conditions of “weak turbulence” to
10−13 m-2/3 or more when the turbulence is “strong”.
⃗⃗ The three-dimensional spatial power spectrum of the random field Ф푛(푘) forms a 3D
Fourier transform pair with the covariance function [27,37,42]:
( ) ∞ 푖푘⃗⃗.푟⃗ ⃗⃗ 3 퐵푛 푟⃗ = ∭−∞ 푒 Ф푛(푘)푑 푘 , (4.7)
3 1 ∞ −푖푘⃗⃗.푟⃗ 3 Ф (푘⃗⃗) = ( ) ∭ 푒 퐵 (푟⃗)푑 푟 , (4.8) 푛 2휋 −∞ 푛 where 풌 = (푘푥, 푘푦, 푘푧)is the vector wave number ( in units of rad/m). We can introduce spherical coordinated and carry out the angular integration. In spherical coordinates, the
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vector wave number 풌 = (푘, 휃, 휙) and 푑3풌 = 푘2 푠푖푛휃 푑휃 푑휙 푑푘 . Assuming homogeneity and isotropy, these Fourier transform relations reduce to their 1D forms,
3 푖푘⃗⃗.푟⃗ 퐵푛(푟) = ∫ 푑 푘⃗⃗ 푒 Ф푛(푘⃗⃗)
∞ 휋 2휋 2 ( ) 푖푘⃗⃗.푟⃗ ⃗⃗ = ∫−∞ ∫0 ∫0 푑푘 푘 푠푖푛 휃 푑휃 푑휙 푑푘 푒 Ф푛(푘)
4휋 ∞ 퐵 (푟) = ∫ Ф (푘⃗⃗) sin(푘푟) 푘푑푘. (4.9) 푛 푟 0 푛
A Fourier integral also relates the structure function to the isotropic and homogeneous turbulence spectrum [4,45,46]:
1 ∞ Ф (푘⃗⃗) = ∫ 퐵 (푟) sin(푘푟) 푟푑푟 , (4.10) 푛 2휋2푘 0 푛 where 푘 = |푘⃗⃗| is the magnitude of the wave number. The eq. (4.9) and eq. (4.10) represent the one dimension Fourier transform pair.
Consequently, the relation between the spectrum and structure function may be expressed by,
∞ 2 푠푖푛(푘푟) 퐷 (푟) = 2[퐵 (0) − 퐵 (푟)] = 8휋 ∫ 푘 Ф (푘⃗⃗)(1 − )푑푘, (4.11) 푛 푛 푛 0 푛 푘푟
and
1 ∞ 푠푖푛(푘푟) 푑 2 푑 Ф (푘⃗⃗) = ∫ [푟 퐷 (푟)] 푑푟 . (4.12) 푛 4휋2푘2 0 푘푟 푑푟 푑푟 푛
From the eq. (4.6) and eq. (4.11), we get;
2 2⁄ ∞ 2 푠푖푛(푘푟) 퐶 푟 3 = 8휋 ∫ 푘 Ф (푘⃗⃗)(1 − )푑푘. (4.13) 푛 0 푛 푘푟
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Since power law usually reflected in other power law, we try the following [14]:
훾 Ф푛(푘) = 푄푘 .
With a change of variable in (4.13) one see that the two-third power scaling is replicated if 훾 = -5/3 (for one-dimension spectrum) and 훾= -11/3 (for three-dimension spectrum).
After compute the integral, the constant Q =0.033.
4.3 Atmospheric turbulence
It is known from EM propagation theory that optical waves (in the visible) especially are significantly impacted by atmospheric refractive index fluctuations. As an optical wave propagates through the atmosphere, atmospheric turbulence induces both amplitude and phase fluctuations [44]. Unfortunately, the atmosphere is not an ideal communication channel. A better understanding of atmospheric effects on propagating optical beams will enable applications of free space optical communications, remote imaging, surveillance, etc. Since the work presented in this research originated with the investigation of EM propagation through a random medium with phase turbulence, we introduce an overview of atmospheric turbulence and the most common models used to represent the random behavior of the atmospheric turbulence. We will focus in particular the MVKS phase turbulence model which used in this research. We also discuss numerical generation of the random phase screen which describes the random behavior of the turbulence over a thin transverse layer in the propagation direction. The characteristics of the optical wave transmitted through atmospheric turbulence can undergo dramatic changes resulting in potential system performance degradation. In standard turbulence modeling, three parameters characterizing atmospheric turbulence play an important role in describing medium behavior: the refractive index structure
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2 parameter (C푛) which describes the strength of the atmospheric turbulence, and the inner
(ℓ표) and outer (L0) scales of turbulence eddies.
4.3.1 Atmospheric turbulence models
The most common atmospheric turbulence models are the Kolmogorov, Tatarskii, and the von Karman [37]. The current work essentially centers upon a phase turbulence model defined as the modified von Karman model (which for all practical purposes is indistinguishable from the standard von Karman model). Kolmogorov proposed that in the inertial subrange, where Lo > ℓ > ℓo, turbulence is isotropic and may be transferred from eddy to eddy without loss. When the diameter of a decaying eddy reaches ℓo, the energy of the eddy is dissipated as heat energy through viscosity processes as illustrated in Fig. 4.1 [14].
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Fig. 4.1. Depiction of the process of turbulent decay, showing the energy cascade and subsequent division of turbulent eddies in the atmosphere.
i) Kolmogorov Spectrum
Turbulence is by nature a random process, and as such may be described using statistical quantities. In 1941, Kolmogorov first developed a universal description of atmospheric turbulence by developing a structure tensor to describe a mean square velocity difference between two points in the atmosphere. Kolmogorov defined the power spectral density for refractive index fluctuations over the inertial range by [39];
−11 2 ⁄3 Ф푛(푘) = 0.033퐶푛 푘 , 1/퐿표 ≤ 푘 ≤ 1/ℓ표 (4.14a) or
−11 −5⁄ ⁄3 3 Ф푛(푘) = 0.023푘 푟표 , (4.14b)
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where k is the spatial frequency, and 푟0 is the Fried parameter. The Kolmogorov spectrum is used when the inner scale is zero and the outer scale is infinite, or as long as the wave number is within the inertial subrange. ii) Tatarski Spectrum
When the inner or outer scale effect cannot be ignored, the Kolmogorov power law spectrum in eq. (4.14a) needs to be modified. Tatarski suggested the extension of eq. (4.14a) into the dissipation range 푘 > 1/ℓ 표 through the introduction of a Gaussian function that essentially truncates the spectrum at high wave number [47];
−11 2 2 ⁄3 −푘 Ф푛(푘) = 0.033퐶푛 푘 exp ( ⁄ 2 ) , 푘 ≥ 1/퐿표 (4.15) −푘푚
푘 = 5.92⁄ Where 푚 ℓ표 . This method was developed for the purpose of mathematical convenience. However, eq. (4.15) suggests a singularity at k = 0 for the limiting case
1/Lo = 0. This implies that the structure function 퐷푛(푟) exists but the covariance function 퐵푛(푟) does not. iii) Von Karman spectrum
The power spectrum density of the von Karman Spectrum (also called the MVKS in the form shown) is given by [39]:
푘2 exp (− 2 ) 2 푘푚 훷푛(푘) = 0.033 퐶푛 11 , 0 ≤ 푘 ≪ ∞ (4.16) 2 2 6 (푘 +푘표)
2 5.92 where C푛 is the medium structure parameter, 푘푚 = ⁄ℓ표 is an equivalent wavenumber
2휋 related to the inner scale, 푘0 = ⁄ is a wavenumber related to the outer scale, and k is 퐿표 the unbounded non-turbulent wavenumber in the medium. In the above equation, 훷푛(푘) represents the so-called power spectral density (PSD) of the refractive index of the
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medium. Additional models such as the Modified Atmospheric Spectrum also exist in the literature [39]. The Modified Atmospheric model has not been considered since it is more complicated than the MVKS.
4.3.2 Thin phase screen generation
In this subsection we discuss the generation of a phase screen to mimic the statistical behavior of the phase fluctuations due to a turbulent atmosphere using a discrete grid and generating the phase screen from the given spectrum based on fast
Fourier transform (FFT) techniques. The purpose of a phase screen is to simulate the random phase perturbations resulting from random index fluctuations in narrow and extended atmospheric turbulence [44]. The generated random phase screen (either planar
2 or extended) in this research is characterized by several different parameters: C푛 (or Fried parameter 푟0), inner and outer scales, ℓ표 and L0 respectively, and the incremental spatial frequencies 훥푘푥, 훥푘푦.
The procedure of phase screen generation is as follows: beginning with the MVKS model with given parameters as mentioned, and by using a standard scheme based on Fourier transform generation, a set of random complex numbers (following a Gaussian distribution) is generated on the chosen grid. Following eq. (4.17), the random numbers are multiplied by the square root of the phase power spectrum (PPS) wherefrom an inverse Fourier transform produces the phase screen. The real part of the result is taken to be the random phase function 휑(푥, 푦) due to atmospheric fluctuations based on the
MVKS model. The discrete phase distribution in 2D is given as:
휑푖푗 = 푅푒{퐼퐹퐹푇((푎 + 푗푏)√훷푝훥푘푥훥푘푦)}, (4.17)
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where IFFT represents the inverse fast Fourier transform operation, 훥푘푥, 훥푘푦 is the incremental spatial frequencies, 훷푝 is the power spectral density (given below) evaluated in the transverse plane, and (a and b) are random numbers generated in order to appropriately mimic the random noise-like characteristics of the von Karman phase.
Correspondingly, the MVKS model in the spatial domain is expressed as:
푘2 5 exp (− ) − 푘2 3 푚 훷푝(푘) = 0.23 푟0 11 . (4.18) 2 2 6 (푘 +푘표)
Propagation through thin phase screens specified by eq. (4.17) is numerically explored using the split-step algorithm, whereby the incident EM wave is alternately transmitted across the screen and thereafter diffracted over incremental distances before encountering a possible subsequent sequence of phase screens (representing an extended phase turbulence), or being propagated directly to a receiving or image plane using standard
Fresnel-Kirchhoff diffraction. The 2D and 3D random phase screen distribution profile are shown in Figs. 4.2 and 4.3.
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Fig. 4.2. 2D random phase screen distribution profile.
Fig. 4.3. 3D random phase screen distribution profile.
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4.4 Fried parameter, scintillation index, and fringe visibility
The model used in subsequent sections relies heavily upon the Fried parameter
(r0). The Fried parameter and structure constant are intricately related; however, the structure constant is more frequently used. Thus, it is important not only to discuss the relation of the parameters, but the manner in which they affect the results obtained in our
2 simulated models. The structure constant 퐶푛 describes the index of refraction fluctuations and is used to characterize the strength of these fluctuations in the atmosphere. The Fried parameter describes the transverse coherence length, and the inner and outer scales. The relationship connecting the structure constant to the Fried parameter is as follows [44]:
3⁄ 4휋2 5 푟0 = 푞 [ 2 2] , (4.19) 푘 퐿 퐶푛 where q is a dimensionless quantity which is numerically 0.185 for plane waves and 3.69 for spherical waves, and L is the distance of propagation within the turbulence [44].
For free space optical communications it is important to understand the effects of atmosphere on propagating beams, because the atmosphere creates beam wander and scintillation. The SI quantifies the irradiance variance of an optical wave propagated through atmospheric turbulence. The SI is the “normalized variance of irradiance” and mathematically defined as [15]:
〈퐼2(푟,퐿)〉 휎2(푟, 퐿) = − 1, (4.20) 〈퐼(푟,퐿)〉2 where 〈 〉 denotes the ensemble average for mean-square irradiance, while 〈 〉2 denotes the square of the mean, 퐼(푟, 퐿) is the local irradiance, L is the propagation distance of the beam, and r is the radial distance from the symmetry axis. The turbulence is weak if the
SI is less than unity.
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For two identical Gaussian beams, separated by a given transverse spatial distance, the beams constructively and destructively interfere resulting in fringes as the beams propagate to the far-field. The contrast and quality of the fringes is generally a measure of the optical coherence and defined with a metric commonly referred to as FV. FV, mathematically defined in terms of the intensity on the observation plane, is given by:
퐼푚푎푥−퐼푚푖푛 퐹푉 = . (4.21) 퐼푚푎푥+퐼푚푖푛
4.5 Propagation of profiled beam through narrow turbulence
In this scenario, the propagation may be visualized as consisting of the passage of a profiled beam across a planar input aperture, followed by subsequent passage through a narrow region where turbulence occurs. The narrow turbulence, when modeled as a random phase fluctuation (as in the MVKS model), may be represented as a planar phase screen which may be placed anywhere between the aperture and the image plane, depending on the placement of the turbulence itself relative to the propagation. This is indicated in Fig. 4.4, where the phase screen has been placed at a distance L1 from the aperture at z = 0, and the image plane is at z = L.
Phase screen 푒푗휑(푥,푦) Aperture plane Image plane
Einc(r)
Einc(r)
Fresnel-Kirchhoff diffraction integral Fresnel -Kirchhoff diffraction integral H(kx, ky,Δz1) H(kx, ky,Δz2)
Linear Linear
Z=L Z=0 L1 L2 L
Fig. 4.4. Schematic illustration of propagation through narrow turbulence.
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From the above figure, the total propagation distance is divided in two segments (L1 and
L2) where L=L1+L2. The EM beam first travels the longitudinal distance (L1) which is free from turbulence. In this region, the field is subjected to the familiar Fresnel-
Kirchhoff diffraction integral as discussed in chapter 3. The diffracted EM wave subsequently reaches the random phase screen defined by the MVKS model (훷푛(푘)) defined in the spatial frequency domain. It is then processed via a series of transformations so that we finally obtain a spatial phase distribution φ푖푗, where the subscripts (i, j) imply spatial coordinates of points on the chosen grid within which the phase distribution is applied. Once the random phase screen representing MVKS turbulence has been generated, the wave is then transmitted through such a (planar) screen placed at z = L1. The phase perturbations caused by refractive index fluctuations arising from MVKS turbulence across the planar phase screen may be represented by multiplying the input diffracted field by the phase function 푒푗휑(푥,푦) as follows:
푗휑(푥푖,푦푖) 푈표푢푡(푥푖, 푦푖) = 푈푖푛(푥푖, 푦푖) 푒 , (4.22) where 푈표푢푡(푥푖, 푦푖) is the field amplitude immediately after random phase screen, and
푈푖푛(푥푖, 푦푖) is the field before random phase screen. The propagation from L1 to the image plane using the planar random screen therefore appropriately models the narrow turbulence along with associated Fresnel-Kirchhoff diffraction until the observation (or image) plane is reached.
In the following, we place the random phase screen at the locations ( 0, 0.5L, and L) and the field is evaluated at the observation plane. A Gaussian beam is then propagated first from the object plane to the phase screen, and thereafter, following passage through the screen, once again via the diffraction integral to the image plane. The resulting beam
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profile, phase distribution and the scalar field amplitudes are then determined at the image plane. Some of the parameters are kept constant during the simulation process.
These parameters consist of: number of sample points (grid resolution) = 512x512, wavelength λ=0.5µm, physical size of grid = 500mmx500mm, inner scale ℓ표 =
2 10푚푚 , and outer scale L0 = 1푘푚 respectively. The other parameters (퐶푛 or 푟0,
w0, and L) are varied depending on weak or strong turbulence, wide or narrow Gaussian beam, and different propagation path lengths.
Case I. Turbulence strength
In standard atmospheric turbulence literature, it is known that the numerical
2 ranges of the structure parameter C푛 defining strong, intermediate and weak turbulence as in Table 4.1.
Table 4.1. Structure parameter, Fried parameter, and turbulence strength.
Structure parameter Fried parameter Turbulence strength 2 −14 -2/3 C푛 > 10 m r0 = 0.01mm Strong −16 2 −14 -2/3 10 < 퐶푛 < 10 m r0 = 0.5 푚푚 Moderate 2 −17 -2/3 C푛 < 10 m r0=10 mm Weak
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(a) Weak turbulence
Using the above information, one may arrive at a weak turbulence regime defined by:
2 -18 -2/3 푟0=10 mm or C푛=1.067x10 m .
Fig.4.5. Gaussian beam propagation to distance z=L (phase screen at the object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Fig. 4.6. Gaussian beam propagation to distance z=L (phase screen at 0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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Fig. 4.7. Gaussian beam propagation to distance z=L (phase screen at L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
(b) Strong turbulence
2 -13 -2/3 Similarly, for this case, we choose 푟0 = 0.01mm or 퐶푛 = 1.067x10 m .
Fig.4.8. Gaussian beam propagation to distance z=L (phase screen at the object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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Fig. 4.9. Gaussian beam propagation to distance z=L (phase screen at 0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Fig. 4.10. Gaussian beam propagation to distance z=L (phase screen at L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
From Figs. (4.5 - 4.10), we observe that the Gaussian beam suffers more amplitude distortion and likely more phase fluctuations for strong turbulence compared to weak turbulence. Regarding the random phase screen placement, we find that the phase
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fluctuations increase when the phase screen is placed at the beginning of the propagation distance (at 0L) or the middle (0.5L) compared to the end (L) relative to the propagation path. An intuitive interpretation might be as follows. Thus, when the phase screen is placed at 0L, the propagated beam passes through the random phase screen (turbulence) before any kind of self-diffraction along the propagation path, while when the phase screen at the image plane (at z = L), the propagated beam is subjected to self-diffraction before it reaches the random phase screen at the end. In other words, when an EM beam acquires a random phase profile, the resulting phase fluctuations are more pronounced as the beam propagates over an arbitrary distance under self-diffraction. On the other hand, when the beam initially self-diffracts with a deterministic phase profile, and encounters a random phase at the end, the exiting beam does not experience a comparable rate of phase fluctuations. We may also note that under weak turbulence, while the phase fluctuations may still be pronounced, the amplitude or intensity profile remains relatively unaffected. Also, under weak turbulence, the incident Gaussian undergoes peak amplitude (or intensity) decay during propagation. Since the medium is considered lossless, this decay simply implies a lowering of the Gaussian peak as the profile broadens due to diffraction. Under strong turbulence, on the other hand, for earlier screen placement, the diffracted beam not only splits and consequently distorts, it also experiences apparent peak intensity increase. This might seem to be contradictory for a lossless propagation; however, we note that the increased intensity peak is probably misleading, and more likely there occurs split output amplitudes of opposite polarities that would still conserve net energy and power. Since the plots show intensity (which is amplitude-squared), this feature is not visible in the plots.
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Case II. Waist of the Gaussian beam
In this case, we consider moderate turbulence ( 푟0 = 0.5 푚푚, or
2 -16 -2/3 퐶푛 =1.5725x10 m ). We consider also the physical size of the Gaussian beam, which we define in the following as narrow or wide. In this series, both narrow and wide will imply Gaussian beams whose spot size (2w0) fits well within the size of the diffraction grid. Thus, narrow and wide beams are only defined in terms of the relative spot sizes of the beams, and not necessarily in terms of comparison with the aperture.
(a) Narrow Gaussian beam (w0=10 mm)
Fig.4.11. Gaussian beam propagation to distance z=L (phase screen at the object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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Fig. 4.12. Gaussian beam propagation to distance z=L (phase screen at 0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Fig. 4.13. Gaussian beam propagation to distance z=L (phase screen at L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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(b) Wide Gaussian beam (w0=100 mm )
Fig. 4.14. Gaussian beam propagation to distance z=L (phase screen at the object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Fig. 4.15. Gaussian beam propagation to distance z=L (phase screen at 0.5L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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Fig. 4.16. Gaussian beam propagation to distance z=L (phase screen at L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Figs.(4.11 - 4.16) show that for narrow Gaussian beams under moderate turbulence, placement of the phase screen early in the propagation likely once again creates greater overall phase fluctuation profiles in the output beam. Additionally, there is possibly some beam amplitude splitting that happens at the output for early phase screen placements compared with placements further along the diffraction path. For wider
Gaussian beams, similar amplitude-splitting behavior is once again evident; moreover, we also observe a phase “clustering” effect around the edges of the grid when the phase screen is placed well before the end of the propagation path. As before, there is also peak amplitude or intensity decay in the output beams.
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Case III. Propagation distance (L)
In this case, we consider moderate turbulence ( 푟0 = 0.5 푚푚 or
2 -16 -2/3 퐶푛 =7.8625x10 m ). For the cases studies presented here, the Fraunhofer or far-field distance (D2/λ, where D is the grid size) happens to be about 250 km. Thus, all propagation cases reported herein apply only to near-field diffraction. In the report presented here, we simply discuss the effect of propagation distance upon the diffracted
EM beam within the Fresnel regime, except that we compare relatively short versus longer longitudinal propagation distances.
(a) L=1km
Fig.4.17. Gaussian beam propagation to distance z=L (phase screen at object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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Fig.4.18. Gaussian beam propagation to distance z=L (phase screen at 0.5 L). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Fig.4.19. Gaussian beam propagation to distance z=L (phase screen at L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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(b) L=10 km
Fig.4.20. Gaussian beam propagation to distance z=L (phase screen at object plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
Fig.4.21. Gaussian beam propagation to distance z=L (phase screen at 0.5L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
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Fig.4.22. Gaussian beam propagation to distance z=L (phase screen at L or image plane). (a) 3D Gaussian beam, (b) Gaussian beam cross-section, (c) Gaussian beam profile, (d) Random phase screen profile, and (e) Gaussian beam phase distribution in transverse output plane.
From Figs.4.17 - 4.22, we find that for propagation to shorter distances, the output EM beam suffers very little amplitude distortion or decay, and likely a small amount of diffractive broadening. On the other hand, the phase fluctuations are once again greater for earlier screen placements compared with placements later along the propagation path.
For propagation to longer distances, we observe greater diffractive broadening, as expected. Additionally, there is also greater amplitude decay and greater phase fluctuation for earlier screen placement, as before. There is likely also a small amount of diffractive beam splitting in the output for early phase screen placement.
4.6 Temporal statistics
We note at this stage that investigating the propagation of EM waves through atmospheric turbulence in the current work has been motivated by the problem of chaotic wave propagation through a turbulent layer. While the majority of turbulence models are in the spatial or inverse domains, the chaotic A-O first-order light is characterized by
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time-dependent chaos. As a result, it becomes necessary to develop the means to track the turbulence as a function of time. One way to accomplish this is to realize that the random phase fluctuations associated with MVKS is also inherently a time-dependent phenomenon. Thus, the phase of an EM wave will not only fluctuate randomly in space
(via inner and outer scales), but also with time. In the simulations presented here, this time fluctuation enter directly into the process because the random numbers used as part of the generation of the MVKS phase function will also automatically change randomly with time. Thus, one may evaluate temporal statistical measures of the output field in the system under study for both amplitude and phase in order to develop an appropriate temporal model. In this section, the temporal mean and variance of the amplitude and phase of the diffracted field are computed by placing the single random phase screen at arbitrary positions along the propagation path (say in steps of 0.1L from the object to the image planes) and evaluating the diffracted Gaussian beam at image plane (z = L). For each phase screen position, the on-axis mean and variance of the amplitude and phase of
2 -13 -2/3 the field are calculated. A turbulence strength of r0 = 0.01mm or 퐶푛 = 2.7 x 10 m ,
Gaussian beam waist of w0=50mm, propagation distance L=5km, and number of sample points (grid resolution) of 513x513 are used for this computation. The number of samples (iterations) used to calculate the mean and variance of the amplitude and phase of the field is 500 times (N=500), i.e., the random phase screens across which the EM wave propagates are re-created 500 times. Fig.4.23 shows the mean and variance of the amplitude and phase of the diffracted Gaussian beam at z =L for different random phase screen positions.
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4
3.5
3
2.5 Amplitude(mean) Amplitude(variance) 2 Phase(mean) Phase(variance) 1.5
Mean and variance and Mean 1
0.5
0
-0.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized phase screen location
Fig.4.23. On-axis mean and variance of the amplitude and phase of diffracted Gaussian beam at Z=L for different random phase screen positions.
We note here that temporal statistics for points in the paraxial region as well as sufficiently off-axis have also been computed. Clearly, the overall scales of the time statistics in terms of corresponding inner and outer scales in time need to be examined and derived further.
4.7 Propagation of profiled beam through extended turbulence
The extended case (via multiple random phase screens) is an extension of narrow turbulence, where the propagation distance L is divided into arbitrarily small segments
Δz = L ⁄ n , where n is an integer equal to the number of phase screens as illustrated in
Fig.4.24. The diffracted EM beam is propagated through subsequent narrow phase screens multiple times as needed, and the same procedure is applied as mentioned in the narrow turbulence case. All the simulation results presented in this work were carried out using BPM either in the narrow or extended turbulence regime.
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phase screen 푒푗휑(푥,푦) Aperture plane Image plane Z=2Δz Z=3Δz Einc(r) Einc(r)
H(k , k ,Δz) H(kx, ky,Δz) x y H(kx, ky,Δz) … … … … … … .
Linear Linear Linear
Z=L Z=0 Δz Δz Δz L= n Δz
Fig.4.24. Schematic illustration of propagation through extended turbulence.
In the extended turbulence case, additionally, the incremental distance Δz representing the physical distance between two adjacent phase screens has to be accounted for. We note here that the extended turbulence modeled here consists of multiple narrow turbulences spatially separated by Δz, which in our numerical simulations, is not necessarily infinitesimally small. This implies that the “extended” medium is idealized as narrow turbulences followed by short regions of pure diffraction and thereafter repeated multiple times. A true extended turbulence problem would require Δz to be arbitrarily small (likely smaller than the turbulence scale sizes); however, in many cases, the problem is analyzed with somewhat larger Δz values, thereby implying discretely separated turbulence
We next consider passage of a profiled Gaussian beam through an extended turbulent medium represented by multiple phase screens. Of course the number of used phase screens is governed by the incremental distance Δz (preferably very small) and the propagation distance. The Gaussian beam 3D view, its transverse plane intensity distribution, 2D intensity profile, the random phase screen distribution profile, and the
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field phase angle distribution in 3D are evaluated at z = Δz, z = 0.5 L, and z = L as illustrated in the following figures. We note here that ideally the split-step method requires that non-turbulent diffraction-limited propagation occurs between alternate phase screens, and thus in order to make this assumption mathematically viable, it is necessary that the inter-screen distance Δz be infinitesimally small (i.e., Δz→0). In reality, however, it turns out that making Δz too small significantly increases the computation time. As a result, the values reported in this research are based on optimal choices of Δz that yield reasonable results. Moreover, the values selected are compatible with those commonly used in the literature.
Case I. Turbulence strength
Since, Δz =10 m, and L=5 km, so the number of random phase screens used is n = 500.
2 -18 -2/3 a) Weak turbulence 푟0 = 10 mm or 퐶푛 =1.067x10 m .
Fig. 4.25. Gaussian beam propagation to incremental distance Δz. (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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Fig. 4.26. Gaussian beam propagation to distance Z=0.5 L (250 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
Fig.4.27. Gaussian beam propagation to distance Z= L (500 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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2 -13 -2/3 b) Strong turbulence ( 푟0= 0.01 mm or 퐶푛 =1.067x10 m ).
Fig.4.28. Gaussian beam propagation to incremental distance Δz. (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
Fig. 4.29. Gaussian beam propagation to distance Z=0.5 L (250 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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Fig. 4.30. Gaussian beam propagation to distance Z= L (500 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
From the Figs. (4.25 - 4.27), we note that the transverse phase fluctuations increase as the propagation distance increases. In Fig. 4.25(e) (when the beam characteristics are evaluated at the first increment), the phase fluctuations are concentrated mainly around the corners of the grid. Also, in Fig. 4.26(e) (when the beam characteristics are evaluated halfway), the phase fluctuations become higher compared with the previous case.
Following this trend, the maximum phase fluctuations occur when the field reaches image plane, as shown in Fig. 4.27(e). The corresponding peak field intensities drop from 1 W/m2 (at Z=Δz) to 0.9 W/m2 (at Z=0.5L) and finally to 0.69 W/m2 (at Z=L). These results also are in accord with intuition, as seen earlier. In the case of strong turbulence, the results are more dramatic for the extended turbulence problem than were seen for the corresponding narrow turbulence cases. This is evident from the plots in Figs.4.29 and
4.30. Thus, we find severe profile distortion in the propagating beam already halfway along the path (Fig.4.29(c)). We observe multiple nested peaks that are spatially
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separated along with side lobes. Evidently, such beam splitting will have serious detection errors in any receiver system. At full propagation distance (Fig.4.30), we find a complete breakdown of the original profile, and the received field therefore has an essentially scrambled profile (Fig.4.30(c)). These observations underscore the importance of studying beam propagation through (phase) turbulence, and the need for finding methods and strategies to minimize such distortions. Incidentally, numerical computations of the corresponding SI (presented later) further corroborate these findings in terms of the effects of both strong and more extended turbulence.
Case II. Propagation distance (L)
In this case, we choose moderate turbulence ( 푟0= 0.5 mm or
2 -16 -2/3 퐶푛 =7.8625x10 m ). a) L=1 km
As Δz =10 m, and L=1 km, the number of random phase screens is n = 100 phase screens.
Fig. 4.31. Gaussian beam propagation to incremental distance Δz. (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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Fig. 4.32. Gaussian beam propagation to distance Z=0.5 L (50 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
Fig. 4.33. Gaussian beam propagation to distance Z= L (100 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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b) L=10 km
For Δz =10 m, and L=10 km, the number of random phase screens is 1000.
Fig. 4.34. Gaussian beam propagation to incremental distance Δz. (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
Fig. 4.35. Gaussian beam propagation to distance Z=0.5 L (500 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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Fig. 4.36. Gaussian beam propagation to distance Z= L (1000 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
We next note from Figs.4.31 and 4.34 that the results are essentially identical regardless of the propagation distance (1km and 10km) because the incident wave has traveled only to the distance Δz. We observe also that there is a tendency for phase fluctuations to accumulate near the corners of the grid at this stage. When the propagation is examined at the halfway distances (as seen in Figs. 4.32 and 4.35), we find that there is a tendency towards beam-splitting (near the peak in Fig.4.32 and also around the middle in Fig.4.35) even for moderate turbulence, and additionally the beam is increasingly broadened as the propagation distance increases, accompanied by corresponding reductions in the peak intensity. These results corroborate intuition and expectation. At full propagation distance, we find that the incident beam undergoes highly visible beam-splitting (Figs.
4.33 and 4.36), and correspondingly, measurably higher broadening as well, up to 85 mm at a distance of 10 km (which closely matches the expected depth of focus of about
7.8 km for this diffraction problem).
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Case III: Incremental distance (Δz)
In this case once again we choose moderate turbulence ( 푟0= 0.5mm or
2 -16 -2/3 퐶푛 =1.5725x10 m ). As intuitively expected, the accuracy of the split-step algorithm requires the increment Δz to be arbitrarily small. However, as was discussed, making Δz too small will increases the computation time considerably. In this section, we present two plots (measured at the full propagation distance) that clearly exemplify the effect of choosing smaller (versus larger) incremental distances in evaluating the overall propagation through turbulence with greater accuracy. a) Δz = 1 m
With Δz =1 m, and L=5 km, the number of random phase screens is 5000 for this case.
Fig. 4.37. Gaussian beam propagation to distance Z= L (5000 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
We note that with the relatively small Δz chosen, the final beam profile exhibits much of the turbulence-induced splitting and phase behavior, as seen in Fig. 4.37.
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b) Δz =50 m
The number of random phase screens is 100 for the much larger Δz chosen for this case.
Fig. 4.38. Gaussian beam propagation to distance Z= L (100 Δz). (a) 3D Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
From Fig. 4.38, we note that even at full propagation distance the diffracted profile under moderate turbulence exhibits only minimal amplitude (or intensity) splitting, a result which is obviously inaccurate when compared with the same propagation with a much smaller Δz, as was seen in Fig.4.37. As a final remark, we observe that in order to execute reliable numerical simulations via the split-step algorithm, there needs to be a tradeoff between accuracy and the acceptable computational time invested. Thus, it is very likely that by reducing the distance Δz even below 1m, the accuracy gained may not justify the substantially higher computational time required.
Case IV: Scintillation index with extended turbulence
In this part, a single Gaussian beam is propagated through extended phase turbulence, represented by a number of random phase screens. The corresponding SI is
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calculated at the image plane using the eq. (4.20). The beam width (w0), Fried parameters (r0), inner (ℓ0), and outer (L0) scales are all set to be constant during the propagation. Two turbulence conditions (weak and strong) are used in this subsection with results as follows:
2 -18 -2/3 a) Weak turbulence (퐶푛 ≈ 10 m )
Shown in Fig.4.39 is the plot for the SI versus the number of phase screens for 4 different choices of propagation distance.
0.9
0.8
0.7
0.6
0.5 1km 2km 0.4 5km Scintillation Index (SI) 10km 0.3
0.2
0.1 10 20 30 40 50 60 70 80 90 100 Number of phase screens
Fig. 4.39. The scintillation index plotted as a function of number of phase screens. The (constant) parameters are: w0=30mm, r0=10mm, ℓ0=10mm, and L0=1km. The propagation distance is given in the legend in Kilometers.
2 -13 -2/3 b) Strong turbulence (퐶푛 ≈ 10 m )
Once again, Fig.4.40 shows the SI as a function of the number of phase screens corresponding to the same 4 propagation distances as in Fig.4.39.
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2
1.8
1.6
1.4
1.2
1
Scintillation Index (SI) 0.8 1km 0.6 2km 5km 0.4 10km
0.2 10 20 30 40 50 60 70 80 90 100 Number of phase screens
Fig. 4.40. The scintillation index plotted as a function of number of phase screens. The (constant) parameters are: w0=30mm, r0=0.01mm, ℓ0=10mm, and L0=1 km. The propagation distance is given in the legend in Kilometers.
Generally, as expected, the SI increases with the propagation distance. Also, the SI increases in the case of strong turbulence compared with weak turbulence. An important observation here has to do with the asymptotic trends in the set of SI plots. We first note that in the case of weak turbulence (Fig.4.39), the SI curve invariably rises up, but eventually appears to converge asymptotically towards a steady-state value. This trend is readily explained by the fact that as the number of phase screens (n) increases, the distance Δz goes down, thereby increasing the computational accuracy. As a result, below a threshold value of Δz, further increase in the number of phase screens will not enhance the accuracy (or the SI for these plots) appreciably, and therefore, the graphs will converge to a steady-state. On the other hand, for strong turbulence, accuracy demands a much smaller Δz for similar parameter values as the weak turbulence case since greater beam fluctuations are expected. This trend is also observed in the plots of Fig.4.39,
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where within a range of n of up to 100 (which is rather small from the accuracy perspective with strong turbulence), the SI curves continue to be on the rise. It is expected that if n were increased sufficiently, the curves would once again begin to saturate, indicating that an accuracy threshold for Δz has been reached.
Case V: Fringe visibility with extended turbulence
By using two identical spatially separated Gaussian beams, the FV is determined using eq. (4.21) at the observation plane for different values of propagation distance for a given number of random phase screens. The beam width (w0), Fried parameters (r0), inner (ℓ0), outer (L0) scales, and the separation distance between the two beams are all set to be constant during the propagation. Again, in this simulation, weak and strong turbulence cases are considered to analyze the effect of the phase turbulence on the FV.
Fig. 4.41(a). Double Gaussian beam propagation to distance Z= L through weak turbulence. (a) 3D Double Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile.
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Fig. 4.41(b). Double Gaussian beam propagation to distance Z= L through strong turbulence. (a) 3D Double Gaussian beam, (b) its transverse plane intensity distribution, (c) 2D intensity profile.
2 -18 -2/3 a) Weak turbulence (퐶푛 ≈ 10 m )
1
0.9 1km 2km 0.8 5km 10km 0.7
0.6
0.5
Fringe VisibilityFringe (FV) 0.4
0.3
0.2
0.1 10 20 30 40 50 60 70 80 90 100 Number of phase screens
Fig. 4.42. The Fringe visibility plotted as a function of number of phase screens. The parameters that held constant are: w0=30mm, r0=10mm, beam separation = 70 mm, ℓ0=10mm, and L0=1km. The propagation distance is given in the legend in Kilometers.
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2 -13 -2/3 b) Strong turbulence (퐶푛 ≈ 10 m )
0.45
0.4 1km 2km 0.35 5km 10km 0.3
0.25
0.2
Fringe VisibilityFringe (FV) 0.15
0.1
0.05
0 10 20 30 40 50 60 70 80 90 100 Number of phase screens
Fig. 4.43. The Fringe visibility plotted as a function of number of phase screens. The parameters that held constant are: w0=30mm, r0=0.01mm, beam separation = 70 mm, ℓ0=10mm, and L0=1km. The propagation distance is given in the legend in Kilometers.
From the Figs. 4.42 and 4.43, in both cases (weak and strong turbulence) the FV decreases with the increasing of the propagation distance and with increasing of the number of phase screens. As expected, the FV has lower values in the case of strong turbulence compared with weak turbulence regime. As mentioned in the last subsection, the incremental distance (Δz) between the two adjacent phase screens can be calculated using the same procedure. Also, the simulation processing time increases as the propagation distance and the number of phase screens increase.
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CHAPTER 5
A TRANSFER FUNCTION BASED FREQUENCY MODEL FOR PROPAGATION
OF A CHAOS WAVE THROUGH MODIFIED VON KARMAN TURBULENCE
UNDER VARIOUS CHAOS TURBULENCE CONDITIONS
5.1 Introduction
Turbulence has large influence EM wave propagation. Turbulence effects arise from atmospheric refractive index fluctuations that cause both large and small variations of EM wave propagation [48]. In recent research, propagation of plane EM waves through a turbulent medium with modified von Karman phase characteristics was modeled and numerically simulated using transverse planar apertures representing narrow phase turbulence along the propagation path [49-52]. Gaussian beam propagation through a turbulent layer has been studied using a split-step methodology. Accordingly, the beam is alternately propagated (a) through a thin Fresnel layer, and hence subjected to diffraction; and (b) across a thin modified von Karman phase screen which is generated using the PSD of the random phase obtained via the corresponding PSD of the medium refractive index for MVKS turbulence. The random phase screen in the transverse plane is generated from the phase PSD by incorporating (Gaussian) random numbers representing phase noise.
The goal of the research is to examine two random phenomena: (a) atmospheric turbulence due to von Karman-type phase fluctuations, and (b) chaos generated in an A-O
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Bragg cell under hybrid feedback. The latter problem has been thoroughly examined for its nonlinear dynamics and applications in secure communications. For the turbulence problem, an optical beam passing through an input aperture is propagated through a random phase screen (placed at different locations) to a desired distance (typically near- field) under different levels of turbulence strength. The resulting spatial intensity profile is then averaged and the process repeated over a (large) number of pre-specified time intervals. From this data, once again, the turbulence PSD is calculated via the Fourier spectra of the average intensity snapshots. The results for the two systems are compared.
The work introduced in this chapter analyzes the overall propagation problem within the framework of a linear system and thereby develops an equivalent transfer function model derived from cross- and self-spectral densities using both chaos and turbulence, as will be shown. Thereafter, the transfer function formalism is applied to track propagation of modulated chaos waves through modified von Karman phase turbulence; the demodulated signal is examined vis-àvis performance relative to turbulence strength in comparison with non-chaotic propagation. The transfer function model has been derived using cross-correlation and cross-power spectral density functions using both unmodulated chaos and turbulence. Characteristics of information-encrypted chaos are examined in this chapter.
5.2 Turbulence power spectral density
In this part, the aim is to estimate the PSD of the atmospheric turbulence using the diffracted intensity profiles at the output plane repeated over discrete time intervals.
From each time sample, an average intensity is computed. The process is repeated over a selected number of intervals (say 1000), whereupon the resulting average intensity is
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plotted versus time. The Fourier transform of this time snapshot of intensity is a measure of the PSD of the turbulence provided that the incident optical wave propagating through the input aperture is equivalent to a point source (i.e., the aperture is treated as an ideal pinhole). In the first series, the PSD is computed under different atmospheric turbulence conditions (weak, moderate, and strong) by placing the phase screen midway between the source and the screen. In the second series, the turbulence is assumed to be moderate, and the PSD is calculated at two different phase screen positions (z = 0 and z = L). In order that the observed time plot and its Fourier transform reflect the PSD characteristics of the turbulence, advantage is taken of the fact that the turbulent system may be regarded as linear. Therefore, the output (spatial) spectrum is readily described as the product of the input spectrum and the equivalent transfer function of the system. Hence, assuming propagation of either a point source or an optical wave through a pinhole leads to an input spectrum that is spatially uniform. Therefore, the resulting diffracted spectrum is a direct measure of the PSD of the turbulent system. The spatial spectral data is thereafter converted into time data by evaluating the average spatial spectra over discrete time intervals. The parameters used for the calculations are: grid size = 500mm x 500 mm; grid resolution = 512x512 pixels; total propagation distance L = 5 km; point source function exiting aperture plane= 훿(푥0)훿(푦0) in the transverse input plane x0 and y0 ; inner scale ℓ표 = 1mm; and outer scale Lo = 1 km.
In pursuing the derivation of the time waveform and the corresponding PSD of the turbulence, as was mentioned, 1000 spectral samples have been used (via discrete time iterations) for each case. For each iteration, we calculate the intensity within the square
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window centered at origin of the grid (widow size=100x100 pixels), the total time for time waveform is adjusted to be 100 seconds (thus a time interval of 0.1 sec.).
5.2.1 Atmospheric turbulence strength
As discussed, the random phase screen is placed at the halfway point along the propagation distance and the field is evaluated at the observation plane. All other
2 parameters mentioned above are kept constant except for the structure parameter 퐶푛 (or
푟0) which characterizes the strength of the turbulence.
2 -18 -2/3 a) Weak turbulence ( 푟0= 10mm or 퐶푛 =1.067x10 m )
Fig.5.1. Point source propagation to distance z = L. (a) 3D point source; (b) transverse plane intensity distribution at z = L, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution in output plane.
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4
3.5
3
2.5
2
PSD( PSD( W / Hz) 1.5
1
0.5
0 0 2 4 6 8 10 12 14 16 18 20 frequency (Hz) (a) (b) Fig.5.2. (a) Time snapshot of the average spatial turbulence intensity, and (b) PSD of turbulence derived from spectrum shown in (a).
2 -15 -2/3 (b) Moderate turbulence ( 푟0= 0.1 mm or 퐶푛 =2.299x10 m )
Fig.5.3. Point source propagation to distance z = L. (a) 3D point source; (b) transverse plane intensity distribution at z = L, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution in output plane.
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2.5
2
1.5
PSD( PSD( W / Hz) 1
0.5
0 0 2 4 6 8 10 12 14 16 18 20 frequency (Hz) (a) (b) Fig.5.4. (a) Time snapshot of the average spatial turbulence intensity, and (b) PSD of turbulence derived from spectrum shown in (a).
2 -13 -2/3 c) Strong turbulence ( 푟0= 0.01mm or 퐶푛 =1.067x10 m )
Fig.5.5. Point source propagation to distance z = L. (a) 3D point source; (b) transverse plane intensity distribution at z = L, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution in output plane.
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1
0.8
0.6 PSD( PSD( W / Hz) 0.4
0.2
0 0 2 4 6 8 10 12 14 16 18 20 frequency (Hz) (a) (b) Fig.5.6. (a) Time snapshot of the average spatial turbulence intensity, and (b) PSD of turbulence derived from spectrum shown in (a).
From Figs.5.1, 5.3, and 5.5, point source excitation has little impact by way of the intensity spectrum for different turbulence strengths, except for the amplitudes seen in the
PSD plots of Figs. 5.2(b), 5.4(b) and 5.6(b). The latter figures indicate an amplitude lowering (from about a peak of 3.8 to 0.95) as the turbulence changes from weak to strong. This makes sense intuitively, since higher turbulence implies greater attenuation and scattering of the EM wave. The PSD in all cases appears to be strongest around a rather low frequency (about 2 Hz); however, this is simply related to the time scale (or interval) over which the scattered intensity is measured. Reducing the time interval is likely to increase this frequency.
5.2.2 Phase screen position
In this part, we assume moderate turbulence ( 푟0 = 0.1 mm or
2 -15 -2/3 퐶푛 =2.299x10 m ) and hold all other parameters constant. The random phase screen is located at the beginning and at the end of the propagation path. The PSD of the field is evaluated from measurements in the image plane.
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a) Phase screen at z = 0
Fig.5.7. Point source propagation to distance z = L. (a) 3D point source; (b) transverse plane intensity distribution at z = L, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution in output plane.
0.16
0.14
0.12
0.1
0.08
PSD( PSD( W / Hz) 0.06
0.04
0.02
0 0 2 4 6 8 10 12 14 16 18 20 frequency (Hz) (a) (b) Fig.5.8. (a) Time snapshot of the average spatial turbulence intensity, and (b) PSD of turbulence derived from spectrum shown in (a).
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b) Phase screen at z = L
Fig.5.9. Point source propagation to distance z = L. (a) 3D point source; (b) transverse plane intensity distribution at z = L, (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution in output plane.
4
3.5
3
2.5
2
PSD( PSD( W / Hz) 1.5
1
0.5
0 0 2 4 6 8 10 12 14 16 18 20 frequency (Hz) (a) (b) Fig.5.10. (a) Time snapshot of the average spatial turbulence intensity, and (b) PSD of turbulence derived from spectrum shown in (a).
There is effectively no difference between the Figs. 5.8(b) and 5.10(b) except for a much lower amplitude for the case where the phase screen is at z = 0 (so that the field propagates through the turbulence much earlier and then diffracts to the position L) than when it is placed at z = L (for very similar reasons).
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5.3 Turbulence time waveform ET(t) derived from spatial model
In order to generate the turbulence time waveform (needed for developing the aforementioned transfer function), we propagate a uniform plane EM wave over a distance (L) using the MVKS model whereby a narrow phase screen is placed half way between the object and image planes. For each sampling time interval (∆T), the on-axis field at (0,0, z) (E (0,0, z)) is evaluated. We repeat this process for pre-selected intervals
∆T in order to generate the on-axis transmitted field as a function of time. The following figures (Fig.5.11 and Fig.5.12) show the propagation of uniform plane wave starting from the aperture plane to image plane passing through homogenous and random phase turbulence and the resultant time wave form of the turbulence. The turbulence frequency
(fT) is determined by taking the reciprocal of the repetition interval ∆T. we can summarize the generation of turbulence time waveform as follows:
Propagate a uniform plane EM wave over a distance (L) using the MVKS model
whereby a narrow phase screen is placed in between the object and image planes.
For each pre-selected sampling time interval (∆T), the on-axis field in the image
plane (E(0,0,L )) is evaluated.
We repeat this process for pre-selected intervals ∆T in order to generate the on-
axis transmitted field as a function of time.
The effective turbulence frequency (fT = 1/ ∆T) is determined by taking the
reciprocal of the repetition interval ∆T.
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Fig.5.11. Propagation of uniform plane wave through narrow turbulence region.
1
0.8
0.6 ( t ) T
E 0.4
0.2
0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time (s) ( b ) Fig.5.12. Time waveform of the turbulence.
5.4 Cross-correlation and cross-power spectral density functions
For two independent random processes (hereafter referred to by C and T for chaos and turbulence respectively), the cross-correlation RCT(τ) and cross-power spectral density SCT(f) functions may be given as [53,54]:
∗ 푅퐶푇(휏) = 퐸(퐸퐶(푡)퐸푇(푡 + 휏)) (5.1) and
SCT (f ) = F {RCT (τ)} , (5.2)
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where E denotes expectation operation, EC and ET denote the scalar chaos and turbulence field amplitudes, and F denotes Fourier transform.
The effective transfer function (HT( f )) between the chaos and the turbulence may be found using the following formula [55,56]:
푆퐶푇( 푓 ) 퐻푇( 푓 ) = , (5.3) 푆퐶( 푓 )
where SCT ( f ) is the cross-power spectral density defined above, and SC( f ) is the power spectral density of the chaos wave.
In [54], the HT( f ) was derived using an unmodulated chaos wave EC(t) with specific values of the average chaos and effective turbulence frequencies. Next, a modulated chaos wave as the EM input field is incorporated where the power spectral density of the modulated chaos wave (SCm( f )) is calculated via an auto-correlation function (RCm(τ)).
Defining a corresponding transfer function HT( f ) based on the spectral densities we compute (a) the cross-power spectral density ((STCm(f))); and (b) the cross-correlation function (RTCm(τ)) between the modulated chaos and turbulence using the following relations on an equivalent linear systems basis:
푆푇퐶푚(푓) = 퐻푇(푓)푆퐶푚(푓) (5.4)
−1 푅푇퐶푚(휏) = 퐹 {푆푇퐶푚(푓)} , (5.5) where 퐹−1 denotes inverse Fourier transform.
5.5 Results and discussion for specific cases of chaos and turbulence
In this section, we will examine the propagation of a modulated chaos wave with average frequency fch (= 500 kHz and 2 MHz) through strong turbulence with effective frequency fT (= 10Hz and 100Hz) as shown in Figs.5.14 -5.21. The parameters used in
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this simulation are chosen as: inner scale ℓ0=1mm; outer scale Lo = 1km; phase screen grid size and resolution 500mmx500mm and 513x513 pixels respectively; the propagation distance L = 5km.
Fig.5.13. Uniform plane wave propagation to distance z=L (phase screen at half way between aperture and image planes). (a) 3D uniform plane wave; (b) its transverse plane intensity distribution at the image plane; (c) 2D intensity profile, (d) random phase screen distribution profile, and (e) 3D field phase angle distribution.
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Case I: f =500kHz (T =1 µs) , and f =10 Hz ) ch D T
Fig.5.14. (a) chaos time waveform, (b) turbulence time waveform, (c) the cross correlation function, (d) the cross-power spectral density function, (e) chaos power spectral density, (f) transfer function magnitude, (g) phase of the transfer function, and (h) inverse Fourier transform of the transfer function.
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Fig.5.15. (a) modulated (encrypted) chaos time waveform, (b) its auto-correlation function, (c) power spectral density of the modulated chaos, (d) the cross-power spectral density function, (e) the cross correlation function.
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Case II: f =500kHz (T =1 µs) , and f =100 Hz ch D T
Fig.5.16. (a) chaos time waveform, (b) turbulence time waveform, (c) the cross correlation function, (d) the cross-power spectral density function, (e) chaos power spectral density, (f) transfer function magnitude, (g) phase of the transfer function, and (h) inverse Fourier transform of the transfer function.
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Fig.5.17. (a) modulated (encrypted) chaos time waveform, (b) its auto-correlation function, (c) power spectral density of the modulated chaos, (d) the cross-power spectral density function, (e) the cross correlation function.
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Case III: f =2MHz ( T =0.25 µs) , and f =10 Hz ch D T
Fig.5.18. (a) chaos time waveform, (b) turbulence time waveform, (c) the cross correlation function, (d) the cross-power spectral density function, (e) chaos power spectral density, (f) transfer function magnitude, (g) phase of the transfer function, and (h) inverse Fourier transform of the transfer function.
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Fig.5.19. (a) modulated (encrypted) chaos time waveform, (b) its auto-correlation function, (c) power spectral density of the modulated chaos, (d) the cross-power spectral density function, (e) the cross correlation function.
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Case IV: f =2MHz (T =0.25 µs) , and f =100 Hz ch D T
Fig.5.20. (a) chaos time waveform, (b) turbulence time waveform, (c) the cross correlation function, (d) the cross-power spectral density function, (e) chaos power spectral density, (f) transfer function magnitude, (g) phase of the transfer function, and (h) inverse Fourier transform of the transfer function.
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Fig.5.21. (a) modulated (encrypted) chaos time waveform, (b) its auto-correlation function, (c) power spectral density of the modulated chaos, (d) the cross-power spectral density function, (e) the cross correlation function.
In recent work, simulation results have been presented indicating the above cross- correlation functions for chaos waves under simple (AM) modulations upon passage through turbulence [57]. Extracting the transmitted information signal from the above cross-correlation (RTCm(τ)) presents additional challenges which are currently under further investigation.
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CHAPTER 6
DIFFRACTIVE PROPAGATION AND RECOVERY OF MODULATED
(INCLUDING CHAOTIC) EM WAVES THROUGH UNIFORM AND TURBULENT
MEDIUM
6.1 Introduction
In the past few decades, the impact of atmospheric turbulence on both the amplitude and phase of propagating EM signals have been investigated extensively [24].
The consequence of (random) phase fluctuations of the transmitted signal in distorting the amplitude of the received signal may be established both experimentally and via numerical simulation. Usually, phase turbulence is modeled using one or more random phase screen(s) placed at certain locations between the aperture plane and the observation plane [25,26]. The traditional models for turbulence (including Kolmogorov, Tatarski, von Karman, and modified von Karman) assume homogenous and isotropic [27]. In the previous chapter, transfer function formalism was applied to examine the propagation of modulated chaos waves through atmospheric turbulence [55]. The turbulence was modeled via a random phase screen placed half way between the aperture and image planes. This random phase screen is created using the MVKS model.
The overall propagation problem is analyzed within the framework of a linear system and thereby an equivalent transfer function model is derived from cross- and self- (power)
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spectral densities using chaos and turbulence simultaneously. Later, as reported in [57], a modulated chaos wave as the input EM field was incorporated where the PSD of the modulated chaos wave (SCm(f)) was calculated via an auto-correlation function (RCm(τ)) using the (effectively random) chaos field amplitude. By defining a corresponding turbulence transfer function HT(f) derived from [55] based on spectral densities, a cross- power spectral density (STCm(f)) and cross-correlation function (RTCm(τ)) between the modulated chaos and turbulence were obtained. Recovery of the actual information from the above cross-correlation poses additional physical and mathematical complexity. In a parallel approach, the diffraction of a time-modulated plane EM wave through both a uniform and a phase-turbulent atmosphere is currently under study, and some results from this approach are presented in this chapter. Specifically, an input EM wave is treated as a modulated optical carrier represented by use of a sinusoidal phasor with a
SVEA due to the (time-dependent, non-sinusoidal) information signal. Using a combination of phasors and spatial Fourier transforms, the resulting complex phasor wave is then transmitted across the propagation path (with or without turbulence) through the dual use of the Kirchhoff-Fresnel integral and the random phase screen. The objective is to ascertain if the presence of turbulence imparts any amplitude and/or phase distortion in the embedded message carried by the EM carrier. In the follow-up work, the transmitted EM wave is assumed to be an encrypted chaotic carrier, for which a similar phasor-Fourier transform approach may be applied to determine the recovery via demodulation of the information encrypted on the chaos as it propagates through the turbulence. Some preliminary results are presented via comparisons between non-chaotic and chaotic information transmission through atmospheric turbulence, outlining thereby
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any possible improvement in system performance by utilizing the robust features of chaos.
6.2 Spectral approach to propagate the (non-chaotic) EM waves through turbulence
using SVEA and Fourier transforms
In this section, we examine the propagation of (non-chaotic) EM waves through atmospheric turbulence and their characteristics in the image plane. Starting with a profiled beam represented by a time-harmonic carrier:
퐸(푥, 푦, 푧, 푡) = 퐸표(푥, 푦, 푧) 푐표푠(휔표푡 − 푘푧) , (6.1)
̅ with 푘 = 푘푧푎̂푧 , and
2 푤 −푟 ⁄ 퐸 (푥, 푦, 푧) = 퐴 표 푒 푤2(푧) , 표 푤(푧) (6.2)
2 2 2 푧 2 2 2 where 푤 (푧) (= 푤표 (1 + 2)) defines the beam spot size; 푟 = 푥 + 푦 ; 휔o is the carrier 푧표 frequency; w0 is the beam waist, and zo is the depth of focus of the (Gaussian) beam
[58,59].
By substituting eq. (6.2) in (6.1), we get
2 푤 −푟 ⁄ 퐸(푥, 푦, 푧, 푡) = 퐴 표 푒 푤2(푧) 푐표푠(휔 푡 − 푘푧) 푤(푧) 표 . (6.3)
The (AM) modulated EM wave may be expressed as:
퐸퐴푀(푥, 푦, 푧, 푡) = 퐸표(푥, 푦, 푧)[1 + 푚푠(푡)] 푐표푠(휔표푡 − 푘푧) , (6.4) where m is the modulation index and s(t) is the modulating signal (message). The above then represents a directly modulated optical carrier.
By substituting eq. (6.2) in (6.4), we get
2 푤 −푟 ⁄ 퐸 (푥, 푦, 푧, 푡) = 퐴 표 푒 푤2(푧) [1 + 푚푠(푡)] 푐표푠(휔 푡 − 푘푧) 퐴푀 푤(푧) 표 . (6.5)
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2 푤 −푟 ⁄ 표 푒 푤2(푧) [1 + 푚푠(푡)] Note that the term A푤(푧) is a time-varying spatial envelope defined by the signal waveform.
As we mentioned above, in our approach we will use the SVEA approximation where the envelope varies slowly compared with the rate of change of the optical carrier (i.e., envelope BW (B << 휔o)). This method involves expressing the optical carrier via phasor representation wherein the envelope (containing the signal waveform) is slowly varying in time. The analysis then proceeds with the application of the diffraction integral acting upon the above complex spatial phasor, and the resulting propagation is tracked numerically to the receiver or “image” plane.
Accordingly, we next propagate the modulated EM wave (퐸퐴푀(푥, 푦, 푧, 푡)) converted to the phasor domain (while retaining the SVEA envelope) by applying the Fresnel-
Kirchhoff diffraction integral along the propagation path. An inverse phasor transform restores the optical carrier. A photodetector recovers the modulated envelope by retaining the intensity profile (which removes the carrier). For a directly modulated optical carrier, the signal is recovered by using appropriate filtering and scaling. On the other hand, if the optical carrier has an envelope containing an RF modulated carrier (such as a modulated chaos wave), recovery would require additional use of heterodyne AM demodulation followed by filtering. At the receiver, the scalar output is proportional to the PD current at the point location (say at 0, 0, zi for on-axis detection) where the PD intercepts the input field:
2 2 퐼푃퐷(푡) ∝ |퐸표푢푡(0, 0, 푧푖)| [1 + 푚푠(푡)] , (6.6) where 퐸표푢푡(0, 0, 푧푖) is equivalent to the spatial Fourier transform of
−푗푘푧푧 퐸표(푥표, 푦표)푒 푔(푥표, 푦표) in the far-field on-axis with 퐸표(푥표, 푦표) being the profile of the
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input wave and 푔(푥표, 푦표) being the aperture function through which the wave is transmitted at the “object” plane.
The message signal may be retrieved from the photocurrent via appropriate electronics.
This establishes the fact that under homogeneous propagation, a modulated carrier may be demodulated directly using time-domain strategies without having to consider spatial diffraction and other effects. On the other hand, when the intervening medium is turbulent (in phase or otherwise), there is a likelihood that the random spatial effects would affect the signal riding on the carrier wave; hence, one of the goals of this study is to examine the effects of such atmospheric turbulence on the transmitted signal.
6.3 Numerical simulations, results and interpretations
In this section, we will examine the propagation of a modulated (non-chaotic) wave through uniform and turbulent media (weak and strong) with effective frequency fT (= 20, 50 and 100Hz). The modulated signal is sent from the aperture plane to the image plane passed through a random phase screen located mid-way of the propagation path. In this approach, to accommodate the temporal variations of the turbulence, the propagation algorithm is applied one pass at a time; thereafter, the resulting spatial output is calculated in the image plane corresponding to that pass. The process is repeated over pre-selected time intervals (representing the average time scale of the turbulence). In this manner, an average time profile of the output is generated taking into account the multiple passes, representing the random output profile influenced by the von Karman turbulence. The parameters used in this simulation are chosen as: inner scale ℓ0 =1mm; outer scale L0 = 1km; phase screen grid size and resolution 500mmx500mm and 513x513 pixels respectively; the propagation distance L = 5km.
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6.3.1 A uniform (non-turbulent) propagation prototype
To test the behavior under turbulence, we first present below results from a simple homogeneous, diffraction-limited propagation of a modulated (optical) EM wave through the use of the Kirchhoff-Fresnel diffraction integral. Accordingly, a Gaussian profile beam (Fig.6.1(a)) is modulated by a sinc-type pulse (Fig.6.1(b)), and is received as a diffracted Gaussian (Fig.6.1(c)) with a circular symmetry (Fig.6.1(d)) and a broadened profile (Fig.6.1(e)). Fig.6.1(f) shows the final recovered signal after carrying out the necessary demodulation operation using the photodetector output.
Fig.6.1. Propagation of EM waves through uniform medium. (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D Gaussian beam output (w0 ≈ 80mm); (d) cross-section of the output Gaussian intensity; (e) 2D output Gaussian profile; and (f) recovered signal.
The above figures demonstrate typical spatial transmission and recovery of an EM wave through a homogeneous medium whereby the profiled EM wave (here Gaussian) undergoes diffractive spread; however, the time signal embedded within the optical
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carrier is simply recovered using standard detection electronics (see Fig.6.1(b) and (f)).
We include this relatively trivial result here simply to enable further understanding as to the consequences on the homogeneous prototype when medium turbulence is incorporated into the problem.
6.3.2 Propagation through weak turbulence
For weak turbulence, we choose Fried parameter r0 = 10 mm with structure
2 -18 -2/3 parameter 퐶푛 =1.067x10 m . Note that turbulence is typically modeled as a spatial effect; however, in reality it also is subject to random time variations. In our analysis, we incorporate the temporal variation of turbulence by repetitive insertion of the thin phase screen over pre-selected time intervals which thereby defines an equivalent mean
(temporal) turbulence frequency.
(a) Propagation through weak turbulence with mean frequency fT = 20 Hz
In this section, we repeat the procedure described in 6.3.1 above for a narrow, weakly turbulent medium. To represent the time fluctuations of the medium phase, the narrow phase screen is inserted into the propagation algorithm (at the half-way distance) repeatedly every (1/20) second, thereby generating random results at a mean turbulence frequency of 20 Hz.
Fig.6.2. Propagation of EM waves through weak turbulence ( fT = 20Hz). (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; and (c) random phase screen distribution profile.
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Fig.6.3. Propagation of EM waves through weak turbulence ( fT = 20Hz). (a) 3D Gaussian beam output (w0=80mm ); (b) cross-section of the output Gaussian intensity; (c) 2D output Gaussian profile; and (d) recovered signal.
The results above show that under weak phase turbulence, the recovered signal upon averaging the repeated time profiles for each pass over the 20 Hz turbulence is beginning to exhibit some distortion (Fig.6.3 (d)). b) Propagation through weak turbulence with mean frequency fT = 50 Hz
In this section, we repeat the same procedure for 50 Hz turbulence and the simulation results are shown in the following two figures.
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Fig.6.4. Propagation of EM waves through weak turbulence (fT = 50Hz). (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; and (c) random phase screen distribution profile.
Fig.6.5. Propagation of EM waves through weak turbulence (fT = 50Hz). (a) 3D Gaussian beam output (w0 ≈ 80mm); (b) cross-section of the output Gaussian intensity; (c) 2D output Gaussian profile; and (d) recovered signal.
We note from Figs. 6.4 and 6.5 that when the turbulence frequency increases, the received Gaussian profile begins to undergo some peak amplitude splitting (Fig.6.5(c)); additionally, the recovered superposed signal waveform shows even greater distortion
(Fig.6.5 (d)).
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c) Propagation through weak turbulence with mean frequency fT = 100 Hz
Here we increase the turbulence frequency to 100 Hz. The numerical results are shown in Figs.6.6 and 6.7.
Fig.6.6. Propagation of EM waves through weak turbulence (fT = 100Hz). (a) 3D Gaussian beam input (w0 = 30mm) ; (b) modulating signal; and (c) random phase screen distribution profile.
Fig.6.7. Propagation of EM waves through weak turbulence (fT = 100Hz). (a) 3D Gaussian beam output (w0 ≈ 80mm); (b) cross-section of the output Gaussian intensity; (c) 2D output Gaussian profile; and (d) recovered signal.
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Again, from the Figs.6.6 and 6.7, we observe that as the turbulence frequency increases, the Gaussian profile nominally broadened according to the Kirchhoff-Fresnel diffraction model begins to exhibit a small splitting in the transverse direction (Fig.6.7(c)).
Moreover, the recovered signal likewise begins to get more distorted (Fig.6.7 (d)). This directly demonstrates the damaging effect of turbulence (with higher mean frequency) on a modulated EM wave even when the turbulence is relatively weak.
6.3.3 Propagation through strong turbulence
In this section, we examine the EM wave propagation for the case where the turbulence is made several orders of magnitude stronger. Accordingly, we choose
2 -13 -2/3 r0 = 0.01 mm with 퐶푛 =1.067x10 m .
a) Propagation through strong turbulence with mean frequency fT = 20 Hz
In this case, we propagate the modulated EM waves along the propagation path with
(mean) turbulence frequency 20 Hz and the simulation results are shown in Figs.6.8 and
6.9.
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Fig.6.8. Propagation of EM waves through strong turbulence (fT = 20Hz). (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D field before the phase screen; and (d) 3D field after the phase screen.
Fig.6.9. Propagation of EM waves through strong turbulence (fT = 20Hz). (a) 3D Gaussian beam output (w0≈ 80mm); (b) cross-section of the output Gaussian intensity; (c) random phase screen distribution profile; (d) 2D output Gaussian profile; and (e) recovered signal.
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From Fig.6.8, we now notice the impact of the random phase screen (representing strong turbulence) on the field profile before and after traversal across the screen. As seen in
Fig.6.8(d), the profile has undergone some amplitude distortion around the peak. In
Fig.6.9, the EM profile undergoes even more radial splitting (Fig. (6.9d)). Another interesting feature to be noted is that following traversal through turbulence (whereby a peak distortion had occurred), the subsequent diffraction broadening has likely
“smoothened” out the peak distortion features immediately following the turbulence region (see Fig.6.9(a)). Also, the recovered signal suffers even greater amplitude distortion compared with the weak turbulence under the same turbulence frequency, as illustrated in Fig.6.9(e). b) Propagation through strong turbulence with mean frequency fT = 50 Hz
In the next series of simulation results, the mean turbulence frequency is increased to 50
Hz. The simulation results are shown in Figs.6.10 and 6.11.
Fig.6.10. Propagation of EM waves through strong turbulence (fT = 50Hz ). (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D field before the phase screen; and (d) 3D field after the phase screen.
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Fig.6.11. Propagation of EM waves through strong turbulence (fT = 50Hz). (a)3D Gaussian beam output (w0 ≈ 80mm); (b) cross-section of the output Gaussian intensity; (c) random phase screen distribution profile; (d) 2D output Gaussian profile; and (e) recovered signal.
The profile of the Gaussian beam gets more distorted now following the phase screen compared with the profile before the screen in a manner similar to the 20 Hz case
(Figs.6.10(c) and (d)). In addition, we observe that the received Gaussian once again exhibits smoothening near the peak area; however, there occurs some additional distortion in the radially outward portion of the waveform (Fig.6.11 (a)). Also, the recovered signal distortion and Gaussian profile splitting scenarios indicate progressively greater magnitude when the mean turbulence frequency increases, as shown in
Figs.6.11(d) and (e). c) Propagation through strong turbulence with mean frequency fT = 100 Hz
Here we examine the effect of the propagation of modulated EM wave through strong turbulence with mean frequency 100Hz.
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Fig.6.12. Propagation of EM waves through strong turbulence (fT = 100Hz). (a)3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) 3D field before the phase screen; and (d) 3D field after the phase screen.
Fig.6.13. Propagation of EM waves through strong turbulence (fT = 100Hz). (a) 3D Gaussian beam output; (b) cross-section of the output Gaussian intensity; (c) random phase screen distribution profile; (d) 2D output Gaussian profile; and (e) recovered signal.
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The significance of the damaging effects of strong phase turbulence on the propagating
Gaussian beam is self-evident from Figs.6.12 and 6.13. Thus, there is even greater peak distortion following passage across the phase screen (Fig.6.12(d)); similarly the received
Gaussian beam now is split into spatially separated humps with peak smoothening and radial distortion (Fig.6.13(a) and (d)); finally, the sinc-signal waveform is distorted almost beyond recognition and is essentially pure amplitude noise (Fig.6.13(e)).
6.4 Spectral approach to encrypted chaotic wave propagation through
turbulence using SVEA and Fourier transforms
In this section, we propagate a modulated chaotic wave through atmospheric turbulence. The AM modulation using chaos frequency may be expressed as [12,13]:
푆푐ℎ(푡) = 퐴푐[1 + 푚푠(푡)] 푐표푠 휔푐ℎ 푡 , (6.7) where m is the modulation index, and 휔ch is an equivalent chaos frequency. It has been verified that in an A-O Bragg cell with first-order feedback, the time-dependent chaos wave generated is approximately amplitude modulated when the signal wave is applied via the RF (sound) input [12].
We note that the chaotic waveform in eq. (6.7) is essentially RF in nature, and manifests itself in the feedback loop as a current in the PD [12]. However, the chaos wave in turn rides on the optical carrier at the output of the Bragg cell (here assumed to be in first order), and is therefore manifested as envelope modulation of the optical carrier.
The modulated optical wave at the output of the Bragg cell may therefore be approximately written as (assuming AM/envelope modulation):
퐸퐴푀(푥, 푦, 푧, 푡) = 퐸표(푥, 푦, 푧) [1 + 푚̃ 푆푐ℎ(푡)] cos(휔표푡 − 푘푧) , (6.8)
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where 푚̃ is the optical modulation index, 휔o is the optical frequency, 퐸표(푥, 푦, 푧) is the spatial profile of the optical wave as discussed earlier, and k in the unbounded wavenumber in the medium of propagation.
By substituting eqs.6.2 and 6.7 in 6.8, we get (assuming Gaussian optical profile):
−(푥2+푦2) 푤 퐸 (푥, 푦, 푧, 푡) = 퐴 표 푒 푤2(푧) {1 + 퐴 푚̃ [1 + 푚푠(푡)] cos 휔 푡} cos(휔 푡 − 푘푧) 퐴푀 푤(푧) 푐 푐ℎ 표 . (6.9)
As mentioned, we have here two levels of AM modulation. We will follow the same process by propagating the chaos-modulated optical carrier along the propagation distance from the transmitter to the receiver, and include the random phase screen located mid-way when incorporating turbulence. Also, we will use SVEA approximation because the optical carrier is orders of magnitude faster than either the chaotic or the signal waveforms. As described, a sinc-type message waveform is therefore embedded onto the chaotic carrier at the output of a Bragg cell (not shown here; see ref. [12] for details).
The resulting numerical data for the encrypted chaos wave is used to amplitude-modulate the optical carrier. It must be noted that for the case where information is used to directly modulate the optical carrier (as described in section 6.2), photo detection automatically transfers the intensity corresponding to the envelope, and the optical carrier is eliminated.
Thereafter, signal recovery involves simple electronic processing. For the case of a chaotic envelope riding the optical carrier, however, photo detection will at first result in a scaled version of the modulated chaos to be detected via the photocurrent. Eventually, the heterodyne receiver strategy with appropriate cut-off frequency may be used to retrieve the baseband (message) signal.
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6.5 Numerical simulations, results and interpretations
6.5.1 A uniform (non-turbulent) propagation prototype
In this case, we examine the propagation of chaotic EM waves through a uniform medium. Normally, transmission and recovery of time-modulated EM wave through uniform or homogeneous media is fairly standard and well-established, and does not require any examination as such. However, the purpose of including this propagation problem prior to introducing turbulence is three-fold. First, the time-modulated signal is being propagated over a homogenous region wherein we incorporate the diffractive effects along the path. Since diffraction is essentially a spatial phenomenon, our interest is to ascertain the effect of spatial diffraction on the time signal carried within the modulated chaotic carrier. Second, we wish to determine whether standard demodulation strategies will enable extraction of the message from the photo-detector response at the receiver. Third, the recovery in this case relates to a message carried within a chaotic wave; retrieval of the message therefore needs special consideration vis-à-vis the properties of the chaos wave as opposed to a standard sinusoidal carrier. The simulation results are shown in the following figures.
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(d) (e) (f)
Fig.6.14. Propagation of modulated chaotic waves through a uniform medium. (a) 3D Gaussian beam input (w0= 30mm); (b) modulating signal; (c) modulated signal; (d) 3D Gaussian beam output; (e) 2D output Gaussian profile; and (f) recovered signal.
From Fig.6.14, we observe that the profile of the EM beam is broadened due to diffraction during passage through the (homogenous) medium (Fig.6.14 (a) and (d)).
Also, Fig.6.14 (f) shows that the received signal is recovered without any distortion using a simple heterodyne receiver used to filter out the chaos frequency and its harmonics in the photocurrent, thereby enabling recovery of the message signal [12].
6.5.2 Chaotic propagation through weak turbulence with mean frequency fT = 50 Hz
Here, we propagate the chaotic EM wave through narrow and weak turbulence with mean frequency of 50 Hz. For weak turbulence, r0 =10 mm with
2 -18 -2/3 C푛=1.067x10 m ; the corresponding simulation results are illustrated in Fig.6.15.
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(d) (e) (f) Fig.6.15. Propagation of modulated chaotic waves through weak turbulence ( fT = 50Hz). (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) modulated signal; (d) 3D Gaussian beam output; (e) 2D output Gaussian profile; and (f) recovered signal.
In the presence of turbulence (here weak), we find that the diffracted beam profile
(Fig.6.15(d) and (e)) begins to exhibit radial splitting (rather small for the weak case; see
Fig. 6.15(e)). As may be recalled, even under weak turbulence, transmission of a directly modulated EM wave results in significant distortion of the recovered signal (see
Fig.6.5(d)). When the message is embedded in a chaotic carrier, it is seen that even after passage though relatively weak chaos, the message is recovered with noticeably higher integrity (see Fig.6.15(f)). Although this finding is relatively limited in scope, it nevertheless holds out the promise that a message wave may be secured and protected from a turbulent environment by packaging it inside a chaos wave via the RF encryption methodology which has been developed by our group [12,13,60]. As is shown later, when the turbulence becomes stronger, the recovery of the message from the chaos wave
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is subject to some degree of distortion, and is not entirely distortion-free. However, the results indicate that packaging a message in a chaos wave does still offer some immunity from even stronger turbulence compared with non-chaotic transmission.
6.5.3 Chaotic propagation through strong turbulence with mean frequency
fT =50 Hz
2 -13 -2/3 In the strong turbulence, we choose r0 = 0.01 mm with C푛=1.067x10 m .
(d) (e) (f) Fig.6.16. Propagation of modulated chaotic waves through strong turbulence ( fT = 50Hz). (a) 3D Gaussian beam input (w0 = 30mm); (b) modulating signal; (c) modulated signal; (d) 3D Gaussian beam output; (e) 2D output Gaussian profile; and (f) recovered signal.
Fig.6.16 shows the simulation results for propagation of the modulated chaotic wave over a medium under strong turbulence. It is evident that under strong turbulence, the diffracted profiled is even further split in the radial direction (see Fig.6.16(e)). For this case, the signal recovered using the heterodyne strategy is noticeably distorted
(Fig.6.16(f)); however, the degree of distortion is considerably lower than that obtained for non-chaotic propagation (see Fig.6.11(e) for comparison). The above further
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reinforces the observation that chaotic encryption likely immunizes a message waveform from the damaging effects of turbulence.
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CHAPTER 7
CONCLUDING REMARKS AND FUTURE WORK
In this work, we have studied the influence of atmospheric turbulence on EM beam propagation over uniform and turbulent atmospheres. A spatial refractive index spectrum (MVKS) was used to represent the random behavior of the atmospheric turbulence via one or more random phase screen(s) representing narrow and extended turbulence. The BPM involving Fresnel-Kirchhoff diffraction combined with transmission through a thin (single or multiple) random phase screen(s) was applied. The numerical split-step propagational analysis approach is followed in order to track the evolution of the field along the propagation path by using the diffraction integral within the (homogenous) medium and the random phase fluctuations within the turbulent
(inhomogeneous) medium.
In the research reported, we have studied the far-field diffraction properties of uniform and profiled EM beams transmitted through binary and random-phase apertures undergoing diffractive spreading. A comparison between the BPM and direct Fresnel-
Kirchhoff diffraction integral confirmed the validity of the BPM algorithm. This validated use of this approach when applied to profiled beam propagation through random phase turbulence. To establish this, diffraction patterns have been studied for uniform, Gaussian and Bessel beams incident upon a planar aperture. Numerical plots compared with the analytic plots have yielded essentially compatible results. Following
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diffractive propagation, the diffraction patterns in the far field have been obtained and plotted along the transverse horizontal and vertical axes. The random phase distribution have been derived using the MVKS model and the Fried parameter of turbulence, and subsequently used in the numerical simulations using the SSA strategy.
For profiled Gaussian beam propagating through turbulent regions, numerical simulations were presented for a narrow region of turbulence. Diffracted output fields were derived for different system parameters consisting of turbulence strength, Gaussian beam waist and the propagation distance. In addition, propagation of profiled Gaussian beams through extended turbulence in relation to several parameters (varied individually) were examined. In order to replicate extended turbulence more accurately, the distance between the two adjacent random phase screens should be very small; however, using infinitesimally small incremental distances (Δz) for the computation leads to higher complexity and considerably more processing time. To avoid unacceptably large computation times, a tradeoff between accuracy and processing time has been applied while ensuring reasonable results. The SI has been determined for the propagation of the
Gaussian beam through both weak and strong turbulence. The FV has been calculated for the propagation of adjacent Gaussian beams through both weak and strong turbulence.
Overall, the effect of strong turbulence on the profile and energy of the propagated beam
(usually leading to serious distortion and loss of energy) is rather evident from all the simulations; on the other hand, propagation over longer turbulence regimes under moderate levels of turbulence appears to primarily affect the beam widths more than the profiles themselves. On balance, the investigation allows one to adequately track the
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random amplitudes and phases of profiled EM waves at an arbitrary output plane by use of the split-step technique and the MVKS modeling.
Due to some limitations in extracting the transmitted message from the numerically derived cross correlation between the turbulence and chaos, an alternative approach was used to examine the propagation of non-chaotic and chaotic waves through both non- turbulent and phase-turbulent atmospheres. This was based on standard modulation theory. The propagation of an optical carrier (with a non-chaotic or chaotic envelope) was investigated using the SVEA approximation. The propagation methodology including the use of (spatial) Fourier transforms and the Kirchhoff-Fresnel integral was implemented on the basis of complex phasors representing the scalar fields. The propagation of a modulated EM wave through random phase turbulence was examined under different atmospheric conditions, viz., uniform (non-turbulent) and turbulent (weak or strong), the latter also under different time fluctuations of the turbulence. In the case of direct modulation of the EM carrier (without chaotic envelope), the received signal following photodetection was obtained using standard electronics. When encrypted chaos was incorporated into the optical carrier, a heterodyne receiver strategy with appropriate cut-off frequency was used to extract the received (message) signal from the chaotic carrier in the photocurrent. For propagation in a uniform atmosphere, results indicated received signals which were scaled versions of the transmitted signal under direct, non- chaotic modulation. When turbulence was introduced, and its average temporal frequency was increased (assuming 20, 50 and 100 Hz), the EM output beam began to experience radial splitting, and the recovered signal became increasingly distorted. Moreover, the simulations showed that the recovered signal underwent greater distortion when the
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turbulence was made stronger. Thus, non-chaotic signals riding an optical carrier suffered greater distortion under turbulence of both higher average temporal frequency and higher strength. More significantly, when chaos was introduced as an encrypted carrier, it was found that the recovered signal suffered considerably less distortion under similar turbulence and medium conditions. These results indicate that embedding a message signal inside a chaos wave prior to transmission via an EM wave over a (phase) turbulent medium may reduce the degree of signal distortion which otherwise occurs under non-chaotic transmission.
As of spring 2016, this work has resulted in four conference presentations and corresponding proceedings publications, two topical digests and one journal paper.
Another journal manuscript is currently under review. There are many possibilities for extending the current work:
The propagation of EM wave through thin random phase screen(s) (transverse (x,y)
plane) can be extended to thick random phase screen (x,y,z) in order to represent more
realistic atmospheric turbulence.
A different spatial power spectrum model representing random phase turbulence with
a high wave number spectral bump may be used instead of MVKS.
2 In this research, 퐶푛 is height independent (h) (height from the earth surface) because
we consider horizontal propagation; in reality, this structure function is a function of
h, and therefore beam propagation angles would need to be incorporated into the
analyses in order to model the h-dependent structure parameter.
Extraction of the transmitted signal from the cross correlation obtained via the use of
the transfer function formalism is a task of ongoing interest.
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Modulating the chaos with sound, still images, video signals or PCM signals would
make the application more realistic and potentially useful.
Alternative space and time-dependent turbulence modeling may also be an area of
further study.
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156
APPENDIX A
ANALYTICAL PLOTS OF DIFFRACTION PATTERN
In this appendix, we introduce the analytic plots of field diffraction due to different aperture configurations to compare with the simulation results provided in chapter 3.
1. Very narrow horizontal double-slit aperture
Fig. A1. Very narrow horizontal double-slit geometry.
The aperture amplitude transmittance function is given by:
푥0 푔(푥 , 푦 ) = 푟푒푐푡( )[훿(푦 − 푏) + 훿(푦 + 푏)]. 0 0 2푎 0 0
The scalar field of the far-field is proportional to Fourier transform of aperture function:
157
푈(푥푖, 푦푖) ∝ 푠푖푛푐(2푎푓푥푖)푐표푠(2휋푏푓푦푖) ,
where 푥푖 and 푦푖 are related to the spatial frequencies 푓푥푖 and 푓푦푖 as: 푥푖 = 푓푥푖휆푧푖 ,
푦푖 = 푓푦푖휆푧푖 .
The field intensity is given by:
2 2 퐼(푥푖, 푦푖) ∝ 푠푖푛푐 (2푎푓푥푖)푐표푠 (2휋푏푓푦푖)
1 1 ) ) 2 0.8 2 0.8
0.6 0.6
0.4 0.4
0.2 0.2 Normalized Intensity (w m Intensity / Normalized Normalized Intensity (w m / Intensity Normalized
-100 -50 0 50 100 -100 -50 0 50 100 x [m] y [m] Fig.A2. Analytic plots of the far-field due to very narrow horizontal double slit.
2. Circular aperture
The aperture amplitude transmittance function is given by:
휌0 1 , 휌0 ≤ 푎 푈(휌 ) = 푐푖푟푐 ( ) = { , 0 푎 0 , 푒푙푠푒푤ℎ푒푟푒
2 2 where 휌0 = √푥표 + 푦표 .
The scalar field of the far field diffraction is proportional to Fourier Bessel transform of aperture function:
퐽1(2휋푎휌푖) 푈(휌푖) ∝ | | , 2휋푎휌푖
2 2 where 휌푖 = √푥푖 + 푦푖 , and the intensity:
158
2 J1(2πaρi) I(ρ ) ∝ | | , i 2πaρi which is the familiar Airy-disk function.
1 1 ) ) 2 2 0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2 Normalized Intensity (w m Intensity / Normalized Normalized Intensity (w m / Intensity Normalized
0 0 -100 -50 0 50 100 -100 -50 0 50 100 x [m] y [m] Fig.A3. Analytic plots of the far-field due to circular aperture.
3. Annular aperture with radii a and b (b > a)
Fig.A4. Annular aperture with radii a and b.
The aperture amplitude transmittance function is given by:
휌0 휌0 푈(휌 ) = 푐푖푟푐 ( ) − 푐푖푟푐 ( ) , 0 푏 푎
2 2 where 휌0 = √푥표 + 푦표 .
The scalar field of the far field is proportional to Fourier Bessel transform of aperture function:
퐽 퐽 1(2휋푏휌푖) 1(2휋푎휌푖) 푈(휌푖) ∝ | | − | | , 2휋푏휌푖 2휋푎휌푖 and the intensity:
159
2 퐽1(2휋푏휌푖) 퐽1(2휋푎휌푖) 퐼(휌푖) ∝ [| | − | |] 2휋푏휌푖 2휋푎휌푖
1 1 ) ) 2 0.8 2 0.8
0.6 0.6
0.4 0.4
0.2 0.2 Normalized Intensity (w m Intensity / Normalized (w m Intensity / Normalized
0 0 -100 -50 0 50 100 -100 -50 0 50 100 x [m] y [m] Fig.A5. Analytic plots of the far-field due to annular aperture.
4. Infinitely narrow annular aperture
Fig.A6. Infinitely narrow annular aperture. The aperture amplitude transmittance function is given by:
휌0 1 , 휌 = 푎 푈(휌 ) = 훿 ( ) = { 0 , 0 푎 0 , 푒푙푠푒푤ℎ푒푟푒
2 2 where 휌0 = √푥표 + 푦표 .
The scalar field of the far field is proportional to Fourier Bessel transform of aperture function:
푈(휌푖) ∝ 퐽0(2휋푎휌푖) ,
160
and the intensity:
2 퐼(휌푖) ∝ |퐽0(2휋푎휌푖)| .
1 1 ) ) 2 2 0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2 Normalized Intensity (w m Intensity / Normalized Normalized Intensity (w m / Intensity Normalized 0 0 -100 -50 0 50 100 -100 -50 0 50 100 x [m] y [m] Fig.A7. Analytic plots of the far-field due to infinitely narrow annular aperture.
5. Thin sinusoidal amplitude grating aperture
Fig.A8. Thin sinusoidal amplitude grating aperture.
The aperture amplitude transmittance function is given by:
1 푚 푥0 푦0 푈(푥 , 푦 ) = [ + 푐표푠(2휋푓 푥 )] 푟푒푐푡( )푟푒푐푡( ) . 0 0 2 2 0 0 푎 푎
161
This aperture is periodic in the x-direction but not the y-direction, and is bounded by a square of width a. The parameter m denotes the peak-to-peak amplitude change (m ≤ 1), and f0 is the spatial frequency of the grating.
As in the previous cases; the scalar field of the far field diffraction is proportional to
Fourier transform of aperture function:
푥0 푦0 F {푟푒푐푡( )푟푒푐푡( )} ∝ 푠푖푛푐(푎푓 )푠푖푛푐(푎푓 ) . (1) 푎 푎 푥푖 푦푖
1 푚 1 푚 푗2휋푓 푥 −푗2휋푓 푥 F { + 푐표푠 (2휋푓 푥 )} = F { + (푒 0 0 + 푒 0 0 )} 2 2 0 0 2 4
1 푚 푚 = 훿(푓 , 푓 ) + 훿(푓 + 푓 , 푓 ) + 훿(푓 − 푓 , 푓 ) (2) 2 푥푖 푦푖 4 푥푖 0 푦푖 4 푥푖 0 푦푖
Convolving eq. (1) with eq. (2) using the shifting property of the δ function, we get:
푈(푥푖, 푦푖) =F {푈(푥표, 푦표)} 훼 푠푖푛푐(푎푓푦푖){푠푖푛푐(푎푓푥푖) +
푚 푚 푠푖푛푐(푎(푓 + 푓 )) + 푠푖푛푐(푎(푓 − 푓 ))} 2 푥푖 0 2 푥푖 0 and the intensity: 푚2 퐼(푥 , 푦 ) ∝ 푠푖푛푐2(푎푓 ) {푠푖푛푐2(푎푓 ) + 푠푖푛푐2(푎(푓 + 푓 )) 푖 푖 푦푖 푥푖 4 푥푖 0 푚2 + 푠푖푛푐2(푎(푓 − 푓 ))} 4 푥푖 0
1 1 ) ) 2 2 0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2 Normalized Intensity (w m Intensity / Normalized Normalized Intensity (w m / Intensity Normalized
-100 -50 0 50 100 -100 -50 0 50 100 x [m] y [m] Fig. A9. Analytic plots of the far-field due to thin sinusoidal amplitude grating.
162
APPENDIX B
PUBLICATIONS RESULTING FROM THIS WORK
[1] M.R. Chatterjee and F.H.A. Mohamed, “ Investigation of chaotic profiled beam
propagation through a turbulent layer and output temporal statistics using a
modified von Karman phase screen,” Proc. SPIE 8971, 897102-1-16 (2014).
[2] M. R. Chatterjee, and F. H. A. Mohamed , “A Numerical Examination of the
Diffraction Properties of Profiled Beam Transmission through Binary Apertures
and Random Phase Screens Using Fresnel-Kirchhoff Diffraction Theory”,
NAECON, Dayton, (2014).
[3] M.R. Chatterjee and F.H.A. Mohamed, “Split-step approach to electromagnetic
propagation turbulence using the modified von Karman spectrum and planar
apertures through atmospheric turbulence,” Opt. Eng., (2014).
[4] M.R. Chatterjee and F.H.A. Mohamed, “ Modeling of Power Spectral Density of
Modified von Karman Atmospheric Phase Turbulence and Acousto-Optic Chaos
Using Scattered Intensity Profiles Over Discrete Time Intervals,” Proc. SPIE 9224,
922404-1 (2014).
163
[5] M.R. Chatterjee and F.H.A. Mohamed, “A Transfer Function Based Frequency
Model for Propagation of a Chaos Wave through Modified von Karman Turbulence
under Various Chaos and Turbulence Conditions”, OSA Topical Meeting on
Imaging and Applied Optics, (Arlington, VA, 2015).
[6] M.R. Chatterjee and F.H.A. Mohamed, “ Spectral and Performance Analysis for
the Propagation and Retrieval of Signals from Modulated Chaos Waves
Transmitted through Modified von Karman Turbulence”, OSA Annual Meeting
(FiO/LS),(San Jose, CA, 2015).
[7] M.R. Chatterjee and F.H.A. Mohamed,“ Diffractive propagation and recovery of
modulated (including chaotic) electromagnetic waves through uniform atmosphere
and modified von Karman phase turbulence”, Proc. SPIE 9833, 98330F-1-16
(2016).
164