Optical Wave Propagation In Discrete Waveguide Arrays

185 Pages·2017·13.4 MB·English

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Electronic Theses and Dissertations, 2004-2019
2005
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Jared Hudock
University of Central Florida
Part of the Electromagnetics and Photonics Commons, and the Optics Commons
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https://stars.library.ucf.edu/etd/569
OPTICAL WAVE PROPAGATION IN DISCRETE
WAVEGUIDE ARRAYS
by
Jared Hudock
B.S. Lehigh University, 2000
A dissertation submitted in partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in the College of Optics and Photonics
at the University of Central Florida
Orlando, Florida
Fall Term
2005
Major Professor: Demetrios N. Christodoulides
(cid:13)c 2005 Jared Hudock
ii
ABSTRACT
The propagation dynamics of light in optical waveguide arrays is characteristic of that en-
countered in discrete systems. As a result, it is possible to engineer the diffraction properties
of such structures, which leads to the ability to control the flow of light in ways that are
impossible in continuous media.
In this work, a detailed theoretical investigation of both linear and nonlinear optical wave
propagation in one- and two-dimensional waveguide lattices is presented.
The ability to completely overcome the effects of discrete diffraction through the mutual
trapping of two orthogonally polarized coherent beams interacting in Kerr nonlinear arrays
of birefringent waveguides is discussed. The existence and stability of such highly localized
vector discrete solitons is analyzed and compared to similar scenarios in a single birefringent
waveguide. This mutual trapping is also shown to occur within the first few waveguides of
a semi-infinite array leading to the existence of vector discrete surface waves.
Interfaces between two detuned semi-infinite waveguide arrays or waveguide array hetero-
junctions and their possible applications are also considered. It is shown that the detuning
between the two arrays shifts the dispersion relation of one array with respect to the other.
Consequently, these systems provide spatial filtering functions that may prove useful in
future all-optical networks. In addition by exploiting the unique diffraction properties of
discrete arrays, diffraction compensation can be achieved in a way analogous to dispersion
compensation in dispersion managed optical fiber systems.
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Finally, it is demonstrated that both the linear (diffraction) and nonlinear dynamics
of two-dimensional waveguide arrays are significantly more complex and considerably more
versatile than their one-dimensional counterparts. As is the case in one-dimensional arrays,
the discrete diffraction properties of these two-dimensional lattices can be effectively altered
depending on the propagation Bloch k-vector within the first Brillouin zone. In general, this
diffraction behavior is anisotropic and as a result, allows the existence of a new class of dis-
creteellipticsolitonsinthenonlinearregime. Moreover, sucharrayssupporttwo-dimensional
vector soliton states, and their existence and stability are also thoroughly explored in this
work.
iv
Dedicated to my family, friends, and all those with a dream
v
ACKNOWLEDGMENTS
Accomplishments such as earning a Ph.D. degree are not achieved on ones own. It takes
the help, love and support of numerous people to maintain the discipline to see it all the
way through. I was no different, and I would like to take this opportunity express my most
heartfelt thanks to those who made this possible.
Over the past five years, I have had the honor and privilege of working closely with
Professor Demetrios Chistodoulides. I would like to thank him for this opportunity and for
standing by me even when my focus drifted from where it needed to be. Not only has he
taught me many things about optics, but about life as well. I am extremely fortunate to
have him as an adviser, a mentor, and most importantly a friend.
IwouldalsoliketothankProfessorGeorgeStegemanforextendingtometheopportunity
to work with him, his students, and in his labs as well as teaching me some of the more
practical sides of optics.
I would also like to thank all of the people from the University of Central Florida, par-
ticularly those from the College of Optics and Photonics/CREOL, for providing such a fine
institution where optics research can be readily undertaken.
During my studies, I also had the pleasure of meeting and making many new friends
whom I will never forget: Nikos, Alexandra, Eugenia, Kostas, Erdem, Bobby, and Joachim.
Good luck to you in all your life’s pursuits.
I, of course, would like to thank my family and friends, especially my mom. Her unwa-
vi
vering love and devotion for her children lead her to sacrifice many things in her life so that
we wouldn’t have to. For this, I will be forever in her dept. Thanks Mom for always being
there and always allowing me to dream... you’re my hero!
Last, but certainly not least, I would like express my gratitude and love to my wife,
Lindsay. Her unconditional love, support, and encouragement gives me the strength and will
to never give up and to always complete the journey. She has taught me to cherish life’s
broken road and keeps me grounded while still allowing me to fly. She is, and always will
be, my roots and wings!
vii
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
CHAPTER 1: INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
CHAPTER 2: OPTICAL WAVE PROPAGATION IN DIELECTRIC WAVEGUIDE
ARRAYS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Optical Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 The Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.3 Vector Phasors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.4 The Poynting Vector Theorem . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Wave Propagation in Linear Media . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Propagation in a Linear Waveguide . . . . . . . . . . . . . . . . . . . 16
2.3 Propagation in Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Propagation in a Nonlinear Waveguide . . . . . . . . . . . . . . . . . 24
2.4 Coupled Mode Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Directional Coupler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Linear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
viii
2.5.2 Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Waveguide Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.1 Linear Diffraction Properties . . . . . . . . . . . . . . . . . . . . . . . 39
2.6.2 Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 44
CHAPTER3: DISCRETEVECTORSOLITONSINAlGaAsWAVEGUIDEARRAYS 49
3.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.1 Vector Beam Instabilities in a Single Waveguide . . . . . . . . . . . . 52
3.2 Discrete Vector Soliton Solutions . . . . . . . . . . . . . . . . . . . . . . . . 55
CHAPTER 4: DISCRETE VECTOR SURFACE WAVES . . . . . . . . . . . . . . 66
4.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Discrete Vector Surface Wave Solutions . . . . . . . . . . . . . . . . . . . . . 68
CHAPTER 5: WAVEGUIDE ARRAY HETERO-JUNCTIONS . . . . . . . . . . . 77
5.1 Introduction to Array Hetero-junctions . . . . . . . . . . . . . . . . . . . . . 77
5.2 Array Hetero-junctions as Spatial Filters . . . . . . . . . . . . . . . . . . . . 80
5.2.1 Band Gap Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.2.2 Reflection and Transmission Coefficients . . . . . . . . . . . . . . . . 82
5.3 Array Hetero-junctions as Diffraction Compensators . . . . . . . . . . . . . . 91
5.3.1 The Transfer Function of a 1D Array . . . . . . . . . . . . . . . . . . 93
5.3.2 Diffraction Compensation . . . . . . . . . . . . . . . . . . . . . . . . 95
5.4 Hybrid Discrete Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.4.1 Hybrid Discrete Soliton Solutions and Stability . . . . . . . . . . . . 98
ix