modeling and analysis of photonic crystal waveguides

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



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


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


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


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


(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


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


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


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


[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


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


[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


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


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


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


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


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


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


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


�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


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


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) =


] ⎡⎣ cos (νϕ)

− sin (νϕ)

⎤⎦ (2.20c)

and

H ir (r, ϕ) = −jk0

q2i

⎧⎨⎩n2

i ν

Z0r


]+

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


]⎫⎬⎭⎡⎣ cos (νϕ)

− sin (νϕ)

⎤⎦ (2.21b)

24


H iz (r, ϕ) =


] ⎡⎣ 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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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




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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


• 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

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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.

122

modeling and analysis of photonic crystal waveguides

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