modeling and analysis of photonic crystal waveguides
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Modeling and Analysis of Photonic Crystal Waveguides
Mhd. Rachad Albandakji
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Electrical Engineering
Advisory Committee:
Ahmad Safaai-Jazi : Chair
Roger Stolen : Member
Sedki Riad : Member
Ioannis Besieris : Member
Ira Jacobs : Member
Randy Heflin : Member
April 27, 2006
Blacksburg, Virginia
Keywords: Photonic Crystal Fibers, Tapered Fibers, Fresnel Fibers
c© Copyright 2006, M. R. Albandakji
Modeling and Analysis of Photonic Crystal Waveguides
Mhd. Rachad Albandakji
Abstract
In this work, we investigate several aspects of photonic crystal waveguides through
modeling and simulation. We introduce a one-dimensional model for two-dimensional
photonic crystal fibers (PCFs), analyze tapered PCFs, analyze planar photonic crystal
waveguides and one-dimensional PCFs with infinite periodic cladding, and investigate
transmission properties of a novel type of fiber, referred to as Fresnel fiber.
A simple, fast, and efficient one-dimensional model is proposed. It is shown that the
model is capable of predicting the normalized propagation constant, group-velocity
dispersion, effective area, and leakage loss for PCFs of hexagonal lattice structure
with a reasonable degree of accuracy when compared to published results that are
based on numerical techniques.
Using the proposed model, we investigate tapered PCFs by approximating the tapered
section as a series of uniform sections along the axial direction. We show that the
total field inside the tapered section of the PCF can be evaluated as a superposition
of local normal modes that are coupled among each other. Several factors affecting
the adiabaticity of tapered PCFs, such as taper length, taper shape, and number of
air hole rings are investigated. Adiabaticity of tapered PCFs is also examined.
A new type of fiber structure, referred to as Fresnel fiber, is introduced. This fiber can
be designed to have attractive transmission properties. We present carefully designed
Fresnel fiber structures that provide shifted or flattened dispersion characteristics,
large negative dispersion, or large or small effective area, making them very attractive
for applications in fiber-optic communication systems.
To examine the true photonic crystal modes, for which the guidance mechanism is
not based on total internal reflection, photonic crystal planar waveguides with infinite
periodic cladding are studied. Attention will be focused on analytical solutions to
the ideal one-dimensional planar photonic crystal waveguides that consist of infinite
number of cladding layers based on an impedance approach. We show that these
solutions allow one to distinguish clearly between light guidance due to total internal
reflection and light guidance due to the photonic crystal effect.
The analysis of one-dimensional PCFs with infinite periodic cladding is carried out
in conjunction with an equivalent T-circuits method to model the rings that are close
to the core of the fiber. Then, at sufficiently large distance from the core, the rest of
the cladding rings are approximated by planar layers. This approach can successfully
estimate the propagation constants and fields for true photonic crystal modes in both
solid-core and hollow-core PCFs with a high accuracy.
iii
Dedication
To my mom and dad who are my source of support and inspiration...
To my lovely sisters who are my source of love and passion...
May Allah bless you all...
iv
Acknowledgments
First of all, my praises and thankfulness are for Allah for his guidance and blessing
with a family that has never stopped giving me love and encouragement.
It gives me a great honor to thank my academic advisor Dr. Ahmad Safaai-Jazi for
his endless support and advice during my research years. Also I would like to thank
Dr. Roger Stolen for his contributions and suggestions in my research. I am also
very greatful to my other advisory committee members who gave me guidance and
precious comments on my research.
I would like to especially thank Dr. Ali Nayfeh and Dr. Ziad Masoud from the Engi-
neering Science and Mechanics Department for their support.
Also, I would like to thank my friends for the wonderful time and brotherhood we lived
together during our studies at Virginia Tech. I would like to mention Mohammad
Daqaq, Sameer Arabasi, Saifuddin Rayyan, Qasem Al-Zoubi, Fadi Mantash, and
Basel Al-Sultan, they were really the best company. I am also very greatful to my
american mother Sonja Murrell for her unlimited care, support, and kindness.
v
Table of Contents
Abstract ii
Dedication iv
Acknowledgments v
Table of Contents vi
List of Figures x
List of Tables xiv
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Photonic Crystals: An Overview . . . . . . . . . . . . . . . . . . . . . 3
1.3 Photonic Crystal Fibers: An Overview . . . . . . . . . . . . . . . . . 5
1.3.1 Solid-Core Photonic Crystal Fibers . . . . . . . . . . . . . . . 7
1.3.2 Hollow-Core Photonic Crystal Fibers . . . . . . . . . . . . . . 9
vi
1.4 Recent Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Scope of the Proposed Research . . . . . . . . . . . . . . . . . . . . . 15
2 Analysis of One-Dimensional Photonic Crystal Fibers 18
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2 Field Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Solution of the Wave Equation . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Boundary Conditions and Dispersion Relation . . . . . . . . . . . . . 25
3 Analysis of Two-Dimensional Photonic Crystal Fibers 28
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 The Proposed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Model Testing, Comparison, and Accuracy . . . . . . . . . . . . . . . 31
3.3.1 Normalized Propagation Constant . . . . . . . . . . . . . . . . 31
3.3.2 Group-Velocity Dispersion (GVD) . . . . . . . . . . . . . . . . 32
3.3.3 Effective Area . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.4 Leakage Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Analysis of Tapered Photonic Crystal Fibers 40
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Analysis of Tapered Photonic Crystal Fibers . . . . . . . . . . . . . . 41
4.3 Adiabaticity of Tapered Photonic Crystal Fibers . . . . . . . . . . . . 44
4.3.1 Taper Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 Taper Length . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3.3 Number of Air Hole Rings . . . . . . . . . . . . . . . . . . . . 53
5 Fresnel Fibers 56
vii
5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.2 Analysis of Fresnel Fibers . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Special Fresnel Fiber Designs . . . . . . . . . . . . . . . . . . . . . . 58
5.3.1 Dispersion-Shifted Fibers . . . . . . . . . . . . . . . . . . . . . 58
5.3.2 Dispersion-Flattened Fibers . . . . . . . . . . . . . . . . . . . 59
5.3.3 Dispersion Compensating Fibers . . . . . . . . . . . . . . . . . 63
6 Analysis of Planar Photonic Crystal Waveguides 65
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 Analysis of PPCW with Finite Number of Cladding Layers . . . . . . 66
6.3 Analysis of PPCW with Infinite Number of Periodic Cladding Layers 70
6.4 Comparison between PPCWs with Finite and Infinite Number of CladdingLayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.5 True Photonic Crystal Modes in PPCWs . . . . . . . . . . . . . . . . 76
6.5.1 High-Index Core PPCWs . . . . . . . . . . . . . . . . . . . . . 77
6.5.2 Low-Index Core PPCWs . . . . . . . . . . . . . . . . . . . . . 81
7 Analysis of Ideal One-Dimensional Photonic Crystal Fibers 85
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3.1 Solid-Core PCF . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.3.2 Hollow-Core PCF . . . . . . . . . . . . . . . . . . . . . . . . . 93
8 Conclusions and Directions for Future Work 102
8.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.2 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . . 105
viii
A Matrix Coefficients 107
B Material Constants 109
Bibliography 110
Vita 122
ix
List of Figures
1.1 Examples of 1-D (left), 2-D (center), and 3-D (right) photonic crystals. 4
1.2 Various PCF structures reported in the literature: (a) hexagonal solid-core PCF, (b) cobweb PCF, (c) hexagonal hollow-core PCF, and (d)honeycomb PCF. [Used with permission from [14]]. . . . . . . . . . . 6
1.3 Summary of the techniques used for the analysis of PCFs. . . . . . . 13
2.1 Index profile of a 1-D PCF (ring fiber). . . . . . . . . . . . . . . . . . 19
2.2 Geometry and coordinates for a 1-D PCF. . . . . . . . . . . . . . . . 20
3.1 Transforming 2-D PCF into 1-D PCF. . . . . . . . . . . . . . . . . . 29
3.2 Hexagonal ring in a 2-D PCF. . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Index profile of the suggested PCF model. . . . . . . . . . . . . . . . 30
3.4 Normalized propagation constant as a function of normalized wave-length for different normalized hole diameter: dots are FEM resultsin [33] and solid lines are results from the proposed PCF model. . . . 31
3.5 Contour plots of the percentage error in the normalized propagationconstant using the proposed PCF model when compared to the FEMin [33]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
x
3.6 GVD comparisons: solid line is measured GVD in [69], dashed line isnumerically calculated GVD in [69], and dotted line is predicted GVDusing the proposed PCF model. . . . . . . . . . . . . . . . . . . . . . 33
3.7 GVD comparisons: solid line is measured GVD from [70], dashed lineis predicted GVD using the proposed PCF model, and dotted line ispredicted GVD using multipole method [45]. . . . . . . . . . . . . . . 34
3.8 Waveguide dispersion comparison: solid and dashed lines are from [33]and dotted lines are predicted waveguide dispersion using the proposedPCF model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.9 Normalized effective area comparison: solid lines are from [72] andcircles are from the proposed PCF model. . . . . . . . . . . . . . . . 36
3.10 Leakage loss versus Λ for 4-ring PCFs with different air-filling fractionsat λ = 1.55 μm: solid lines are reported in [74] and dashed lines arepredicted by the proposed PCF model. . . . . . . . . . . . . . . . . . 38
3.11 Leakage loss versus Λ for d/Λ = 0.9 PCFs with different number ofrings at λ = 1.55 μm: solid lines are reported in [74] and dashed linesare predicted by the proposed PCF model. . . . . . . . . . . . . . . . 39
4.1 Cross section of a typical fiber taper. . . . . . . . . . . . . . . . . . . 41
4.2 Modeling of a fiber taper using a series of cascaded uniform sections. 42
4.3 Approximating a finite taper section by a cylindrical structure of uni-form cross section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Three different taper shapes: (a) linear taper, (b) raised cosine taper,and (c) modified exponential taper. . . . . . . . . . . . . . . . . . . . 46
4.5 The power coupled to HE12 mode in different PCF taper shapes. . . . 47
4.6 Local taper length-scale (zt) in a tapered fiber. . . . . . . . . . . . . . 48
4.7 Linear down-tapered fiber showing the taper angle. . . . . . . . . . . 49
4.8 The normalized propagation constants of the first three modes for ataper length of 100 μm at 1.55 μm wavelength. . . . . . . . . . . . . 50
4.9 Variation of coupled power for the first three modes in the linear taperwith a length larger than adiabatic length. . . . . . . . . . . . . . . . 51
4.10 Power density distribution inside the linear taper with a length largerthan adiabatic length. . . . . . . . . . . . . . . . . . . . . . . . . . . 51
xi
4.11 Variation of coupled power for the first three modes in the linear taperwith a length smaller than adiabatic length. . . . . . . . . . . . . . . 52
4.12 Power density distribution inside the linear taper with a length smallerthan adiabatic length. . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.13 Variation of power of the first three modes in a 1-ring PCF taper. 2-and 3-ring PCF tapers have almost the same power variation. . . . . 54
4.14 Power density distribution inside a 1-ring PCF taper. . . . . . . . . . 54
4.15 Power density distribution inside a 2-ring PCF taper. . . . . . . . . . 55
4.16 Power density distribution inside a 3-ring PCF taper. . . . . . . . . . 55
5.1 Index profile of a Fresnel fiber. . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Effective area for the fibers listed in Table 5.1. . . . . . . . . . . . . . 59
5.3 Dispersion-flattened Fresnel fiber no. 11. . . . . . . . . . . . . . . . . 61
5.4 Dispersion-flattened Fresnel fiber no. 12. . . . . . . . . . . . . . . . . 61
5.5 The effect of changing the core radius of Fresnel fiber no. 11 on dis-persion curve. Core radii used are 1.54, 1.57, and 1.6 μm (bottom totop). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.6 The effect of changing the ring area of Fresnel fiber no. 11 on dispersioncurve. Ring areas used are 0.8, 1.0, and 1.2 μm (bottom to top). . . . 62
5.7 Dispersion compensating Fresnel fiber no. 13. . . . . . . . . . . . . . 64
5.8 Dispersion compensating Fresnel fiber no. 14. . . . . . . . . . . . . . 64
6.1 A planar photonic crystal waveguide. . . . . . . . . . . . . . . . . . . 66
6.2 Index profile of a planar photonic crystal waveguide. . . . . . . . . . . 66
6.3 Planar semi-infinite periodic structure with Zin and Zin shown. . . . . 70
6.4 Dispersion curves for TE0 mode for different number of cladding layers.The PPCW parameters are d0 = 1 μm, d1 = 0.5 μm, d2 = 0.5 μm,material 1 is M11, and material 2 is M12. . . . . . . . . . . . . . . . . 76
6.5 Normalized propagation constant for several TE and TM modes of anideal PPCW with d0 = 2 μm, d1 = 1 μm, d2 = 1 μm, material 1 is M1,and Material 2 is M5. Gray area is the region of allowed modes whenthe PPCW has a finite number of cladding layers. . . . . . . . . . . . 78
xii
6.6 Field distributions for TE2 mode at λ = 1.3 μm: (a) Ey, (b) Hx, and(c) Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.7 Field distributions for TE2 mode at λ = 1.55 μm: (a) Ey, (b) Hx, and(c) Hz. In this case, the mode is a true photonic crystal mode withβ < n2 < n1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.8 Normalized propagation constant for several TE and TM modes of anideal PPCW with d0 = 2 μm, d1 = 1 μm, d2 = 1 μm, material 1 isM12, and Material 2 is M11. Gray area is the region of possible modesin practical PPCW if n1 and n2 were interchanged. . . . . . . . . . . 82
6.9 Field distributions for TM1 mode at λ = 1.3 μm: (a) Hy, (b) Ex, and(c) Ez. In this case, the mode is a true photonic crystal mode withβ < n1 < n2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.1 Equivalent transmission-line circuit of an optical fiber. . . . . . . . . 86
7.2 Equivalent circuits for the analysis of 1-D PCF structure with infinitenumber of rings: (a) actual structure and (b) equivalent circuit model. 88
7.3 Periodic coaxial fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.4 Index profile of the discussed periodic coaxial fiber. . . . . . . . . . . 91
7.5 Transverse field distribution for TM01 mode when λ = 4.9261 μm: (a)Hϕ, (b) Er, and (c) Ez. . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7.6 Power density distribution for TM01 mode when λ = 4.9261 μm. . . . 93
7.7 Hollow-core PCF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.8 Transverse field distribution for TE01 mode when k0 = 1.2: (a) Eϕ, (b)Hr, and (c) Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.9 Power density distribution for TE01 mode when k0 = 1.2. . . . . . . . 97
7.10 Transverse field distribution for TM01 mode when k0 = 1.2: (a) Hϕ,(b) Er, and (c) Ez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.11 Power density distribution for TM01 mode when k0 = 1.2. . . . . . . . 99
7.12 Transverse field distribution for TM01 mode when λ = 1.3 μm: (a) Hϕ,(b) Er, and (c) Ez. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.13 Power density distribution for TM01 mode when λ = 1.3 μm. . . . . . 101
xiii
List of Tables
5.1 Several single-mode Fresnel fiber designs with nearly zero dispersionat λ = 1.55 μm. Materials 1 and 2 are included in Appendix B. . . . 58
5.2 Fresnel fiber designs with flat dispersion around λ = 1.55 μm. . . . . 60
5.3 Fresnel fiber designs with large negative dispersion at λ = 1.55 μm. . 63
7.1 Impedance method compared to three different techniques studied in [95]. 95
B.1 Sellmeier coefficients for several materials. . . . . . . . . . . . . . . . 109
xiv
M. R. Albandakji Chapter 1. Introduction
cation networks where data speed, security, and reliability are essential. As an exam-
ple, long distance landline telephone wire bundles, typically consisting of thousands
of bulky copper wire pairs, were replaced by a single optical fiber. That is because
a single strand of optical fiber is capable of carrying much more voice conversations
with much better sound quality than a traditional copper wire pair.
There are many nontelecom applications for fibers too. In fact, the first commercial
application of fiber optics was in medicine where bundled fibers were used to deliver
light to internal, hard-to-reach parts of the body to capture images for diagnostic
analysis. Recently, optical fibers have been used as a compact and light weight
instruments to deliver high-power laser beams to patients with virtually no invasive
surgery involved.
Optical fibers have also been used as chemical sensors and biosensors. Such optical
sensors are prepared by immobilizing indicators that change their optical properties
(index of refraction) on interacting with analytes. The main advantages of using
optical sensors over their electrochemical counterparts include freedom from EMI,
lack of the need for direct electrical connections to the solution being analyzed or for
a reference sensor, and the potential for transmitting a higher density of information
using multi-wavelength transmission [2].
Because of the growing influence of the optical fiber technology on our lives, there
has been considerable interests from many engineers and scientists all over the globe
to improve optical fiber design and performance. In particular, they are devoting a
great deal of efforts to develop novel types of optical fibers that possess enhanced
optical properties and cost less when compared to conventional fibers. During 1980s,
fiber-optics researchers envisioned synthesizing a new type of structured materials
that are periodic on the optical wavelength scale (on the order of a micrometer),
2
M. R. Albandakji Chapter 1. Introduction
known today as photonic crystals. The attractive optical properties of these materials
have led to extensive research activities to study them in two and three dimensional
configurations and to use them later in building new type of optical devices and fibers.
Such fibers, known as phonic crystal fibers (PCFs), possess the unique electromagnetic
properties of photonic crystals and thus allow performance levels that can not be
achieved using conventional optical fiber waveguides [1]. Photonic crystals and PCFs
will be discussed in more details in the following two sections.
1.2 Photonic Crystals: An Overview
A crystal is a homogeneous material composed of repetitive arrangement of atoms
or molecules. A crystal lattice is formed when a small group of atoms or molecules
is repeated in three dimensional space of the matter. The way a crystal lattice is
formed determines many of its electrical and optical properties. In particular, certain
geometries of crystal lattices might prevent electrons of specific energy levels from
moving in specific directions. If the crystal lattice prevents electrons from moving in
all directions, then a complete band gap is formed. Semiconductors are examples of
crystal lattices that have complete band gaps between their valence and conduction
energy bands.
The optical analogy for the physical crystal is the photonic crystal, which can be
defined as a low-loss periodic dielectric medium [3]. In synthetic photonic crystals, the
periodicity of the structure occurs on the macroscopic level instead of the microscopic
level in crystal lattices. This periodic structure might result in photonic band gaps
that prevent light with specific energies (frequencies) from propagating in certain
directions; therefore, photonic crystals can be used in light control and manipulation.
3
M. R. Albandakji Chapter 1. Introduction
For instance, light might be guided to propagate through low-loss photonic crystals
rather than through optical fibers, resulting in reduced optical losses. Also, new
optical devices, designed based on photonic crystals, have a great potential to be
key components in building fast communication networks and high speed optical
computers. However, there is a huge challenge in fabricating photonic crystals because
the lattice constant; i.e. periodicity distance, of the photonic crystal must be on the
order of magnitude of the wavelength of the light propagating through the crystal.
As an example, laser light used in many optical communication systems has a typical
wavelength in the micrometer range. Therefore, the photonic crystal lattice constant
must be on the order of a micrometer, which introduces an overwhelming challenge
in fabrication [4].
Photonic crystals can be classified, according to their degree of periodicity, into one-
dimensional (1-D), two-dimensional (2-D), or three-dimensional (3-D), as shown in
Fig. 1.1.
Figure 1.1: Examples of 1-D (left), 2-D (center), and 3-D (right) photonic crystals.
One-dimensional photonic crystals, also known as Bragg mirrors or multi-layer films,
are the simplest photonic crystal structure, because they are periodic in one direc-
tion only. They are usually manufactured using a stack of two alternating dielectric
materials. When designed with appropriate layer thicknesses and refractive indices,
they can exhibit many important phenomena, such as photonic band gaps and lo-
calized modes around defects. However, because the index contrast is only along
4
M. R. Albandakji Chapter 1. Introduction
one direction, the band gaps and the localized modes are limited to that direction.
Nevertheless, this simple system illustrates most of the physical features and basic
behaviors of the more complicated 2-D and 3-D photonic crystal structures.
Two-dimensional photonic crystals are periodic in two dimensions only. They are
usually made of either parallel dielectric rods in air, or through drilling or etching
holes in a dielectric material. These systems can have photonic band gaps in the
plane of periodicity and localized modes in the plane of the defect.
Three-dimensional photonic crystals are periodic along three axes. It is remarkable
that such a system can have a complete photonic band gap so that no propagat-
ing modes are allowed in any direction in the crystal. They are more difficult to
manufacture, although several techniques for their fabrication have been developed
and applied with varying degrees of success, including silicon micromachining [5],
wafer fusion bonding [6], holographic lithography [7], self-assembly [8], angled-etching
[9], micromanipulation [10], glancing-angle deposition [11], and auto-cloning [12, 13].
These crystals can allow localization of light at point defects (optical resonators) and
propagation along linear defects.
1.3 Photonic Crystals Fibers: An Overview
The most common PCFs reported in the literature have a structure that takes the
form of hexagonal, honeycomb, or cobweb geometry, as shown in Fig. 1.2. PCFs with
hexagonal lattice structures are made with a solid core or a hollow core, whereas
cobweb microstructures usually have a solid core and honeycomb PCFs usually have
a hollow core.
5
M. R. Albandakji Chapter 1. Introduction
(a) (b)
(c) (d)
Figure 1.2: Various PCF structures reported in the literature: (a) hexagonal solid-core PCF, (b)
cobweb PCF, (c) hexagonal hollow-core PCF, and (d) honeycomb PCF. [Used with permission from
[14]].
In solid-core PCFs, light is guided inside the fiber based on the average index effect.
The core region of the solid-core PCF is formed by disturbing the periodicity of the
lattice; usually by removing a single air hole from the periodic structure, which itself
forms the cladding region. Therefore, the refractive index of the core region becomes
higher than the average refractive index of the cladding region [15] and light is guided
by the effective index difference between the high-refractive index core region and the
low-refractive index cladding region. On the other hand, light guidance solely due
6
M. R. Albandakji Chapter 1. Introduction
to the band gap effect can be achieved when the photonic crystal lattice is made
with large air holes. Guidance of light through air has been observed in hollow-core
PCFs. Since the core index is lower than the average cladding index, light guidance
is primarily because of the photonic band gap effect in the transverse direction [16].
Researchers have devoted extensive efforts in recent years to better understand and
further advance the technology of PCFs, being motivated by their unique transmission
properties. The most useful feature of PCFs is that they can be fabricated using one
material only in contrast to conventional single-mode fibers which require two or more
materials. This unique feature does not only simplify the manufacturing process
of the fiber, but it also reduces fiber losses due to material absorption. Another
remarkable feature of PCFs is the wide wavelength range available for single- mode
operation. This feature allows PCFs to be single-mode and have anomalous waveguide
dispersion at the same time, whereas step-index fibers (SIFs) are usually multi-mode
when the waveguide dispersion is anomalous. Therefore, one can use PCFs to shift
the wavelength of zero group-velocity dispersion (GVD) to less than 1.27 µm, where
material dispersion is normal. This could be significant for soliton transmission in
the 1.3 µm window, dispersionless transmission at shorter wavelengths where fiber
amplifiers may be more readily available, and phase matching in nonlinear optics [17].
PCFs can also be manufactured with very small core sizes in order to obtain high
nonlinearity, which, if combined with an appropriate GVD, can be used for generating
a supercontinuum that can extend from the infrared to the visible region [18].
1.3.1 Solid-Core Photonic Crystal Fibers
Solid-core PCFs are usually fabricated by surrounding a solid glass rod, which forms
the core, by a group of hollow glass tubes, which form the cladding. The whole
7
M. R. Albandakji Chapter 1. Introduction
structure is then drawn in a conventional optical fiber drawing tower. Examples of
solid-core PCFs are shown in Figs. 1.2a and 1.2b.
Light in solid-core PCFs propagates mainly in the core region by the virtue of effective
refractive index difference between the core and the cladding regions. This unique
light propagation mechanism has opened the door for many new potential applications
for the PCF. As an example, PCFs designed with large effective area can be used in
applications that require delivery of high-power laser light, whereas PCFs designed
with small effective area can be used in novel nonlinear optical devices. Also, cladding
dimensions can be varied to achieve flat dispersion, which is useful in wavelength
division multiplexed (WDM) communication systems, or dispersion compensation,
which is useful in upgrading the already installed 1.31 µm optical fiber links to operate
at 1.55 µm.
Usually, light can be guided in the solid-core PCF by the average index effect, which
means periodicity of the air holes is not that critical [15, 19]. This is because the
solid-core region has higher refractive index than the effective refractive index of
the surrounding periodic cladding region, so light can be guided by total internal
reflection. The effective refractive index difference between the core and the cladding
in the PCF shows a high correlation with wavelength. This is because when the light
wavelength is increased, the modal field starts to spread into the periodic region and,
hence, reduces the effective index of the cladding. PCFs with small air holes have
been predicted to be single-mode over a wide wavelength range [15]. It has also been
shown that the air hole arrangement in a holey fiber does not have to be regular in
order to guide light. Besides that, many of the unique characteristics, such as the
endless single-mode, that are present in periodic holey fibers have also been found in
holey fibers with randomly arranged holes [20, 21].
8
M. R. Albandakji Chapter 1. Introduction
1.3.2 Hollow-Core Photonic Crystal Fibers
Guidance of light due to the photonic band gap effect can be achieved if the air holes
size in the photonic crystal lattice is made large enough [22]. A full 2-D photonic band
gap has been predicted by numerical simulations when the air holes are arranged in
hexagonal distribution, as shown in Fig. 1.2c. Based on the simulation results, it was
found that it is only possible for a band gap to form when the size of the air holes is
larger than the hole separation by at least 43% [23]. The large air hole made inside
the crystal lattice, shown in the center of the photonic crystal in Fig. 1.2c, causes a
localized mode to be trapped in the photonic band gap region and, therefore, light
can be guided inside an air-core fiber. This new mechanism of light guidance inside
hollow-core PCFs can lead to a large variety of applications. For instance, these fibers
can be used to carry large amounts of power or they can be used as sensing elements
in gas sensors with an increased effective length of interaction between the light and
the gas [24].
1.4 Recent Advances
In 1995, the first PCF with solid-core was proposed as a thin silica glass fiber made
with a periodic arrangement of circular air holes running along the entire length of
the fiber [25]. Knight et al. [15] reported the fabrication of the first solid-core PCF
made with a regular hexagonal arrangement of air holes. They also photographed the
near and far field patterns at different wavelengths, discovering that the PCF has the
ability to support only one mode over a large wavelength range. Later, they used an
effective index model to confirm that the PCF can be single-mode at all wavelengths
although this is practically limited by bending loss at small and large wavelengths
9
M. R. Albandakji Chapter 1. Introduction
[26].
Ferrarini et al. [16] reported that lossless propagation in PCFs is only possible if the
air holes arrangement is of an infinite extent and, of course, if a lossless material is
used. In practice, only a finite number of holes can be made; therefore, the modes of
such fibers are, strictly speaking, leaky. Furthermore, the material introduces losses
due to absorption and Rayleigh scattering. Thus a PCF can be seen practically as
confinement lossless if the leakage loss is negligible compared with material losses.
Tajima et al. [27] were successful in fabricating a 10-km long PCF with 0.37 dB/km
loss at 1.55 µm. They used highly pure silica glass made with the vapor-phase axial
deposition (VAD) technique and tried to enhance the polishing and etching process
to reduce the loss caused by the irregularities in the interior surfaces of the holes.
PCFs can be properly designed to obtain unusual optical properties, such as large or
small chromatic dispersion and/or large or small effective areas needed in linear and
nonlinear applications. Several dispersion-tailored PCF designs have been reported
[28-33]. Ferrando et al. [28, 29] proposed using PCFs to obtain flattened dispersion
characteristics near the wavelength of 0.8 µm and a nearly zero flat dispersion around
1.13 µm. They also reported a procedure for designing PCFs with nearly zero and
ultra-flattened dispersion around the wavelength of 1.55 µm. Their idea was based
on starting from an arbitrary PCF configuration, then using scale transformation
to shift the waveguide dispersion until it overlaps with the negative of the material
dispersion [30]. A similar procedure was reported in [31]. The idea of using PCFs
for dispersion compensation was suggested by Birks et al. [32]. They used a simple
silica rod in air to model a PCF with large air holes, claiming that silica core of the
PCF is well isolated as it is only connected to the rest of the fiber by the small silica
sections between the holes. They reported a total dispersion of -2000 ps/nm.km at
10
M. R. Albandakji Chapter 1. Introduction
µm. Recently, empirical relations based on numerical analyses have been developed
for evaluating the chromatic dispersion of PCFs [33].
Photonic crystal fibers with high birefringence have also been thoroughly investigated
in recent years [34-39]. Highly birefringent PCFs are used to eliminate polarization-
mode coupling and polarization-mode dispersion. This is usually achieved by reducing
the axial symmetry of the fiber by varying the size of air holes near the core area [34,
35], deforming the air hole shape from circular into elliptical [36], or deforming the
core shape [37]. Several PCF designs that combined large mode area and maintained
polarization of light have also been reported [38, 39].
In order to assess the transmission properties of PCFs and optimize their design,
accurate modeling tools are necessary. Birks et al. [26] applied an average index
model to evaluate PCFs with hexagonal hole structures. They replaced the entire
microstructure cladding with an averaged-index cladding and used a circular unit cell
approximation which allowed them to obtain rough approximations to some of the
propagation properties of such PCFs. After realizing, through experimental observa-
tion, that the guided modes are localized inside the core region, they expressed each
modal field as a sum of Hermite-Gaussian orthogonal basis functions [17]. However,
this method required some prior knowledge of the solution, which might not be always
available.
Ferrando et al. [40] used a full-vector method to study PCFs. Their main goal
was to reduce the complexity of solving a system of differential equations into a
simpler problem of solving a system of algebraic equations by using a set of complex
exponential functions to represent the modal fields.
Monro et al. [19] described a hybrid approach that combined the best features of
11
M. R. Albandakji Chapter 1. Introduction
[17], which is high efficiency, and [40], which is high accuracy. In their approach,
the electric field and the defect in the core region were decomposed into localized
Hermite-Gaussian functions, while the air holes lattice was represented by periodic
functions.
Another technique based on plane-wave expansion, in which the solution is expressed
as a plane-wave modulated by a periodic function that has the same periodicity as
that of the photonic crystal structure , was suggested in [41]. However, this technique
models the finite PCF structure as an infinite structure and, therefore, it is not capable
of predicting the confinement loss. Also, it does not take into account the geometry
of inclusions, thus not an efficient method.
Several investigators have employed an imaginary-distance beam-propagation tech-
nique that accounts for polarization effect in calculating the field profiles and propa-
gation constants of the modal fields in the transverse directions of the fiber [42, 43].
Guan et al. [44] used the vector form of the boundary element method (BEM) to ex-
amine the guided modal fields of PCFs. In this method, the curved edges between the
silica structure and the air holes are modeled as tiny linear segments, then Green’s
theorem is used to solve the eigenvalue equation of the unknowns assigned to the
segments. Spurious solutions of the BEM were avoided by formulating the eigenvalue
problem using the transverse magnetic field components instead of the longitudinal
components of the electric and magnetic fields. Resonances were also suppressed by
introducing two observation points for each boundary segment instead of one point.
White et al. [45] extended the multipole formulations for multi-core conventional
fibers to treat PCFs. In their method, they divide the cross-section of the fiber into
homogeneous regions where the wave equation decomposes into two scalar Helmholtz
equations that, in turn, lead to a matrix equation which is solved by an iterative
12
M. R. Albandakji Chapter 1. Introduction
technique. This method takes into account the rotational symmetry of PCFs to
increase computational efficiency.
Finally, Saitoh and Koshiba [46] developed a full-vector imaginary-distance beam
propagation method based on a finite element scheme to analyze the bound and
leaky modes of PCFs. Like the multipole method, the cross-section of the fiber is
divided into homogeneous regions where the wave equation, formulated as a matrix
eigenvalue system, is solved numerically.
A summary of all these techniques is shown in Fig. 1.3.
PCF Analysis Techniques
Analytical/Numerical Experimental Modeling
Field Expansion Numerical Effective index
- Plane-Wave - Hermite-Gaussian - Biorthogonal Modal
- Finite Element - Finite Difference - Boundary Element - Multipole
Figure 1.3: Summary of the techniques used for the analysis of PCFs.
Optical fiber tapers have also been extensively investigated theoretically [47-50] and
experimentally [51-53]. Marcuse [47] used the scalar-wave approximation to investi-
gate the conversion of a single-mode to multi modes in optical fibers with step-index
and parabolic-index profiles whose radii increased monotonically along their length.
His calculations showed that if the change in the fiber radius is gradual, the dominant
mode can adapt itself to that change.
13
M. R. Albandakji Chapter 1. Introduction
Hermansson et al. [48] used the beam propagation method to analyze slowly and
rapidly varying tapers. Li et al. [49] studied the transmission properties of multi-mode
tapered fibers by deriving a formula that governs the propagation of rays inside the
tapered fiber. Burns et al. [50] studied the loss mechanisms in tapered fibers. Safaai-
Jazi and Suppanitchakij [54] studied a parabolic-index taper and used it for enhancing
the coupling efficiency of light sources to optical fibers. On the experimental side,
several researchers have suggested using optical fiber tapers as fiber-to-fiber [51, 52]
and laser-to-fiber [53] couplers.
More recently, research work has been reported on the analysis and manufacturing of
microstructure optical fiber and PCF tapers. Chandalia et al. [55] used beam prop-
agation method to study the propagation of modal fields in a tapered microstructure
optical fiber. e. They demonstrated an adiabatic down-taper from 132 µm to 10
µm over 6 mm length. Huntington et al. [56] used atomic force microscopy (AFM)
to demon- strate a tapered PCF with several hundred nanometers in diameter with
maintained hole array structures. Magi et al. [57] demonstrated a tapered PCF with
a pitch of less than 300 nm, allowing them to observe a Bragg reflection in the visible
spectrum. Nguyen et al. [58] reported a loss of signal at long wavelengths as the ta-
pered PCF diameter is decreased, relating it cladding modes as the fiber dimensions
contract.
Some research has also been performed to analyze theoretically PCF structures with
infinite periodic cladding since they are capable of allowing propagation of true guided
modes and providing zero leakage loss. Mirlohi [59] has carried out an exact analysis
of such a structure with planar geometry; that is, a slab waveguide with infinite
number of cladding layers of periodically varying index. Xu et al. [60] used an
asymptotic matrix approach to analyze both Bragg fibers [60-62] and dielectric coaxial
14
M. R. Albandakji Chapter 1. Introduction
waveguides [63-65] by treating an arbitrary number of inner rings exactly using a
matrix formulation, whereas the outside cladding structure was approximated in the
asymptotic limit.
1.5 Scope of the Proposed Research
As discussed in the previous section, several theoretical and experimental techniques
have been proposed and utilized to study the transmission properties of PCFs. How-
ever, most of these techniques are very time consuming and require extensive com-
putational resources. Researchers are still trying to develop fast and low cost tools
for the analysis and design of PCFs. It would be very beneficial if one can utilize a
simplified model that can predict the transmission properties of PCFs without the
need for complicated computer simulations or physical experiments, both of which
are time consuming and costly.
One of the objectives of this research is to develop a simple model for the analysis
of PCFs based on a periodic dielectric ring structure for which an exact analytical
solution exists. Such a model allows for efficient calculation of transmission proper-
ties, including axial propagation constant, dispersion, effective area, and leakage loss.
Konorov et al. [66] have mentioned a similar model for studying the spectrum of
guided modes and the spatial distribution of radiation intensity in hollow-core PCFs.
The proposed model reduces a 2-D PCF, in which every ring of air holes is a hexag-
onal inhomogeneous ring consisting of silica and air, into a 1-D periodic ring fiber
structure. We model the hexagonal ring as a circular homogeneous ring with an ef-
fective refractive index that is related to the ratio of air holes area to the total ring
15
M. R. Albandakji Chapter 1. Introduction
area. In order to solve the 1-D circular ring structure, we use a rigorous full-vector
field analysis in which the solutions for the axial components of the electric and mag-
netic fields are first found, then the transverse components of the fields are evaluated
by expressing them in terms of the axial components, and finally boundary condi-
tions are applied to determine the modal dispersion equation. Then, the dispersion
equation is solved numerically for the axial propagation constant and the results are
used to calculate the group-velocity dispersion, field distributions, and effective area.
The model predictions for the transmission properties of hexagonal lattice PCFs are
found in a good agreement with the corresponding results reported in the literature
by different authors.
The proposed model can also aide in analyzing tapered photonic crystal fibers. The
analysis is based on approximating the tapered PCF by a series of small cylindrical
PCF sections where the modal fields are essentially independent of the axial coordi-
nate. Every section is then analyzed by the proposed model so that adiabaticity of
the taper can be investigated.
One useful aspect of the analysis of the 1-D circular ring fiber structure is that it can
also be utilized to investigate different types of fibers that do not necessarily have
rings of equal thickness; i.e., non-periodic fiber structures; therefore, we used the
same analysis to investigate a special type of fibers referred to as Fresnel fiber. These
fibers are ring fibers with rings of equal area instead of equal thickness, and they
are found to possess unique optical properties, such as flat dispersion, large negative
dispersion, and/or large or small effective area. The formulation developed here is
general enough so that the analysis of Fresnel fibers can be carried out conveniently.
Another aim of this research is to analyze the 1-D photonic crystal ring fiber with
infinite number of rings; a problem that, to our knowledge, has not yet been solved
16
M. R. Albandakji Chapter 1. Introduction
satisfactorily. The ring fiber with an infinite number of cladding rings is an ideal
1-D PCF which can allow propagation of truly guided modes with zero leakage loss.
The main objective in studying this ideal PCF structure is to gain better insight and
understanding of true photonic crystal modes. This ideal structure may also serve as
a more accurate model of PCFs with large number of rings. An exact analysis of 1-D
planar photonic crystal structure with infinite number of cladding layers has been
obtained using an impedance approach [59]. We attempt to extend this analysis to a
1-D photonic crystal structure of cylindrical geometry. In doing so, we found out that
representing the optical fiber as a transmission line circuit, proposed in [67], can be
useful in solving this problem. Accordingly, we use the transmission line equivalent
circuit to represent the cladding rings that are close to the core region, then, at a
sufficiently large radius, the rest of the cylindrical rings are approximated as planar
layers and are analyzed using the impedance approach.
17
Chapter
2 Analysis of One-Dimensional
Photonic Crystal Fibers
2.1 Overview
One-dimensional (1-D) cylindrical photonic crystal fibers (PCFs), also known as ring
fibers, have periodic index variations in the radial direction only. These fibers con-
sist of a central core surrounded by a multi-layer cladding composed of alternating
refractive indices and alternating equal-thickness rings. The refractive index profile
for this geometry is shown in Fig. 2.1. An advantage of such 1-D PCFs is that exact
analytical solutions exist for them. This allows more accurate simulations of transmis-
sion properties of such fibers. Furthermore, these fibers can serve as an approximate
model for the more complicated two-dimensional (2-D) PCFs. The cladding of the
model is formed by an effective refractive index as we will discuss in Chapter 3.
18
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
r
n1
r1 r2
…
r3 rN-1
n2
n(r)
Figure 2.1: Index profile of a 1-D PCF (ring fiber).
2.2 Field Analysis
In this section, we will determine the electromagnetic fields in a 1-D PCF that has
a geometry similar to the one shown in Fig. 2.2. In our preliminary formulation
of the problem, we will assume that the fiber is straight and the core and cladding
materials are linear, isotropic, homogeneous, and lossless. Since the index profile is
periodic, the refractive index ni can assume one of two values only: n1 or n2 < n1.
Because of the circularly cylindrical shape of the 1-D PCF, we choose a cylindrical
coordinate system (r, ϕ, z) in which the z-axis coincides with the PCF axis, as shown
in Fig. 2.2. It is emphasized that the core and claddings are dielectric materials with
a permeability equal to μ0. Furthermore, we consider time harmonic fields that vary
with time as ejωt; ω being the angular frequency. Our interest is in the guided modes
traveling along the z-axis; therefore, the z dependence of the fields is assumed to take
the form of e−jβz; β being the propagation constant. Accordingly, a mode in the fiber
can be described by a set of fields ( �E, �H) that satisfy Maxwell’s equations and take
the following form
�E (r, ϕ, z) = �e (r, ϕ) e−jβz (2.1a)
�H (r, ϕ, z) = �h (r, ϕ) e−jβz (2.1b)
19
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
z
y
r
n1
n2
r1
x
ninN-1
r2rirN-1
Figure 2.2: Geometry and coordinates for a 1-D PCF.
In the ith region with εi = ε0n2i , Maxwell’s equations can be expressed in complex
phasor forms as
�∇× �E = −jωμ0�H (2.2a)
�∇× �H = jωεi�E (2.2b)
By substituting Eqs. (2.1a) and (2.1b) into Maxwell’s equations (2.2a) and (2.2b), we
obtain the following equations
�∇× [�e (r, ϕ) e−jβz]
= −jωμ0�h (r, ϕ) e−jβz (2.3a)
�∇×[�h (r, ϕ) e−jβz
]= jωεi�e (r, ϕ) e−jβz (2.3b)
It is convenient to express the transverse components of the fields in terms of their
axial components. This approach reduces the mathematical derivation because we
just need to solve the wave equation for the axial field components. In doing so, we
20
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
decompose the del-operator(
�∇)
and the modal fields �e and �h into transverse and
axial components as follows
�∇ = �∇t +
(∂
∂z
)az (2.4)
�e = �et + ezaz (2.5a)
�h = �ht + hzaz (2.5b)
where the subscripts t and z stand for transverse and axial components, respectively.
The transverse components in cylindrical coordinates are expressed as follows
�∇t =
(∂
∂r
)ar +
1
r
(∂
∂ϕ
)aϕ (2.6)
�et = erar + eϕaϕ (2.7a)
�ht = hrar + hϕaϕ (2.7b)
Hence, Eqs. (2.3a) and (2.3b) can be rewritten as(�∇t − jβaz
)× [(�et + ezaz) e−jβz
]= −jωμ0
[�ht + hzaz
]e−jβz (2.8a)(
�∇t − jβaz
)×[(
�ht + hzaz
)e−jβz]
= jωεi [�et + ezaz] e−jβz (2.8b)
It is obvious that the exponential term e−jβz cancels out from both sides of Eqs. (2.8a)
and (2.8b). These two equations can be then expanded and written as
�∇t × �et + �∇t × (ezaz) − jβaz × �et = −jωμ0�ht − jωμ0hzaz (2.9a)
�∇t ×�ht + �∇t × (hzaz) − jβaz ×�ht = jωεi�et + jωεiezaz (2.9b)
By equating the transverse and axial components of both sides we obtain
ez = − j
ωεiaz ·(
�∇t ×�ht
)(2.10a)
21
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
�et = − j
ωεi
[�∇t × (hzaz) − jβaz ×�ht
](2.10b)
hz =j
ωμ0az ·(
�∇t × �et
)(2.10c)
�ht =j
ωμ0
[�∇t × (ezaz) − jβaz × �et
](2.10d)
From these equations, the transverse components can be obtained in terms of the
axial components, leading to the following results
�et =j
k2i
[ωμ0az × �∇thz − β�∇tez
](2.11a)
�ht = − j
k2i
[ωε0n
2i az × �∇tez + β�∇thz
](2.11b)
where k2i = k2
0n2i − β2; k0 = ω
√μ0ε0. In the cylindrical coordinate system, more
explicit relations are obtained as follows
er = − j
k2i
[β
∂ez
∂r+
ωμ0
r
∂hz
∂ϕ
](2.12a)
eϕ = − j
k2i
[β
r
∂ez
∂ϕ− ωμ0
∂hz
∂r
](2.12b)
hr = − j
k2i
[−ωε0n
2i
r
∂ez
∂ϕ+ β
∂hz
∂r
](2.12c)
hϕ = − j
k2i
[ωε0n
2i
∂ez
∂r+
β
r
∂hz
∂ϕ
](2.12d)
2.3 Solution of the Wave Equation
Now we try to find the solutions for the axial components of the fields. We start by
decoupling Maxwell’s equations in order to obtain an equation in terms of �E or �H
only. This leads to what is known as the vector wave equation
∇2 �E + ω2μ0εi�E = 0 (2.13a)
∇2 �H + ω2μ0εi�H = 0 (2.13b)
22
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
Now by substituting Eqs. (2.1a) and (2.1b) into the wave equations (2.13a) and (2.13b),
respectively, we obtain
(∇2 + k20n
2i
) [�e (r, ϕ) e−jβz
]= 0 (2.14a)(∇2 + k2
0n2i
) [�h (r, ϕ) e−jβz
]= 0 (2.14b)
where ω2μ0εi = k20n
2i is used. These two equations can be simplified further by
rewriting the Laplacian operator as ∇2 = ∇2t + (−jβ)2. Doing so, Eqs. (2.14a)
and (2.14b) can now be expressed as
∇2t�e (r, ϕ) + k2
i �e (r, ϕ) = 0 (2.15a)
∇2t�h (r, ϕ) + k2
i�h (r, ϕ) = 0 (2.15b)
Each of these equations can be split into three scalar wave equations in terms of
the Cartesian components of the fields. As was mentioned earlier, it suffices to solve
for the axial components of the fields ez and hz, both of which satisfy the following
general scalar wave equation
∇2t Ψ + k2
i Ψ = 0 (2.16)
where Ψ represents ez or hz. The transverse Laplacian operator in the cylindrical
coordinate system can be expressed as
∇2t =
∂2
∂r2+
1
r
∂
∂r+
1
r2
∂2
∂ϕ2(2.17)
Therefore, Eq. (2.16) assumes the following second order partial differential equation
form∂2Ψ
∂r2+
1
r
∂Ψ
∂r+
1
r2
∂2Ψ
∂ϕ2+ k2
i Ψ = 0 (2.18)
This equation can be solved using the separation of variables method leading to the
following general solutions
23
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
Ψ (r, ϕ) =
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
[AiJν (kir) + AiYν (kir)
] ⎡⎣ sin (νϕ)
cos (νϕ)
⎤⎦ ; k2i > 0
[AiIν (|ki| r) + AiKν (|ki| r)
] ⎡⎣ sin (νϕ)
cos (νϕ)
⎤⎦ ; k2i < 0
(2.19)
In these solutions, Ai and Ai are constant coefficients, Jν and Yν are Bessel functions
of the first and second kind, respectively, whereas Iν and Kν are modified Bessel
functions of the first and second kind, respectively. The parameter ν is an integer
constant that represents the order of the Bessel or modified Bessel function. In the
core region (r < r1), Yν and Kν must be excluded from the solution because they are
undefined at the fiber axis (r = 0). Also, in the outermost cladding layer (r > rN−1),
the field solution must just include Kν function in order to ensure that the fields
remain finite as r approaches infinity. Using this approach, the six components of the
electric and magnetic fields in the i th layer are expressed as
Eir (r, ϕ) = −jk0
q2i
⎧⎨⎩ βki
[AiF
′ν,i (kir) + BiF
′ν,i (kir)
]+
Z0νr
[CiFν,i (kir) + DiFν,i (kir)
]⎫⎬⎭⎡⎣ cos (νϕ)
− sin (νϕ)
⎤⎦ (2.20a)
Eiϕ (r, ϕ) =
jk0
q2i
⎧⎨⎩βνr
[AiFν,i (kir) + BiFν,i (kir)
]+
kiZ0
[CiF
′ν,i (kir) + DiF
′ν,i (kir)
]⎫⎬⎭⎡⎣ sin (νϕ)
cos (νϕ)
⎤⎦ (2.20b)
Eiz (r, ϕ) =
[AiFν,i (kir) + BiFν,i (kir)
] ⎡⎣ cos (νϕ)
− sin (νϕ)
⎤⎦ (2.20c)
and
H ir (r, ϕ) = −jk0
q2i
⎧⎨⎩n2
i ν
Z0r
[AiFν,i (kir) + BiFν,i (kir)
]+
βki
[CiF
′ν,i (kir) + DiF
′ν,i (kir)
]⎫⎬⎭⎡⎣ sin (νϕ)
cos (νϕ)
⎤⎦ (2.21a)
H iϕ (r, ϕ) = −jk0
q2i
⎧⎨⎩n2
i ki
Z0
[AiF
′ν,i (kir) + BiF
′ν,i (kir)
]+
βνr
[CiFν,i (kir) + DiFν,i (kir)
]⎫⎬⎭⎡⎣ cos (νϕ)
− sin (νϕ)
⎤⎦ (2.21b)
24
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
H iz (r, ϕ) =
[CiFν,i (kir) + DiFν,i (kir)
] ⎡⎣ sin (νϕ)
cos (νϕ)
⎤⎦ (2.21c)
where
q2i = ηik
2i (2.22a)
ηi =
⎧⎨⎩ +1 ; ni > β
−1 ; ni < β(2.22b)
Fν,i (ki, r) =
⎧⎨⎩ Jν (kir) ; ni = n1
Iν (kir) ; ni = n2
(2.22c)
Fν,i (ki, r) =
⎧⎨⎩ Yν (kir) ; ni = n1
Kν (kir) ; ni = n2
(2.22d)
where β = β/k0 is the normalized propagation constant, Z0 is the free-space charac-
teristic impedance, ki = k0
√∣∣n2i − β2
∣∣, and Ai, Bi, Ci, and Di are constant amplitude
coefficients that can be evaluated by imposing the boundary conditions for the fields
and knowing the source power.
2.4 Boundary Conditions and Dispersion Relation
In general, boundary conditions demand the continuity of the tangential field com-
ponents at the interface between two different layers. Hence, for an N -layer ring
fiber, we have N −1 interfaces, and at every interface we have 4 boundary conditions
(continuity of Eϕ, Ez, Hϕ, and Hz). Therefore, we obtain 4N − 4 equations with
4N − 4 unknown coefficients plus the normalized propagation constant β, which is
also unknown. Applying the boundary condition on Ez at r = ri, yields
AiFν,i (Ui) + BiFν,i (Ui) = Ai+1Fν,i+1 (Wi) + Bi+1Fν,i+1 (Wi) (2.23)
25
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
where Ui = kiri and Wi = ki+1ri. Similarly, applying the boundary condition on Hz
at r = ri, results in
CiFν,i (Ui) + DiFν,i (Ui) = Ci+1Fν,i+1 (Wi) + Di+1Fν,i+1 (Wi) (2.24)
The continuity of Eϕ at r = ri leads to
1kiηi
{βνUi
[AiFν,i (Ui) + BiFν,i (Ui)
]+ Z0
[CiF
′ν,i (Ui) + DiF
′ν,i (Ui)
]}=
1ki+1ηi+1
{βνWi
[Ai+1Fν,i+1 (Wi) + Bi+1Fν,i+1 (Wi)
]+
Z0
[Ci+1F
′ν,i+1 (Wi) + Di+1F
′ν,i+1 (Wi)
]} (2.25)
Finally, the continuity of Hϕ at r = ri leads to
1kiηi
{n2
i
Z0
[AiF
′ν,i (Ui) + BiF
′ν,i (Ui)
]+ βν
Ui
[CiFν,i (Ui) + DiFν,i (Ui)
]}=
1ki+1ηi+1
{n2
i+1
Z0
[Ai+1F
′ν,i+1 (Wi) + Bi+1F
′ν,i+1 (Wi)
]+
βνWi
[Ci+1Fν,i+1 (Wi) + Di+1Fν,i+1 (Wi)
]} (2.26)
Equations. (2.23) to (2.26) can be rewritten into matrix form as⎡⎢⎢⎢⎢⎢⎢⎣Ai+1
Bi+1
Ci+1
Di+1
⎤⎥⎥⎥⎥⎥⎥⎦ =
⎡⎢⎢⎢⎢⎢⎢⎣γi
11 γi12 γi
13 γi14
γi21 γi
22 γi23 γi
24
γi31 γi
32 γi33 γi
34
γi41 γi
42 γi43 γi
44
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣Ai
Bi
Ci
Di
⎤⎥⎥⎥⎥⎥⎥⎦ (2.27)
where the expressions for γi are included in Appendix A.
Applying the boundary conditions at all the interfaces r = ri; i = 1, 2, ..., N − 1,
leads to the following expression which relates the amplitude coefficients of the outer
cladding region to those of the central core⎡⎢⎢⎢⎢⎢⎢⎣AN
BN
CN
DN
⎤⎥⎥⎥⎥⎥⎥⎦ =
N−1∏i=1
⎡⎢⎢⎢⎢⎢⎢⎣γi
11 γi12 γi
13 γi14
γi21 γi
22 γi23 γi
24
γi31 γi
32 γi33 γi
34
γi41 γi
42 γi43 γi
44
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣A1
B1
C1
D1
⎤⎥⎥⎥⎥⎥⎥⎦ =
⎡⎢⎢⎢⎢⎢⎢⎣Γ11 Γ12 Γ13 Γ14
Γ21 Γ22 Γ23 Γ24
Γ31 Γ32 Γ33 Γ34
Γ41 Γ42 Γ43 Γ44
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣A1
B1
C1
D1
⎤⎥⎥⎥⎥⎥⎥⎦(2.28)
26
M. R. Albandakji Chapter 2. Analysis of One-Dimensional Photonic Crystal Fibers
In order to insure that the fields are finite at r = 0, B1 and D1 must be set to zero.
Also, the fields must decay exponentially in the outer cladding region; therefore, AN
and CN must be set to zero too. These requirements result in the following set of
equations
Γ11A1 + Γ13C1 = 0 (2.29a)
Γ31A1 + Γ33C1 = 0 (2.29b)
In order to avoid a trivial solution for the above system of equations, the determinant
of the coefficients of this system must be zero, that is
Γ11Γ33 − Γ13Γ31 = 0 (2.30)
Equation (2.30) is what is referred to as the characteristic equation, the dispersion
equation, or the eigenvalue equation. It can be solved for the normalized propagation
constant(β). Thus, all amplitude coefficients can be expressed in terms of one am-
plitude coefficient chosen as the independent coefficient, which is eventually obtained
from the source condition. This equation may also be written as
f(λ, β, ν, ni, ri; i = 1, 2, ..., N − 1
)= 0 (2.31)
Equation (2.31) is solved numerically to obtain the normalized propagation constant(β)
which can then be used to evaluate other transmission properties of the fiber,
such as phase and group velocities, dispersion, and effective area. It is worth men-
tioning that the suggested model is able to directly account for the dependency of
the refractive index of silica on wavelength using Sellmeier’s equation [68]
n1 (λ) =
√√√√1 +3∑
j=1
Ajλ2
λ2 − λ2j
(2.32)
where Aj and λj are material constants listed in Appendix B. In the next chapter,
the formulation presented here will be used to model the 2-D PCFs and calculate
their transmission properties.
27
Chapter
3 Analysis of Two-Dimensional
Photonic Crystal Fibers
3.1 Overview
As mentioned in Chapter 1, many techniques have been proposed for simulating the
relatively complicated structure of the two-dimensional (2-D) photonic crystal fibers
(PCFs). All these techniques require very long processing time and large amounts
of computer memory. Here, we introduce an analysis technique that is based on
modeling the complicated 2-D PCF structure with a simpler one-dimensional (1-D)
structure in order to predict the basic transmission properties. In Chapter 2, analyt-
ical solutions for the 1-D PCFs were obtained. These solutions, in conjunction with
the model presented here, allow for approximate, yet reasonably accurate evaluation
of transmission properties of 2-D PCFs, including propagation constant, chromatic
dispersion, effective area, and an estimation of leakage loss.
28
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
3.2 The Proposed Model
The modeling of a 2-D PCF is illustrated in Fig. 3.1. Each hexagonal inhomogeneous
ring of air holes in the 2-D PCF is replaced by a circular homogeneous ring with
certain effective refractive index and specific radius. Assuming that the PCF is made
Figure 3.1: Transforming 2-D PCF into 1-D PCF.
entirely of silica glass, this transformation suggests that the core of the model is
made of silica with an equivalent radius equals to Λ−d/2; where Λ is the hole-to-hole
spacing and d is the hole diameter in the 2-D PCF. Also, this model suggests two
types of rings: a silica ring and an effective index ring. The silica ring has a thickness
of Λ − d, whereas the effective index ring has a thickness of d and an effective index
calculated by evaluating the air-filling fraction (f), which is defined as the ratio of
the air holes area to the overall area of one ring. Referring to Fig. 3.2, the effective
29
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
d
Figure 3.2: Hexagonal ring in a 2-D PCF.
index is calculated as
neff = nsilica (1 − f) + f (3.1)
where
f =π
2√
3
d
Λ(3.2)
In general, the field decays to very small values at the outermost cladding, so the
index of the outermost cladding has no significant impact on the analysis of the PCF.
Therefore, we chose the index of the outermost cladding to be navg, given by
navg = 0.5 [nsilica + neff ] (3.3)
Fig. 3.3 shows the index profile of the suggested 1-D model of the 2-D PCF.
n(r)
r
nsilica
r1 r2
d -d-d/2
…
r3 rN-1
navg
neff
Figure 3.3: Index profile of the suggested PCF model.
30
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
3.3 Model Testing, Comparison, and Accuracy
3.3.1 Normalized Propagation Constant
In order to be able to test the proposed model, some comparisons with published
results are made. First, we examine the accuracy of the model in predicting the
normalized propagation constant (β). In doing so, we need to compare the results
of the model with highly accurate ones that have been calculated using one of the
numerical techniques. Saitoh and Koshiba [33] have investigated what they refer to
as the effective index (β) using the finite element method (FEM), which is considered
to be a powerful tool capable of handling any kind of geometry. Fig. 3.4 compares
the normalized propagation constant obtained from this model with those calculated
using the FEM.
Figure 3.4: Normalized propagation constant as a function of normalized wavelength for different
normalized hole diameter: dots are FEM results in [33] and solid lines are results from the proposed
PCF model.
31
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
d/Λ
λ/Λ
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Error(%)
Figure 3.5: Contour plots of the percentage error in the normalized propagation constant using
the proposed PCF model when compared to the FEM in [33].
It is noted that the results based on the proposed model agree very well for small
λ/Λ. An error analysis shows that for λ/Λ ≤ 0.5, the model agrees within 0.2 %
when compared to the FEM, whereas for λ/Λ ≤ 1.0, the model agrees within 0.7 %.
Fig. 3.5 shows contour plots of the percentage error.
3.3.2 Group-Velocity Dispersion (GVD)
GVD is another parameter that can be used to check the accuracy of the proposed
model. It can be calculated using the following relation
GVD = −λ
c
d2β (λ)
dλ2(3.4)
where c is the speed of light in free space.
32
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
Researchers have investigated GVD both numerically and experimentally. Wadsworth
et al. [69] fabricated a two-ring PCF with Λ = 1.8 μm and d/Λ = 0.8. They measured
the GVD by low-coherence interferometry using a white light source. They also mod-
eled the PCF numerically by using the plane-wave expansion method and considering
the structure as an infinite periodic array of round holes with a single hole removed.
Fig. 3.6 shows the good agreement between the GVD obtained from this model and
that reported in Wadsworth’s paper both numerically and experimentally.
Figure 3.6: GVD comparisons: solid line is measured GVD in [69], dashed line is numerically
calculated GVD in [69], and dotted line is predicted GVD using the proposed PCF model.
Recently, Nakajima et al. [70] reported the fabrication of a low-loss PCF with Λ =
5.6 μm, d/Λ = 0.5, and 60 air holes. They measured the GVD using pulse delay mea-
surements with supercontinuum light. We simulated the same structure twice; once
using our proposed model, and another time using the multipole method [45]. Using
33
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
this method, the fiber cross-section is devided into homogeneous subspaces where
the wave equation is solved iteratively. We noticed that the model showed better
agreement with the multipole method because no measurement error was involved.
Results are shown in Fig. 3.7.
Figure 3.7: GVD comparisons: solid line is measured GVD from [70], dashed line is predicted GVD
using the proposed PCF model, and dotted line is predicted GVD using multipole method [45].
Saitoh and Kashiba [33] provided empirical relations to calculate the normalized
propagation constant, which were used to predict waveguide dispersion only. They
reported results for pitch values of 2, 2.5, and 3 μm with different values of d/Λ
over a wide wavelength range. They used the FEM to test the accuracy of their
empirical relations. We simulated the same cases using our model which showed
good agreements especially at small wavelengths. This is because the accuracy of the
model decreases as the ratio λ/Λ increases. The results for Λ = 3 μm are shown in
Fig. 3.8.
34
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
Empirical relation FEMProposed PCF model
Figure 3.8: Waveguide dispersion comparison: solid and dashed lines are from [33] and dotted lines
are predicted waveguide dispersion using the proposed PCF model.
3.3.3 Effective Area
Effective area is an important parameter used as a measure of nonlinearities in optical
fibers. Small effective areas are indicative of significant nonlinear effects inside the
core of the fiber. Effective area is also related to confinement loss, micro-bending loss,
macro-bending loss, and splicing loss of the fiber [71]. Generally speaking, the larger
the effective area the higher the above mentioned losses.
The effective area (Aeff ) can be calculated using the following formula
Aeff =
[∫ 2π
0
∫∞0
∣∣∣ �Et (r, ϕ)∣∣∣2 rdrdϕ
]2∫ 2π
0
∫∞0
∣∣∣ �Et (r, ϕ)∣∣∣4 rdrdϕ
(3.5)
35
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
where �Et is the transverse electric field.
We used our model to predict the effective area for different values of d/Λ and λ/Λ.
The results, shown as circles in Fig. 3.9, show good agreement with those published
in [72]. In particular, we notice that the agreement between both sets increases as λ/Λ
decreases. This is because the accuracy of the model in calculating the normalized
propagation constant increases as discussed earlier in Section 3.3.1.
FEM data PCF model
Figure 3.9: Normalized effective area comparison: solid lines are from [72] and circles are from the
proposed PCF model.
36
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
3.3.4 Leakage Loss
In practice, PCFs have a finite number of air hole rings and the outermost layer
is a relatively thick layer of glass. In this case, the guided modes of the PCF are
leaky and the confinement of power is not perfect so it leaks out of the guiding
structure. Leakage loss is an important parameter and needs to be estimated for
practical applications.
We used the model proposed here to estimate the leakage loss of the PCF based on
the method described in [73]. Provided that the real part of the jacket refractive
index is not too different from the cladding index, then the ratio between the radial
and the axial power flow densities at the outermost cladding interface becomes [73]
χ =
√(ncladding/β
)2 − 1 (3.6)
Also, the axial power flow density is given by
paxial =1
2�e{ �E × �H∗} · aze
−αz (3.7)
= p0 sin2 ϕe−αz (3.8)
where p0 is the average axial power flow density and α is the leakage loss coefficient.
The lost power due to leakage can be expressed as
Ploss =
∫ 2π
0
∫ L
0
pradialRdϕdz (3.9)
= πχp0R(1 − e−αL
)/α (3.10)
where R is the interface radius and L is the length of the fiber segment. On the other
hand, the lost power Ploss can be associated with an attenuation constant α and the
power launched into the fiber P0 through the following equation
Ploss = P0
(1 − e−αL
)(3.11)
37
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
By equating Eqs. (3.10) and (3.11), α in dB/m is calculated as
α = 4.34πχRp0
P0(3.12)
Fig. 3.10 shows the leakage loss as a function of pitch (Λ) for a range of different
4-ring PCF structures estimated using the model (dashed lines) compared to the
results reported in [74] (solid lines). Each curve represents results for a given fiber
profile scaled to a range of different dimensions. We notice that the leakage loss
always decreases when larger air holes are used because the mode is always more
tightly confined to the core region for larger air-filling fractions, which is similar to
the behavior of step-index fibers. Next, we estimated the leakage loss versus Λ for
Figure 3.10: Leakage loss versus Λ for 4-ring PCFs with different air-filling fractions at λ = 1.55 μm:
solid lines are reported in [74] and dashed lines are predicted by the proposed PCF model.
a fixed air-filling fraction and different number of rings. The results (dashed lines)
38
M. R. Albandakji Chapter 3. Analysis of Two-Dimensional Photonic Crystal Fibers
were also compared to those reported in [74] (solid lines), as shown in Fig. 3.11. As
expected, for all the values of Λ, increasing the number of rings decreases the leakage
loss because the holey cladding extends over a larger region. It is noted that the
Figure 3.11: Leakage loss versus Λ for d/Λ = 0.9 PCFs with different number of rings at λ =
1.55 μm: solid lines are reported in [74] and dashed lines are predicted by the proposed PCF model.
model generally underestimates the leakage loss. This is believed to be due to the
fact that the openings between the holes, which are the main cause for the leakage
of power, are replaced with closed rings. This behavior has also been observed by
others who used another ring model to analyze microstructure fibers with circularly
arranged holes [75].
39
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
In addition, tapered PCF can be used to generate a high-intensity optical energy,
which can significantly enhance the efficiency of nonlinear optical devices used in
supercontinuum generation [83]. They can also help in fabricating small size band
gap photonic crystal structures inside relatively small fibers [56].
An OFT is fabricated by heating the fiber then gently stretching it to form a structure
consisting of two regions; the taper waist, which is the narrow stretched section in the
middle of the taper, and the taper transition, which comprises of two conical tapered
sections attached to the waist, as shown in Fig. 4.1. The taper transitions transform
the local fundamental mode from a core mode in the untapered fiber to a cladding
mode in the taper waist, and this is the basis of many of its applications. However, if
this transformation is to be accompanied by small loss of light from the fundamental
mode, the shape of the taper transitions must be sufficiently gradual. On the other
hand, it is desirable for the transition to be as short as possible, allowing the resulting
component to be compact and insensitive to environmental degradations [84].
TaperWaist
TaperTransition
Unstretched Fiber
Unstretched Fiber
TaperTransition
Figure 4.1: Cross section of a typical fiber taper.
4.2 Analysis of Tapered Photonic Crystal Fibers
One way to analyze tapered PCFs is by approximating the tapered section as a series
of uniform sections along the axial direction, as shown in Fig. 4.2.
41
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
z z
Figure 4.2: Modeling of a fiber taper using a series of cascaded uniform sections.
Each PCF section can be modeled using the 1-D ring structure discussed in Chapter 2.
Using this approach, the index profile becomes independent of z within each section;
therefore, the modes within the finite section can be approximated by the modes of
an infinitely long fiber, as shown in Fig. 4.3. These modes are referred to as local
modes, which are considered to be an excellent approximation for slowly varying
tapers although, in fact, they are not exact solutions. Following this approach, it
becomes feasible to express the actual field inside the tapered fiber as a superposition
of the local normal modes, which are coupled among each other [47]. Therefore, the
total electric field inside the taper can be expressed as [85]
�E =
∞∑ν=0
cν (z)�Eνe
−j� z0
βν(z′)dz
′(4.1)
where cν is the expansion coefficient,�Eν is the local mode, and βν(z) is the z-
dependent propagation constant, all evaluated at the νth section. Inside a fiber taper,
the local modes are not independent from each other; therefore, the expansion coeffi-
cients are also coupled to each other and they satisfy the coupled wave equations [85]
dcν
dz=∑
μ, μ�=ν
Rνμcμej� z0
(βν−βμ)dz′
(4.2)
42
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
z
Figure 4.3: Approximating a finite taper section by a cylindrical structure of uniform cross section.
where the coupling coefficients are given as
Rνμ =ωε0
4P0 [βν − βμ]
∫ 2π
0
∫ ∞
0
∂n2(r, z)
∂z�Eν(r, φ) · �
E∗μ(r, φ)rdrdφ (4.3)
In Eq. 4.3, ω is the angular frequency, ε0 is the free-space permittivity, Z0 is the
free-space characteristic impedance, and P0 is the mode power normalized to unity
and is given by
P0 =1
2�e
{∫ 2π
0
∫ ∞
0
(�E × �
H∗)· z rdrdφ
}(4.4)
Using the chain rule, the z derivative of n2(r, z) can be written as
∂n2(r, z)
∂z=
∂n2(r, z)
∂r
∂r
∂z(4.5)
Since the refractive index n(r, z) is either n1 or n2, then its radial derivative is zero
everywhere except at the ring boundaries, so it can be expressed as
∂n2(r, z)
∂r=
N−1∑i=1
(−1)i[n2
1 − n22
]δ (r − ri) (4.6)
43
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
where δ (r − ri) is the dirac-delta function. It is noted that the local modes couple
only to those local modes that share the same azimuthal symmetry with them. Using
ωε0 = k0/Z0 and the fact that all the modes are orthonormal modes, Eq. (4.3) becomes
Rνμ =k0 [n2
1 − n22]
4Z0 [βν − βμ]
∫ 2π
0
∫ ∞
0
N−1∑i=1
(−1)iδ (r − ri)∂r
∂z�Eν(r, φ) · �
E∗μ(r, φ)rdrdφ (4.7)
By evaluating the integral in Eq. (4.7) and after some algebraic manipulation, the
coupling coefficients can be expressed in a simpler form as
Rνμ =π [n2
1 − n22]
4Z0
[βν − βμ
] N−1∑i=1
(−1)iri(z)∂ri(z)
∂z
[�Eν · �
E∗μ
]r=ri(z)
(4.8)
The summation term in Eq. (4.8) represents the contribution of all the layers of the
tapered PCF in the coupling among different local modes. We notice that the main
contribution comes from the taper shape, taper slope, and the field values at the ring
boundaries.
4.3 Adiabaticity of Tapered Photonic Crystal Fibers
Tapered PCFs, similar to tapered conventional fibers, can be either adiabatic or
nonadiabatic. Several factors may affect adiabaticity of tapered PCFs, such as taper
shape, taper length, and number of air hole rings. The following sections discuss
these factors in more details.
4.3.1 Taper Shape
Taper shape has a direct effect on the coupling among local modes inside the taper,
as suggested by Eq. (4.8) through ri(z) and its radial derivative. Many mathematical
44
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
models have been used to represent the actual taper shape, such as the linear, raised
cosine, and modified exponential taper [47]. Assuming the length of the taper is
represented by L, initial core radius ri, and final core radius rf , one can define the
linear taper shape as
r(z) = ri + (rf − ri)z
L, (4.9)
the raised cosine taper shape as
r(z) =1
2
[(rf + ri) − (rf − ri) cos
(π
z
L
)], (4.10)
and the modified exponential taper shape as
r(z) =
⎧⎨⎩ ri + a[cos(
3πL
z)− 1]
; 0 ≤ z ≤ 0.8L
rfe(z−L)b ; 0.8L ≤ z ≤ L
(4.11)
where a =0.2rf
1−cos(0.6π)and b = 1
0.8Lln(
rf
0.8ri
). These taper shapes are shown in
Figs. 4.4(a)-(c).
In order to analyze the effect of the taper shape on the coupling among modes,
we simulated three different taper shapes for a down-tapered single-ring PCF at
λ = 1.55 μm with initial pitch and holes diameter of 15 and 6 μm, respectively,
taper ratio of 6:1, and a length of 100 μm. Figure 4.5 shows how the power of
the second mode HE12 changes along the taper. It is noted that, in general, the
modified exponential taper has the largest coupled power, while the linear taper has
the smallest one. This is because the linear taper has smoother transition over the
entire taper length than the other two tapers. These results are also similar to those
studied in [47] for the step-index fiber taper.
45
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
0 50 1000
5
10
15
20
z [μm]
r [μ
m]
(a)
0 50 1000
5
10
15
20
z [μm]
r [μ
m]
(b)
0 50 1000
5
10
15
20
z [μm]
r [μ
m]
(c)
Figure 4.4: Three different taper shapes: (a) linear taper, (b) raised cosine taper, and (c) modified
exponential taper.
46
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
0 20 40 60 80 1000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
z [μm]
Nor
mal
ized
Pow
er
linearraised cosinemodified exponential
Figure 4.5: The power coupled to HE12 mode in different PCF taper shapes.
4.3.2 Taper Length
Taper length has a direct effect on the adiabaticity of the taper. When the taper
length is sufficiently long, the taper angle becomes small enough to ensure that the
power lost from the fundamental mode to the other modes is negligible. However,
for practical taper devices, there is a limit on the smallest taper angle that can be
achieved over a significant taper length. Furthermore, long taper devices are difficult
to package and are more susceptible to environmental effects [86]. Therefore, it is
necessary for the taper length to exceed the minimum taper length that guarantees
the fundamental mode is adiabatic along the entire length of the taper.
One method has been suggested in [86] to estimate the minimum adiabatic taper
length for a conventional fiber taper. Basically, this method compares the taper
length-scale to the coupling length-scale, imposing a bound on the fraction of power
47
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
coupled from the fundamental mode to the next higher order mode that has the same
azimuthal symmetry. Since this method is not inherently specific to a certain fiber
structure, it can be applied to analyze tapered PCFs.
When the taper length-scale is much larger than the coupling length-scale between the
fundamental mode and the next higher order mode, the lost power due to coupling is
sufficiently small [87]. The taper length-scale is defined as the height of a right circular
cone with base coincident with the local core cross-section and apex angle equal to
the local taper angle (Φ = tan−1 |dr/dz|) [86], as shown in Fig. 4.6. In practice, the
zt
(z)
z
y
x
r(z)
Figure 4.6: Local taper length-scale (zt) in a tapered fiber.
taper angle Φ is very small; therefore, the taper length-scale can be approximated by
zt =r
Φ(4.12)
Also, the local coupling length-scale is defined as
zc =2π
β1 − β2
(4.13)
When zt >> zc everywhere along the taper, the fundamental mode propagates almost
adiabatically along the taper. On the other hand, when zt << zc, there will be
significant coupling between the fundamental mode and the next higher order mode.
48
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
Therefore, the condition zt = zc provides an approximate margin for adiabaticity, so
the taper angle can be written as
Φ =r [β1 − β2]
2π(4.14)
For the linear down-tapered fiber shown in Fig. 4.7, it can be easily shown that its
taper angle is given by (Φ << 1)
Φ =ri − rf
L(4.15)
z
L
ri
rf
Figure 4.7: Linear down-tapered fiber showing the taper angle.
As shown in Fig. 4.8, the maximum local coupling length-scale occurs at z = 0 since
β1 (0) − β2 (0) is minimum. By equating Eq. (4.14) and Eq. (4.15), we obtain the
minimum taper length requirement for adiabatic linear taper, given by
Lmin =ri − rf
ri
[β1(0) − β2(0)
]λ (4.16)
Now, we discuss some of the properties observed by inspecting the numerical solutions
for a linear 1-ring PCF taper that has an initial pitch of 5.6 μm, initial hole diameter
of 2.8 μm, and a taper ratio of 3:1. Figure 4.8 shows the normalized propagation
49
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
0 20 40 60 80 100
1.25
1.3
1.35
1.4
1.45
z [μm]
β
HE11
EH11
HE12
Figure 4.8: The normalized propagation constants of the first three modes for a taper length of
100 μm at 1.55 μm wavelength.
constants variation along the axial dimension of the fundamental mode HE11 and the
next two modes EH11 and HE12 for a taper length of 100 μm at 1.55 μm wavelength.
Using Eq. (4.16), the minimum length requirement for an adiabatic field propagation
is 40 μm. Figure 4.9 shows the normalized power coupled between the fundamental
mode HE11 and the next two modes EH11 and HE12 for a taper length of 100 μm.
We see that the power coupled to these two mode is almost negligible since the taper
length of this taper is much larger than the minimum length of 40 μm. Figure 4.10
shows the 3-D plot of the power density inside the taper. We notice that the field
propagates almost adiabatically as predicted. Figure 4.11 shows the normalized power
of the three modes for a taper length of 20 μm. We observe that there is significant
amount of power coupled to EH11 and HE12 modes since the taper length is less than
the minimum length predicted by Eq. (4.16). Figure 4.12 shows the 3-D plot of the
power density inside the nonadiabatic taper.
50
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
0 20 40 60 80 1000
0.2
0.4
0.6
0.8
1
1.2
1.4
z [μm]
Nor
mal
ized
Pow
er
0 20 40 60 80 1000
0.05
0.1
HE11
EH11
HE12
Figure 4.9: Variation of coupled power for the first three modes in the linear taper with a length
larger than adiabatic length.
Figure 4.10: Power density distribution inside the linear taper with a length larger than adiabatic
length.
51
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [μm]
Nor
mal
ized
Pow
er
HE11
EH11
HE12
Figure 4.11: Variation of coupled power for the first three modes in the linear taper with a length
smaller than adiabatic length.
Figure 4.12: Power density distribution inside the linear taper with a length smaller than adiabatic
length.
52
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
4.3.3 Number of Air Hole Rings
The effect of the number of air hole rings on the degree of adiabaticity of tapered
PCFs comes from the contribution of air hole rings in evaluating the power coupling
coefficients of the local fields in the summation term in Eq. (4.8). In order to illustrate
this effect, we assume that only one mode, which is the fundamental mode HE11, is
launched inside a raised cosine PCF taper, then we compare the coupled power to the
higher order modes for 1-ring, 2-ring, and 3-ring PCFs. The tapers are simulated at
λ = 1.55 μm and are assumed to have initial Λ = 5 μm, initial d = 2 μm, L = 50 μm,
and a taper ratio of 5:2. As expected, the effect of increasing the air hole rings
on the coupled power to higher order modes is almost negligible. In other words,
the variation of coupled power to higher order modes for the three different tapers is
almost the same, as shown in Fig. 4.13. To illustrate this behavior further, we plot the
power density distribution inside the three tapers in Figs. 4.14, 4.15, and 4.16. From
these plots we notice that the three tapers are nonadiabatic and they also behave
similarly since there is almost no difference in the power density distribution among
them. Therefore, the number of air hole rings has, in general, no significant impact
on tapered PCF adiabaticity since the modal fields decay to very insignificant values
at rings located away from the core region.
53
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
0 10 20 30 40 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
z [μm]
Nor
mal
ized
Pow
er
HE11
EH11
HE12
Figure 4.13: Variation of power of the first three modes in a 1-ring PCF taper. 2- and 3-ring PCF
tapers have almost the same power variation.
Figure 4.14: Power density distribution inside a 1-ring PCF taper.
54
M. R. Albandakji Chapter 4. Analysis of Tapered Photonic Crystal Fibers
Figure 4.15: Power density distribution inside a 2-ring PCF taper.
Figure 4.16: Power density distribution inside a 3-ring PCF taper.
55
Chapter
5
Fresnel Fibers
5.1 Overview
Fresnel fibers are a special type of ring fibers whose rings have a constant area instead
of a constant thickness. This structure makes the thickness of rings shrink along the
fiber’s radial direction, as shown in Fig. 5.1. To our knowledge, this fiber has not
been analyzed before and it is likely for this unique fiber to possess special transmis-
sion properties [88]. The formulation developed in Chapter 2 is, in fact, applicable to
any circularly cylindrical dielectric waveguide with arbitrary refractive index profile;
therefore, it can be used to analyze Fresnel fibers. In this chapter, we will show that
Fresnel fibers can be properly designed to achieve shifted, flattened, or large negative
chromatic dispersion, making them very attractive in broadband fiber-optic commu-
nication systems. Fresnel fibers can also be designed to possess either large effective
area that makes them suitable for use in very long distance fiber-optic communica-
tion links, or small effective area that makes them suitable for use in many fiber-optic
applications, such as solitons and nonlinear fiber devices.
56
M. R. Albandakji Chapter 5. Fresnel Fibers
rrcore r1 r2 r3 r4 r5 r6
…
rN
n2 ( )
n(r)
n1( )
Figure 5.1: Index profile of a Fresnel fiber.
5.2 Analysis of Fresnel Fibers
Assuming that Fresnel fibers are made of two materials only, there will be five para-
meters that can be changed during the design process of Fresnel fibers: the refractive
index of the first material n1, the refractive index of the second material n2, the core
radius rcore, the ring area Ar, and the number of rings N . By enforcing the equal
area condition on the first two rings we obtain,
π[r21 − r2
core
]= Ar (5.1)
π[r22 − r2
1
]= Ar (5.2)
Therefore,
π[r22 − r2
core
]= 2Ar (5.3)
In general,
π[r2i − r2
core
]= iAr (5.4)
Therefore,
ri =
√iAr
π+ r2
core (5.5)
Equation(5.5) can be used in conjunction with the formulation developed in Chapter 2
to analyze the Fresnel fiber and obtain the axial propagation constant, chromatic
dispersion, and effective area.
57
M. R. Albandakji Chapter 5. Fresnel Fibers
5.3 Special Fresnel Fiber Designs
Here we introduce special Fresnel fiber designs that include the design of dispersion-
shifted, dispersion-flattened, and dispersion compensating fibers. The design ap-
proach is based on that described in [89].
5.3.1 Dispersion-Shifted Fibers
Dispersion-shifted fibers provide nearly zero chromatic dispersion at 1.55 μm, which
is the wavelength of minimum attenuation in silica-based fibers. These fibers are
mainly used in fiber-optic communication systems because they provide minimum
attenuation and dispersion. Table 1 lists several single-mode Fresnel fiber designs
that are capable of providing nearly zero dispersion at 1.55 μm with effective areas
ranging from 15.6 μm2 to 152.7 μm2. Figure 5.2 shows the variations of the effective
area versus wavelength for these fibers.
Table 5.1: Several single-mode Fresnel fiber designs with nearly zero dispersion at λ = 1.55 μm.
Materials 1 and 2 are included in Appendix B.Fresnel Material Material Core Radius Ring Area Number of Effective Area at Dispersion at λ = 1.55 μm
Fiber 1 2 [ μm] [ μm2] Rings λ = 1.55 μm [ μm2] [ps/nm.km]
1 M11 M2 1.7 1 8 15.6 -0.2
2 M4 M6 1.6 3 4 22.1 -0.05
3 M3 M2 1.5 4 4 24 0.3
4 M3 M5 1.7 3 4 28.85 -0.08
5 M3 M6 1.7 3 4 33.35 0.1
6 M11 M10 2.0 6 6 45.5 0.34
7 M5 M1 1.75 1 3 73.5 0.15
8 M5 M3 0.8 2 5 92.5 0.3
9 M7 M8 1.45 2 4 101.15 0.09
10 M9 M12 2.75 1 2 152.7 2
58
M. R. Albandakji Chapter 5. Fresnel Fibers
1.5 1.55 1.60
20
40
60
80
100
120
140
160
180
λ [μm]
Aef
f [μm
2 ]Fiber 1Fiber 2Fiber 3Fiber 4Fiber 5Fiber 6Fiber 7Fiber 8Fiber 9Fiber 10
Figure 5.2: Effective area for the fibers listed in Table 5.1.
During the design process of dispersion-shifted fibers, it was observed that increasing
the number of rings increases the effective area of the fiber, and, therefore, reduces
the nonlinearities of the fiber but, at the same time, it increases the cutoff wavelength
of the second mode, which may cause the fiber to become multi-mode. Therefore,
careful selection of the number of rings is necessary in the design of small nonlinearity
dispersion-shifted fibers in order to minimize nonlinearity while keeping the fiber
single-mode.
5.3.2 Dispersion-Flattened Fibers
Dispersion-flattened fibers have small and flat (nearly constant) chromatic dispersion
over an extended range of wavelengths. These fibers are suitable for use in wave-
division-multiplexed (WDM) optical fiber systems in which several, or even more,
59
M. R. Albandakji Chapter 5. Fresnel Fibers
optical channels are transmitted in the same fiber.
Two interesting dispersion-flattened single-mode fiber designs were obtained. The
first fiber has a chromatic dispersion of 0.26 ps/nm.km and a dispersion slope of
0.004 ps/nm2.km both at 1.55 μm wavelength. In addition, the dispersion is within
±5 ps/nm.km over a wide wavelength range (1.32 to 1.97 μm), as shown in Fig. 5.3.
The second fiber has a chromatic dispersion of 0.025 ps/nm.km and a dispersion slope
of 0.002 ps/nm2.km both at 1.55 μm wavelength. Moreover, the dispersion is within
±3 ps/nm.km over even a wider wavelength range (1.34 to 2.05 μm), as shown in
Fig. 5.4.
Table 5.2: Fresnel fiber designs with flat dispersion around λ = 1.55 μm.
Fresnel Material Material Core Radius Ring Area Number of Effective Area at
Fiber 1 2 [ μm] [ μm2] Rings λ = 1.55 μm [ μm2]
11 M1 M4 1.57 1 2 10.25
12 M1 M9 1.25 0.25 10 6.5
During the design process of dispersion-flattened fiber, it was observed that increasing
the core radius has the effect of shifting the dispersion curve up, as shown in Fig. 5.5,
while increasing the ring area shifts the dispersion curve up and to the right, as shown
in Fig. 5.6. Therefore, the design of dispersion-flattened fiber requires searching for
the optimal values of the core radius and ring area for the specific materials and
number of rings used.
60
M. R. Albandakji Chapter 5. Fresnel Fibers
1 1.2 1.4 1.6 1.8 2−45
−40
−35
−30
−25
−20
−15
−10
−5
0
5
λ [μm]
D [p
s/(n
m.k
m)]
Figure 5.3: Dispersion-flattened Fresnel fiber no. 11.
1 1.2 1.4 1.6 1.8 2−40
−35
−30
−25
−20
−15
−10
−5
0
5
λ [μm]
D [p
s/(n
m.k
m)]
Figure 5.4: Dispersion-flattened Fresnel fiber no. 12.
61
M. R. Albandakji Chapter 5. Fresnel Fibers
1 1.2 1.4 1.6 1.8 2−45
−40
−35
−30
−25
−20
−15
−10
−5
0
5
λ [μm]
D [p
s/(n
m.k
m)]
optimum core radiussmall core radiuslarge core radius
Figure 5.5: The effect of changing the core radius of Fresnel fiber no. 11 on dispersion curve. Core
radii used are 1.54, 1.57, and 1.6 μm (bottom to top).
1 1.2 1.4 1.6 1.8 2−45
−40
−35
−30
−25
−20
−15
−10
−5
0
5
λ [μm]
D [p
s/(n
m.k
m)]
optimum ring areasmall ring arealarge ring area
Figure 5.6: The effect of changing the ring area of Fresnel fiber no. 11 on dispersion curve. Ring
areas used are 0.8, 1.0, and 1.2 μm (bottom to top).
62
M. R. Albandakji Chapter 5. Fresnel Fibers
5.3.3 Dispersion Compensating Fibers
Dispersion compensating fibers have very large negative chromatic dispersion at
1.55 μm. These fibers are used in upgrading the older 1.3 μm fiber-optic systems,
which have a fairly large positive dispersion at 1.55 μm. Therefore, dispersion com-
pensating fibers can compensate for the accumulated positive dispersion over the
fiber’s link and, at the same time, obtain the benefit of the small attenuation at
1.55 μm.
Table 5.3 shows two dispersion compensating single-mode fiber designs. One design
provides a total dispersion of -134 ps/nm.km and dispersion slope of 0.003 ps/nm2.km
between 1.5 and 1.6 μm, as shown in Fig. 5.7. The second fiber design provides a
total dispersion of -170 ps/nm.km, as shown in Fig. 5.8.
Table 5.3: Fresnel fiber designs with large negative dispersion at λ = 1.55 μm.
Fresnel Material Material Core Radius Ring Area Number of Effective Area at
Fiber 1 2 [ μm] [ μm2] Rings λ = 1.55 μm [ μm2]
13 M1 M9 0.8 0.2 2 7.2
14 M1 M2 0.5 0.5 2 11.6
63
M. R. Albandakji Chapter 5. Fresnel Fibers
1 1.2 1.4 1.6 1.8 2−135
−130
−125
−120
−115
−110
−105
−100
−95
λ [μm]
D [p
s/(n
m.k
m)]
Figure 5.7: Dispersion compensating Fresnel fiber no. 13.
1 1.2 1.4 1.6 1.8 2−240
−220
−200
−180
−160
−140
−120
−100
−80
−60
−40
λ [μm]
D [p
s/(n
m.k
m)]
Figure 5.8: Dispersion compensating Fresnel fiber no. 14.
64
Chapter
6 Analysis of Planar Photonic
Crystal Waveguides
6.1 Overview
Planar photonic crystal waveguides (PPCW) are periodic one-dimensional (1-D) struc-
tures consisting of a core layer bound between two semi-infinite alternating layers of
materials with two different refractive indices. Figure 6.1 illustrates the geometry of
a 1-D planar photonic crystal waveguide. The analysis of this structure is straightfor-
ward when it has a finite number of cladding layers, but is by no means obvious when
the number of cladding layers approaches infinity. In this chapter, we will first present
a systematic procedure to solve for the fields in a planar waveguide with an arbitrary,
but finite, number of cladding layers. Then, we will review the analysis of the ideal
structure with infinite number of periodic cladding layers based on a novel impedance
approach [59, 90]. We will focus attention on true photonic crystal modes and show
that these modes can be supported in both high- and low-index core regions.
65
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
.
.
.
.
.
.-
Figure 6.1: A planar photonic crystal waveguide.
6.2 Analysis of PPCW with Finite Number of
Cladding Layers
Let us consider a 1-D PPCW consisting of 2N − 2 cladding layers stacked along the
x-axis with alternating refractive indices n1 and n2, and thicknesses d1 and d2. We
also assume that the waveguide is symmetric about the x = 0 plane, and has a core
layer with thickness 2d0 and a refractive index n1. Figure 6.2 shows the index profile
of the structure.
n(x)
x
n1
d1
+d0
…n2…
d0
1
N1
N
d1d2 d2
Figure 6.2: Index profile of a planar photonic crystal waveguide.
Similar to the analysis of the 1-D photonic crystal fiber presented in Chapter 2, we
can solve the wave equation for axial field components and then obtain the transverse
66
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
components from the axial components at the ith layer using the following relations
eiy =
jZ0
k0
(n2
i − β2) dhi
z
dx(6.1a)
hix =
−jβ
k0
(n2
i − β2) dhi
z
dx(6.1b)
eix =
jβ
k0
(n2
i − β2) dei
z
dx(6.1c)
hiy =
−jn2i
Z0k0
(n2
i − β2) dei
z
dx(6.1d)
In the above equations, Z0 is the free space characteristic impedance = 120π Ω, k0 is
the free space wave number = 2π/λ, β is the normalized propagation constant, and
ni is the refractive index of the ith layer. It is noted that the field components in
the above equations can be easily separated into TE and TM modes. The TE modes
include hx, hz, and ey, while the TM modes include ex, ez, and hy field components.
In general, the solutions of the wave equation for the TE mode in the ith layer can
be expressed as
hiz = AiFi (uix) + BiFi (uix) (6.2a)
hix =
−j(−1)i+1k0β
ui
[AiF
′i (uix) + BiF
′i (uix)
](6.2b)
eiy =
j(−1)i+1k0Z0
ui
[AiF
′i (uix) + BiF
′i (uix)
](6.2c)
and the solutions for the TM mode in the ith layer can be expressed as
eiz = AiFi (uix) + BiFi (uix) (6.3a)
eix =
j(−1)i+1k0β
ui
[AiF
′i (uix) + BiF
′i (uix)
](6.3b)
hiy =
−j(−1)i+1k0n2i
Z0ui
[AiF
′i (uix) + BiF
′i (uix)
](6.3c)
67
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
where
Fi (ui, x) =
⎧⎨⎩ sin (uix) ; β < ni
e−uix ; β > ni
(6.4a)
Fi (ui, x) =
⎧⎨⎩ cos (uix) ; β < ni
euix ; β > ni
(6.4b)
F′i (ui, x) =
⎧⎨⎩ cos (uix) ; β < ni
−e−uix ; β > ni
(6.4c)
F′i (ui, x) =
⎧⎨⎩ − sin (uix) ; β < ni
euix ; β > ni
(6.4d)
with
ui = k0
√∣∣n2i − β2
∣∣ (6.5)
and Ai and Bi are constant amplitude coefficients obtained by applying the boundary
conditions. The boundary conditions at the interface of two dielectric regions require
the continuity of tangential components of the electric and magnetic fields. Accord-
ingly, the field components ey, ez, hy, and hz must be continuous at x = d0, d0 + d1,
d0 + d1 + d2, etc. Doing so, we obtain two equations at every interface, leading us to
a matrix expression that relates the coefficients of the N th layer to those of the first
layer, which is given by⎡⎣ AN
BN
⎤⎦ =N−1∏i=1
⎡⎣ γi11 γi
12
γi21 γi
22
⎤⎦⎡⎣ A1
B1
⎤⎦ =
⎡⎣ Γ11 Γ12
Γ21 Γ22
⎤⎦⎡⎣ A1
B1
⎤⎦ (6.6)
where
γi11 =
⎧⎨⎩12
[sin (Ui) +
(n2
i
n2i+1
)mWi
Uicos (Ui)
]eWi ; β < ni[
sin (Wi) +(
n2i
n2i+1
)mWi
Uicos (Wi)
]e−Ui ; β > ni
(6.7a)
68
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
γi12 =
⎧⎨⎩12
[cos (Ui) −
(n2
i
n2i+1
)mWi
Uisin (Ui)
]eWi ; β < ni[
sin (Wi) −(
n2i
n2i+1
)mWi
Uicos (Wi)
]eUi ; β > ni
(6.7b)
γi21 =
⎧⎨⎩12
[sin (Ui) −
(n2
i
n2i+1
)mWi
Uicos (Ui)
]e−Wi ; β < ni[
cos (Wi) −(
n2i
n2i+1
)mWi
Uisin (Wi)
]e−Ui ; β > ni
(6.7c)
γi22 =
⎧⎨⎩12
[cos (Ui) +
(n2
i
n2i+1
)mWi
Uisin (Ui)
]e−Wi ; β < ni[
cos (Wi) +(
n2i
n2i+1
)mWi
Uisin (Wi)
]eUi ; β > ni
(6.7d)
with
Ui = uixi (6.8a)
W = ui+1xi (6.8b)
For TE modes m = 0, whereas for TM modes m = 1. Now to solve for the even
modes, B1 is set to zero in order to eliminate the cosine term in the axial components
of the fields and A1, chosen as the independent field coefficient, can be set to 1. On
the other hand, the odd modes are obtained by setting A1 to zero in order to eliminate
the sine term in the axial components of the fields and B1, chosen as the independent
field coefficient, can be set to 1. Also, in order to have guided (non-radiating) fields,
they have to be exponentially decaying in the outermost cladding layer, so BN in
Eq. (6.6) must be set to zero leading to the following results
Γ21 = 0 ; for even TE or TM modes (6.9a)
Γ22 = 0 ; for odd TE or TM modes (6.9b)
Equations (6.9a) and (6.9b) are the characteristic or eigenvalue equations from which
β of the finite PPCW structure can be obtained.
69
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
6.3 Analysis of PPCW with Infinite Number of
Periodic Cladding Layers
In the analysis of 1-D planar photonic crystal waveguide, we assume that the structure
has an index profile similar to the one shown in Fig. 6.2, where each cladding is a
semi-infinite periodic structure. The thicknesses of the core and cladding layers and
their indices remain the same.
For the semi-infinite periodic structure shown in Fig. 6.3, we can notice that the input
impedance seen at x = 0 is equivalent to the input impedance seen at x = d1 + d2,
where d1 and d2 are the thicknesses of the alternating layers of the cladding. This
means that the impedance of the structure beyond x = d1 + d2 can be replaced by
the input impedance seen at x = 0, which is expressed as
Zin = Z2Zin + jZ2 tan (β2d2)
Z2 + jZin tan (β2d2)(6.10)
Zin is the impedance of the structure beyond x = d1 and it is expressed as
Zin = Z1Zin + jZ1 tan (β1d1)
Z1 + jZin tan (β1d1)(6.11)
inZ inZ inZx
z
d1 d2
Figure 6.3: Planar semi-infinite periodic structure with Zin and Zin shown.
70
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
where
Z1 =
⎧⎨⎩ Z0u/n21 ; for TM modes
Z0/u ; for TE modes(6.12a)
u =√
n21 − β2 (6.12b)
β1 = k0u (6.12c)
with β = β/k0. Now, if n2 < β < n1, we obtain
Z2 =
⎧⎨⎩ jZ0w/n22 ; for TM modes
−jZ0/w ; for TE modes(6.13a)
w =√
β2 − n22 (6.13b)
β2 = jk0w (6.13c)
otherwise, if β < n1 < n2 or β < n2 < n1, we obtain
Z2 =
⎧⎨⎩ Z0w/n22 ; for TM modes
Z0/w ; for TE modes(6.14a)
w =√
n22 − β2 (6.14b)
β2 = k0w (6.14c)
Substituting Eq. (6.11) into (6.10) and after some algebraic simplification, we obtain
the following solution for Zin
Zin =−b ∓√
b2 − 4ac
2a(6.15)
where
a = Z1 tan (β2d2) + Z2 tan (β1d1) (6.16a)
b = j[(
Z21 − Z2
2
)tan (β1d1) tan (β2d2)
](6.16b)
c = −Z1Z2 [Z1 tan (β1d1) + Z2 tan (β2d2)] (6.16c)
71
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
For the case when n2 < β < n1, we combine Eqs. (6.12b) and (6.13b) with Eqs. (6.16a),
(6.16b), and (6.16c) to obtain
a = jZ0a = jZ0
⎧⎨⎩un2
1tanh (W ) + w
n22tan (U) ; for TM modes
1u
tanh (W ) − 1w
tan (U) ; for TE modes(6.17a)
b = −Z20 b = −Z2
0
⎧⎪⎨⎪⎩[(
un2
1
)2+(
wn2
2
)2]tan (U) tanh (W ) ; for TM modes[(
1u
)2+(
1w
)2]tan (U) tanh (W ) ; for TE modes
(6.17b)
c = −jZ30 c = −jZ3
0
⎧⎨⎩uw
n21n2
2
[un2
1tan (U) − w
n22tanh (W )
]; for TM modes
− 1uw
[1u
tan (U) + 1w
tanh (W )]
; for TE modes(6.17c)
where
U = k0d1u (6.18a)
W = k0d2w (6.18b)
For the case when β < n1 < n2 or β < n2 < n1, we combine Eqs. (6.12b) and (6.14b)
with Eqs. (6.16a), (6.16b), and (6.16c) to obtain
a = Z0a = Z0
⎧⎨⎩ un2
1tan(W)
+ wn2
2tan (U) ; for TM modes
1u
tan(W)
+ 1w
tan (U) ; for TE modes(6.19a)
b = jZ20 b = jZ2
0
⎧⎪⎨⎪⎩[(
un2
1
)2−(
wn2
2
)2]tan (U) tan
(W)
; for TM modes[(1u
)2 − ( 1w
)2]tan (U) tan
(W)
; for TE modes
(6.19b)
c = −Z30 c = −Z3
0
⎧⎨⎩uw
n21n2
2
[un2
1tan (U) + w
n22tan(W)]
; for TM modes
1uw
[1u
tan (U) + 1w
tan(W)]
; for TE modes(6.19c)
where
W = k0d2w (6.20)
72
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
We can simplify Zin further by using the normalized coefficients a, b, and c or a, b,
and c defined in the above equations to obtain the following relations
Zin =
⎧⎨⎩ −jZ0b∓√
b2−4ac
2a; n2 < β < n1
−jZ0b±√
b2−4ac2a
; β < n2 < n1 or β < n1 < n2
(6.21)
The characteristic equation for the guided modes can be obtained by using the trans-
verse resonance condition. According to this condition, the total phase change in the
transverse direction in the core region for one complete cycle of a ray representing
a guided mode must be an integer multiple of 2π. This condition is, in fact, the
requirement for the constructive interference of the ray with itself after reflection.
Mathematically, this condition is expressed as [91]
−2k0
∫ d0
−d0
√n2 (x) − β2dx + 2θΓ = 2νπ (6.22)
where θΓ is the phase angle of the reflection coefficient (Γ) at x = −d0 and x = d0
and ν is an integer. The integral term in the transverse resonance equation (6.22)
can be easily evaluated leading to the following expression
θΓ = νπ + 2U0 (6.23)
where U0 = k0d0u.
We can use the theory of transmission lines and wave impedance to obtain θΓ. It
can be easily shown that the reflection coefficient (Γ) for an incident wave on the
core-cladding interface is given by
Γ =Zin − Z1
Zin + Z1(6.24)
Now θΓ can be evaluated from Eq. (6.24) with Zin substituted from Eq. (6.21). Doing
73
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
so, we find
Γ =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩
−j b∓√
b2−4ac2a
−Z1
−j b∓√
b2−4ac2a
+Z1
; n2 < β < n1
−j b±√
b2−4ac2a
−Z1
−j b±√
b2−4ac2a
+Z1
; β < n2 < n1 or β < n1 < n2
(6.25)
where Z1 = Z1/Z0. The phase of the reflection coefficient (θΓ) can be expressed as
θΓ =
⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩−2 tan−1
[b∓√
b2−4ac
2aZ1
]; n2 < β < n1
−2 tan−1
[b±√
b2−4ac
2aZ1
]; β < n2 < n1 or β < n1 < n2
(6.26)
Substituting θΓ from Eq. (6.26) into Eq. (6.23) and after some algebraic simplifica-
tions, yields(aZ2
1 − c)cos (2U0 + νπ) + bZ1 sin (2U0 + νπ) +
(aZ2
1 + c)
= 0 ; n2 < β < n1
(6.27a)
(aZ2
1 − c)cos (2U0 + νπ) + bZ1 sin (2U0 + νπ) +
(aZ2
1 + c)
= 0 ;β < n2 < n1
β < n1 < n2
(6.27b)
Equations (6.27a) and (6.27a) are the characteristic equations that can be solved
numerically to find the unknown normalized propagation constant β for a given
wavelength λ. These equations may result in different solutions for a specific wave-
length representing different modes. To solve for the TE (or TM) modes, these
equations are used with coefficient values that correspond to the TE (or TM) modes
in Eqs. (6.17a)−(6.17c) or Eqs. (6.19a)−(6.19c). Furthermore, solving Eq. (6.27a)
or (6.27b) with even integer values for ν results in the solutions for even modes, while
solving it with odd values results in the solutions for odd modes.
74
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
6.4 Comparison between PPCWs with Finite and
Infinite Number of Cladding Layers
In this section, we will compare the PPCW that has a finite number of cladding layers
with a similar PPCW that has an infinite number of cladding layers. In particular,
we want to show that the technique presented in Section 6.3 leads to the same results
obtained using the traditional matrix approach presented in Section 6.2, provided that
the number of cladding layers is sufficiently large. It is noted that the characteristic
equations (6.27a) and (6.27b) can be used to solve for all possible modes in the infinite
structure, whereas the characteristic equations (6.9a) and (6.9b) can only be used to
solve for modes that have n2 < β < n1 in the finite structure.
As an example, let us consider a high-index core PPCW with d0 = 1 μm, d1 = 0.5 μm,
d2 = 0.5 μm, material 1 is M11 (13.5 m/0 GeO2, 86.5 m/0 SiO2) with index n1 (λ),
and material 2 is M12 (SiO2) with index n2 (λ). The β values are calculated for a
range of wavelengths varying between 1.0 to 2.0 μm, in increments of 0.01 μm. For
the infinite-layer cladding case (N = ∞), β is calculated from Eq. (6.27a), whereas
for the finite number of layers cases, β is calculated from Eq. (6.9a). In both cases,
we used the interval-halving technique with roots of accuracy in the order of ±10−14
to calculate the first even TE and TM modes (TE0 and TM0). Figure 6.4 shows
the dispersion curves for TE0 mode for different number of cladding layers. The
results for TM0 are very similar to TE0 so they are not shown here in order to avoid
repetition. As expected, the dispersion curves converge to the infinite case as the
total number of layers (N) increases. We can notice that the difference is small
for small wavelengths, but it becomes larger as λ increases because the accuracy of
approximating the infinite structure by the finite structure with large N decreases as
75
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
λ increases. Similar behavior is also observed when the structural parameters d0, d1,
and d2 are decreased and when the index difference of the materials used is increased.
1 1.2 1.4 1.6 1.8 21.445
1.45
1.455
1.46
1.465
1.47
λ [μm]
β
N = ∞N = 2N = 4N = 6N = 8N = 10
Figure 6.4: Dispersion curves for TE0 mode for different number of cladding layers. The PPCW
parameters are d0 = 1 μm, d1 = 0.5 μm, d2 = 0.5 μm, material 1 is M11, and material 2 is M12.
6.5 True Photonic Crystal Modes in PPCWs
In this section we will investigate the modes of ideal PPCWs with infinite number of
cladding layers. We will show that these ideal structures can support true photonic
crystal modes in both high-index and low-index cores. In the high-index core case,
the supported modes are guided inside the core region due to total internal reflection
and possibly the photonic band gap effect when n2 < β < n1. We will show that
true photonic crystal modes for which β < n2 < n1 can exist due to the perfect
76
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
photonic band gap that is formed in the ideal structure. Such modes do not exist in
conventional planar waveguides. Also, for the low-index core case where β < n1 < n2,
we will show that guidance is only allowed for the true photonic crystal modes inside
the photonic band gap of the structure.
6.5.1 High-Index Core PPCWs
As an example, let us consider a high-index core ideal PPCW with d0 = 2 μm,
d1 = 1 μm, d2 = 1 μm, material 1 is M1 (16.9 m/0 Na2O, 50.6 m/0 SiO2, 32.5
m/0 B2O3) with index n1 (λ), and material 2 is M5 (Quenched SiO2) with index
n2 (λ). The β values are calculated for a range of wavelengths varying between 1.0
to 2.0 μm, in increments of 0.01 μm. For the infinite-layer cladding, the solutions
for β are calculated from Eq. (6.27a) when n2 < β < n1 and from Eq. (6.27b) when
β < n2 using the interval-halving root search technique with accuracies in the order
of ±10−14 for both TE and TM modes. Figure 6.5 shows several modes supported
by the waveguide. In this figure, we also show the region of the allowed modes when
the number of cladding layers is finite and the index of the outermost cladding layer
is n2. It is noted that there are modes with β < n2 (λ), which means that this ideal
structure is able to support modes that can not be supported in structures with finite
number of cladding layers since a perfect photonic band gap can be formed in the
infinite structure. At λ = 1.3 μm, a guided TE2 mode is allowed to propagate due to
both total internal reflection and the photonic band gap effect, while at λ = 1.55 μm
the same mode is supported due to the photonic band gap effect only. The fields
corresponding to the first case are shown in Figs. 6.6(a), 6.6(b), and 6.6(c), while the
fields of the second case are shown in Figs. 6.7(a), 6.7(b), and 6.7(c).
77
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
λ [μm]
β
1 1.2 1.4 1.6 1.8 21.36
1.38
1.4
1.42
1.44
1.46
1.48
1.5
1.52
TE0
TM0
TM1
TE1
TE2
TM2
TE3
TM3
TE modeTM mode
Figure 6.5: Normalized propagation constant for several TE and TM modes of an ideal PPCW
with d0 = 2 μm, d1 = 1 μm, d2 = 1 μm, material 1 is M1, and Material 2 is M5. Gray area is the
region of allowed modes when the PPCW has a finite number of cladding layers.
0 5 10 15 20−1000
−800
−600
−400
−200
0
200
400
600
800
1000
x [μm]
Ey
(a)
78
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
0 5 10 15 20−4
−3
−2
−1
0
1
2
3
4
x [μm]
Hx
(b)
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x [μm]
Hz
(c)
Figure 6.6: Field distributions for TE2 mode at λ = 1.3 μm: (a) Ey , (b) Hx, and (c) Hz.
79
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
0 5 10 15 20−1000
−800
−600
−400
−200
0
200
400
600
800
1000
x [μm]
Ey
(a)
0 5 10 15 20−4
−3
−2
−1
0
1
2
3
4
x [μm]
Hx
(b)
80
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
0 5 10 15 20−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x [μm]
Hz
(c)
Figure 6.7: Field distributions for TE2 mode at λ = 1.55 μm: (a) Ey, (b) Hx, and (c) Hz. In this
case, the mode is a true photonic crystal mode with β < n2 < n1.
6.5.2 Low-Index Core PPCWs
Here we will consider a low-index core ideal PPCW with d0 = 2 μm, d1 = 1 μm,
d2 = 1 μm, material 1 is M12 (pure SiO2) with index n1 (λ), and material 2 is M11
(13.5 m/0 GeO2, 86.5 m/0 SiO2) with index n2 (λ). Again, β values are calculated
for a range of wavelengths varying between 1.0 to 2.0 μm, in increments of 0.01 μm,
and the solutions for β are calculated from Eq. (6.27b) since β < n1 < n2 for this
structure. We used the interval-halving root search technique with accuracies in the
order of ±10−14 to search for both TE and TM modes. Figure 6.8 shows several modes
supported by the waveguide. It is noted that all these modes have their β < n1 < n2,
which means that this ideal structure is able to guide modes in a low-index core
81
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
λ [μm]
β
1 1.2 1.4 1.6 1.8 21
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
1.45
1.5
TE0
TE1
TE3
TM0
TM1
TM3
TM4
TE4
TE5
TM5
TM6
TE6
Figure 6.8: Normalized propagation constant for several TE and TM modes of an ideal PPCW
with d0 = 2 μm, d1 = 1 μm, d2 = 1 μm, material 1 is M12, and Material 2 is M11. Gray area is the
region of possible modes in practical PPCW if n1 and n2 were interchanged.
because of the perfect photonic band gap that can be formed in the infinite structure.
Also, we notice that TE and TM modes become degenerate to each other in this type
of structure. On the same Figure, we show the region of possible modes in practical
PPCW if n1 and n2 were interchanged. As an example, we plot the fields of the TM1
mode at λ = 1.3 μm in Figs. 6.9(a), 6.9(b), and 6.9(c). It is noted that the fields in
the low-index core need more layers to decay to insignificant values when compared
to the modes with β < n2 in the high-index core case.
82
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
0 5 10 15 20 25 30 35 40−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
x [μm]
Hy
(a)
0 5 10 15 20 25 30 35 40−5
−4
−3
−2
−1
0
1
2
3
4
x [μm]
Ex
(b)
83
M. R. Albandakji Chapter 6. Analysis of Planar Photonic Crystal Waveguides
0 5 10 15 20 25 30 35 40−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x [μm]
Ez
(c)
Figure 6.9: Field distributions for TM1 mode at λ = 1.3 μm: (a) Hy, (b) Ex, and (c) Ez. In this
case, the mode is a true photonic crystal mode with β < n1 < n2.
84
Chapter
7 Analysis of Ideal One-Dimensional
Photonic Crystal Fibers
7.1 Overview
Practical one-dimensional (1-D) photonic crystal fibers (PCFs) are manufactured with
a finite number of rings; therefore, the modes inside these structures are inherently
leaky modes since the outermost cladding layer is usually made from silica. It is very
attractive to study theoretically the ideal 1-D PCF structure which has an infinite
number of cladding rings. This is because this structure can allow for the propagation
of truly guided modes; i.e., guided modes that have no leakage loss. Also, this ideal
structure may serve as a more accurate model for two-dimensional (2-D) PCFs with
large number of air hole rings.
85
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
7.2 Method of Analysis
Here, we use a new idea to analyze the infinite cladding 1-D PCF problem based on an
impedance approach and a recent publication that suggests modeling the optical fiber
as a transmission-line with a series of cascaded T-circuits connected in tandem [67],
as shown in Fig. 7.1.
ZB (r) ZB (r)
ZP (r)
rr r+ r
Figure 7.1: Equivalent transmission-line circuit of an optical fiber.
The equivalent T-circuit has two impedances given by [67]
ZB = sinh (γδr) tanh
(γδr
2
)Zp (7.1a)
Zp =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩γZ0
jnrk30
�β2+
�ν
k0r
�2�
sinh(γδr); HE/EH modes
γZ0
jrk30β2 sinh(γδr)
; TE modes
γZ0
jn2rk30β2 sinh(γδr)
; TM modes
(7.1b)
where
γ = k0
[β2 − n2 +
(ν
k0r
)2
∓ 2νnβ
ν2 +(k0βr)2]1/2
;− for HE modes
+ for EH modes(7.2)
In the above equations, β stands for the normalized propagation constant to be cal-
culated, ν is the azimuthal mode number, n is the refractive index of the layer at
86
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
distance r from the axis of the fiber, δr is the radial distance in the fiber cross-section
(the length of the transmission line), k0 is the free space wave number, and Z0 is the
free space characteristic impedance. The cascaded circuits are terminated with the
characteristic impedance of the medium of the core at r = 0, and the characteristic
impedance of the outer cladding at r = ∞.
When a guided mode propagates inside the fiber, the optical energy becomes trapped
inside the core and the cascaded equivalent T-circuits resonate [92]. This resonance
condition occurs when the sum of the input impedance from r = 0 up to the core-
cladding boundary, Zin, and the output impedance from r = ∞ up to the core-
cladding boundary, Zout, equals to zero. This means that at resonance Ztotal =
Zin + Zout = 0, which occurs only at β value of a guided mode. So we can use a root
searching technique to locate the roots of Ztotal of the equivalent T-circuits to obtain
β.
Theoretically, a confined mode is obtained if the alternating index cladding is infinite
in thickness. However, it is found that the field decay is nearly complete in several
pairs of cladding layers so that practical structures with, say ten pairs of cladding
layers, are good approximation to the infinite alternating index cladding [61]. There-
fore, we can use the equivalent T-circuits to model the fiber rings that are close to the
core, then at a sufficiently large distance from the core, say R, the rest of the cylin-
drical cladding rings can be well approximated by planar layers. The main advantage
of this approach is that it allows us to model the remaining cladding structure by a
single equivalent impedance since the impedance seen at R becomes equivalent to the
impedance seen at R + d1 + d2, where d1 + d2 is the period of the alternating layers,
as shown in Fig. 7.2(a).
In Fig. 7.2(b), ZA, ZB, and ZC are the impedances of the equivalent T-circuit for
87
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
r
Zeq
Zeq
…
d1 d2
R
n1 n1n2 n2
d1 d2
…
(a)
rR
Zeq
ZA ZB ZD ZE
ZC ZF Zeq
(b)
Figure 7.2: Equivalent circuits for the analysis of 1-D PCF structure with infinite number of rings:
(a) actual structure and (b) equivalent circuit model.
88
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
one layer, whereas ZD, ZE, and ZF are the impedances of the equivalent T-circuit for
the other layer. Each impedance represents the equivalent impedance for one layer
and it is calculated by combining the infinitesimal impedances using circuit theory
techniques including a series of Y-Δ and Δ-Y impedance transformations. From basic
circuits theory, Zeq is given by
Zeq =
[(Zeq+ZE)ZF
Zeq+ZE+ZF+ ZB + ZD
]ZC
(Zeq+ZE)ZF
Zeq+ZE+ZF+ ZB + ZC + ZD
+ ZA (7.3)
Solving for Zeq, we obtain the following quadratic equation
AZ2eq + BZeq + C = 0 (7.4)
where
A = ZB + ZC + ZD + ZF (7.5a)
B =ZEZF + (ZE + ZF ) (ZB + ZC + ZD)−ZA (ZB + ZC + ZD + ZF ) − ZC (ZB + ZD + ZF )
(7.5b)
C =−{ZA [ZEZF + (ZE + ZF ) (ZB + ZC + ZD)] +
ZCZEZF + ZC (ZB + ZD) (ZE + ZF )}(7.5c)
We used Zeq to represent the semi-infinite structure beyond radius R to calculate
Zout, then we used the interval halving technique as a root searching method to
determine the roots of Ztotal of the equivalent T-circuits and then obtain β. Once β
is calculated, the field and power distributions can be easily found using the matrix
approach discussed in Chapter 2. The results of this technique are presented in the
following section.
89
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
7.3 Results
7.3.1 Solid-Core PCF
The solid-core PCF, also known as periodic coaxial fiber, can be viewed as a step-
index fiber that is surrounded by a cylindrical 1-D photonic crystal structure, as
shown is Fig. 7.3. The photonic crystal structure is designed so that light of certain
frequency incident from the low-index medium is totally reflected back, no matter
what the incident angle and polarization are [93]. In other words, the light frequency
should be within the photonic band gap of the photonic crystal structure in order to
achieve guidance inside the coaxial region. This makes the structure very similar to
the metallic coaxial cables, although there is a substantial amount of power penetrated
through the high-index core and the photonic crystal cladding [64].
Figure 7.3: Periodic coaxial fiber.
In order to illustrate the accuracy of the impedance approach, we used the periodic
coaxial fiber discussed in [63] which has an index profile similar to the one shown in
Fig. 7.4. The fiber was found to be capable of supporting photonic crystal modes
inside the photonic band gap of the photonic crystal structure surrounding the coaxial
region. When λ = 4.9261 μm, the fiber can support a fundamental TM photonic
90
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
n(r)
r
…1.6 1.0
4.6
1.0 m
0.4 m
1/3m
2/3m
Figure 7.4: Index profile of the discussed periodic coaxial fiber.
crystal mode with β = 0.9852 (less than unity). Using the impedance approach with
δr = 0.005 μm and 15 periodic layers, β was found to be equal to 0.99, which is
0.48 % different from the reported value. The fields and power of the fundamental
TM mode are shown in Figs. 7.5 and 7.6, respectively. It is noted that the main
power is confined inside the coaxial region between r = 0.4 μm and r = 1.4 μm.
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.2
0
0.2
0.4
0.6
0.8
1
r [μm]
Nor
mal
ized
Hφ
(a)
91
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.2
0
0.2
0.4
0.6
0.8
1
r [μm]
Nor
mal
ized
Er
(b)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.2
0
0.2
0.4
0.6
0.8
1
r [μm]
Nor
mal
ized
Ez
(c)
Figure 7.5: Transverse field distribution for TM01 mode when λ = 4.9261 μm: (a) Hϕ, (b) Er,
and (c) Ez.
92
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
Figure 7.6: Power density distribution for TM01 mode when λ = 4.9261 μm.
7.3.2 Hollow-Core PCF
The hollow-core PCF consists of a low-index core surrounded by a cylindrical 1-D
photonic crystal structure, as shown in Fig. 7.7. The special feature of this fiber is
that it is capable of guiding light inside its core even if the core index is smaller than
the average cladding index. This happens when the frequency of the propagating
wave is within the photonic band gap of the photonic crystal structure surrounding
the core, so the wave will be totally reflected back due to Bragg reflection [94]. This
guiding mechanism has attracted the attention of many researchers recently because it
offers lower material absorption and higher power threshold for nonlinear effects [61].
We used the impedance approach discussed in the previous section to study the
photonic crystal modes that can be guided inside the hollow-core PCF structure. As
93
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
Figure 7.7: Hollow-core PCF.
an example, we simulated the PCF reported in [95], which has an air core radius
of 5 μm, refractive indices of the alternating layers n1 = 1.0 and n2 = 2.0, along
with widths d1 = 1.0 μm and d2 = 1.0 μm, respectively. The number of layers was
chosen to be 30 and the infinitesimal radial distance δr = 0.01 and 0.005 μm. When
δr = 0.01 μm and the wave number k0 = 1.2; i.e., λ = 5.236 μm, β was found to be
0.7863 for the first TE mode and 0.5282 for the first TM mode but when we increased
δr to 0.005 μm, β was found to be 0.7861 for the first TE mode and 0.5276 for the
first TM mode, which means that larger δr provides higher accuracy as expected. We
notice that β values for both modes are less than unity, which is a unique feature of
guided modes in hollow-core PCFs.
Figures. 7.8(a), 7.8(b), and 7.8(c) show the calculated fields for the first TE mode,
whereas Figs. 7.10(a), 7.10(b), and 7.10(c) show the calculated fields for the first
TM mode when k0 = 1.2 and δr = 0.005 μm. We notice that azimuthal and axial
fields components are continuous, whereas the radial components have discontinuities
at the boundaries because of the boundary conditions. Also, we notice that all the
fields decay to almost zero within a few pairs of the cladding layers, which ensures
that these modes are truly guided modes inside the hollow-core PCF. Figures. 7.9
and 7.11 show the power of the first TE and TM modes, respectively. We notice that
94
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
the power is almost totally confined inside the core region for both modes. Table 7.1
compares these results with three different techniques studied in [95]. We can see
that the impedance method provides a very small percentage error when compared
to the transfer matrix method.
Table 7.1: Impedance method compared to three different techniques studied in [95].
TE01 % Error TM01 % Error
Transfer matrix method 0.7859080 - 0.5270 -
Asymptotic method 0.79935 1.7 0.5785 9.8
Galerkin method 0.7858 0.014 0.5335 1.2
Impedance method (δr = 0.01 μm) 0.7863 0.05 0.5282 0.23
Impedance method (δr = 0.005 μm) 0.7861 0.02 0.5276 0.11
Also, we simulated a more practical case at λ = 1.3 μm for a hollow-core PCF
structure with a core radius of 2 μm, refractive indices of the alternating layers n1 =
1.4504 (Material 12) and n2 = 1.4716 (Material 11), along with widths d1 = 1.0 μm
and d2 = 1.0 μm, respectively. The number of layers was chosen to be 30 and the
infinitesimal radial distance δr = 0.005. β was found to be 1.44536 for the first TE
mode and 1.44335 for the first TM mode. We notice that β values for both modes
are less than n1. The fields and power distribution for both modes are very similar
so, in order to avoid repetition, we show those corresponding to the TM mode only
in Figs. 7.12(a), 7.12(b), 7.12(c), and 7.13.
95
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 5 10 15 20 25−1
−0.5
0
0.5
r [μm]
Nor
mal
ized
Eφ
(a)
0 5 10 15 20 25−0.5
0
0.5
1
r [μm]
Nor
mal
ized
Hr
(b)
96
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 5 10 15 20 25−0.5
0
0.5
1
r [μm]
Nor
mal
ized
Hz
(c)
Figure 7.8: Transverse field distribution for TE01 mode when k0 = 1.2: (a) Eϕ, (b) Hr, and (c)
Hz.
Figure 7.9: Power density distribution for TE01 mode when k0 = 1.2.
97
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 5 10 15 20 25−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
r [μm]
Nor
mal
ized
Hφ
(a)
0 5 10 15 20 25−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
r [μm]
Nor
mal
ized
Er
(b)
98
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 5 10 15 20 25−0.5
0
0.5
1
r [μm]
Nor
mal
ized
Ez
(c)
Figure 7.10: Transverse field distribution for TM01 mode when k0 = 1.2: (a) Hϕ, (b) Er , and (c)
Ez .
Figure 7.11: Power density distribution for TM01 mode when k0 = 1.2.
99
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 5 10 15 20 25 30−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
r [μm]
Nor
mal
ized
Hφ
(a)
0 5 10 15 20 25 30−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
r [μm]
Nor
mal
ized
Er
(b)
100
M. R. Albandakji Chapter 7. Analysis of Ideal One-Dimensional Photonic Crystal Fibers
0 5 10 15 20 25 30−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
r [μm]
Nor
mal
ized
Ez
(c)
Figure 7.12: Transverse field distribution for TM01 mode when λ = 1.3 μm: (a) Hϕ, (b) Er, and
(c) Ez .
Figure 7.13: Power density distribution for TM01 mode when λ = 1.3 μm.
101
Chapter
8 Conclusions and Directions for
Future Work
8.1 Conclusions
In this work, several aspects of photonic crystal waveguides have been investigated,
including one-dimensional (1-D) modeling of two-dimensional (2-D) photonic crystal
fibers (PCFs), analysis of tapered PCFs, analysis of 1-D PCFs and planar photonic
crystal waveguides with infinite periodic cladding, and investigation of transmission
properties of a novel type of fibers referred to as Fresnel fiber.
We presented an exact vector-wave solution for the 1-D PCFs, which was used to
model 2-D PCFs. We showed that this model can predict transmission properties
of 2-D PCFs with a relatively high degree of accuracy and much less processing
time and computer storage when compared to conventional numerical techniques,
such as the finite element method and the multipole method. In particular, the
normalized propagation constant agreed within 0.2 % for λ/Λ ≤ 0.5 and within
0.7 % for λ/Λ ≤ 1.0. For PCFs with small values of normalized wavelengths; i.e.,
102
M. R. Albandakji Chapter 8. Conclusions and Directions for Future Work
λ/Λ ≤ 0.5, the model yielded dispersion results that agreed very well with published
results. Also, for large values of normalized holes sizes; i.e., d/Λ ≥ 0.5, the results
for effective area showed good agreement with those obtained using the finite element
method. However, the leakage loss was generally underestimated by the model. This
is believed to be due to the fact that the openings between the holes, which are the
main cause for the leakage of power, are replaced with closed rings.
Tapered PCFs were then investigated using the same model and by approximating
the tapered section as a series of uniform sections along the axial direction. We
showed that the total field inside the tapered section of the PCF can be evaluated as
a superposition of local normal modes that are coupled among each other. We also
studied the evolution of power density inside the taper and concluded that adiabaticity
depends mainly on two factors: the taper length and the taper shape. We showed
that when the tapered PCF is smooth and long enough, the propagating mode will
be capable of modifying itself to evolve adiabatically inside the taper. Otherwise, the
mode will start to couple to other modes and the propagation becomes nonadiabatic.
We also showed that the number of air hole rings has a minimum effect on taper
adiabaticity.
Also, a novel type of fibers, called Fresnel fiber, was analyzed. We showed that Fresnel
fibers can be properly designed to achieve fibers with shifted, flattened, or large
negative chromatic dispersion. In particular, we obtained two dispersion-flattened
single-mode fiber designs. The first design has a flat dispersion characteristic within
±5 ps/nm.km over a wide wavelength range from 1.32 μm to 1.97 μm. The second
fiber design has almost the same flat dispersion within ±5 ps/nm.km over even a wider
wavelength range from 1.28 μm to more than 2 μm. Two large negative dispersion
designs were also presented. One design provides a total dispersion of −134 ps/nm.km
103
M. R. Albandakji Chapter 8. Conclusions and Directions for Future Work
with almost flat dispersion between 1.5 μm and 1.6 μm wavelengths. The second fiber
design provides a total dispersion of −170 ps/nm.km at 1.55 μm wavelength. Also,
we showed that a wide range of effective areas can be achieved, making Fresnel fibers
very attractive in long distance fiber-optic communication links where large effective
area is desirable, or in soliton systems where small effective area is desirable. In
particular, we obtained a large effective area design with an effective area as high as
152.7 μm2 and nearly zero dispersion at 1.55 μm wavelength. Also, a small effective
area design with an effective area as small as 15.6 μm2 and nearly zero dispersion at
1.55 μm wavelength was achieved.
Also, we examined the ideal 1-D planar photonic crystal waveguides that consist of
infinite number of cladding layers based on an impedance approach. We presented
results which allow one to distinguish clearly between light guidance due to total
internal reflection and light guidance due to the photonic crystal effect.
Finally, we introduced a new approach for analyzing 1-D PCFs with infinite periodic
cladding. We used an equivalent T-circuits method to model the rings that are close to
the core of the fiber. Then, at sufficiently large distance from the core, the rest of the
cladding rings were approximated by planar layers. We showed that this approach can
successfully estimate the propagation constants and fields for true photonic crystal
modes in both solid-core and hollow-core PCFs with a very high accuracy.
In summary, the main contributions of this research are:
• A simple model for predicting transmission properties of PCFs was proposed [96–
98]. The model provides reasonably accurate estimate of propagation constant,
dispersion, and effective area.
• Adiabaticity of tapered PCFs was analyzed by modeling the PCF using the
104
M. R. Albandakji Chapter 8. Conclusions and Directions for Future Work
proposed model [99]. An estimate for the minimum taper length for a linear
PCF taper was presented. The effect of taper shape, taper length, and the
number of rings was investigated.
• A new type of optical fiber, referred to as Fresnel fiber, was proposed [100,101].
We showed that this type of fibers can be carefully designed to obtain desirable
dispersion and/or effective area properties, making them very suitable for use
in communications and sensing applications.
• Extended the analytical solutions of the ideal 1-D planar photonic crystal wave-
guides with infinite cladding to hollow-core structures [102]. By comparing the
results obtained from the presented approach with those obtained from the
conventional matrix approach, we showed that both results converge provided
that there are sufficient number of cladding layers in the finite structure. Results
for propagation characteristics and modal field distributions in both solid-core
and hollow-core ideal planar photonic crystal structures were presented.
• A new approach for solving for the true photonic crystal modes in PCFs with in-
finite periodic cladding was proposed. We showed that these modes can appear
in both solid-core and hollow-core ideal PCF structures.
8.2 Directions for Future Work
There are a number of issues that require further investigations. These include:
• The ring model discussed in this work has been developed for PCFs with hexag-
onal hole arrangement. Extending the modeling effort to other hole arrange-
ments, such as rectangular and triangular, would be useful.
105
M. R. Albandakji Chapter 8. Conclusions and Directions for Future Work
• The present model provides a crude estimate of leakage loss. Appropriate mod-
ifications in the outer cladding layer of the ring model is expected to improve
the estimation of leakage loss.
• Several dispersion-shifted, dispersion-flattened, dispersion compensating and
large/small effective area Fresnel fiber designs made from silica-based mate-
rials have been analyzed. Investigating other Fresnel fiber designs made from
polymers or nonsilica-based materials might reveal Fresnel fiber designs with
attractive transmission properties as well.
• A closed form solution for the 1-D PCF with infinite periodic cladding, similar
to that presented for the 1-D planar waveguide, is desirable.
106
Appendix
A
Matrix Coefficients
The matrix coefficients in Eq. (2.27) are given by
γi11 = q0,i+1W
2i
[(ηi+1n
2i
ηin2i+1
)1
UiF
′ν,i (Ui) Fν,i+1 (Wi) − 1
WiFν,i (Ui) F
′ν,i+1 (Wi)
](A.1a)
γi12 = q0,i+1W
2i
[(ηi+1n
2i
ηin2i+1
)1
UiF
′ν,i (Ui) Fν,i+1 (Wi) − 1
WiFν,i (Ui) F
′ν,i+1 (Wi)
](A.1b)
γi13 =
Z0νβζi
n2i+1
q0,i+1W2i Fν,i (Ui) Fν,i+1 (Wi) (A.1c)
γi14 =
Z0νβζi
n2i+1
q0,i+1W2i Fν,i (Ui) Fν,i+1 (Wi) (A.1d)
γi21 = q0,i+1W
2i
[1
WiFν,i (Ui) F
′ν,i+1 (Wi) −
((ηi+1n
2i
ηin2i+1
)1
UiF
′ν,i (Ui) Fν,i+1 (Wi)
](A.1e)
γi22 = q0,i+1W
2i
[1
Wi
Fν,i (Ui) F′ν,i+1 (Wi) −
(ηi+1n
2i
ηin2i+1
)1
Ui
F′ν,i (Ui) Fν,i+1 (Wi)
](A.1f)
γi23 = −Z0νβζi
n2i+1
q0,i+1W2i Fν,i (Ui) Fν,i+1 (Wi) (A.1g)
γi24 = −Z0νβζi
n2i+1
q0,i+1W2i Fν,i (Ui) Fν,i+1 (Wi) (A.1h)
107
M. R. Albandakji Appendix A. Matrix Coefficients
γi31 =
(ni+1
Z0
)2
γi13 (A.1i)
γi32 =
(ni+1
Z0
)2
γi14 (A.1j)
γi33 = q0,i+1W
2i
[(ηi+1
ηi
)1
UiF
′ν,i (Ui) Fν,i+1 (Wi) − 1
WiFν,i (Ui) F
′ν,i+1 (Wi)
](A.1k)
γi34 = q0,i+1W
2i
[(ηi+1
ηi
)1
Ui
F′ν,i (Ui) Fν,i+1 (Wi) − 1
Wi
Fν,i (Ui) F′ν,i+1 (Wi)
](A.1l)
γi41 =
(ni+1
Z0
)2
γi23 (A.1m)
γi42 =
(ni+1
Z0
)2
γi24 (A.1n)
γi43 = q0,i+1W
2i
[1
Wi
Fν,i (Ui)F′ν,i+1 (Wi) −
(ηi+1
ηi
)1
Ui
F′ν,i (Ui)Fν,i+1 (Wi)
](A.1o)
γi44 = q0,i+1W
2i
[1
WiFν,i (Ui)F
′ν,i+1 (Wi) −
(ηi+1
ηi
)1
UiF
′ν,i (Ui)Fν,i+1 (Wi)
](A.1p)
where
q0,i+1 =
⎧⎨⎩ −π/2 ; ηi+1 = 1
1 ; ηi+1 = −1(A.2)
and
ζi = −[
1
U2i
+1
W 2i
](A.3)
108
Appendix
B
Material Constants
Material dispersion is accounted for using Sellmeier’s equation [68]
n (λ) =
√√√√1 +3∑
j=1
Ajλ2
λ2 − λ2j
(B.1)
with the following coefficients:
Table B.1: Sellmeier coefficients for several materials.Material Material A1 A2 A3 λ1 λ2 λ3
Symbol Name
M1 16.9 m/0 Na2O, 50.6 m/0 SiO2, 32.5 m/0 B2O3 0.796468 0.497614 0.358924 0.094359 0.0933865 5.999652
M2 1.0 m/0 F, 99.0 m/0 SiO2 0.691116 0.399166 0.890423 0.068227 0.11646 9.993707
M3 9.1 m/0 P2O5, 90.9 m/0 SiO2 0.69579 0.452497 0.712513 0.061568 0.119921 8.656641
M4 13.5 m/0 GeO2, 86.5 m/0 SiO2 0.71104 0.451885 0.704048 0.06427 0.129408 9.425478
M5 Quenched SiO2 0.69675 0.408218 0.890815 0.069066 0.115662 9.900559
M6 2.2 m/0 GeO2, 94.5 m/0 SiO2, 3.3 m/0 B2O3 0.699339 0.4111269 0.9035275 0.0617482 0.1242404 9.896158
M7 7.9 m/0 GeO2, 92.1 m/0 SiO2 0.7136824 0.4254807 0.8964226 0.0617167 0.1270814 9.896161
M8 3.1 m/0 GeO2, 96.9 m/0 SiO2 0.7028554 0.4146307 0.897454 0.0727723 0.1143085 9.896161
M9 4.03 m/0 GeO2, 86.27 m/0 SiO2, 9.7 m/0 B2O3 0.7042042 0.41289413 0.95238253 0.067974973 0.12147738 9.6436219
M10 7.0 m/0 GeO2, 93.0 m/0 SiO2 0.6869829 0.44479505 0.79073512 0.078087582 0.1155184 10.436628
M11 13.5 m/0 GeO2, 86.5 m/0 SiO2 0.73454395 0.42710828 0.82103399 0.08697693 0.11195191 10.84654
M12 SiO2 0.6961663 0.4079426 0.8974794 0.068043 0.1162414 9.896161
109
Bibliography
[1] J. C. Knight, “Photonic crystal fibres,” Nature, vol. 424, pp. 847–851, 2003.
[2] S. M. Barnard and D. R. Walt, “A fibre-optic chemical sensor with discrete
sensing sites,” Nature, vol. 353, pp. 338–340, 1991.
[3] J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling
the Flow of Light. New Jersey: Princeton University Press, 1995.
[4] J. D. Joannopoulos, “Self-assembly lights up,” Nature, vol. 414, pp. 257–258,
2001.
[5] J. G. Fleming and S.-Y. Lin, “Three-dimensional photonic crystal with a stop
band from 1.35 to 1.95 μm,” Opt. Lett., vol. 24, no. 1, 1999.
[6] S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, “Full three-dimensional
photonic bandgap crystals at near-infrared wavelengths,” Science, vol. 289,
no. 5479, pp. 604–606, 2000.
110
M. R. Albandakji BIBLIOGRAPHY
[7] M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turber-
field, “Fabrication of photonic crystals for the visible spectrum by holographic
lithography,” Nature, vol. 404, pp. 53–56, 2000.
[8] Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, “On-chip natural assembly
of silicon photonic bandgap crystals,” Nature, vol. 414, pp. 289–293, 2001.
[9] C. C. Cheng and A. J. Scherer, “Fabrication of photonic band-gap crystals,” J.
Vacuum Sci. Technol. B, vol. 13, no. 6, pp. 2696–2700, 1995.
[10] K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, N. Shinya, and
Y. Aoyagi, “Three-dimensional photonic crystals for optical wavelengths assem-
bled by micromanipulation,” Appl. Phys. Lett., vol. 81, no. 17, pp. 3122–3124,
2002.
[11] S. R. Kennedy, M. J. Brett, O. Toader, and S. John, “Fabrication of tetragonal
square spiral photonic crystals,” Nano Lett., vol. 2, no. 1, pp. 59–62, 2002.
[12] E. Kuramochi, M. Notomi, T. Kawashima, J. Takahashi, C. Takahashi,
T. Tamamura, and S. Kawakami, “A new fabrication technique for photonic
crystals: nanolithography combined with alternating-layer deposition,” Opt.
Quant. Elec., vol. 34, pp. 53–61, 2002.
[13] T. Sato, K. Miura, N. Ishino, Y. Ohtera, T. Tamamura, and S. Kawakami,
“Photonic crystals for the visible range fabricated by autocloning technique
and their application,” Opt. Quant. Elec., vol. 34, pp. 63–70, 2002.
[14] P. Russell, “Photonic crystal fibers,” Science, vol. 299, no. 5605, pp. 358–362,
2003.
111
M. R. Albandakji BIBLIOGRAPHY
[15] J. C. Knight, T. A. Briks, P. S. J. Russell, and D. M. Atkin, “All-silica single-
mode optical fiber with photonic crystal cladding,” Opt. Lett., vol. 21, no. 19,
pp. 1547–1549, 1996.
[16] D. Ferrarini, L. Vincetti, and M. Zoboli, “Leakage properties of photonic crystal
fibers,” Opt. Express, vol. 10, no. 23, pp. 1314–319, 2002.
[17] D. Mogilevtsev, T. A. Birks, and P. S. J. Russell, “Group-velocity dispersion in
photonic crystal fibers,” Opt. Lett., vol. 23, no. 21, pp. 1662–1664, 1998.
[18] J. M. Dudley, L. Provino, N. Grossard, H. Maillotte, R. Windeler, B. Eggleton,
and S. Coen, “Supercontinuum generation in airsilica microstructured fibers
with nanosecond and femtosecond pulse pumping,” J. Opt. Soc. Am. B, vol. 19,
no. 4, pp. 765–771, 2002.
[19] T. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey
optical fibers: An efficient modal model,” J. Lightwave Technol., vol. 17, no. 6,
pp. 1093–1102, 1999.
[20] T. M. Monro, P. J. Bennett, N. G. R. Broderick, and D. J. Richardson, “Holey
fibers with random cladding distributions,” Opt. Lett., vol. 25, no. 4, pp. 206–
208, 2000.
[21] G. Pickrell, D. Kominsky, R. Stolen, F. Ellis, J. Kim, A. Safaai-Jazi, and
A. Wang, “Microstructural analysis of random hole optical fibers,” IEEE Pho-
ton. Technol. Lett., vol. 16, no. 2, pp. 491–493, 2004.
[22] J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap
guidance in optical fibers,” Science, vol. 282, no. 5393, pp. 1476–1478, 1998.
[23] M. D. Nielsen and N. A. Mortensen, “Photonic crystal fiber design based on
the V-parameter,” Opt. Express, vol. 11, no. 21, pp. 2762–2768, 2003.
112
M. R. Albandakji BIBLIOGRAPHY
[24] J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gerard, D. Maystre, and A. Tchel-
nokov, Photonic Crystals Towards Nanoscale Photonic Devices. Springer, 2005.
[25] P. S. J. Russell, T. A. Birks, and F. D. Lloyd-Lucas, Photonic Bloch Waves and
Photonic Band Gaps. New York: Plenum Press, 1995.
[26] T. A. Birks, J. C. Knight, and P. S. J. Russell, “Endlessly single-mode photonic
crystal fiber,” Opt. Lett., vol. 22, no. 13, pp. 961–963, 1997.
[27] K. Tajima, J. Zhou, K. Nakajima, and K. Sato, “Ultralow loss and long length
photonic crystal fiber,” IEEE J. Lightwave Technol., vol. 22, no. 1, pp. 7–10,
2004.
[28] A. Ferrando, E. Silvestre, and P. Andres, “Designing the properties of
dispersion-flattened photonic crystal fiber,” Opt. Express, vol. 9, no. 13,
pp. 687–697, 2001.
[29] A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andres, and
P. S. J. Russell, “Designing a photonic crystal fiber with flattened chromatic
dispersion,” Electron. Lett., vol. 35, no. 4, pp. 325–327, 1999.
[30] A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Nearly zero ultraflattened
dispersion in photonic crystal fibers,” Opt. Lett., vol. 25, no. 11, pp. 790–792,
2000.
[31] L. P. Shen, W. P. Huang, and S. S. Jian, “Design of photonic crystal fibers for
dispersion-related applications,” J. Lightwave Technol., vol. 21, no. 7, pp. 1644–
1651, 2003.
[32] T. A. Birks, D. Mogilevtsev, J. C. Knight, and P. S. J. Russell, “Dispersion com-
pensation using single material fibers,” IEEE Photon. Technol. Lett., vol. 11,
no. 6, pp. 674–676, 1999.
113
M. R. Albandakji BIBLIOGRAPHY
[33] K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic
crystal fibers,” Opt. Express, vol. 13, no. 1, pp. 267–274, 2005.
[34] A. Ortigosa-Blanch, J. C. Knight, W. J. Wadsworth, J. Arriaga, B. J. Mangan,
T. A. Birks, and P. S. J. Russell, “Highly birefringent photonic crystal fibers,”
Opt. Lett., vol. 25, no. 18, pp. 1325–1327, 2000.
[35] H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Ab-
solutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett.,
vol. 16, no. 1, pp. 182–184, 2004.
[36] M. J. Steel and J. R. M. Osgood, “Elliptical-hole photonic crystal fibers,” Opt.
Lett., vol. 26, no. 4, pp. 229–231, 2001.
[37] T. P. Hansen, J. Broeng, S. E. B. Libori, E. Knudsen, A. Bjarklev, J. R. Jensen,
and H. Simonsen, “Highly birefringent index-guiding photonic crystal fibers,”
IEEE Photon. Technol. Lett., vol. 13, no. 6, pp. 588–590, 2001.
[38] J. R. Folkenberg, M. D. Nielsen, N. A. Mortensen, C. Jakobsen, and H. R.
Simonsen, “Polarization maintaining large mode area photonic crystal fiber,”
Opt. Express, vol. 12, no. 5, pp. 956–960, 2004.
[39] J. R. Folkenberg, M. D. Nielsen, and C. Jakobsen, “Broadband single-
polarization photonic crystal fiber,” Opt. Lett., vol. 30, no. 12, pp. 1446–1448,
2005.
[40] A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, “Full-vector analysis of a
realistic photonic crystal fiber,” Opt. Lett., vol. 24, no. 5, pp. 276–278, 1999.
[41] J. Broeng, T. Søndergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev,
“Waveguidance by the photonic bandgap effect in optical fibers,” J. Opt. A:
Pure Appl. Opt., vol. 1, no. 4, pp. 477–482, 1999.
114
M. R. Albandakji BIBLIOGRAPHY
[42] B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and
G. L. Burdge, “Cladding-mode-resonances in air-silica microstructure optical
fibers,” J. Lightwave Technol., vol. 18, no. 18, pp. 1084–1100, 2000.
[43] J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta
air-silica microstructure optical fibers,” Opt. Lett., vol. 25, no. 11, pp. 796–797,
2000.
[44] N. Guan, S. Habu, K. Takenaga, K. Himeno, and A. Wada, “Boundary element
method for analysis of holey optical fibers,” J. Lightwave Technol., vol. 21,
no. 8, pp. 1787–1792, 2003.
[45] T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez,
C. M. de Sterke, and L. C. Botten, “Multipole method for microstructured
optical fibers. I. Formulation,” J. Opt. Soc. Amer., vol. 19, no. 10, pp. 2322–
2330, 2002.
[46] K. Saitoh and M. Koshiba, “Full-vectorial imaginary-distance beam propaga-
tion method based on a finite element scheme: application to photonic crystal
fibers,” IEEE J. Quantum Electron., vol. 38, no. 7, pp. 927–933, 2002.
[47] D. Marcuse, “Mode conversion in optical fibers with monotonically increasing
core radius,” J. Lightwave Technol., vol. LT-5, no. 1, pp. 125–133, 1987.
[48] B. Hermansson, D. Yevick, and P. Danielsen, “Propagating beam analysis of
multimode waveguide tapers,” IEEE J. Quantum Electron., vol. QE-19, no. 8,
pp. 1246–1251, 1983.
[49] Y. Li and J. W. Y. Lit, “Transmission properties of a multimode optical-fiber
taper,” J. Opt. Soc. Amer. A, vol. 2, no. 3, pp. 462–468, 1985.
115
M. R. Albandakji BIBLIOGRAPHY
[50] W. Burns, M. Abebe, C. Villarruel, and R. Moeller, “Loss mechanisms in single-
mode fiber tapers,” J. Lightwave Technol., vol. 4, no. 6, pp. 608–613, 1986.
[51] N. Amitay, H. M. Presby, F. V. Dimarcello, and K. T. Nelson, “Optical fiber
tapers — a novel approach to self-aligned beam expansion and single mode
hardware,” J. Lightwave Technol., vol. LT-5, no. 1, pp. 70–76, 1987.
[52] H. M. Presby, N. Amitay, F. V. Dimarcello, and K. T. Nelson, “Optical fiber
tapers at 1.3 μm for self-aligned beam expansion and single-mode hardware,”
J. Lightwave Technol., vol. LT-5, no. 8, pp. 1123–1128, 1987.
[53] H. M. Presby, N. Amltay, R. Scotti, and A. F. Benner, “Laser-to-fiber coupling
via optical fiber up-tapers,” J. Lightwave Technol., vol. 7, no. 2, pp. 274–278,
1989.
[54] A. Safaai-Jazi and V. Suppanitchakij, “A tapered graded-index lens: Analysis of
transmission properties and applications in fiber-optic communication systems,”
IEEE J. Quantum Electron., vol. 33, no. 12, pp. 2159–2166, 1997.
[55] J. K. Chandalia, B. J. Eggleton, R. S. Windeler, S. G. Kosinski, X. Liu, and
C. Xu, “Adiabatic coupling in tapered air-silica microstructured optical fiber,”
IEEE Photon. Technol. Lett., vol. 13, no. 1, pp. 52–54, 2001.
[56] S. T. Huntington, J. Katsifolis, B. C. Gibson, J. Canning, K. Lyytikainen, J. Za-
gari, L. W. Cahill, and J. D. Love, “Retaining and characterising nano-structure
within tapered air-silica structured optical fibers,” Opt. Express, vol. 11, no. 2,
pp. 98–104, 2003.
[57] E. C. Magi, P. Steinvurzel, and B. Eggleton, “Tapered photonic crystal fibers,”
Opt. Express, vol. 12, no. 5, pp. 776–784, 2004.
116
M. R. Albandakji BIBLIOGRAPHY
[58] H. C. Nguyen, B. T. Kuhlmey, M. J. Steel, C. L. Smith, E. C. Magi, R. C.
McPhedran, and B. J. Eggleton, “Leakage of the fundamental mode in photonic
crystal fiber tapers,” Opt. Lett., vol. 30, no. 10, pp. 1123–1125, 2005.
[59] S. Mirlohi, “Exact solutions of planar photonic crystal waveguides with infinite
cladding,” Master’s thesis, Virginia Polytechnic Institute and State University,
ECE Department, Blacksburg, VA, 2003.
[60] Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of
bragg fibers,” J. Lightwave Technol., vol. 20, no. 3, pp. 428–440, 2002.
[61] P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am.,
vol. 68, no. 9, pp. 1196–1201, 1978.
[62] Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of bragg fibers,” Opt.
Lett., vol. 25, no. 24, pp. 1756–1758, 2000.
[63] M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, and J. D. Joannopoulos, “An
all-dielectric coaxial waveguide,” Science, vol. 289, no. 5478, pp. 415–419, 2000.
[64] Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of dielectric coaxial
fibers,” Opt. Lett., vol. 27, no. 12, pp. 1019–1021, 2002.
[65] J. A. Monsoriu, E. Silvestre, A. Ferrando, and P. Andres, “High-index-core
bragg fibers: dispersion properties,” Opt. Express, vol. 11, no. 12, pp. 1400–
1405, 2003.
[66] S. O. Konorov, A. B. Fedotov, O. A. Kolevatova, V. I. Beloglazov, N. B. Skibina,
A. V. Shcherbakov, and A. M. Zheltikov, “Waveguide modes of hollow photonic-
crystal fibers,” JETP Lett., vol. 76, no. 6, pp. 341–345, 2002.
117
M. R. Albandakji BIBLIOGRAPHY
[67] X. Qian and A. C. Boucouvalas, “Propagation characteristics of single-mode op-
tical fibers with arbitrary complex index profiles,” IEEE J. Quantum Electron.,
vol. 40, no. 6, pp. 771–777, 2004.
[68] M. J. Adams, An Introduction to Optical Waveguides. Wiley, 1981.
[69] W. J. Wadsworth, J. C. Knight, A. Ortigosa-Blanch, J. Arriaga, E. Silvestre,
and P. S. J. Russell, “Soliton effects in photonic crystal fibers at 850 nm,”
Electron. Lett., vol. 36, no. 1, pp. 53–55, 2000.
[70] K. Nakajima, J. Zhou, K. Tajima, K. Kurokawa, C. Fukai, and I. Sankawa,
“Ultrawide-band single-mode transmission performance in a low-loss photonic
crystal fiber,” J. Lightwave Technol., vol. 23, no. 1, pp. 7–12, 2005.
[71] N. Mortensen, “Effective area of photonic crystal fibers,” Opt. Express, vol. 10,
no. 7, pp. 341–348, 2002.
[72] M. Koshiba and K. Saitoh, “Structural dependence of effective area and mode
field diameter for holey fibers,” Opt. Express, vol. 11, no. 15, pp. 1746–1756,
2003.
[73] D. Gloge, “Propagation effects in optical fibers,” IEEE Trans. Microwave The-
ory Tech., vol. 23, no. 1, pp. 106–120, 1975.
[74] V. Finazzi, T. M. Monro, and D. J. Richardson, “The role of confinement loss
in highly nonlinear silica holey fibers,” IEEE Photon. Technol. Lett., vol. 15,
no. 9, pp. 1246–1248, 2003.
[75] A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, M. C. J. Large, and J. Za-
gari, “Ring structures in microstructured polymer optical fibers,” Opt. Express,
vol. 9, no. 13, pp. 813–820, 2001.
118
M. R. Albandakji BIBLIOGRAPHY
[76] L. C. Bobb and P. M. Shankar, “Tapered optical fiber components and sensors,”
Microwave J., vol. 35, no. 5, pp. 218–228, 1992.
[77] H. Murata, Handbook of Optical Fibers and Cables. New York: Marcel Dekker,
2nd ed., 1996.
[78] D. K. Mynbaev and L. L. Scheiner, Fiber-Optic Communications Technology.
New York: Prentice Hall, 2001.
[79] T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, “Supercontinuum generation
in tapered fibers,” Opt. Lett., vol. 25, no. 19, pp. 1415–1417, 2000.
[80] G. Brambilla, V. Finazzi, and D. J. Richardson, “Ultra-low-loss optical fiber
nanotapers,” Opt. Express, vol. 12, no. 10, pp. 2258–2263, 2004.
[81] I. G. Koprinkov, I. K. Ilev, and T. G. Kortenski, “Transmission characteristics
of wide-aperture optical fiber taper,” J. Lightwave Technol., vol. 10, no. 2,
pp. 135–141, 1992.
[82] R. P. Kenny, T. A. Birks, and K. P. Oakley, “Control of optical fiber taper
shape,” Electron. Lett., vol. 27, no. 18, pp. 1654–1656, 1991.
[83] G. E. Town and J. T. Lizier, “Tapered holey fibers for spot size and numerical
aperture conversion,” Opt. Lett., vol. 26, no. 14, pp. 1042–1044, 2001.
[84] T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol.,
vol. 10, no. 4, pp. 432–438, 1992.
[85] D. Marcuse, Theory of Dielectric Optical Waveguides. New York: Academic,
1974.
119
M. R. Albandakji BIBLIOGRAPHY
[86] J. D. Love, W. M. Henry, W. J. Stewart, R. J. Black, S. Lacroix, and
F. Gonthier, “Tapered single-mode fibers and devices. Part 1: Adiabaticity
criteria,” IEE Proceedings - J, vol. 138, no. 5, pp. 343–354, 1991.
[87] J. D. Love, “Application of a low-loss criterion to opitcal waveguides and de-
vices,” IEE Proc. J., vol. 136, no. 4, pp. 225–228, 1989.
[88] R. Stolen and M. R. Albandakji, “Personal communications,” April 2005.
[89] H. T. Hattori and A. Safaai-Jazi, “Fiber designs with significantly reduced
nonlinearity for very long distance transmission,” Appl. Opt., vol. 37, no. 15,
pp. 3190–3197, 1998.
[90] A. Safaai-Jazi and M. R. Albandakji, “Research discussion on formulation of
the solution for 1-D planar photonic crystal waveguides with infinite number of
cladding layers,” February 2006.
[91] A. W. Snyder and J. D. Love, Optical Waveguide Theory. New York: Chapman
and Hall, 1983.
[92] A. C. Boucouvalas and C. D. Papageorgiou, “Cutoff frequencies in optical fibers
of arbitrary refractive index profile using the resonance technique,” IEEE J.
Quantum Electron., vol. 18, no. 12, pp. 2027–2031, 1982.
[93] Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L.
Thomas, “A dielectric omnidirectional reflector,” Science, vol. 282, no. 5394,
pp. 1679–1682, 1998.
[94] T. Kawanishi and M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Ex-
press, vol. 7, no. 1, pp. 10–22, 2000.
120
[95] S. Guo, S. Albin, and R. S. Rogowski, “Comparative analysis of Bragg fibers,”
Opt. Express, vol. 12, no. 1, pp. 198–207, 2004.
[96] M. R. Albandakji, A. Safaai-Jazi, and R. Stolen, “Simple models for predicting
transmission properties of photonic crystal fibers.” To appear in Microwave
Opt. Technol. Lett., vol. 48, no. 7, 2006.
[97] M. R. Albandakji, A. Safaai-Jazi, and R. H. Stolen, “Simple models of photonic
crystal fibers,” Optics in the Southeast – OSA Conference, November 2004.
[98] A. Safaai-Jazi, R. Stolen, M. R. Albandakji, and J. Kim, “1-D model for pre-
dicting propagation properties of photonic crystal and holey fibers,” Frontiers
in Optics – The 88th OSA Annual Meeting, October 2004.
[99] M. R. Albandakji, A. Safaai-Jazi, and R. H. Stolen, “Tapered photonic crystal
fibers.” Submitted to Optics East – SPIE Conference, October 2006.
[100] M. R. Albandakji, A. Safaai-Jazi, and R. H. Stolen, “A novel optical fiber with
applications in sensing and communications.” Submitted to Optics East – SPIE
Conference, October 2006.
[101] M. R. Albandakji, A. Safaai-Jazi, and R. H. Stolen, “Fresnel optical fibers,”
Optics in the Southeast – OSA Conference, October 2005.
[102] A. Safaai-Jazi, M. R. Albandakji, and S. Mirlohi, “Exact analytical solutions
of planar photonic crystal waveguides.” Submitted to Optics East – SPIE Con-
ference, October 2006.
121
Vita
Mhd Rachad Albandakji was born on April 4, 1979 in Damascus, Syria. He received
his Bachelor’s degree in Electrical Engineering from the University of Jordan, Amman,
Jordan in 2001. In 2003, he received his Master’s degree in Electrical Engineering
from Virginia Tech, Blacksburg, Virginia and continued to pursue his Ph.D. degree
in Electrical Engineering from Virginia Tech. His main research interests are in
communication systems and optical waveguides.
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