Simulations of Step-Like Bragg Gratings in Silica Fibers Using Comsol

SIMULATIONS OF STEP-LIKE BRAGG GRATINGS IN SILICA FIBERS USING COMSOL

A Thesis Presented to The Graduate Faculty of The University of Akron

In Partial Fulfillment of the Requirements for the Degree Master of Science

Rasika Dahanayake May, 2016

SIMULATIONS OF STEP-LIKE BRAGG GRATINGS IN SILICA FIBERS USING COMSOL

Rasika Dahanayake

Thesis

Approved: Accepted:

______Advisor Dean of the College Dr. Sergei F. Lyuksyutov Dr. John Green

______Faculty Reader Dean of the Graduate School Dr. Robert R. Mallik Dr. Chand Midha

______Faculty Reader Date Dr. Alper Buldum

______Department Chair Dr. David Steer

ABSTRACT

The purpose of the research was to build a model for simulations of Fiber

Bragg Grating (FBG) sensors under harsh conditions. In this thesis we were studying the 2D model of a fiber under different wavelength EM radiation (530 nm, 1310 nm, and 1530 nm), also we introduced a novel 3D model of FBG. The

3D model of the FBG is used to study the modes of the fiber under varying grating period (1310 nm, 908.62 nm, and 454.31 nm) and with varying number of slabs in the grating (10, 50, and 100). The novelty of this work is an introduction of step-like profile in FBGs. Potentially, this approach may be used for experimental high temperature sensing under harsh conditions.

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ACKNOWLEDGEMENTS

First I would like to thank Dr. Sergei Lyuksyutov for being my advisor and taking me into your research group. I was able to learn and grow under your guidance and advice during the project and throughout my time at the

University of Akron.

Then I would like to thank Professor Robert Mallik and Professor Alper

Buldum for being kind enough to be on my graduate thesis committee and especially for valuable advice on improvements on my thesis.

I would also like to thank colleagues Jeff McCausland, Sajeevi Withanage,

Pedram Esfahani and Liudmyla Barabanova for the help and guidance during my time at the University of Akron and on my research.

I am grateful to my parents for all the love and guidance they have given me throughout my life.

Finally, I would like to thank my wife Ruvini for the love and support. You make everything in my life better.

TABLE OF CONTENTS

Page

LIST OF TABLES………………………………………………..….……...….….…vii

LIST OF FIGURES……………………………………………………....…....……viii

CHAPTER

I. INTRODUCTION……………………………………………………………….…...1

II. BACKGROUND…………………………………………………..………………...3

2.1. Optical Fibers………………………………………………………….…..3

2.2. Geometrical Optics…………………………………………….………….5

2.2.1. Numerical Aperture………………………………..…………...7

2.3. Wave Optics……………………………………………….…………….…8

2.3.1. Waveguides…………………………………………….……….11

2.3.2. Coupled-Mode theory………………………………………....15

2.4. Fiber Bragg Gratings………………………………………………..…..20

2.4.1. Origin and Inscribing FBG’s………………………………....20

2.4.2. Types of FBG’s…………………………………….…………...25

2.4.2.1. By the varying conditions of gratings production.25

2.4.2.2. By the Structure (composition) of the gratings….27

2.4.3. Photosensitivity………………………………….….………....31

2.4.4. Applications of FBG’s……….………….…………….…...…..35

2.5. Finite Element Analysis and COMSOL Multiphysics programming……………………………………………………….………….37

III. PROCEDURE…………………………………………………………………….43

3. Simulations Using COMSOL……………………………..………………43

3.1. 2D Step index fiber Model……………………...………….…………....44

3.2. 3D Model for Fiber Bragg grating………………….…….…………....49

IV. RESULTS & DISCUSSIONS……………………………………….……….....61

4.1. Results of the 2D model…………………………………………...……61

4.2. Results of the 3D model……………………………………...…………69

V. CONCLUSIONS…..………………………………………………...... …………82

REFERENCES…………………………………………………………….....………84

LIST OF TABLES

Table Page

3.1 Table of Parameters for the 2D fiber Simulation……………..…………….44

3.2 Table of Parameters for the 3D fiber Simulation……………………..…….50

4.1 Stable modes sustained by the fiber in the 2D model……………..…………68

4.2 Simulation results of the fiber with 10 gratings for different grating periods……………………………………………………………………...…………..78

4.3 Simulation results of fiber’s with grating period Λ = 454.31 nm, and different number of slabs…..…………………………………………...……………79

4.4 Simulation results for the 3D model……………………………………………80

vii

LIST OF FIGURES

Figure Page

2.1.1 A schematic representation of an optical fiber, with a core of refractive index n1 and a cladding of refractive index n2, the fiber is in air, which has a refractive index of n0……………………………………………….………………..…..………...4

2.2.1 A schematic representation of light wave propagating along a fiber…..…5

2.2.2 Optical fiber with Cartesian coordinates (x,y,z) and Cylindrical coordinates (r,θ,z)………………………………………………………………………9

2.3.1 These are several of the electric field patterns for different LP modes in a fiber. Red is for positive and blue is for negative values of the electric field [22]...... ………19

2.4.1 The Original setup used by Meltz, Morey and Glenn [20].………………..22

2.4.2 The interference pattern created inside of the fiber [21].……………..….22

2.4.3 Phase Mask technique used to fabricate gratings in a fiber [23].………..24

2.4.4 Representation of uniform refractive index profile FBGs [30]..…………28

2.4.5 Apodized refractive index profile FBGs [30].…………….………………..30

2.4.6 The 3D Tetrahedral crystal structure of silica [35].………………...... …..32

2.4.7 Graphical representation of “wrong bonds” broken under exposure of UV light. (a) Wrong bonds structure, (b) GeE’ structure, (c) oxygen vacant silicon atom, and (d) free electron released from the reaction…………………………..33

viii

2.4.8 The defect Ge(1) in germania-doped silica fiber……………………………34

2.4.9 The defect Ge(2) in germania-doped silica fiber….………………………...34

2.4.10 Schematic representation of the FBG experimental setup [38]………...35

2.4.11 Diagram of the area (A) in the interval between -1 to 1. The area under the function y is found by the integration [36]…………….…………………...…38

2.4.12 Graphical representation of numerical integration of the calculation of A, changing with the number of subsections used [36].………………………… 40

3.1 Basic model of the 2D fiber. Center circle is the core and the bigger circle is the cladding……………………………………………………………………...….45

3.2 A close up of the triangular shaped mesh used for the 2D model………….48

3.3 The 3D model of the fiber……………………………………………………...…53

3.4 The 3D model of the single grating with thickness Λ =454.31 nm……...….54

3.5 The 3D model of the FBG with 10 slabs….………………………….…..…….55

3.6 The 3D model of the FBG with 50 slabs….…………………….……...... 56

3.7 The 3D model of the FBG with 100 slabs…..………………….…….………...56

3.8 The grating with of the FBG with 10 slabs. Where the slabs in the core with even number is assigned with the core material and the odd number is assigned with grating material. The outer was cylinder was assigned with the material of the cladding……………………………………………...…..….……….57

3.9 The grating with of the FBG with 50 slabs with each even numbered disk assigned with the core material and the odd numbered disk assigned with grating material ……………………………………………………...……...... 58

3.10 The grating with of the FBG with 100 slabs with each even numbered disk assigned with the core material and the odd numbered disk assigned with grating material …………………………………………………………….………...58

3.11 The tetrahedral mesh on the 3D model. Since the light propagates through the core, it has a smaller mesh to increase the accuracy of calculations. Cladding was given a larger mesh to decrease the unnecessary use of the computer power……………………………….………………………………………59

3.12 A close up of the tetrahedral shaped mesh used for the 3D model……....60

4.1 LP01 mode experimental [39] on the left compared to simulated result on the right………………………………………………….………………………..……62

4.2 LP12 mode experimental [39] on the left compared to simulated result on the right………………………………………………….……………………………..62

4.3 LP21 mode experimental [39] on the left compared to simulated result on the right………………………………………………….……………………………..62

4.4 LP01* (donut shape) mode experimental [40] on the top left compared to simulated results of a few modes on the top right and on bottom, the simulated results have the same outer diameter but the inner diameter is changing………………………………………………………………………………..63

4.5 LP41 mode experimental [39] on the left compared to simulated result on the right ………………………………………………………………………………..63

4.6 LP51 mode experimental on the left compared to simulated result on the right……………………………………………………………………………………..64

4.7 LP01 mode experimental [39] on the left compared to simulated result on the right…………………………………………………….……………………..……65

4.8 LP02 mode experimental [39] on the left compared to simulated result on the right…………………………………………………….……………………..……65

4.9 LP12 mode experimental [39] on the left compared to simulated result on the right…………………………………………………….………………….....……65

4.10 LP01* (donut shape) mode experimental [40] on the top left compared to simulated results of few modes on the top right and on bottom……………………………………………………………………………..……66

4.11 LP01 mode experimental [39] on the left compared to simulated result on the right……………………………………………………..…………..………..……67

4.12 LP02 mode experimental [39] on the left compared to simulated result on the right……………………………………………………..…………..………..……67

4.13 Incident wave of the fiber with no FBG…………………………………...….70

4.14 LP11 mode experimental [39] on the left compared to simulated result of the transmitted wave on the right. The resultant mode indicates instability in the mode……………………………………………………………..…………………70

4.15 Power distribution of the transmitted for the fiber with no FBG………..70

4.16 The experimental LP12 [39] on the right, incident wave on the middle and the transmitted wave on the right. Both incident and transmitted wave modes are identical……………………………………………….……………...……………71

4.17 Power distribution of the single grating of period 2Λ………………………71

4.18 The experimental LP12 [39] on the right, incident wave on the middle and the transmitted wave on the right. Both incident and transmitted wave modes are identical……………………………………………..…………………...………..72

4.19 Power distribution of the single grating with width Λ……………………..72

4.20 Incident wave of the fiber with 10 slabs and period 1310nm…….……….73

4.21 LP11 mode experimental [39] on the left compared to simulated result of the fiber with 10 slabs and grating period 1310nm. The sustained mode indicates instability.……………….………………………………….….…………..73

4.22 Power distribution of the transmitted wave of the fiber with 10 slabs and grating period 1310nm……………………………………….………………………73

4.23 Incident wave of the fiber with 10 slabs and grating period 2Λ………….74

4.24 LP11 mode experimental [39] on the left compared to simulated result of the fiber with 10 slabs and grating period 2Λ.…………………………….………74

4.25 Power distribution of the transmitted wave of the fiber with 10 slabs and grating period 2Λ………………………………………...……………………………74

4.26 Incident wave of the fiber with 10 slabs and grating period Λ……...... …75

4.27 LP12 mode experimental [39] on the left compared to simulated result of the fiber with 10 slabs and grating period Λ.……………………………..………75

4.28 Power distribution of the transmitted wave of the fiber with 10 slabs and grating period Λ…………………………………………………………………….....75

4.29 For the fiber with 50 slabs and grating period Λ, incident wave on left. The transmitted wave (right) seems to form a combination of LP12 and LP21 modes.………………………...……………...………...……..…………….…………76

4.30 Power distribution of the transmitted wave of the fiber with 50 slabs and grating period Λ……………………………………………………………………….76

4.31 For the fiber with 100 slabs and grating period Λ, incident wave on left. The transmitted wave (right) seems to form a combination of LP12 and LP21 modes.……………………………...……………………….…………………………..77

4.32 Power distribution of the transmitted wave of the fiber with 100 slabs and grating period Λ……………………………………………………...………………..77

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CHAPTER I

INTRODUCTION

First developed in 1970s, Fiber optics communication systems have revolutionized the telecommunication industry. Made to be transparent and flexible from dielectric materials such as silica (Glass) or plastic, it’s a fiber with diameters in the micrometer range. By sending pulses of light through these fibers information can be transmitted. These fibers possess the ability to transmit information over longer distances and in higher bandwidths than wire cables.

Due to their immunity to electromagnetic interference optical fibers have the ability to carry information with higher conservation compared to regular cable wires. This has caused the telecommunication companies to replace the cable wires with optical fibers.

In 1978 Ken Hill demonstrated the first Fiber Bragg Grating (FBG). A

FBG is a type of Bragg reflector that reflects particular wavelengths of light and transmit others [7], Therefore FBG’s can be used as optical filters or as wavelength-specific reflectors. Initially FBG’s were fabricated using a visible laser propagation along a fiber core. After the discovery of external imprinting

1 technique by Gerald Meltz in 1989, FBG fabrication became possible at an industrial level.

FBGs have many applications, one of the main application is as sensors.

They could be used to measure physical parameters such as temperature, pressure, strain, etc. Our research is focused on building a simulation model for a high temperature sensor. In this thesis we will be using finite element analysis to build a basic model to study FBGs. The next step would be to study

FBGs under different temperature conditions.

CHAPTER II

BACKGROUND

2.1. Optical Fibers

The basic structure of an optical fiber consists of three parts, the Core, the Cladding, and the Buffer. The Core is made of cylindrical dielectric materials, through which light propagates. As mentioned in Introduction the core is generally made out of silica or plastic with diameters in micrometer range. If we described the core to have a diameter of 2a the radius would be “a” and the refractive index could be described as n1.

The cladding is the material that surrounds the core, even though the light can be propagated through the core without the cladding, it is necessary for the proper functioning of the fiber. We can describe the refractive index of the cladding as n2. The functions of the cladding can be listed as,

 Adding Mechanical strength.

 Reducing the loss of light from the core to the air from transmission.

 Reduce the loss of light from scattering at the surface.

 Protect the fiber from the surrounding surface contaminants.

The buffer is an additional coating that is made out of plastic; it is used on

Optical fibers to protect it from physical damage such as abrasions. It also helps to prevent loss of light due to scattering from microbends.

The relationship between the refractive indices of cladding and core can be shown as,

n1 > n2

This difference between the refractive indices is usually a small value.

Figure 2.1.1: A schematic representation of an optical fiber, with a core of refractive index n1 and a cladding of refractive index n2, the fiber is in air, which has a refractive index of n0.

2.2. Geometrical Optics

Geometrically the light ray entering the fiber should have an angle small enough with the normal of the input face so that it would undergo total internal reflection at the core/cladding interface (Figure 2.1.2). This angle is called the acceptance angle, θa.

Figure 2.2.1: A schematic representation of a light wave traveling along an optical fiber with total internal reflection.

For the light to be guided along the fiber by total internal reflection the angle θ2 must be bigger than the critical angle θc of the core to cladding interface[3].

푛2 푆푖푛휃푐 = 푛1

Therefore for the fiber, θ2 ≥ θc;

That is,

푛2 푆푖푛휃2 ≥ 푛1

Now from the Figure 2.1.2, we can see Cosθ1 = Sinθ2;

Therefore,

푛2 퐶표푠휃1 ≥ 푛1

Now by using the Snell’s law for the input face interface, we can determine

Sinθ1,

푛0푆푖푛휃0 = 푛1푆푖푛휃1

By rearranging the equation,

푛0 푆푖푛휃1 = 푆푖푛휃0 푛1

Now, we can use the relationship of Sin2θ + Cos2θ = 1 to find a value for Sinθ0,

2 2 2 2 푛2 푛0 2 퐶표푠 휃1 + 푆푖푛 휃1 = 1 ≥ ( ) + ( ) 푆푖푛 휃0 푛1 푛1

Now by solving for Sinθ0,

2 2 푛1 − 푛2 푆푖푛휃0 ≤ √ 2 (2.2.1) 푛0

2 2 −1 푛1 − 푛2 휃0 ≤ 휃푎 = 푆푖푛 √ 2 (2.2.2) 푛0

2.2.1. Numerical Aperture

Numerical Aperture (NA) of a fiber is the dimensionless number that defines the range of angles which the fiber can accept or emit light. In other words it is the number that shows the Capacity the fiber has to gather light.

The higher the Numerical aperture is more light the fiber can gather.

The Numerical aperture of a fiber is defined as shown below,

푁퐴 = 푛0푆푖푛휃푎

By using the equation (2.1.1.1) we can get and expression for NA,

2 2 푁퐴 = √푛1 − 푛2 (2.2.3)

2.3. Wave optics in fibers

Here we consider the propagation of light as propagation of a wave in the optical fiber. Since light is an electro-magnetic (EM) wave field vectors 퐸̅

(electric field) and 퐻̅ (magnetic field) are used.

퐷̅ = 휀0퐸̅ + 푃̅ (2.3.1)

퐵̅ = 휇0퐻̅ (2.3.2)

Where: 퐷̅ is the electric displacement vector.

퐵̅ is the magnetic flux vector.

푃̅ is the induced polarization.

휀0 is the dielectric permittivity constant.

휇0 is the magnetic permeability.

The induced polarization can be written in terms of electric field vector and linear susceptibility휒푖푗, where 휒푖푗 is a second rank tensor and is related to the permittivity tensor 휀푖푗.

푃̅ = 휀푖푗휒푖푗퐸̅ (2.3.3)

Where: 휀푖푗 = 1 + 휒푖푗

Now we can consider the optical fiber (Figure 2.1.3). We can assume the optical fiber to be source free and also we know it is free of ferromagnetic materials, there for we can write,

∇ ∙ 퐷̅ = 0

∇ ∙ 퐵̅ = 0

Figure 2.2.2: Optical fiber with Cartesian coordinates (x,y,z) and Cylindrical coordinates (r,θ,z).

The Electric field 퐸̅ can be written as,

1 퐸̅ = [퐸푒푖(휔푡−훽푍) + 퐸푒−푖(휔푡−훽푧)] 2

Now by using the Maxwell’s equations,

휕퐵̅ ∇ × 퐸̅ = − (2.3.4) 휕푡

휕퐷̅ ∇ × 퐻̅ = + 퐽 ̅ (2.3.5) 휕푡

Where: 퐽 ̅ is the displacement current.

Now by using equations (2.3.4) and (2.3.1) we can get

휕퐸̅ 휕푃̅ ∇ × 퐻̅ = 휀 + (2.3.6) 0 휕푡 휕푡

Now by using all the equations above, we get the relationship,

휕2퐸̅ 휕2푃̅ ∇2퐸̅ = 휇 휀 + 휇 (2.3.7) 0 0 휕푡2 0 휕푡2

Now, by substituting the value of 푃̅, we can get the equation,

휕2퐸̅ ∇2퐸̅ = 휇 휀 휀 (2.3.8) 0 0 푖푗 휕푡2

2.3.1. Waveguides

Now we need to look into the introduction of guided modes of optical fiber in the equation. To describe the optical modes of a fiber we need transverse guided mode amplitude Aµ(z), continuum of radiation mode amplitude Aρ(z) [1], and the corresponding propagation constants βµ and βρ as a summation of n total modes [6].

휇=1 1 퐸 = ∑[퐴 (푧)휉 푒−푖(휔푡−훽휇푧) + 푐푐] 푡 2 휇 휇푡 휇=1

휌=∞ −푖(휔푡−훽휌푧) + ∑ ∫ 퐴휌(푧)휉휌푡푒 푑휌 (2.3.9) 휌=0

th Where:휉휇푡 is the radial transverse field distribution of the µ guided mode.

th 휉휌푡 is the radial transverse field distribution of the ρ radiation mode.

In the equation (2.1.2.7) the polarization has been included by using the transverse subscript, t. The summation before integral is so that we include all the radiation modes. The power of the µth mode can be found to be |Aµt|2 in watts, by using a orthogonally relationship.

+∞ +∞ +∞ +∞ 1 ∗ 1 훽휇 ∗ ∫ ∫ 푒̂푧[휉휇푡 × 휉휈푡]푑푥푑푦 = [ ] ∫ ∫ 휉휇푡휉휈푡 푑푥푑푦 = 훿휇휈 2 −∞ −∞ 2 휔휇0 −∞ −∞

(2.3.10)

Where: 푒̂푧 is the unit vector along the direction of propagation z.

For the guided modes 훿휇휈 is the Kronecker’s delta and is unity for

µ=ν and Zero otherwise.

For the radiation modes 훿휇휈 is the Dirac delta function and is

infinite for µ=ν and Zero otherwise.

(where ν is a positive integer)

From the equation (2.3.10) we can see that for the weakly guided case the longitudinal component of electric field is smaller than the transverse component. There for it is linearly polarized in the transverse direction predominantly compared to the direction of propagation. There for the transverse component of the magnetic field is [2].

휀0휀푟 휕 퐻푡 = √ [푒̂푧 × 휉푡] (2.3.11) 휇0 휕푧

The fields Et and Ht also satisfies the wave optics equations a well as the wave guide equations.

For the waveguide in the optical fiber the mode fields in the core are J-

Bessel functions and for the cladding they are K-Bessel functions. The two sets of orthogonally polarized solutions for the general case are given for the transverse fields for the µth x-polarized mode [1].

For the core (r ≤ a),

푟 cos 휇휃 휉 = 퐶 퐽 (푢 ) ( ) (2.3.12) 푥 휇 휇 휇휌 푎 sin 휇휃

휀0 퐻푦 = 푛푒푓푓√ 휉푥 (2.3.13) 휇0

Similarly for the cladding (r ≥ a),

퐽휇(푢휇휌) 푟 cos 휇휃 휉푥 = 퐶휇 퐾휇 (휔휇휌 ) ( ) (2.3.14) 퐾휇(휔휇휌) 푎 sin 휇휃

휀0 퐻푦 = 푛푒푓푓√ 휉푥 (2.3.15) 휇0

Where the normalization parameters for the equations used above are,

2휋푎 휈 = √푛2 − 푛2 = 푎 √푘2 − 푘2 (2.3.16푎) 휆 1 2 1 2

2휋푎 푢 = √푛2 − 푛2 (2.3.16푏) 휆 1 푒푓푓

휔2 = 휈2 − 푢2 (2.3.16푐)

푛1 − 푛2 푛푒푓푓 = 푛2 [푏 ( ) + 1] (2.3.16푑) 푛2

휔2 푏 = (2.3.16푒) 푢2

Where: 휆 is the wavelength in vacuum.

k1 and k2 are wavenumbers in core and cladding.

푛푒푓푓 is the effective index of the mode.

n = n1 for the core and n = n2 for the cladding.

From the equation (2.3.16a) we can see that the v-number depends on the properties of the fiber and the wavelength of the light wave, which determines the allowed modes in a fiber.

By making the assumption that this is only a single polarization, the y- polarized mode would be zero.

휉푦 = 퐻푥 = 0

Since we know the power carried by the fiber to be |Aµ|2 and by using the relationship of the equation (2.3.10), we can determine the normalization constant Cµ to be,

2휔 √휇0⁄휀0 퐶휇 = √ (2.3.17) 푎휈 푛푒푓푓휋푒휇|퐽휇−1(푢)퐽휇+1(푢)|

Now by using the boundary condition for the core-cladding boundary for the equations (2.3.12) & (2,3,14), we can find the eigenvalue equation for the waveguide.

퐽휇(푢) 퐾휇(휔) = (2.3.18) 푢 × 퐽휇±1(푢) 휔 × 퐾휇±1(휔)

2.3.2. Coupled-Mode theory

The effects of perturbation are needed to derive the coupled mode equations, and we need to assume that the unperturbed modes we derived in the waveguides would remain the same. Now we start with the wave equation

[6, 37].

휕2퐸̅ 휕2푃̅ ∇2퐸̅ = 휇 휀 + 휇 (2.3.19) 0 0 휕푡2 0 휕푡2

The total polarization due the dielectric medium could be divided to two parts, one that is unperturbed and the other perturbed.

푃̅̅푡표푡푎푙̅̅̅̅̅ = 푃̅̅푢푛푝푒푟푡푢푟푏푒푑̅̅̅̅̅̅̅̅̅̅̅̅̅ + 푃̅̅푔푟푎푡푖푛푔̅̅̅̅̅̅̅̅ (2.320)

Where, 푃푢푛푝푒푟푡푢푟푏푒푑 = 휀표휒퐸̅̅휇̅

Therefore using this relationship we can rewrite the equation (2.3.18) as shown below.

2 2 휕 퐸휇푡 휕 푃̅̅푔푟푎푡푖푛푔̅̅̅̅̅̅̅̅̅,휇̅ ∇2퐸 = 휇 휀 휀 + 휇 (2.3.21) 휇푡 0 0 푟 휕푡2 0 휕푡2

Where: µ is the transverse mode number.

Now we need to substitute the modes we got in the wave guides equation

(2.3.9) into the coupled wave equation (2.3.20), and then to further simplify the relationship we can use the relation of the amplitude of the mode change over a distance.

2 휕 퐴휇 휕퐴휇 ≪ 훽 (2.3.22) 휕푧2 휇 휕푧

And then we can come up with the simplified wave equation,

휇=1 2 휕퐴휇 휕 푃̅̅푔푟푎푡푖푛푔̅̅̅̅̅̅̅̅ ∑ [−푖훽 휉 푒−푖(휔푡−훽휇푧) + 푐푐] = 휇 (2.3.23) 휇 휕푧 휇푡 0 휕푡2 휇=1

Here cc is a constant produced after integration. After that by integrating the wave equation (2.3.19) over the cross section of the fiber and using the orthogonally relationship of the Equation (2.3.10), we get wave propagation equation,

휇=1 휕퐴휇 ∑ [−2푖휔휇 푒−푖(휔푡−훽휇푧) + 푐푐] 표 휕푧 휇=1

+∞ 2 휕 푃̅̅푔푟푎푡푖푛푔̅̅̅̅̅̅̅̅ = ∬ 휇0 2 휉휇푡 푑푥푑푦 (2.3.24) −∞ 휕푡

Above equation could be used to describe the coupling of the modes. This includes both forward and backwards propagating modes. The total of the transverse fields is the sum of both individual fields.

We can write the electric and magnetic fields as given below.

1 퐸 = (퐴 휉 푒푖(휔푡−훽푣푧) + 푐푐 + 퐵 휉 푒푖(휔푡+훽휇푧) + 푐푐) (2.3.25) 푡 2 푣 푣푡 휇 휇푡

1 퐻 = (퐴 퐻 푒푖(휔푡−훽푣푧) + 푐푐 − 퐵 퐻 푒푖(휔푡+훽휇푧) − 푐푐) (2.3.26) 푡 2 푣 푣푡 휇 휇푡

Now by substituting the fields into the equation (2.3.22), we can come up with the propagation equation for the coupled wave in a fiber.

휕퐴푣 휕퐵휇 ( 푒−푖(휔푡−훽휇푧) + 푐푐) − ( 푒−푖(휔푡+훽휇푧) + 푐푐) 휕푧 휕푧

+∞ 2 휕 푃̅̅푔̅푟̅̅푎푡푖푛푔̅̅̅̅̅ = ∬ 휇0 2 휉휇푡 푑푥푑푦 (2.3.27) −∞ 휕푡

Now to find the modal power in the core of the fiber we use a term η, which is defined as the ratio of the power confined in the core to the total power.

For the LP modes we can use the closed form given below,

1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ ∫ (푒 × ℎ⃗⃗⃗⃗∗⃗⃗ ) 푧̂푑퐴 2 2 2 2 푐표푟푒 휇휌 휇휌 푢 휔 퐾휇(휔) 휂 = = ( + ) (2.3.28) 1 푣2 푢2 퐾 (휔)퐾 (휔) ∫ (푒 × ℎ⃗⃗⃗⃗∗⃗⃗ ) 푧̂푑퐴 휇−1 휇+1 2 푡표푡푎푙 휇휌 휇휌

From the beginning we have being using two modes for the fiber transverse guided and radiation modes. We used the subscripts µ and ρ to denote these modes. Here µ is a non-negative integer that determine the

17 eigenvalues for the equations (2.3.12) and (2.3.14), for particular v-numbers.

The modes ρ determines which root of the (2.3.18) is to be used as the eigenvalue.

The propagation constant of the fiber 훽휇휌 is determined by the eigenvalue u.

푢 2 훽 = √푘2 − ( ) (2.3.29) 휇휌 1 푎

By looking at the Figure 2.3.1 and from the above equations we can see that there many possible modes, but a fiber cannot sustain all the modes. The modes that a fiber can sustain is determined by the v - number. The mode

LP 휇휌 can only be sustained if the v - number is greater than the cutoff vc.

The cutoff vc values depends on the refractive indices of the fiber, the diameter and the wavelength of the EM wave (equation 2.3.16a).

Figure 2.3.1: These are several of the electric field patterns for different LP modes in a fiber. Red is for positive and blue is for negative values of the electric field [22].

2.4. Fiber Bragg Gratings

Fiber Bragg gratings including their history, preparation, types, photosensitivity and associated experimental aspects are discussed in this subsection.

2.4.1. Origin and Inscribing FBG’s

The Origin of Fiber Bragg Gratings was briefly mentioned in the

Introduction section. It was K.O Hill et al who discovered Fiber Bragg Gratings

(FBG’s) at the Canadian communications research center in 1978 [7]. The

FBG’s were discovered during an experiment that was done using germania- doped silica fiber, and the laser used for this experiment was a visible argon ion laser. During the experiment it was discovered that the light directed coupled in the fiber increases reflectivity as a function of time. They figured that the It was suggested that a standing wave pattern was formed due to the reflected wave from the end of the fiber. This, in turn, modulates the refractive index grating inside of the fiber core. The modulation of the refractive index increases increase reflectivity working as a positive feedback. When the modulation of the refractive index reaches its maximum, an FBG is formed inside of the fiber core.

In the next decade only few researchers studied this interesting phenomena [8,9]. It was believed that the difficulties in experimental setting

20 at the Canadian Communications Research Center were the reasons and at that time.

Other researchers discovered reported different results from the nonlinear phenomenon of sum - frequency generation by Ohmari and Sasaki

[10] and K.O. Hill et al [11]. This phenomenon was the generation of second- harmonic generation germanium-doped silica fibers with zero second order nonlinear coefficient responsible for the second-harmonic generation. U.

Ö sterberg and Margulis [12] demonstrated that efficiency of the second harmonic could be enhanced by exposing germanium-doped silica fiber using infrared radiation, forming an FBG through nonlinear process [13,14]. Stone’s work on how argon laser radiation affects germanium-doped silica [15] showed that germania doped silica can be affected by any argon laser. This discovery opened the path in research of FBGs to generation of second harmonics in optical fibers [16] and to writing single mode gratings in fibers [17]. The two- photon absorption nature of the phenomenon from the fundamental radiation at 488nm in a fiber was first noticed by J. Bures et al [18].

Meltz et al proposed recording FBGs using holographic technique using single-photon absorption at 244nm in 1989[19]. This technique was based on a recording gratings in the fibers using interference of UV beams.

Figure 2.4.1: The Original Setup used by Meltz, Morey and Glenn [20].

Figure 2.4.2: The interference pattern created inside of the fiber [21].

An interference pattern created inside the fiber, which inscribe the grating into the fiber core. The period of the grating, (Λ) is given by,

휆 Λ = 푢푣 (2.4.1) 휗 2 푆푖푛(2)

And the Bragg wavelength, (휆퐵푟푎푔푔) is given by,

푛푒푓푓 휆푢푣 휆 = 2푛 Λ = (2.4.2) 퐵푟푎푔푔 푒푓푓 휗 푆푖푛(2)

Where; 휆푢푣 is the wavelength of the light UV radiation.

푛푒푓푓 is the effective refractive index of the fiber.

휗 is the angle between the two light beams.

Here the 푛푒푓푓 depends on the wavelength of the light and the mode at which the light propagates, this is the element that decided the velocity of the light in the fiber.

In 1992 Hill et al introduced a method to produce gratings using a phase mask. This is a photolithographic method that uses a special phase mask grating made out of silica. When a laser beam is incident on it, the light traveling through the mask creates thus creating a grating inside the fiber core. In this technique the 0th diffraction order is suppressed so the damage in the fiber core is minimized. The grating period can be varied depending on the phase mask period and the distance between the phase mask and the grating.

Figure 2.4.3: Phase Mask technique used to fabricate gratings in a fiber [23].

Another technique to produce gratings in the fibers is the point-to- point method. In this method the laser beam is focused at a point in the fiber core, and then moved to another point. The process repeated and the difference between two corresponding points will determine the grating period.

2.4.2. Types of FBG’s

There are two different ways we can group FBGs: By the conditions under which the FBGs are produced, and by the structure of the gratings.

2.4.2.1. By the varying conditions of gratings production

Below, we consider the temperature difference and the hydrogenation conditions during the FBGs production.

Type I

Type I FBGs are also known as standard gratings are the most common type used in telecommunication, and in temperature sensing for T< 80oC. They are produced at lower temperature and under moderate UV radiation.

Hydrogenated and non-hydrogenated conditions can be used in production.

These FBGs could be erased at T~ 200oC.

Type IA

This type of FBGs are produced under an intense exposure to UV radiation,

Hydrogenated optical fibers are typically used to fabricate this type of FBGs.

They present regenerated gratings [25] created after erasure of Type I FBGs.

A long exposure time is required for accomplish this procedure.

Type IA gratings were introduced in 2001 [24] with the purpose to determine influence of a hydrogen loading on the formation of type IIA gratings in germanosilicate fibers. The effective refractive index of these FBGs is greatly increased and the Bragg wavelength is shifted in the red region.

Type II

These gratings are made under pulsed UV radiation exceeding the damage threshold of the fiber core. The energy laser pulses would melt the core creating areas with modulated refractive index turning in to FBG. Since this type of FBGs based on periodic damage in the fiber core is known as “damaged gratings” type. Type II gratings produced by Archambault et al [26, 27] demonstrated almost 100% reflectivity withstanding temperatures up to 800-

1000oC.

Type IIA

Type IIA gratings also fabricated under prolonged UV radiation in a non-hydrogenated fiber. The first-order grating erased and a second-order grating is created. Unlike to the type IA gratings in these gratings the Bragg wavelength was shifted in the blue region.

Research performed by Xie et al [28,29] demonstrated that type IIA gratings were able to withstand high temperature (~500oC). In the process of

26 fabrication the first order grating was erased and the second order grating was created.

2.4.2.2. Structure (composition) of the gratings

Here we consider the structure of the FBGs inside of the fibers. The parameters we consider are: grating period and the variation (modulation) of the refractive index. The refractive index profile of the grating could be either uniform or apodized.

Uniform refractive index profile gratings have a simple structure compared to the apodized profile. Here the refractive index difference between the grating and the fiber core remains constant, that is if the is the fiber core has a refractive index n1 and the grating has a refractive index n3 the inside the grating the refractive indices would change to n1 or to n3. Inside the fiber only the grating period would change depending on the grating type.

Figure 2.4.4: Representation of uniform refractive index profile FBGs [30].

Figure 2.4.4 illustrates a uniform FBG’ profile with constant period. In the chirped FBG the grating period changes as a linear function with distance.

This gives the grating the ability to reflect multiple frequencies of light. In the

28 tilted FBG, the grating period stays uniform but the grating is tilted at an angle thus affecting light frequency.

In the apodized FBGs the refractive index changes according to an apodizing function. It is a mathematical function where it has zero value outside a chosen interval. Inside the interval it could have different variation depending on the mathematical function.

In the Figure 2.4.5 the uniform positive index profile shows a simple form of the Apodized structure. In the Gaussian-apodized grating the refractive index changes as a Gaussian function inside the grating with a positive offset of the refractive index. In the Raised-cosine-apodized index changes as a Cosine function inside the interval. A phase of the refractive index modulation is shifted the middle of the grating for the discrete phase shift index profile.

Figure 2.4.5: Apodized refractive index profile FBGs [30].

2.4.3. Photosensitivity

The photosensitivity of a fiber refers to its ability locally change the refractive index under exposure to radiation and is related to the defects in the fiber core. Meltz et al suggested that in germanium-doped fibers the photosensitivity peaks at around 240nm (UV radiation) [19] because of germanium-oxygen deficiencies [31].

To produce germanium-doped fibers, three main components are used: germanium tetrachloride (GeCl4), silicon tetrachloride (SiCl2) and oxygen (O2).

Below are the two chemical reactions between the chlorides and the oxygen during the manufacturing process.

SiCl4 + O2 → SiO2 + 2Cl2

GeCl4 + O2 → GeO2 + 2Cl2

GeO2 (liquid) ↔ GeO(gas) + ½O2 (gas)

The core of the fiber made of silica is mixed with germanium. Silica assumes a 3D tetrahedral atomic structure during the doping process when the germanium atoms replace some silicon atoms. Since germanium has 5 electrons in the valance band, only four will be in the bond with either oxygen or silicon, the 5th electron would be unsatisfied single electron.

Figure 2.4.6: The 3D Tetrahedral crystal structure of silica [35].

These defect are the paramagnetic Ge(n) defects [34]. Friebel et al introduced the model based on Ge-Si bonds in the oxygen deficient centers are called “wrong bonds” and these bonds are broken releasing electrons when irradiated by UV light.

GeO3SiO3 + UV → GeO3 + SiO3+ + e-

Figure 2.4.7: Graphical representation of “wrong bonds” broken under exposure of UV light. (a) Wrong bonds structure, (b) GeE’ structure, (c) oxygen vacant silicon atom, and (d) free electron released from the reaction.

There are three different types of defects in the germania-doped silica fibers, in all of these the middle atom is a germanium atom. Germanium atom has five electrons in the outer layer, only four of these electron would be in bond, the 5th would be an unpaired electron. When the germania atom bound to 4 silica it gives Ge(1) defect (Figure 2.4.8). In the Ge(2) defect (Figure 2.4.9) one of the silica would be replaced by germania. In the GeE’ defect (Figure 2.4.7(b)) an oxygen vacant silica would be right next to the germania.

Figure 2.4.8: The defect Ge(1) in germania-doped silica fiber.

Figure 2.4.9: The defect Ge(2) in germania-doped silica fiber

2.4.4. Applications of FBG’s

The FBGs are capable of measuring physical parameters such as temperature, strain, pressure, etc. The experimental procedure described by

Adamovsky et al of the thermo-optic coefficient under different temperature regimes described in [38]. The experimental setup for the study of the thermo- optic coefficient in FBGs contains two light sources (Figure 2.4.10). They include two lasers: one a 20 mW laser emitting light at 532 nm, and the second laser a superluminescent laser diode (SLD) 5 mW laser emitting light at 1310 nm and a minimum bandwidth of 20 nm. A light is coupled into two identical optical fibers with a core of diameter 9.15 µm. Both fibers are placed in ceramic capillary tubes and inside of a 24’’ long split hinge tube furnace heated up to

1200oC. One fiber contains the FBG and the second one is without one. The peak wavelength of the FBG at room temperature (20 oC) is about 1310.135 nm with a bandwidth between 16.8 and 25.7 GHz for 1 and 3 dB.

Figure 2.4.10: Schematic representation of the FBG experimental setup [38].

The radiation reflected by FBG is detected by a photo detector incorporated inside an optical spectrum analyzer. The wavelength of the detected light is subject to change with the temperature. The detected change is displayed on the screen of the optical spectrum analyzer. The visible light passing through the both fibers is observed on the screen. One pattern is from the regular fiber and the other is from the fiber with the FBG. The transmitted light is recorded using a CCD camera in order to study the observed LP modes of the two fibers. The diffraction efficiencies measured using the power ratio of the light reflected on the FBG with respect to the coupled light.

2.5 Finite Element Analysis and COMSOL Multiphysics Programming.

Finite element analysis (FEA) uses numerical techniques to find approximate solutions to problems. This method finds an approximated solution to a problem by solving partial differential equations using numerical integration. Even though traditionally an FEA was a part of the solid mechanics, now it’s used commonly in solving various problems. The FEA can be used for structure analysis, solid mechanics, dynamics, thermal physics, biomaterials, and optics.

The general procedure of the FEA is to study the problem and then simplify the problem to a model. The next step is to build a model for the simulation, add a mesh, the material properties and define the boundary conditions. Physics formalism is the key to define the functions and solve the equations.

The FEA uses numerical techniques to find the solutions for boundary value problems. But solving complex problems including multiple differential equations is impossible. For simplicity the FEA divides the interval of the mathematical problem in two parts.

This can be illustrated by taking a looking at an example [36]. First let’s take a simple integral, and use numerical methods to calculate the solution over a given range.

Let’s take the function,

푦 = 푥2 + 6

Now, to find the area (A) under the function y, between -1 and 1,

1 1 퐴 = ∫ 푦 푑푥 = ∫ (푥2 + 6)푑푥 −1 −1

Figure 2.4.11: Diagram of the area (A) in the interval between -1 to 1. The area under the function y is found by the integration [36].

By using simple integration we can find the value of A to determine the exact value of A.

1 1 1 1 38 퐴 = ∫ (푥2 + 6)푑푥 = |( 푥3 + 6푥)| = 2 ( + 6) = ≈ 12.667 −1 3 −1 3 3

The total area A could be divided into N number of subsections. Then a function is chosen to approximate the variation of y for each of the subsections.

The most simple is a constant that is equal to y at the midpoint of each of the subsections. This constant or the function along with the length of the section provides an approximate value for the area of each subsection. Finally by summing all of these values the total area A could be determined.

Figure 2.4.12 presents the number of subsections with the error 5.26%.

When 8 subsections are used the error is 0.08%. It is clear that the accuracy of the approximation increases with the number of subsections.

The FEA gives an approximation of an exact value. The accuracy depends on the number of the subsection (or the sub-intervals) and the function used during the approximation process. The selection of a mesh produces more accurate results but require computational power.

Figure 2.4.12: Graphical representation of numerical integration of the calculation of A, changing with the number of subsections used [36].

COMSOL Multiphysics is a simulation program that uses FEA. The

COMSOL provides an ability designing specific experiment by setting geometry, mesh, physics settings, boundary conditions, solvers, post- processing, and visualizations of the experimental results. The COMSOL has a friendly interface to design and build simulations for research purpose. The

40 negative side of COMSOL is that the program’s ability solving complex models depends highly on the power of the computer.

PDEs templates make model the systems with either nonlinear or linear equation systems. The COMSOL combines equations to solve higher order differential equation models. For example, in mechanics the COMSOL measures the stress of a metal supporting a beam including a thermal element component to measure how the model will behave with temperature.

It allows a user to simultaneously work in 3D and 2D. Different models could be coupled or used separately. The models could be done in Cartesian, cylindrical, spherical or even in Euler-angle based coordinate systems. The program has a modeling console which has built in basic shapes including the tools to model different shapes.

When designing a model the choice of mesh is very important. COMSOL provides automatic and semi-automatic tools for meshing including tetrahedral meshing and swept meshing. A tetrahedral meshing is a default mesh but the user defines the mesh. The COMSOL is mostly considered as an

FEA based program. Finite volume method (FVM) is very similar to FEA, although in the FVM a 3D model is divided into subsection of small volume.

The calculations are performed as in FEA. The boundary element method uses integration methods to solve the equations, as in our study. Also, the particle tracing method uses physics such as gravity or electromagnetic forces to

41 calculate the trajectory of particles and give the ability simulating interaction in particle-particle, particle-fluid or particle-field events.

COMSOL has a unique feature selecting an appropriate method of numerical analysis. Depending on the simulation all of the above mentioned methods could be coupled and automatically generated by the program during the solving process, since the FEA allows a user to combine different physics modules.

CHAPTER III

PROCEDURE

3. Simulations Using COMSOL

The first step for simulations using COMSOL is to decide which model to build. When using COMSOL for simulations, both the model we build and the physics depend on the dimensions used. Two and three dimensional models were chosen for our simulations. Next we picked the physics module and then we had started building our project.

When building the models, COMSOL gives option to define a “repair tolerance” under the geometry. Here, geometric entities that have a distance less than the absolute repair tolerance are merged.

퐴푏푠표푙푢푡푒 푟푒푝푎푖푟 푡표푙푒푟푎푛푐푒

= 푟푒푙푎푡푖푣푒 푟푒푝푎푖푟 푡표푙푒푟푎푛푐푒

∗ 푚푎푥푖푚푢푚 푐표표푟푑푖푛푎푡푒 표푓 푡ℎ푒 푖푛푝푢푡 표푏푗푒푐푡푠

3.1. 2D Step index fiber Model

The 2D Step index fiber model in the COMSOL library was used as a starting point. The premade model was modified for our simulation purposes.

The physics module for the simulation was “Electromagnetic waves” and

“Frequency domain” modules. In this simulation we used the mode analysis study, thus the program calculated the difference created for a given fiber.

Building the model started with setting the parameters for the fiber.

Parameters such as the refractive indices, wave length of the light and radius of the fiber were chosen.

Table 3.1: Table of Parameters for the 2D fiber Simulation

Name Expression Description

n1 1.4457 refractive index of the core

n2 1.4378 refractive index of the

cladding lambda 530[nm], 1310[nm], wave length of the light

1530[nm] f speed of light /lambda frequency of the light r_core 8 [µm] radius of the core r_cladding 125 [µm] radius of the cladding

After setting the parameters we started to build the exact model. The cladding was built using the COMSOL “geometry builder” to draw a circle, following a smaller circle in the middle. The circles had the same axis with the radii given in table 3.1

Figure 3.1: Basic model of the 2D fiber. Center circle is the core and the bigger circle is the cladding.

The next step was assigning the materials for the fiber core and the cladding. Doped Silica (n1 = 1.4457) was chosen for the core and silica glass (n2

= 1.4378) was chosen for the cladding.

After building the model of the fiber, we set the wave equation and the boundary conditions for the model. Equation 3.1.1 represents the wave equation for the electromagnetic wave for our model.

2 ∇ × (∇ × 퐸) − 푘0 휖푟퐸 = 0 (3.1.1)

퐸(푥, 푦, 푧) = 퐸̃(푥, 푦)푒−푖푧푘푧

The eigenvalues calculated by COMSOL are in the form:

휆 = −푖훽 − 훿 (3.1.2)

2 휖푟 = (푛 − 푖푘)

휔 푘0 = 휔√휀0휇0 = 푐0

Where 푘푧 is the plane wave number in the z-direction; 푧 is the unit vector in z- direction; E is the incident electric field; 훿 is the real part of the solution responsible for the damping; 훽 is the eigenfrequency; 푛 is the real part of the refractive index; 푘 is the imaginary part of the refractive index; 푘0 is he wave number of the free space; 휔 is the angular frequency; 휀0 is the permittivity of free space; 휇0 is the permeability of free space and 푐0 is the speed of light in vacuum.

The boundaries of the fiber were set to be perfect electric conductors.

Thus the EM waves could pass through the boundaries while obeying laws of physics. The initial value of the EM waves inside the Fiber was set to be zero.

Before and at the point start of the simulation of the experiment there would be no EM wave inside the fiber core.

Next the mesh for the model set which also affects the accuracy of the simulation’s results. As presented in chapter 2 section 5, the smaller elements yield higher accuracy but at the cost of requiring more computing power. The mesh contained triangular shaped elements with sizes between 2.5 µm and

0.005 µm. Setting the mesh was the final step of building the model.

After the model was built, mode analysis calculation settings were chosen. For the effective mode index with the green light (f=5.6565E-14), the modes for the model were calculated (effective mode indices were 1.446 and

1.44). The program was set to numerically calculate and find 100 modes around the effective mode index.

Figure 3.2: A close up of the triangular shaped mesh used for the 2D model.

3.2. 3D Model for Fiber Bragg grating

The 3D model was inspired by the experimental research from ref. 38.

The model build for the 3D simulations is genuine and built from the scratch.

For simulation purposes, we build a few models including a fiber without a

FBG and FBG’s with different numbers of gratings. The parameters and materials were remained the same. The fiber without the FBG was used as a control model.

Table 3.2 shows the list of parameters used for the FBG models. In the model the material used were based on the previous experimental research

[38]. For the fiber cladding, the silica grass with refractive index of 1.4378 was chosen. For the fiber core doped silica with refractive index of 1.4457 was chosen with the step index grating made of another doped silica with refractive index 1.4460. To find the period of the grating we rearranged the equation

2.4.2, where the grating period (Λ ) was:

휆퐵푟푎푔푔 Λ = 2푛푒푓푓

Table 3.2: Table of Parameters for the 3D fiber Simulation

Name Expression Value Description

n1 1.4457 1.4457 refractive index of the

core n2 1.4378 1.4378 refractive index of the

cladding n3 1.4460 1.4460 refractive index of the

step index grating n_eff (n1 + n2)/2 1.4417 effective refractive

index lambda 1310 [nm] 1.31E-6 m wave length of the

light

Λ lambda/(2 *n_eff) 4.5431E-7 m grating period f c_const/lambda 5.6565E14 1/s frequency of the light r_core 9 [µm] 9E-6 m radius of the core r_cladding 50 [µm] 5E-5 m radius of the cladding l Λ *x depends on x, length of the fiber

(x is the number of

slabs)

The physics module and the wave equation of the simulation was similar to the 2D model, however, the 3D wave equation was solved. The electric displacement field stayed the same as the 2D model:

2 ∇ × (∇ × 퐸) − 푘0 휖푟퐸 = 0 (3.1.1)

The eigenvalues calculated by COMSOL were in the form:

휆 = −푖훽 − 훿 (3.1.2)

2 휖푟 = (푛 − 푖푘)

휔 푘0 = 휔√휀0휇0 = 푐0

Where: E is the incident electric field; 훿 is the real part of the solution responsible for the damping; 훽 is the eigenfrequency; 푛 is the real part of the refractive index; 푘 is the imaginary part of the refractive index; 푘0 is he wave number of the free space; 휔 is the angular frequency; 휀0 is the Permittivity of free space; 휇0 is the Permeability of free space and 푐0 is the speed of light in vacuum

The boundary of the fiber was set to be perfect electric conductor for the cladding and air boundary. The boundary where the light enters and leaves the fiber had needed a special condition. Here, the scattering boundary condition was used. This boundary condition made the boundary transparent

51 for a scattered wave and the incident wave. The incident light was set to have an electric-field that is a plane wave and is propagated towards the z-direction.

As like the 2D model the initial value of the EM wave was set to be zero. Thus only the incident wave would be present inside the fiber.

푖푘∙푘푑푖푟∙푟 푛 × (∇ × 퐸) − 푖푘푛 × (퐸 × 푛) = −푛 × (퐸0 × (푖푘(푛 − 푘푑푖푟))) 푒 (3.2.1)

−푖푘(푛∙푟) −푖푘(푘푑푖푟∙푟) 퐸 = 퐸푠푐푒 + 퐸0푒 (3.2.2)

−푖푘(푛∙푟) 퐸푠푐 = 퐸푠푐푒

Where: 푘푑푖푟 is the directional vector of the wave; 푘 is the wave number; 퐸 is the incident plane wave; 퐸0 = (−1,1,0) in (x,y,z) directions for the simulation and 퐸푠푐 is the plane scattered wave

As mentioned above several fiber models were designed for the simulations. All of the fibers had the same core and cladding radii and the materials but the length of the fiber was changed according to the length of the grating. The first model was the fiber without the grating. The model was a simple model consisting of two cylinders; one to form the core and the second was the cladding. The cylinders were assigned with the materials mentioned above.

Figure 3.3: The 3D model of the fiber.

Next a model for a single grating was made, which was built exactly as the 3D fiber but with the length of the fiber equal to the period of a grating.

This let us to study the behavior of light when it is passing through a single piece of grating.

Figure 3.4: The 3D model of the single grating with thickness Λ =454.31 nm.

The next step was to build fiber models with the grating. These models were started by building the grating with 10 slabs. 3 models of the fiber with

10 slabs were built with different grating periods (1310 nm, 908.62 nm and

454.31 nm) in order to study the dependence of the wave propagation through a fiber on the grating period. Then the number of slabs was increased to 50 and

100 slabs. The purpose was to study the dependence of the wave propagation through the fiber with the number of the slabs in the grating. The first step was to build the grating, which was shaped to a cylindrical disk with above

54 mentioned radius of the core. The thickness of the disk was set to be the grating period and then the grating was cloned to form the desired number of slabs.

Figure 3.5: The 3D model of the FBG with 10 slabs.

Figure 3.6: The 3D model of the FBG with 50 slabs.

Figure 3.7: The 3D model of the FBG with 100 slabs.

A model was created to simulate a FBG with a step index grating. After the model was built, the even numbered disks were assigned with the material of the core and the odd numbered disks were assigned with the grating material except the first was set for the cladding.

Figure 3.8: The grating with of the FBG with 10 slabs. Where the slabs in the core with even number is assigned with the core material and the odd number is assigned with grating material. The outer was cylinder was assigned with the material of the cladding.

Figure 3.9: The grating with of the FBG with 50 slabs with each even numbered disk assigned with the core material and the odd numbered disk assigned with grating material.

Figure 3.10: The grating with of the FBG with 100 slabs with each even numbered disk assigned with the core material and the odd numbered disk assigned with grating material.

The mesh of the 3D model was set similar for all the models. Fine mesh settings result with high accuracy. The mesh elements were tetrahedral shaped with sizes in between 3.5 µm and 0.15 µm. The smallest possible element size was chosen to increase the accuracy of the results.

Figure 3.11: The tetrahedral mesh on the 3D model. Since the light propagates through the core, it has a smaller mesh to increase the accuracy of calculations.

Cladding was given a larger mesh to decrease the unnecessary use of the computer power.

Figure 3.12: A close up of the tetrahedral shaped mesh used for the 3D model.

A study for the 3D model was to calculate “Boundary mode analysis” and

“Frequency domain”. For the boundary mode analysis, the effective analysis frequency was selected to be the frequency of EM waves in the infrared region and in the visible range. The simulations were done for different EM waves.

The modes calculated were the modes around the effective refractive index. As in the 2D model the program was set to find 100 modes around the effective mode index. The frequency domain could be set to have a range of frequencies but for our simulation the same frequency that was used in the boundary mode analysis was used.

CHAPTER IV

RESULTS & DISCUSSIONS

4.1. Results of the 2D model

The COMSOL simulations provide modes that are physically possible in the step index fiber. The simulations provide 100 possible modes for the particular selected model. Of these 100 possible modes some are identical and some of them are not stable modes. Therefore only few of the 100 modes could be chosen to represent the LP modes. The models chosen for the simulations were all identical except for the wavelength of the electromagnetic waves used.

The experimental results of the fiber bragg grating modes represent the power distribution. Therefore the simulated modes were plotted also to represent the power distribution. The equation used to plot is shown below,

2 2 2 푃 = √푃푥 + 푃푦 + 푃푧 (4.1.1)

Where; P is the total power

Pi is the power in the x, y, and z directions in a cartesian

coordinate

Simulation results for the 530 nm light waves.

Figure 4.1: LP01 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.2: LP12 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.3: LP21 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.4: LP01* (donut shape) mode experimental [40] on the top left compared to simulated results of a few modes on the top right and on bottom, the simulated results have the same outer diameter but the inner diameter is changing

Figure 4.5: LP41 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.6: LP51 mode experimental on the left compared to simulated result on the right.

Simulations of the 2D fiber for 530 nm wavelength light were able to sustain a variety of modes. The majority of the results were duplicate solutions of the same stable mode. There were 6 unique stable modes and the highest stable mode was LP51. Interestingly, the 530 nm wavelength light was not able to sustain LP02 mode, while the 1310 nm and 1530 nm were.

Simulation results for the 1310 nm IR waves.

Figure 4.7: LP01 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.8: LP02 mode experimental [39] on the left compared to simulated

result on the right.

Figure 4.9: LP12 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.10: LP01* (donut shape) mode experimental [40] on the top left compared to simulated results of few modes on the top right and on bottom.

Similar to the simulation results at 530 nm wavelength, simulations at

1310 nm and 1530 nm were able to sustain a variety of mode. But the majority of the modes were duplicate results of the same mode. The 1310 nm wavelength was able to sustain 4 different modes. The highest mode sustained by 1310 nm wavelength was LP12.

Simulation results for the 1530 nm IR waves.

Figure 4.11: LP01 mode experimental [39] on the left compared to simulated result on the right.

Figure 4.12: LP02 mode experimental [39] on the left compared to simulated result on the right.

The simulations of1530 nm wavelength sustained only 2 modes. Both 1310 nm and 1530 nm wavelengths were able to sustain the LP02 mode, which was the highest mode sustained by the 1530 nm wavelength.

The simulation results for the 2D model are summarized in the table given below.

Table 4.1: Stable modes sustained by the fiber in the 2D model

Wave Stables modes sustained Number of different

modes

530 nm LP01 , LP12 , LP01* , LP21 , LP41 , LP51 6

1310 nm LP01 , LP02 , LP01* , LP12 4

1530 nm LP01 , LP02 2

From table 4.1 we can see that the shortest wavelength (530 nm) was able to sustain 6 different stable modes and it was able to sustain stable modes up to LP51. But as the wavelength was increased the number of modes sustained by the fiber was decreased. The second lowest wavelength (1310 nm) sustained 4 different modes and the highest wavelength (1530 nm) was able to sustain only 2 different modes. Similarly the highest mode possible by the fiber was also decreased as the wavelength increased. The second lowest wavelength

(1310 nm) sustained modes up to LP12 mode and the highest wavelength (1530 nm) was able to sustain modes up to LP02 mode.

4.2. Results of the 3D model

As was the case in the 2D model simulations, the 3D model COMSOL simulations provide modes that are possible in the 3D fiber. The simulation was set to provide 100 possible modes for the selected model. An interesting result for the 3D model was that each fiber model was able to sustain only one numerically stable mode. In 3D simulations a fiber with no FBG was first simulated as a control. Then the rest of the model was modifies by the number of the slabs and the grating period. IR light with wavelength 1310 nm was used for the simulation.

In a similar manner to the 2D model, the 3D model results were plotted to represent the power distribution. The equation used to plot is the power shown below,

2 2 2 푃 = √푃푥 + 푃푦 + 푃푧 (4.1.1)

Where; P is the total power

Pi is the power in the x, y, and z directions in a cartesian

coordinates

Simulation results of the Fiber without a FBG.

Figure 4.13: Incident wave of the fiber with no FBG.

Figure 4.14: LP11 mode experimental [39] on the left compared to simulated result of the transmitted wave on the right. The resultant mode indicates instability in the mode.

Figure 4.15: Power distribution of the transmitted for the fiber with no FBG

Simulation results for a single grating with grating period, 2Λ = 908.62 nm,

Figure 4.16: The experimental LP12 [39] on the right, incident wave on the middle and the transmitted wave on the right. Both incident and transmitted wave modes are identical.

Figure 4.17: Power distribution of the single grating period 2Λ.

Simulation results for a single grating with grating period, Λ = 454.31 nm,

Figure 4.18: The experimental LP12 [39] on the right, incident wave on the middle and the transmitted wave on the right. Both incident and transmitted wave modes are identical.

Figure 4.19: Power distribution of the single grating period Λ.

Simulations of single gratings with periods 2Λ = 908.62 nm (Figure 4.19) and Λ = 454.31 nm (Figure 4.22) resulted in LP12 mode. When the length of a fiber was changed to the size of a grating the mode sustained by the fiber was clearly changed from PL11 to LP12. Even without adding a grating the change of the length of the fiber affected the stable mode sustained by the fiber.

Simulation results of the fiber with grating period of 1310nm and 10 slabs.

Figure 4.20: Incident wave of the fiber with 10 slabs and grating period

1310nm

Figure 4.21: LP11 mode experimental [39] on the left compared to simulated result of the fiber with 10 slabs and grating period 1310nm. The sustained mode indicates instability.

Figure 4.22: Power distribution of the transmitted wave of the fiber with 10 slabs and grating period 1310nm.

Simulation results of the fiber with grating period, 2Λ = 908.62 nm and 10 slabs.

Figure 4.23: Incident wave of the fiber with 10 slabs and grating period 2Λ

Figure 4.24: LP11 mode experimental [39] on the left compared to simulated result of the fiber with 10 slabs and grating period 2Λ.

Figure 4.25: Power distribution of the transmitted wave of the fiber with 10 slabs and grating period 2Λ.

Simulation results of the fiber with grating period, Λ = 454.31 nm and 10 slabs.

Figure 4.26: Incident wave of the fiber with 10 slabs and grating period Λ.

Figure 4.27: LP12 mode experimental [39] on the left compared to simulated result of the fiber with 10 slabs and grating period Λ.

Figure 4.28: Power distribution of the transmitted wave of the fiber with 10 slabs and grating period Λ.

Simulation results of the Fiber with grating period, Λ = 454.31 nm and 50 slabs.

Figure 4.29: For the fiber with 50 slabs and grating period Λ, incident wave on left. The transmitted wave (right) seems to form a combination of LP12 and LP21 modes.

Figure 4.30: Power distribution of the transmitted wave of the fiber with 50 slabs and grating period Λ.

Simulation results of the Fiber with grating period, Λ = 454.31 nm and 100 slabs.

Figure 4.31: For the fiber with 100 slabs and grating period Λ, incident wave on left. The transmitted wave (right) seems to form a combination of LP12 and

LP21 modes.

Figure 4.32: Power distribution of the transmitted wave of the fiber with 100 slabs and grating period Λ.

Table 4.2: Simulation results of the fiber with 10 slabs for different grating periods Grating Incident wave Transmitted wave LP Mode Period (experimental)

1310nm

2Λ = 908.62 nm

Λ = 454.31 nm

From table 4.2 we can see that as the period of the grating is decreased and fulfills the Bragg condition the mode became more stable. Furthermore the mode of the fiber change from unstable LP11 to a stable LP12.

Table 4.3: Simulation results of fibers with grating period Λ = 454.31 nm, and different number of slabs.

Number Incident wave Transmitted LP Mode of slabs wave (experimental)

50 Combination of LP12 and LP21 modes

100 Combination of LP12 and LP21 modes

Table 4.3 shows that the number slabs does have an effect on the stable mode of the fiber. As the number of the slabs got higher the mode changed from

LP12 to combination of LP12 and LP21. The fibers with 50 and 100 slabs show a mode that is changing to form a combination of LP12 and LP21 modes, but hasn’t fully formed the mode.

As mentioned earlier for the 3D model the simulations were only able to sustain one stable mode for the each simulation. The simulation results for the

3D model are summarized in Table 4.4 below

Table 4.4: Summary of simulation results for the 3D model

3D model mode Comments

sustained

Fiber with no FBG LP11 The observed mode is not sharply

defined in shape. This could be due

to instability in the simulation

result.

Single grating of LP12 The mode is sharp and stable period 2Λ

Single grating of LP12 The mode is sharp, stable and period Λ brighter than single slit with

width 2Λ. which indicates higher

stability

Fiber with 10 slabs LP11 The mode sustained is not defined and Grating period sharply in shape. which indicates

1310nm instability in the simulation

result.

Fiber with 10 slabs LP11 The mode sustained is sharper and and Grating period 2Λ stable.

Fiber with 10 slabs LP12 The mode sustained is sharp and and Grating period Λ stable. The mode that is sustained

is a higher order mode.

Fiber with 50 slabs LP12 + LP21 The mode sustained resemble the and Grating period Λ LP21 mode. Increasing the number

of slabs from 10 to 50 seems to

change the stable mode from LP12

to combination of LP12 and LP21

modes.

Fiber with 100 slabs LP12 + LP21 The mode sustained appears to be and Grating period Λ a combination of LP12 and LP21

modes. The mode stability of the

100 slabs versus the 50 slabs

appears to be increased.

CHAPTER V

CONCLUSIONS

Stable modes sustained by 2D COMSOL simulation have been compared with theoretical and experimental [39, 40] results and found to be in a good agreement. LP modes manifest more stability at lower wavelengths. It appears that lower wavelengths are able to sustain higher order modes and a higher number of stable modes. The highest possible mode for 530nm light was LP51 and it was able to sustain 6 modes.

A novel use of the 3D COMSOL model in this thesis was the introduction of a step like grating composed of 10, 50 and 100 slabs of refractive index varied between n1 = 1.4457 and n3 = 1.4460 introduced in the silica fibers. Each slab of the grating is a cylinder with radius of rcore = 9 µm and height of grating periods 1310nm, 908.62 nm and 454.31 nm. Each slab is placed consecutively with refractive indices alternating between n1 & n3 to form the grating. Each model in the 3D simulations was set to calculate 100 possible modes, but each model was able to sustain only one stable mode. It was found that the fiber without a FBG was able to sustain only the LP11 mode. The single grating simulation was able to sustain only the LP12 mode.

For the Bragg grating composed of 10 slabs when the grating period was 1310 nm and 908.62 nm, it was able to sustain LP11 mode, but when the grating period was changed to 454.31 nm (to satisfy the Bragg condition) the stable mode sustained was LP12 and with higher stability. The stability of the modes strongly depends on the grating period. Tuning the period close to the Bragg condition produced higher stability in the mode. Increasing the number of slabs in the grating from 10 to 50 to 100 produced a combination of LP12 and LP21 modes versus LP12. The number of slabs in the grating affected the stable mode that could be sustained by the fiber.

Finally, scope exists for future work to introduce variable temperature and strain modules the 3D model in order to simulate real-world temperature and strain sensors using fiber Bragg gratings.

REFERENCES

1. D. Marcuse, in Theory of Dielectric Optical Waveguides. Academic Press, New York, 1994.

2. D. Gloge, “weakly guiding fibres”, Appl. Opt. 10(10), 2252-2258 (1971)

3. F.L. Pedrotti, L.S. Pedrotti, geometrical optics and fiber optics. Prentice- Hall. Inc. 1987, Ch. 10.

4. A.W. Snyder and J.D. Love, in Optical Waveguide Theory. Chapman-Hall, London, 1983.

5. Ajoy Ghatak and K. Thyagarajan, Fundamentals of photonics- module 1.7 optical waveguides and Fibers, Department of Physics , Indian Institute of Technology , New Delhi, India.

6. Raman Kashyap, Fiber Bragg Gratings, Academic Press, 1999, Ch. 1-4.

7. K.O. Hill, Y. Fuji, D.C. Johnson, and B.S. Kawasaki “Photosensitivity in optical waveguides: Application to reflection filter fabrication”, Appl. Phys. Lett. 32(10), 647 (1978).

8. J. Bures, J. Lapiere, and D. Pascale, “Photosensitivity effect in Optical fibers: A model for the growth of an interference filter fabrication” Appl. Phys. Let. 37(10), 860 (1980).

9. D.W.K. Lam and B.K. Garside, “Characterisation of Single-mode Optical fiber filters”, Appl, Opt. 20(3), 440 (1981).

10. Y. Ohomari and Y. Sasaki, “Phase matched sum frequency generation in optical fibers”, Appl, Phys. Lett. 39, 466-468 (1981)

11. Y. Fuji, B.S. Kawasaki, K.O. Hill, and D.C. Johnson, ‘Sum frequency generation in optical fibers”, Opt. Lett. 5, 48-50 (1980)

12. U. Ö sterberg and W. Margulis, “Efficient second harmonic in an Optical fiber”, in Technical Digest of XIV internat. Quantum Electron. Conf., paper WBB1 (1986)

13. R.H Stolen and H.W.K. Tom, “Self-organized phase-matched harmonic generation in optical fibers”, Opt, Lett. 12, 585-587 (1987).

14. M.C. Farries, P. St.J. Russell, M.E. Ferman, and D.N. Payne, “Second harmonic generation in an optical fiber by self-writtern χ grating” Electron. Lett. 23, 322-323 (1987).

15. J. Stone, “Photorefractivity in GeO2-doped silica fibers”, J. Appl. Phys. 62(11) 4371 (1987).

16. R. Kashyap, “Photo induced enhancements of second harmonic generations in optical fibers”, Technical digest series, Vol. 2, February 2-4, 1989, Houston (Optical Society of America, Washington, D.C. 1989), pp. 255-258.

17. D.P. Hand and P. St. J. Russell, “Single mode fibre gratings written into a Sagnac loop using photosensitive fibre: transmission filters”, IOOC, Technical Digest, pp. 21C3-4, Japan (1989).

18. J. Bures, S. Lacroix, and J. Lapiere, “Bragg reflector induced by photosensitivity in an optical fibre: model growth and frequency response”, Appl, Opt. 21(19) 3052 (1982)

19. G. Meltz, W.W. Morey, and W.H. Glenn, “Formation of Bragg gratings in optical fibers by transverse holographic method”, Opt. Lett. 14(15), 823 (1989).

20. K.O. Hill, B. Malo, F. Bilodeau, D.C. Johnson, J. Albert, “Bragg gratings fabricated in monomode photosensitive optical fiber by UV exposure through a phase mask”, Appl. Phys. Lett. 62(10), 1035-1037. (1992).

21. Fiber Optics and Non Linear Optics laboratory; Recording of Bragg Gratings in Optical Fibers.

(http://eewebt.technion.ac.il/LABS1/nl/new_nl/Info/projects/Recording/Rec ording.html)

22. Encyclopedia of Laser Physics and Technology

(http://www.rp-photonics.com/fibers.html)

23. A. Othonos, “Fiber Bragg gratings”, Rev. Sci. Instrum. 68 (12), 4309 – 4341. (1997)

24. Liu, Y. (2001), Advanced fiber gratings and their application, Ph.D. Thesis, Aston University.

25. J. Canning, M. Stevenson, S. Bandyopadhyay, K. Cook, "Extreme silica optical fibre gratings". Sensors 8: 1–5, (2008).

26. L. Dong, J. L. Archambault, L. Reekie, P. S. J. Russell, D. N. Payne, "Single- Pulse Bragg Gratings Written During Fibre Drawing". Electronics Letters 29 (17): 1577–1578, (1993).

27. J. L. Archambault, L. Reekie, P. S. J. Russell, "100-Percent Reflectivity Bragg Reflectors Produced in Optical Fibres By Single Excimer-Laser Pulses". Electronics Letters 29 (5): 453–455. (1993).

28. W. X. Xie, P. Niay, P. Bernage, M. Douay, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, "Experimental-Evidence of 2 Types of Photorefractive Effects Occurring During Photo inscriptions of Bragg Gratings Within Germanosilicate Fibres". Optics Communications 104 (1- 3): 185–195. (1993).

29. P. Niay, P. Bernage, S. Legoubin, M. Douay, W. X. Xie, J. F. Bayon, T. Georges, M. Monerie, B. Poumellec, "Behaviour of Spectral Transmissions of Bragg Gratings Written in Germania-Doped Fibres - Writing and Erasing Experiments Using Pulsed or CW UV Exposure". Optics Communications 113 (1-3): 176–192. (1994).

30. Fiber Bragg Gratings – Wikipedia.

(https://en.wikipedia.org/wiki/Fiber_Bragg_grating).

31. H. Hosono, Y. Abe, L. Kindser, R.A. Weeks, K. Mutta, H. Kawazoe, “Nature and Origin of the 5-e band in GeO2:SiO2 glasses”, Phys. Rev. B46, 11445- 11451. (1992).

32. Y. Hibino, H. Hanafusu, “Defect Structure and formation mechanism of drawing induced absorption at 630nm in Silica optical fibers”, J. Appl. Phys. 60, 1797 (1986).

33. P. Kaiser, “Drawing induced coloration in vitreous silica fibers”, J. Opt. Soc. Am. 64, 475 (1974).

34. E.J. Friebele, D.L. Griscom, and G.H. Siegel Jr., “Defect centers im Germania doped silica core optical fiber”,J.Appl.Phys.45,3424-3428. (1974).

35. P. Mégret, S. Bette, C. Crunelle, C Caucheteur, “Fiber Bragg Gratings: fundamentals and applications”, FBG lecture April 2007.

36. H. Qi, “Finite Element Analysis” Lecture notes, fall, 2006 – Univeristy of Colorado.

37. L. Solymar, D.J. Webb, A. Grunnet-Jepsen, The Physics and Applications of Photorefractive materials, Clarendon press, Oxford, 1996. Ch-1.

38. G. Adamovsky, S.F. Lyuksotov, J.R. Mackey, B.M. Floyd, U. Abeywickrama, I. Fedin and M. Rackaitis, “Peculiarities of thermo-optic coefficient under different temperature regimes in optical fibers containing fiber bragg gratings”, Optics Communications 285 (2012) 766-773

39. I Gómez-Castellanos, R.M. Rodríguez-Dagnino, “Intensity distributions and cutoff frequencies of linearly polarized modes for a step-index elliptical optical fiber”, Opt. Eng. 46(4), 045003 (April 10, 2007). doi:10.1117/1.2719698

40. Transverse More – LP modes – Wikipedia.

(https://en.wikipedia.org/wiki/Transverse_mode).