bi-tapered fiber sensor using a supercontinuum light source for a wide measuring range

BI-TAPERED FIBER SENSOR USING A SUPERCONTINUUM LIGHT SOURCE

FOR A BROAD SPECTRAL RANGE

Dissertation

Submitted to

The School of Engineering of the

UNIVERSITY OF DAYTON

In Partial Fulfillment of the Requirements for

The Degree of

Doctor of Philosophy in Electro-Optics

By

Diego Felipe García Mina

UNIVERSITY OF DAYTON

Dayton, Ohio

May, 2017

ii

BI-TAPERED FIBER SENSOR USING A SUPERCONTINUUM LIGHT SOURCE

FOR A BROAD SPECTRAL RANGE

Name: Garcia Mina, Diego Felipe.

APPROVED BY:

_________________________________ ______________________________

Joseph W. Haus, Ph.D. Imad Agha, Ph.D. Advisory Committee Chairman Committee Member Professor, Electro-Optics and Photonics, Assistant Professor, Physics and Electrical and Computer Engineering Electro-Optics and Photonics _________________________________ ______________________________ Andrew Sarangan, Ph.D. Karolyn Hansen, Ph.D. Committee Member Committee Member Professor, Electro-Optics and Photonics, Associate Professor, Biology and Electrical Engineering Department _________________________________ ______________________________ Robert J. Wilkens, Ph.D., P.E. Eddy M. Rojas, Ph.D., M.A., P.E. Associate Dean for Research and Innovation Dean, School of Engineering Professor School of Engineering

iii

© Copyright by

Diego Felipe Garcia Mina

All rights reserved

2017

iv

ABSTRACT

BI-TAPERED FIBER SENSOR USING A SUPERCONTINUUM LIGHT

SOURCE FOR A BROAD SPECTRAL RANGE

Name: Garcia Mina, Diego Felipe University of Dayton

Advisor: Dr. Joseph W. Haus

We describe the fabrication bi-tapered optical fiber sensors designed for shorter

wavelength operation and we study their optical properties. The new sensing system

designed and built for the project is a specialty optical fiber that is single-mode in the

visible/near infrared wavelength region of interest. In fabricating the tapered fiber we

control the taper parameters, such as the down-taper and up-taper rate, shape and length,

and the fiber waist diameter and length. The sensing is mode is via the electromagnetic

field, which is evanescent outside the optical fiber and is confined close to the fiber’s

surface (within a couple hundred nanometers). The fiber sensor system has multiple

advantages as a compact, simple device with an ability to detected tiny changes in the

refractive index.

v

We developed a supercontinuum light source to provide a wide spectral wavelength

range from visible to near IR. The source design was based on coupling light from a

femtosecond laser in a photonic crystal fiber designed for high nonlinearity. The output

light was efficiently coupled into the bi-tapered fiber sensor and good signal to noise was

achieved across the wavelength region.

The bi-tapered fiber starts and ends with a single mode fiber in the waist region

there are many modes with different propagation constants that couple to the environment

outside the fiber. The signals have a strong periodic component as the wavelength is

scanned; we exploit the periodicity in the signal using a discrete Fourier transform analysis

to correlate signal phase changes with the refractive index changes in the local

environment. For small index changes we also measure a strong correlation with the

dominant Fourier amplitude component. Our experiments show that our phase-based signal

processing technique works well at shorter wavelengths and we extract a new feature, the

Fourier amplitude, to measure the refractive index difference.

We conducted experiments using aqueous medium with controlled refractive

index, such as water-glycerol mixtures. We find sensitivity to changes in the refractive

index close to n=2x10-5 in so-called Refractive Index Units (RIUs). That is smaller than

reported in recent literature, but by no means a limiting value. The technique is not

limited to aqueous solutions surrounding the fiber, but it can also be adapted to study

volatile organic compounds. Future improvements in the fiber sensing system are

discussed, including adding thin films to the surface for label-free detection and to draw

the electromagnetic field to the fiber’s surface.

vi

Dedicated to my parents and my brother Pablo Cesar (R.I.P)

vii

ACKNOWLEDGEMENTS

I would first like to thank to God. I am grateful to my thesis advisor Dr. Joseph W.

Haus of Electro-Optics and Photonic Department. I feel admiration and also a great respect

to Dr. Haus; he was the most important reason for me to select University of Dayton for a

Ph.D. The door to Professor Haus office was always open whenever I had a question about

my classes or research. He pointed me in the right direction for my thesis. It was an honor

for me to have worked with him.

After my advisor, I would like to thank the other members of my thesis committee:

Dr. Karolyn Hansen, Dr. Andrew Sarangan, and Dr. Imad Agha, for their encouragement,

significant comments, and interesting questions.

My sincere thanks also goes to Dr. Andy Chong, it was fantastic to have an

opportunity to work closely with him on the supercontinuum project.

I must express my very profound thankfulness to my mother, my family, and my

girlfriend for providing me with unfailing support and continuous encouragement

throughout my years of study and through the process of researching and writing this thesis.

This accomplishment would not have been possible without them. Thank you.

I also expand my great thanks to my roommate Dr. Ujitha Abeywickrema because

he was a wonderful friend and classmate during my Ph.D. I thank my friends from Mexico,

Dr. Daniel Jauregui and Dr. Juan Manuel Sierra because they give important

recommendations at the beginning of my project and they were always aware of my

viii

progress.

A very special gratitude goes out to COLCIENCIAS AND COLFUTURO for

helping, unfailing support, assistance, and providing the funding for the work during my

M.Sc. and Ph.D. I am also grateful to Electro-Optics and Photonic Department at

University of Dayton, faculty staff, and my classmate because they were my second family.

And finally, but not means least, also to everyone to continuous encouragement throughout

my years of study.

ix

TABLE OF CONTENTS

ABSTRACT ....................................................................................................................... iv

DEDICATION……………………………………………………………………………vi

ACKNOWLEDGEMENTS .............................................................................................. vii

LIST OF TABLES ............................................................................................................ xii

LIST OF FIGURES ......................................................................................................... xiii

LIST OF SYMBOLS ...................................................................................................... xvii

CHAPTER 1 BACKGROUND .......................................................................................... 1

Introduction .......................................................................................................... 1

Fiber Optics and Sensing...................................................................................... 2

CHAPTER 2 FIBER OPTICS ............................................................................................ 5

Introduction .......................................................................................................... 5

Mathematical Representation of Optical Fibers ................................................... 6

2.2.1. The Wave Equation....................................................................................... 7

Planar Waveguide ................................................................................................ 9

Eigenvalues of Propagating Waveguide ............................................................ 12

Step Index Fiber ................................................................................................. 15

2.5.1. TM Mode Solutions .................................................................................... 18

2.5.2. TE Mode Solutions ..................................................................................... 19

2.5.3. The Fundamental HE11 Mode ..................................................................... 21

x

2.5.4. The Hybrid Modes ...................................................................................... 22

Important Parameters in Fibers Optics ............................................................... 30

2.6.1. Numerical Aperture .................................................................................... 30

2.6.2. Normalized Frequency ................................................................................ 31

Chapter Summary ............................................................................................... 31

CHAPTER 3 SUPERCONTINUUM LIGHT SOURCE .................................................. 33

Introduction ........................................................................................................ 33

Numerical Modeling .......................................................................................... 35

Basic Numerical Simulation............................................................................... 39

Experimental Results.......................................................................................... 40

Chapter Summary ............................................................................................... 44

CHAPTER 4 FIBER OPTICS SENSORS ........................................................................ 45

Introduction ........................................................................................................ 45

Tapered Fiber Optics .......................................................................................... 45

4.2.1. Tapered Fiber Geometry ............................................................................. 46

Sensing Principles .............................................................................................. 48

4.3.1. Signal Processing ........................................................................................ 49

Bi-Tapered Faber Fabrication ............................................................................ 53

Experimental Sensing in IR Region ................................................................... 55

4.5.1. Results in IR Bands ..................................................................................... 56

Experimental Sensing from Visible to Near Infrared Region ............................ 60

4.6.1. Results from Visible to Near Infrared Region ............................................ 62

Chapter Summary ............................................................................................... 70

xi

CHAPTER 5 SUMMARY AND FUTURE WORK ........................................................ 71

Summary ............................................................................................................ 71

Ongoing Work .................................................................................................... 72

5.2.1. Tapered Fiber with Metal Coating .............................................................. 72

5.2.2. Surface Plasmon Polariton .......................................................................... 74

5.2.3. Optical Properties of Metals ....................................................................... 75

5.2.4. Metal Film Selection ................................................................................... 77

5.2.5. Experimental Results with TiO2 ................................................................. 78

REFERENCES ................................................................................................................. 86

xii

LIST OF TABLES

Table 1. Propagation modes in fiber optics for selected wavelengths with an

air cladding........................................................................................................................ 26

Table 2. Propagation modes in fiber optics for selected wavelengths with a

water cladding. .................................................................................................................. 26

Table 3. Values of the slope from Eq. 2-47 for air and water claddings in

units cm ∙ nm-1. ................................................................................................................. 30

Table 4. Glycerol concentrations in water and their corresponding refractive

indices n. ........................................................................................................................... 57

xiii

LIST OF FIGURES

Figure 1. Tapered fiber profile after heating (not to scale). ................................................ 2

Figure 2. Regular fiber Schematic with core and cladding. ................................................ 6

Figure 3. Slab optical waveguide. ..................................................................................... 10

Figure 4. TE, TM, and k-vetor inside of the planar waveguide. ....................................... 11

Figure 5. Graphical solution of planar waveguide. ........................................................... 14

Figure 6. (a) The five Bessel functions of the first kind, Jν(r), and (b) the modified

Bessel functions Kν(r) of the first kind. .......................................................................... 16

Figure 7. Fiber waveguide with core radius a................................................................... 17

Figure 8. Transverse electric field for radial TM01. .......................................................... 19

Figure 9. Transverse electric field for azimuthal TE01. .................................................... 20

Figure 10. Transverse electric field for the fundamental HE11 mode for two orthogonal

polarizations. ..................................................................................................................... 21

Figure 11. Graphical solution of the Eq. 2-44 for the fundamental HE11 mode for

several wavelengths. ......................................................................................................... 25

Figure 12. Difference between selected Propagation constants versus wavelength for a

10 micron diameter fiber. The inset is the propagation constants for the first six

cladding modes of the fiber with water outside the fiber.................................................. 28

Figure 13. Difference between selected Propagation constants versus wavelength for a

10 micron diameter fiber. The inset is the propagation constants for the first six

xiv

cladding modes of the fiber with air outside the fiber. ..................................................... 29

Figure 14. Photonic crystal fiber profile. .......................................................................... 34

Figure 15. Raman response function hR respect to the time t. .......................................... 37

Figure 16. Input and output spectra for different peak powers. ........................................ 40

Figure 17. Autocorrelation spectrum from Ti:sapphire laser. ........................................... 41

Figure 18. 3D diagram of the experimental setup for a supercontinuum generation. ...... 42

Figure 19. The input spectrum and output supercontinuum spectrum using the

Ti:sapphire laser. ............................................................................................................... 43

Figure 20. Geometry of biconical tapered (aka bi-tapered) fiber. .................................... 47

Figure 21. Cosine wave for two interfering modes, as a function of wavelength; the

fiber diameter is 10 m and it is immersed in water; the fiber length is 12 mm. ............. 51

Figure 22. Spatial frequency spectrum of the cosine wave. ............................................. 52

Figure 23. Phase for the wave frequency 0.55 nm-1 Fourier component versus

wavelength. ....................................................................................................................... 53

Figure 24. Vytran GPX-3000 machine for fabrication of tapered fiber. .......................... 54

Figure 25. Taper fiber image using the Vytran machine. ................................................. 54

Figure 26. Experimental setup of tapered refractive index change. ................................. 55

Figure 27. 3D diagram of the tabletop experimental setup with all the devices. .............. 56

Figure 28. Power transmission spectrum of the tapered fiber sensor as a function of

wavelength. ....................................................................................................................... 57

Figure 29. Spatial frequency spectrum from transmission for tapered fiber with

L=10 mm, d=10 μm. ......................................................................................................... 58

Figure 30. Phase shift versus wavelength for the complex data for a tapered fiber with

xv

L=5 mm, d=10 μm. ........................................................................................................... 59

Figure 31. Phase shift sensitivity versus refractive index differences for tapered fiber

with L=10 mm, d=10 μm. ................................................................................................. 60

Figure 32. (a) . Experimental setup of bi-tapered fiber sensor coupled to a

supercontinuum generating light source. (b) 3D diagram of the experimental setup with

all the elements. ................................................................................................................ 61

Figure 33. Transmission spectrum taken from the tapered fiber sensor immersed in air

generated by the supercontinuum light source.................................................................. 63

Figure 34. Transmission spectrum of the tapered fiber sensor as a function of the

wavelength for different refractive indices. ...................................................................... 64

Figure 35. Spatial frequency spectrum of the transmission for different refractive

indices. .............................................................................................................................. 66

Figure 36. Phase for the dominant Fourier component respect the wavelength for

different refractive indices. ............................................................................................... 67

Figure 37. Difference phase versus the refractive index difference for a bi-tapered

sensor. ............................................................................................................................... 68

Figure 38. Normalized spectral peak as a function of the refractive index difference for

a taper with L = 10 mm and d = 10 μm. ........................................................................ 69

Figure 39. Schematic diagram of the tapered fiber coated with AuNPs. .......................... 74

Figure 40. Kretschmann configuration for SPP excitation as a function of incident

angle. ................................................................................................................................. 75

Figure 41. Real and Imaginary components of gold permittivity as a function of the

wavelength. ....................................................................................................................... 78

xvi

Figure 42. Power transmission spectrum of the tapered fiber sensor as a function of the

wavelength coated and uncoated with 10 nm of TiO2. ..................................................... 79


wavelength for different refractive indices. ...................................................................... 80

Figure 44. Phase for the dominant Fourier component respect to the wavelength for

different refractive indices. Inset: Phase difference versus refractive index

differences. ........................................................................................................................ 81

Figure 45. Normalized spectral peak as a function of the refractive index differences

for a bi-tapered sensor coated with 10 nm of TiO2. .......................................................... 82


wavelength coated with 20 nm of TiO2. ........................................................................... 83

Figure 47. Spatial frequency spectrum from transmission for tapered fiber coated with

20 nm of TiO2. .................................................................................................................. 84

Figure 48. Phase for the dominant Fourier component respect to the wavelength for

different refractive index. Inset: Phase difference versus refractive index difference. ... 85

xvii

LIST OF SYMBOLS

𝐸 Electric field amplitude

𝐻 Magnetic field amplitude

𝐷 Electric flux density

𝐵 Magnetic flux density

𝜌 Charge density

𝜖 Electric permittivity

𝜇 Magnetic permeability

𝑘 𝑘-vector

𝑘0 Vacuum wave vector

𝜔0 Frequency

𝑐 Speed of light in vacuum

𝛽 Propagation coefficient

𝛾 Attenuation coefficient

𝜅 Transverse wave vector

𝑛𝑐𝑜𝑟𝑒 Refractive index of the core

𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 Refractive index of the cladding

xviii

𝑑𝑝 penetration depth

𝐼𝑇 Total intensity

𝐼𝑚 Intensity of the mth mode

Δ𝜙 Phase difference

𝐿 Taper Length

Δ𝑛 Refractive index difference.

𝑚 Mass of the free electron

𝑒 Electron charge

𝑣 Velocity of the free electron

𝜈 Frequency of electron

𝑷 Dipole moment

𝜔𝑝 Plasma frequency

𝑛 Density of electron gas per unit volume

𝐽𝜐 Bessel function

𝐾𝜐 Modified Bessel function

𝑎 Core radius

𝜆 Wavelength

1

CHAPTER 1

BACKGROUND

Introduction

Over the past several decades optical fibers have been modified for fiber sensor

applications. The physical changes required to make an optical fiber into a sensor are

designed to disturb the passage of the light so that it is sensitive to a particular external

variable (temperature, strain, biomolecule presence, etc.). So, fiber sensors analyze the

response or sensitivity of these physical conditions 1. Some of most important advantages

that this kind of sensor present are: high sensitivity, invulnerability to electromagnetic

interference, small size, and easy transmission and detection 2. Fiber sensors are designed

to measure temperature, pressure, electric current, gas species, vibration, torsion, rotation,

and displacement.

Optical fiber sensors are a simple, useful, and inexpensive platform that can be

adapted for applications in biomolecule or volatile organic compound sensing. In this

dissertation we study the bi-taper tapered optical fiber; this kind of sensor the cladding and

core diameter of the fiber are reduced by heating one section of the fiber 3. There is an

extensive literature on biomolecule detection and sensing, also called biosensing.

Biosensing is the subject of many international conferences sponsored by international

2

organizations every year, such as the BIOS symposium at the SPIE Photonics West or the

SPIE Defense, Sensing and Security Conference. Of particular interest are sensing

techniques that rely on optical detection.

Fiber Optics and Sensing

The bi-tapered optical fiber, illustrated in Figure 1, is simple to fabricate and the

fiber stock often comes from widely available and inexpensive optical fibers. The tapered

fiber often starts with a standard optical fiber, which is single-mode at infrared wavelengths

around 1.5 microns. The taper is made by locally heating the fiber close to (but below) the

glass transition temperature and then stretching the fiber ends in a controlled manner to

produce a waist region. The bi-tapered fiber has a down-taper and an up-taper section on

each end of the waist section. In modern glass processing systems the cross-sectional

profile is reproducible from sample to sample 4. Generally, the diameter of the waist for a

taper fiber is between 15 and 2 𝜇m.

Figure 1. Tapered fiber profile after heating (not to scale).

The light is launched at one end of the tapered fiber and passes through the waist

region and back into the fiber. One major problem to solve was the development of a light

source that is broad band and can be coupled into the fiber. We solved this problem by

working with Andy Chong’s group to develop a supercontinuum source using a photonic

3

crystal fiber. In this work we advance an experimental design and signal analysis that

enables the measurement of tiny changes in phase and amplitude of the transmitted optical

signal with respect to refractive indexes using visible to the infrared wavelengths generated

by our supercontinuum light source.

The flow of the dissertation is as follows. We begin by describing Maxwell’s

equations in Chapter 2; it forms the basis for our understanding of electromagnetic

phenomenon in a waveguide optics. The derivation of the wave equation is clearly

described and applied to planar waveguide and circular, step index, fibers. Additionally,

we calculated the solutions for TM, TE, and hybrid modes, which will be further described

in the chapter. In other words our primary goal is to use the calculate fiber modes to

understand our fiber sensor. The theory is developed without applying the weakly guiding

approximation.

Chapter 3 of our dissertation describes the supercontinuum light source in the

visible and near-infrared region. We begin with a literature review about the development

of supercontinuum generation from the 70’s. We explore the numerical modeling by

studying the generalized nonlinear Schrödinger equation and calculated its solution using

the split step Fourier method. In last section of this chapter, we present the experimental

result after design a supercontinuum light source.

In Chapter 4 we introduce the concept of the bi-tapered fiber and develop its use as

a sensor. We describe its geometry and sensing principles. Additionally, we present some

results related to measured changes in refractive index using a tunable laser as a sources in

the infrared band (around 1550 nm) and go on to look at the data using the supercontinuum

light extending from the visible to the near infrared. We present a signal analysis based in

4

the predominant Fourier component of the transmission spectrum.

Finally, in Chapter 5 we present the summary of this dissertation and we include an

extensive discussion of future work. The future work is related to the improvement of the

sensor by adding metal or high 𝜖 dielectric films on the surface of the tapered fibers to

enhance biomolecular sensing. The proposal is based on the ability of a metal (or dielectric)

film to draw the electromagnetic field to the fiber’s surface where the analytes can be

captured. The device sensitivity will be increased by depositing a gold metal film or TiO2

film for a few nanometers in thickness on the surface.

5

CHAPTER 2

FIBER OPTICS

Introduction

In many optical applications, the optical beams are confined in a microscopic region

of the space. There are some special optical systems that confine beams, but allow

propagation through space. One of the most useful and simplest of such structures is the

optical waveguide or fiber that is generally made of two (or more) materials with different

refractive indices. The materials can be crystalline or non-crystalline, for example, glass is

a common amorphous (i.e. non-crystalline) optical fiber material that is simply pulled into

the desired shape without defects. In optical fibers the material is pulled to form a

cylindrical waveguide; as the optical wave propagates through fiber it is largely confined

to the core region. Its basic structure with a central core and an outer coating layer as

illustrated in Figure 2.

6

Figure 2. Regular fiber Schematic with core and cladding.

Mathematical Representation of Optical Fibers

The electromagnetic phenomena can be described from the Maxwell's equations.

These four equations are basically the Ampere’s, law, Faraday’s law, and Gauss’s law.

∇ × 𝑬 = −

𝜕𝑩

𝜕𝑡, ( 2-1)

∇ ×𝑯 = 𝑱 +

𝜕𝑫

𝜕𝑡, ( 2-2)

∇ ∙ 𝑩 = 0, ( 2-3)

∇ ∙ 𝑫 = 𝜌, ( 2-4)

where 𝑬 and 𝑯 are the electric and magnetic field amplitude respectively, which define the

strength of a field at a specific point in space, and 𝜌 is the charge density 5. 𝑫 and 𝑩 are

electric and magnetic flux density respectively.

7

2.2.1. The Wave Equation

Now, from Maxwell’s equations is possible to derive the wave equation, which can

be solved by analytical and computational techniques. For optical wave propagation in a

uniform optical medium we consider the following conditions; it is uniform

(homogeneous), isotropic, non-conducting, and charge free. In this case Equations 2-1 to

2-4 become

∇ × 𝑬 = −

𝜕𝑩

𝜕𝑡, ( 2-5)

∇ × 𝑯 =

𝜕𝑫

𝜕𝑡, ( 2-6)

∇ ∙ 𝑩 = 0, ( 2-7)

∇ ∙ 𝑫 = 0, ( 2-8)

The previous set of equations are simple first order partial differential equations

where in Eq. 2-5 a time varying 𝑩 field can drive the rotation of the 𝑬 field. Well known

mathematical tools are applied to obtain a second-order partial differential equation, called

the wave equation. We derive the wave equation to describe wave propagation in term of

the electric field 𝑬. For Eq. 2-5 we can take the curl of both sides of the equation

∇ × (∇ × 𝑬) = −∇ ×

𝜕𝑩

𝜕𝑡. ( 2-9)

We apply the constitutive relations for the flux densities 𝑫 and 𝑩,

8

𝑩 = 𝜇𝑯, 𝑫 = 𝜖𝑬, ( 2-10)

where 𝜖 is the Electric permittivity and 𝜇 is the magnetic permeability of the medium. Here

for simplicity we assume that the coefficients are time independent or equivalently that the

material responds instantaneously to the electromagnetic fields. Replacing 𝑩 in Eq. 2-9

with 𝑯 we have

∇ × (∇ × 𝑬) = −∇ ×

𝜕𝜇𝑯

𝜕𝑡. ( 2-11)

If the functions are continuous is possible to change the order of the curl and time

derivative; furthermore we can assume that 𝜇 is independent of the time, so we have

∇ × (∇ × 𝑬) = −μ

𝜕

𝜕𝑡(∇ × 𝑯). ( 2-12)

Now, replacing the ∇ × 𝑯 in Eq. 2-12 using Eq. 2-6 and again assuming 𝜖 has an

instantaneous response time we arrive at

∇ × ∇ × 𝑬 = −μϵ

𝜕2𝑬

𝜕𝑡2. ( 2-13)

Using the following mathematical identities, ∇ × ∇ × 𝑬 = ∇(∇ ∙ 𝑬) − ∇2𝑬 and

∇ ∙ 𝑫 = ∇ ∙ (𝜖𝑬) = 𝑬 ∙ ∇𝜖 + 𝜖∇ ∙ 𝑬, and rewriting the divergence of 𝑬

(∇ ∙ 𝑬) = −𝑬 ∙ (

∇𝜖

𝜖), ( 2-14)

then Eq. 2-13 becomes

9

−∇(𝑬 ∙ (

∇𝜖

𝜖)) = ∇2𝑬 − 𝜇𝜖

𝜕2𝑬

𝜕𝑡2. ( 2-15)

By restricting the domain to be a homogeneous medium, the term on the left side vanishes,

so the wave equation can be written as:

∇2𝑬 = 𝜇𝜖

𝜕2𝑬

𝜕𝑡2. ( 2-16)

A similar derivation is used to obtain the wave equation for the magnetic field

∇2𝑯 = 𝜇𝜖𝜕2𝑯

𝜕𝑡2. ( 2-17)

The wave equation is solved by using the Fourier transform of the equations in time. In

this case the constitutive equations can have frequency-dependent coefficients and the

material response is more physically realistic. Important dispersion effects can be included

in the equations.

Planar Waveguide

A planar waveguide is characterize by a rectangular structure that is infinite in

extent with respect to the propagation direction (z) and has different refractive indices with

respect to the transverse direction, say the x-direction, as shown in Figure 3. For

concreteness, we assume that the refractive index of the cover material 𝑛𝑐 is lower with

respect to the refractive index of the substrate material 𝑛𝑠 and the guiding slab 𝑛𝑓, in other

words 𝑛𝑓 > 𝑛𝑠 > 𝑛𝑐.

10

Figure 3. Slab optical waveguide.

One of the characteristics of waveguide structure is that it supports only a certain

numbers of (discrete) modes and the field for each mode is simply characterized and

labeled by the field that is polarized in the direction transverse to the plane of propagation,

i.e. the direction of propagation and the direction perpendicular to the films. The transverse

electric field (TE) has the electric field polarized perpendicular to the plane of propagation

and transverse magnetic field (TM) has the magnetic field so polarized. From the

perspective of plane wave “rays” the 𝑘-vector has a “zig-zag” characteristic as a “ray”

propagates through the waveguide. The ray changes direction after reflection at an

interface. The rays are incident at an angle greater than the critical angle at each interface,

as shown in Figure 4 below.

11

Figure 4. TE, TM, and 𝒌-vetor inside of the planar waveguide.

Now, we can consider that a source is going into the waveguide, so it is excited

with a frequency 𝜔0 them the magnitude of the vacuum wave vector is 𝑘0 =𝜔0

𝑐, where 𝑐 =

1/√휀0𝜇0 is the speed of light in vacuum, where 휀0 is the free space permittivity and 𝜇0 is

the free space permeability. By Fourier transforming time to frequency space we can Eq.

2-16 as

∇2𝑬𝒚 + 𝑛2𝑘0

2𝑬𝒚 = 0, ( 2-18)

where 𝑛 = √휀𝑟𝜇𝑟 is the refractive index of the particular medium; 𝜖 = 휀0휀𝑟 and 𝜇 = 𝜇0𝜇𝑟;

휀𝑟 is the relative permeability and we assume the free space permeability is 𝜇𝑟 = 1. A

solution of Eq. 2-18 can be written as

𝑬𝑦(𝑥, 𝑧) = 𝑬𝑦(𝑥)𝑒−𝑗𝛽𝑧, ( 2-19)

𝛽 is known as the propagation coefficient and it is the component of the wave vector along

the z-direction. Replacing this solution in Eq. 2-18 the new equation will be as

𝜕2𝐸𝑦

𝜕𝑥2+ (𝑛2(𝑥)𝑘0

2 − 𝛽2)𝐸𝑦 = 0, ( 2-20)

Now, for some case relate to 𝛽, 𝑘0, and 𝑛 = 𝑛(𝑥) the solution of the last equation can be

12

𝐸𝑦(𝑥) = 𝐸0𝑒

±𝑗√𝑛2𝑘02−𝛽2𝑥

𝑓𝑜𝑟 𝛽 < 𝑛𝑘0 ( 2-21)

𝐸𝑦(𝑥) = 𝐸0𝑒

±√𝛽2−𝑛2𝑘02𝑥 𝑓𝑜𝑟 𝛽 > 𝑛𝑘0 ( 2-22)

According to the previous parameter is, possible to define the coefficient, 𝛾, and the

transverse wave vector, 𝜅, as

𝛾 = √𝛽2 − 𝑛2𝑘02 and 𝜅 = √𝑛2𝑘02 − 𝛽2. ( 2-23)

Referring to the indices defined in Figure 3 we define 𝜅𝑓 = √𝑛𝑓2𝑘0

2 − 𝛽2 for the film

transverse wave vector and 𝛾𝑐,𝑠 = √𝛽2 − 𝑛𝑐.𝑠2 𝑘02 for the imaginary wave vector in the cover

and substrate.

Eigenvalues of Propagating Waveguide

To find the values of 𝛽 that give us solution of the wave equation we apply the

boundary conditions. We will consider a general TE solution as

𝐸𝑦(𝑥) = {

𝐴𝑒−𝛾𝑐𝑥 0 < 𝑥𝐵𝑐𝑜𝑠(𝜅𝑓𝑥) + 𝐶𝑠𝑖𝑛(𝜅𝑓𝑥) − ℎ < 𝑥 < 0

𝐷𝑒𝛾𝑠(𝑥+ℎ) 𝑥 < −ℎ.

( 2-24)

Applying boundary conditions from Maxwell’s equations we can get the following

transcendental equation for TE and TM case as,

13

TE case:

tan(ℎ𝜅𝑓) =𝛾𝑐+𝛾𝑠

𝜅𝑓[1−𝛾𝑐𝛾𝑠

𝜅𝑓2 ]

, ( 2-25)

Similarly he have a transcendental equation for the TM case:

tan(ℎ𝜅𝑓) =

𝑛𝑓2

𝑛𝑠2𝛾𝑠+

𝑛𝑓2

𝑛𝑐2𝛾𝑐

𝜅𝑓[1−𝑛𝑓4

𝑛𝑠2𝑛𝑐2𝛾𝑐𝛾𝑠

𝜅𝑓2 ]

, ( 2-26)

where 𝜅𝑓 and 𝛾𝑐,𝑠 are defined under Eq. 2-23. MATLAB® is used to graphically and

numerically solve Eqs. 2-25 and 2-26 and find the eigenvalues 𝛽 for TE or TM modes in

the waveguide.

For example, the graphical solution of a planar waveguide with the following

refractive index 𝑛𝑓 = 1.48, 𝑛𝑠 = 1.46, and 𝑛𝑐 = 1.44, the thickness of the guiding core is

10 𝜇𝑚 and incident wavelength is 1.33 𝜇𝑚 is,

14

Figure 5. Graphical solution of planar waveguide.

Figure 5 shows the graphical solution of the previous example for a waveguide with four

eigenvalues for TE (blue dotted line) and TM (red dotted line) modes. For the TE case,

there are four eigenvalues for 𝜅 are 2730, 5444, 8111, and 10652 cm-1. The TM modes are

close to the TE modes. Once the TE modal eigenvalues are determined the mode function

for the transverse electric field can be constructed using Eq. 2-24. The other H-field

functions are constructed using Maxwell’s equations.

0 2000 4000 6000 8000 10000-15

-10

-5

0

5

10

15

(cm-1)

tan(h) Vs

TE

TM

tan(h)

(c+s)/(f(1-cs/f2))

15

Step Index Fiber

In Sections 2.2 and 2.3 we discussed the planar waveguide, now we will focus on a

cylindrical waveguide, since the step index fiber is relevant to the central subject of this

dissertation. A compact treatment of a simple optical fiber is presented in this section; the

fiber consists of a core and cladding material only. An application to a tapered fiber will

also be given and used to subsequently analyze the data. The wave equation in cylindrical

coordinates for step index optical fiber (Figure 2) is given by

∇2𝑬 − 𝜇𝜖

𝜕2𝑬

𝜕𝑡2= 0, ( 2-27)

we can consider time harmonic field, then we have

∇2𝑬 + 𝑘02𝑛2𝑬 = 0, ( 2-28)

we can write the above equation in cylindrical coordinate as,

𝑬(𝑟, 𝜙, 𝑧) = 𝐸𝑟(𝑟, 𝜙, 𝑧)�̂� + 𝐸𝜙(𝑟, 𝜙, 𝑧)�̂� + 𝐸𝑧(𝑟, 𝜙, 𝑧)�̂�. ( 2-29)

The scalar wave equation in cylindrical coordinates respect to 𝐸𝑧 is written as,

1

𝑟

𝜕

𝜕𝑟(𝑟

𝜕𝐸𝑧

𝜕𝑟) +

1

𝑟2𝜕2𝐸𝑧

𝜕𝜙2+𝜕2𝐸𝑧

𝜕𝑧2+ 𝑘0

2𝑛2𝐸𝑧 = 0, ( 2-30)

using the separation of variables technique we can analytically solve Eq. 2-30 and we apply

Maxwell’s Equations to find 𝐸𝑟 , 𝐸𝜙 we can solve 𝐸𝑧 from Eq. 2-30,

𝐸𝑧(𝑟, 𝜙, 𝑧) = 𝑅(𝑟)Φ(𝜙)𝑍(𝑧). ( 2-31)

We assume that 𝑍(𝑧) = 𝑒−𝑗𝛽𝑧 and the 𝜙 variable function satisfies −Φ′′ = 𝜈2Φ, where 𝜈

is the angular mode number. Doing some calculations we can get an ordinary differential

equation (ODE) that only contain 𝑅(𝑟):

𝑟2𝑅′′ + 𝑟𝑅′ + 𝑟2 (𝑘02𝑛2 − 𝛽2 −

𝜈2

𝑟2)𝑅 = 0, ( 2-32)

this particular ODE can be solve for two types of Bessel functions according to the sign of

16

the expression (𝑘02𝑛2 − 𝛽2 −𝜈2

𝑟2). For (𝑘02𝑛2 − 𝛽2 −

𝜈2

𝑟2) > 0 the solution is Bessel

functions of the first kind of order 𝜈 (in the core) 𝐽𝜈(𝜅𝑟). A second solution of the

differential equation called the Neumann diverges at 𝑟 = 0, the transverse wave vector is

𝜅2 = 𝑘02𝑛2 − 𝛽2. ( 2-33)

For (𝑘02𝑛2 − 𝛽2 −𝜈2

𝑟2) < 0 the solutions are Modified Bessel functions of the second kind

of order 𝜈 (𝐾𝜈(𝛾𝑟)). The argument of the Bessel function for this case is

𝛾2 = 𝛽2 − 𝑘02𝑛2 ( 2-34)

Figure 6. (a) The five Bessel functions of the first kind, Jν(r),and (b) the modified Bessel

function Kν(r) of the first kind.

17

The first four Bessel function orders used for solutions of the simple optical fiber

geometry are plotted in Figure 6. The Bessel functions of the first kind are oscillating

functions and the Bessel functions of the second kind are monotonic functions. Solution of

the wave equation for a simple fiber waveguide (Figure 7) the fields in term of the Bessel

functions in both regions is given in terms of Bessel functions. The fiber geometry in Figure

6 is simple with core and cladding materials.

Figure 7. Fiber waveguide with core radius a.

The modal solutions for the geometry shown in Figure 7 have the form

𝑓𝑜𝑟 𝑟 < 𝑎: {

𝐸𝑧(𝑟, 𝜙, 𝑧) = 𝐴𝐽𝜈(𝜅𝑟)𝑒𝑗𝜈𝜙𝑒−𝑗𝛽𝑧 + 𝑐. 𝑐

𝐻𝑧(𝑟, 𝜙, 𝑧) = 𝐵𝐽𝜈(𝜅𝑟)𝑒𝑗𝜈𝜙𝑒−𝑗𝛽𝑧 + 𝑐. 𝑐

𝑓𝑜𝑟 𝑟 > 𝑎: {𝐸𝑧(𝑟, 𝜙, 𝑧) = 𝐶𝐾𝜈(𝛾𝑟)𝑒

𝑗𝜈𝜙𝑒−𝑗𝛽𝑧 + 𝑐. 𝑐

𝐻𝑧(𝑟, 𝜙, 𝑧) = 𝐷𝐾𝜈(𝛾𝑟)𝑒𝑗𝜈𝜙𝑒−𝑗𝛽𝑧 + 𝑐. 𝑐

( 2-35)

By applying boundary conditions we can determine the relationship between the

coefficients A, B, C, and D. Additionally, using Maxwell’s equations in cylindrical

18

coordinates we can find the other field components 𝐸𝑟, 𝐸𝜙, 𝐻𝑟, ad 𝐻𝜙. After some algebra

we get

{

𝐸𝑟 = −

𝑗

𝑘02𝑛2 − 𝛽2

(𝛽𝜕𝐸𝑧𝜕𝑟

+𝜔𝜇

𝑟

𝜕𝐻𝑧𝜕𝜙

) ,

𝐸𝜙 = −𝑗

𝑘02𝑛2 − 𝛽2

(−𝜔𝜇𝜕𝐻𝑧𝜕𝑟

+𝛽

𝑟

𝜕𝐸𝑧𝜕𝜙

) ,

𝐻𝑟 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝛽𝜕𝐻𝑧𝜕𝑟

−𝜔휀

𝑟


) ,

𝐻𝜙 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝜔휀𝜕𝐸𝑧𝜕𝑟

+𝛽

𝑟


) .

( 2-36)

2.5.1. TM Mode Solutions

The TM mode solution for a fiber is the solution when 𝐻𝑧 = 0. For the TM case we

can express the nonzero field components in Eq. 2-36 as,

{

𝐸𝑟 = −

𝑗

𝑘02𝑛2 − 𝛽2

(𝛽𝜕𝐸𝑧𝜕𝑟) ,

𝐸𝜙 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝛽

𝑟


) ,

𝐻𝑟 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝜔휀

𝑟


) ,

𝐻𝜙 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝜔휀𝜕𝐸𝑧𝜕𝑟) .

( 2-37)

The TM01 mode in a fiber optics is has a radial polarization for the electric field. In

Figure 8 we illustrate the transverse TM electric field pointing radially from the central

axis of the fiber. The on-axis amplitude of the electric field is zero to resolve the

polarization ambiguity at that point. The lengths of the arrows indicates to some degree

19

the amplitude of the electric field vector.

Figure 8. Transverse electric field for radial TM01.

2.5.2. TE Mode Solutions

The TE mode is another solution for the case when 𝐸𝑧 = 0. For the TE case we can

express Eq. 2-36 as,

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

TM01

20

{

𝐸𝑟 = −

𝑗

𝑘02𝑛2 − 𝛽2

(𝜔𝜇

𝑟


) ,

𝐸𝜙 = −𝑗

𝑘02𝑛2 − 𝛽2

(−𝜔𝜇𝜕𝐻𝑧𝜕𝑟

) ,

𝐻𝑟 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝛽𝜕𝐻𝑧𝜕𝑟

) ,

𝐻𝜙 = −𝑗

𝑘02𝑛2 − 𝛽2

(𝛽

𝑟


) .

( 2-38)

For the TE01 mode the electric field is azimuthally polarized as illustrated in Figure 9. The

radial profile of the mode is similar to the profile for the TM01 mode.

Figure 9. Transverse electric field for azimuthal TE01.

1.0 0.5 0.0 0.5 1.0

1.0

0.5

0.0

0.5

1.0

TE01

21

2.5.3. The Fundamental HE11 Mode

The fundamental mode HE11 is the only mode that don’t have a cutoff condition. In

standard fibers the core and cladding indices are very close to one another and an

approximate set of modes are adopted called linear polarization or LP modes; the

fundamental linear polarization mode is LP01. The HE11 mode has the special characteristic

mentioned above that it has no cutoff wavelength and is supported by all optical fibers.

The electric field of the HE11 mode is linearly polarized and is illustrated in Figure 10. Two

orthogonal polarizations are shown in Figure 10. For a core of few microns most of the

field is concentrated inside the core; as the core is squeezed further the field eventually

expand into the cladding.

Figure 10. Transverse electric field for the fundamental HE11 mode for two orthogonal

polarizations.

22

2.5.4. The Hybrid Modes

In previous sections we described cases when 𝐸𝑧 𝑎𝑛𝑑 𝐻𝑧 = 0; however, for the HE11

mode 𝐸𝑧 𝑎𝑛𝑑 𝐻𝑧 ≠ 0 and this mode is called a hybrid modes. The modes are labelled as

HE or EH. The 𝐸𝐻 modes have a larger contribution from the 𝐸𝑧 field component (i.e.

A>B in Eq. 2-35) and the 𝐻𝐸 modes have a larger contribution from the 𝐻𝑧 field

component (i.e. B>A in Eq. 2-35).

The propagation constants of each mode are determined by applying the boundary

conditions. Returning to Eq. 2-35, we can applying boundary conditions at 𝑟 = 𝑎, to obtain

the matrix notation shown below,

[

𝐽𝜈(𝜅𝑎) 0 −𝐾𝜈(𝛾𝑎) 00 𝐽𝜈(𝜅𝑎) 0 −𝐾𝜐(𝛾𝑎)

𝛽𝜈

𝑎𝜅2𝐽𝜈(𝜅𝑎)

𝑗𝜔𝜇

𝜅𝐽𝜈′ (𝜅𝑎)

𝛽𝜈

𝑎𝛾2𝐾𝜈(𝛾𝑎)

𝑗𝜔𝜇

𝛾𝐾𝜈′(𝛾𝑎)

−𝑗𝜔𝜀𝑐𝑜𝑟𝑒

𝜅𝐽𝜈′ (𝜅𝑎)

𝛽𝜈

𝑎𝜅2𝐽𝜈(𝜅𝑎)

−𝑗𝜔𝜀𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔

𝛾𝐾𝜈′(𝛾𝑎)

𝛽𝜐

𝑎𝛾2𝐾𝜈(𝛾𝑎)]

[

𝐴𝐵𝐶𝐷

] =0. ( 2-39)

The nontrivial solution is given when the determinant of Eq. 2-39 is equal to zero.

This result represent a well known “characteristic” equation for the step index fiber:

𝛽2𝜈2

𝑎2[1

𝛾2+

1

𝜅2]2

= [𝐽𝜈′(𝜅𝑎)

𝜅𝐽𝜈(𝜅𝑎)+

𝐾𝜈′(𝛾𝑎)

𝛾𝐾𝜈(𝛾𝑎)] [𝑘02𝑛𝑐𝑜𝑟𝑒

2 𝐽𝜈′ (𝜅𝑎)

𝜅𝐽𝜈(𝜅𝑎)+

𝑘02𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔

2 𝐾𝜈′(𝛾𝑎)

𝛾𝐾𝜈(𝛾𝑎)].

( 2-40)

𝐸𝐻 and 𝐻𝐸 have 𝜈 ≠ 0 and there are two different situations discussed in the literature

with respect to the ratio of the refractive indices in the core and the cladding. The first

23

situation often considered occurs when 𝑛𝑐𝑜𝑟𝑒 ≈ 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 = 𝑛 that is known as weakly

guiding approximation. This is the condition for using the LP solutions discussed earlier.

The second, more general, situation is when 𝑛𝑐𝑜𝑟𝑒 and 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 are much different. For

example, when the outer diameter of the fiber is reduce to around 15 or even 4 𝜇𝑚, in this

case all the light inside to the fiber is not confined to the core or cladding region, so the

field interacts with the surrounding medium such as air (𝑛 = 1.0) or water (𝑛 = 1.3). In

this case we can use the glass cladding as a type of core with Δ𝑛 = 𝑛𝑐𝑜𝑟𝑒 − 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 ≈

0.5, 0.2 ≫ 0.005, so we cannot apply the weakly guiding approximation. We are interested

in the second case and we will describe it in greater detail. The modes of the tapered fiber

waist that we solve for are cladding modes in the strictest sense since the Bessel functions

extend throughout the entire fiber. The modified Bessel functions are used outside the fiber

in the surrounding environment.

The derivative of the Bessel functions can be replaced by Bessel functions using

mathematical identities. For instance, 𝐽𝜈′ (𝜅𝑎) satisfies the identity,

𝐽𝜈′ (𝜅𝑎) = ±𝐽𝜈±1(𝜅𝑎) ∓

𝜈

𝜅𝑎𝐽𝜈(𝜅𝑎), ( 2-41)

dividing by 𝜅𝐽𝑣(𝜅𝑎) the Eq. 2-41 will be,

𝐽𝜈′ (𝜅𝑎)

𝜅𝐽𝜈(𝜅𝑎)= ±

𝐽𝜈±1(𝜅𝑎)

𝜅𝐽𝜈(𝜅𝑎)∓

𝜈

𝜅2𝑎, ( 2-42)

24

Similarly the modified Bessel function 𝐾𝜈′(𝛾𝑎) is expressed as,

𝐾𝜈′(𝛾𝑎)

𝛾𝐾𝜈(𝜅𝑎)= ±

𝐾𝜈±1(𝛾𝑎)

𝛾𝐾𝜈(𝛾𝑎)∓

𝜈

𝛾2𝑎 . ( 2-43)

Now, replacing the derivatives in Eq. 2-40 by substituting Eqs. 2-42 and 2-43 for

the lowest order modes with 𝜈 = 1 (𝐸𝐻/𝐻𝐸) and 𝛽2 = 𝑘02𝑛𝑐𝑜𝑟𝑒2 − 𝜅2, we have,

𝑘02𝑛𝑐𝑜𝑟𝑒

2 −𝜅2

𝑎2[1

𝛾2+

1

𝜅2]2

= [−𝐽2(𝜅𝑎)

𝜅𝐽1(𝜅𝑎)+

1

𝑎𝜅2−

𝐾2(𝛾𝑎)

𝛾𝐾1(𝛾𝑎)+

1

𝑎𝛾2] [−𝑘0

2𝑛𝑐𝑜𝑟𝑒2 𝐽2(𝜅𝑎)

𝜅𝐽1(𝜅𝑎)+

1

𝑎𝜅2−𝑘0

2𝑛𝑐𝑙𝑎𝑑𝑑𝑛𝑖𝑔2 𝐾2(𝛾𝑎)

𝛾𝐾1(𝛾𝑎)+

1

𝑎𝛾2].

( 2-44)

We numerically solved Eq. 2-44 and we also graphically determined approximate

solutions as a check on our numerical results using Matlab. Consider an optical fiber with

a waist core radius 𝑎 = 5𝜇𝑚, length 𝐿 = 1.2 𝑐𝑚 core index (internal index), 𝑛𝑐𝑜𝑟𝑒 =

1.45, cladding index (external or surrounding index), 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 = 1.00, and wavelength,

𝜆 from 1480 to 1550 nm. The graphical solution is shown in Figure 11. The red lines are

the right side and the blue lines are the left side of Eq. 2-44 for each wavelength. The

intersections between this two plots (red and blues) are the solution for the previous

equation. The results for smaller values of 𝜅 (transverse wave vector) are the modes of

interest. The modes vary little with wavelength at small values of 𝜅. As the 𝜅 approaches

its maximum value the dispersion of values is more evident by a dispersion of values.

25

Figure 11. Graphical solution of the Eq. 2-44 for the fundamental HE11 mode for several

wavelengths.

The mth fiber mode propagating through the waist of the fiber has a different

propagation constant 𝛽𝑚(𝜆). For simplicity we ignore the core and treat the fiber as a glass

(cladding) center surrounded by an ambient environment (air and/ water). The propagation

constants can be expressed as,

𝛽𝑚(𝜆) =2𝜋

𝜆𝑛𝑚(𝜆),𝑚 = 1,2, …. ( 2-45)

The mode number is index by m and the effective mode index is 𝑛𝑚(𝜆). The free

space wavelength is 𝜆. From Eq. 2-44 we calculated the first six modes for a fiber

26

surrounded by water and air for wavelengths 𝜆 from 740 nm to 910 nm. The propagation

constants of the first six modes for selected wavelengths are shown in Table 1 for a cladding

with an air index and in Table 2for a cladding with a water index.

Table 1. Propagation modes in fiber optics for selected wavelengths with an air cladding.

Air (𝒏 = 𝟏. 𝟎𝟎) 𝝀 (𝒏𝒎) 𝜷𝟏(𝝁𝒎−𝟏) 𝜷𝟐(𝝁𝒎

−𝟏) 𝜷𝟑(𝝁𝒎−𝟏) 𝜷𝟒(𝝁𝒎

−𝟏) 𝜷𝟓(𝝁𝒎−𝟏) 𝜷𝟔(𝝁𝒎

−𝟏) 740.0 12.30 12.27 12.26 12.20 12.19 12.10 741.7 12.27 12.24 12.24 12.17 12.16 12.07 743.4 12.25 12.21 12.21 12.14 12.14 12.04 745.1 12.22 12.19 12.18 12.12 12.11 12.01 746.8 12.19 12.16 12.15 12.09 12.08 11.98 748.5 12.16 12.13 12.12 12.06 12.05 11.96 750.2 12.14 12.10 12.10 12.03 12.02 11.93 751.9 12.11 12.07 12.07 12.00 12.00 11.90 753.6 12.08 12.05 12.04 11.98 11.97 11.87 755.3 12.05 12.02 12.01 11.95 11.94 11.84 757.0 12.03 11.99 11.99 11.92 11.91 11.82 758.7 12.00 11.97 11.96 11.89 11.89 11.79 760.4 11.97 11.94 11.93 11.87 11.86 11.76

Table 2. Propagation modes in fiber optics for selected wavelengths with a water cladding.

Water(𝒏 = 𝟏. 𝟑𝟑𝟐𝟗𝟓) 𝝀 (𝒏𝒎) 𝜷𝟏(𝝁𝒎−𝟏) 𝜷𝟐(𝝁𝒎

−𝟏) 𝜷𝟑(𝝁𝒎−𝟏) 𝜷𝟒(𝝁𝒎

−𝟏) 𝜷𝟓(𝝁𝒎−𝟏) 𝜷𝟔(𝝁𝒎

−𝟏) 740.0 12.30 12.27 12.27 12.20 12.20 12.11 741.7 12.27 12.24 12.24 12.18 12.17 12.08 743.4 12.25 12.22 12.21 12.15 12.14 12.05 745.1 12.22 12.19 12.18 12.12 12.11 12.02 746.8 12.19 12.16 12.15 12.09 12.09 11.99 748.5 12.16 12.13 12.13 12.06 12.06 11.97 750.2 12.14 12.10 12.10 12.04 12.03 11.94 751.9 12.11 12.08 12.07 12.01 12.00 11.91 753.6 12.08 12.05 12.04 11.98 11.97 11.88 755.3 12.05 12.02 12.02 11.95 11.95 11.85 757.0 12.03 11.99 11.99 11.93 11.92 11.83 758.7 12.00 11.97 11.96 11.90 11.89 11.80 760.4 11.97 11.94 11.93 11.87 11.87 11.77

27

The difference in the phase between two modes determines the interference patterns

that are observed in the transmission data. The phase difference between two modes is

written as,

∆𝜙𝑛𝑚 = (𝛽𝑛(𝜆) − 𝛽𝑚(𝜆))𝐿. ( 2-46)

In Figure 12 and Figure 13 we plot the difference between pairs of propagation

constants versus wavelength for the fiber immersed in water and in air, respectively. The

insets in Figure 12 and Figure 13 shows that individually the modes have a slight curvature

over the range of wavelengths, which is chosen based on the collected data. The difference

between two selected modes gives the straight lines in the figures show that the differences

change is nearly linear as a function of lambda over our entire range of values of interest

from 740 nm to 920 nm. Hence, the phase difference between modes in Eq. 2-46 is nearly

a linear function of the wavelength over the region of interest.

28

Figure 12. Difference between selected Propagation constants versus wavelength for a 10 micron

diameter fiber. The inset is the propagation constants for the first six cladding modes of the fiber

with water outside the fiber.

29

Figure 13. Difference between selected Propagation constants versus wavelength for a 10 micron

diameter fiber. The inset is the propagation constants for the first six cladding modes of the fiber

with air outside the fiber.

Using the linear results in Figure 12 and Figure 13, we approximate the phase

difference using a Taylor series expansion to first order

30

∆𝜙𝑛𝑚 ≈ (

𝜕(𝛽𝑛(𝜆0) − 𝛽𝑚(𝜆0))

𝜕𝜆0)∆𝜆𝑛𝑚𝐿, ( 2-47)

and ∆𝜆𝑛𝑚 is the wavelength increment from a center wavelength 𝜆0, i.e. ∆𝜆𝑛𝑚 = 𝜆 − 𝜆0.

The slope of the function is the contribution in parentheses, which we denote as 𝑆𝑛𝑚. The

values of the slopes after averaging the slopes over the range 791 nm to 861 nm are

tabulated in Table 3. We chose three values because they are relevant for our experiments:

𝑆12, 𝑆13, and𝑆34.

Table 3. Values of the slope from Eq. 2-47 for air and water claddings in units (𝑐𝑚 ∙ 𝑛𝑚)-1.

𝑺𝒎𝒏(𝒄𝒎 ⋅ 𝒏𝒎)−𝟏 Air Water 𝑺𝟏𝟐 0.420 0.382 𝑺𝟏𝟑 0.510 0.460 𝑺𝟑𝟒 0.830 0.760

We use Eq. 2-47 to determine a dominant period for the transmitted signal, which, as

we will show below, is closely related to the calculated value 𝑆13. We compare these

calculations with the experimental data in Chapter 4.

Important Parameters in Fibers Optics

Basically, an optical fiber is a cylindrically shaped structure with similar optical

properties previously describe for the planar waveguide. There are some important

parameters in fiber optics (and planar waveguides) that we briefly discuss in this section.

Our definitions here only pertains to optical fibers.

2.6.1. Numerical Aperture

The numerical aperture 𝑁𝐴 in a fiber optics gives us a relation between the sine of the

31

largest acceptance angle for guiding electromagnetic waves and the difference between

refractive index of the core and cladding 6. The largest acceptance angle 𝜃𝑎 for coupling

light into the optical fiber core is

𝑁𝐴 = 𝑠𝑖𝑛𝜃𝑎 = √𝑛𝑐𝑜𝑟𝑒2 − 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔

2 . ( 2-45)

The cladding index is only slightly lower than the core index in standard fibers.

2.6.2. Normalized Frequency

The normalized frequency or V-number is a parameter that characterize a fiber, its

magnitude is used to determine the boundary between a single-mode and multi-mode fiber

and it can be used to approximate numbers of mode in a fiber optics. This parameter is

proportional to the radius of the fiber and numerical aperture. The V-number is defined as

𝑉 =

2𝜋𝑎

𝜆√𝑛𝑐𝑜𝑟𝑒2 − 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔

2 = 𝑘0𝑎𝑁𝐴, ( 2-46)

where 𝑎 is the core radius (Figure 7), and 𝜆 is the incident wavelength. The V-number is

a parameter of central interest when finding the number of modes. The mode cutoff

condition is expressed in terms of the V-number, which is also related to the transverse

wave vector component.

Chapter Summary

We developed and used the characteristic equation for a step index fiber without

the weakly guiding approximation to calculate propagation HE1m modes for several

specific conditions of a core/cladding geometry. For optical fiber with a small cladding

radius ~5𝜇𝑚 and Δ𝑛 = 𝑛𝑐𝑜𝑟𝑒 − 𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 ≈ 0.005, the solutions for waves in the IR are

well established and approximated by linear polarization modes. However, we are

32

studying different regimes for our sensors; the wavelengths of interest for our experiments

vary over the range from 740 nm to 910 nm and the index contrast is much greater. The

difference between the propagation constant of two selected modes are helpful when

calculating the predominant period for a signal and we determined that the functions are

nearly linear as the wavelength is varies. We will apply these results in Chapter 4 when

the experimental results are examined.

33

CHAPTER 3

SUPERCONTINUUM LIGHT SOURCE

Introduction

The supercontinuum generation light source has a broad spectrum a bright spectral

power density. Generally, it is generated from an ultrashort pulse pump beam that

propagates through a nonlinear medium. The first work to report a supercontinuum source

was developed by R. Alfano and S.L. Shapiro in the early 70’s 7. Basically, they observed

two main spectral emission curves from 400 to 530 nm and another from 530 to 700 nm,

after pumping a borosilicate glass (BK-7) with pulses at a center wavelength of 530 nm

with an energy per pulse of 5 mJ, and a pulse length between 4 and 8 psec. This work

initiated many additional publications on source development and applications of

supercontinuum light. Of special notes are the study of different material to improve the

spectral range and the power density of the supercontinuum light.

One of the first reports of supercontinuum generation in optical fibers appeared in

1976 by Ch. Lin and R. H. Stolen 8. They used a N2 laser that was pumped by a dye laser,

the pulse duration was approximately 10 ns, and the output peak power from the laser was

between 10 to 20 kW. Using a silica core fiber with a core diameter of 7 𝜇𝑚 and length

19.5 𝑚 they were able to generate a broadband spectrum with a width between 110 and

180 nm in the visible regime. It was determined by numerical computations that the two

34

principle nonlinear processes responsible for the supercontinuum generation are self-phase

modulation (SPM) and stimulated Raman scattering. At the end of the twentieth century

the development of the photonic crystal fiber (PCF) was another significant step forward

in the development of supercontinuum sources 9. PCFs come in many different geometries,

depending on their particular application, but they have glass separated by air filled

structures, rather than a glass only structure given by conventional fiber optics. The PCF

of interest in our research has a solid core in the center that is surrounded by a matrix

arrangement of air holes along the fiber (as illustrated in Figure 14). The large index

contrast between glass and air strongly confines the field in the core region and thus

increases its nonlinear effect.

Figure 14. Photonic crystal fiber profile.

Some of the advantages of PCF respect to regular fiber are the ability to control

some parameter in the design of the PCF that can enhance nonlinear processes. The

confinement of the laser energy is in the core, surrounded by holes. The nonlinear

coefficient is enhanced because the mode area is much smaller than in conventional fiber.

For ultrashort pulses there is also the advantage of improving short-pulse conversion

35

efficiency by modifying the group velocity dispersion (GVD). The objective of dispersion

management in the fiber is to adjust the phase matching condition to cover a wide range of

wavelengths.

It became apparent that the properties of PCF of confinement and GVD could be

applied for supercontinuum generation. An early experiment using PCFs for

supercontinuum generation used center wavelength of 790 nm, an input pulse duration of

100 fs, and a peak power of 55 W. The PCF silica core diameter was 1.7 𝜇𝑚 with air holes

of 1.3 𝜇𝑚 diameter. As a result, the generation of a supercontinuum from 390 to 1600 nm

using 75 cm of fiber 10. This is a two octave span of the supercontinuum source.

Numerical Modeling

A model of a supercontinuum generation has several orders of dispersion

incorporated into the expression and includes self-phase modulation and stimulated Raman

scattering, The pulse dynamics is governed by the generalized nonlinear Schrödinger

equation (GNSE) 11, which is written in a scalar form as

𝜕𝐴

𝜕𝑧= −

𝛼

2𝐴 − ∑ 𝑖𝑚−1

𝛽𝑚𝑚!

𝜕𝑚𝐴

𝜕𝑡𝑚

𝑀

𝑚≥2

+ 𝑖 (𝛾 + 𝑖𝛾1𝜕

𝜕𝑡) [𝐴(𝑧, 𝑡)∫ 𝑅(𝑡′)|𝐴(𝑧, 𝑡 − 𝑡′)|2𝑑𝑡′

∞

0

],

( 3-1)

where 𝐴(𝑧, 𝑡) is the field envelope, 𝛼 is the linear absorption coefficient, 𝛽𝑚 is a set of

dispersion coefficients up to order M, 𝑅(𝑡′) represent the Raman nonlinear response

36

function, 𝛾 is the nonlinear (Kerr) coefficient and 𝛾1 =𝛾

𝜔0 is the self-steepening coefficient,

which becomes relevant for very short pulse widths. Our fibers are only a few centimeters

in length (25 cm) so that the linear absorption can be neglected (𝛼 ≈ 0). In our case, the

observed broadening is from 416 THz (720 nm) to 326 THz (920 nm). We included self-

steepening since the broadening is more than 20 THz 12. The Eq. 3-1 will be

𝜕𝐴

𝜕𝑧= −∑ 𝑖𝑚−1

𝛽𝑚𝑚!

𝜕𝑚𝐴

𝜕𝑡𝑚

𝑀

𝑚≥2

+ 𝑖𝛾 (1 + 𝑖1

𝜔0

𝜕

𝜕𝑡) [𝐴(𝑧, 𝑡)∫ 𝑅(𝑡′)|𝐴(𝑧, 𝑡 − 𝑡′)|2𝑑𝑡′

∞

0

].

( 3-2)

We can represent the 𝑅(𝑡′) as,

𝑅(𝑡′) = (1 − 𝑓𝑅)𝛿(𝑡′) + 𝑓𝑅ℎ𝑅(𝑡′), ( 3-3)

The first term is the instantaneous electronic contribution to the nonlinearity, usually called

the Kerr nonlinearity, and 𝑓𝑅 is related to the relative strengths of the Kerr and Raman

interaction. ℎ𝑅(𝑡) is the Raman response function associated with the frequency response

of silica molecules 13. An approximate functional form of ℎ𝑅 depends on the phonon

frequency (1 𝜏1⁄ ) and the bandwidth of the Lorentzian line(1 𝜏2⁄ ),

ℎ𝑅(𝑡) =

𝜏12 + 𝜏2

2

𝜏1𝜏22 exp [−𝑡 𝜏2⁄ ]𝑠𝑖𝑛[𝑡 𝜏1⁄ ], ( 3-4)

37

According to 13 with can represent graphically ℎ𝑅 as function of 𝑡 for 𝑓𝑅 = 0.18, 𝜏1 =

0.0122𝑝𝑠, and 𝜏2 = 0.032𝑝𝑠 as show below,

Figure 15. Raman response function hR respect to the time 𝒕.

Now, substituting Eq. 3-3 into Eq. 3-2 we have,

∫ 𝑅(𝑡′)|𝐴(𝑧, 𝑡 − 𝑡′)|2∞

0

= (1 − 𝑓𝑅)|𝐴(𝑡)|2 + 𝑓𝑅∫ ℎ𝑅(𝑡′)|𝐴(𝑧, 𝑡 − 𝑡

′)|2𝑑𝑡′∞

0

, ( 3-5)

then Eq. 3-2 is written as,

39

Using the split step Fourier method (SSFM) we can obtain an approximated solution of

Eq.3-9 if the evolution of the pulse occur in a segment form 𝑧 to 𝑧 + ℎ, the final result is

show below,

𝐴(𝑧 + ℎ, 𝑡) ≈ 𝑒𝑥𝑝 (

ℎ

2𝐿) 𝑒𝑥𝑝 (∫ 𝑁𝐿(𝑧′)𝑑𝑧′

𝑧+ℎ

𝑧

) 𝑒𝑥𝑝 (ℎ

2𝐿)𝐴(𝑧, 𝑡) ( 3-10)

Basic Numerical Simulation

We used the GNSE to simulate the super continuum light source. The wavelength

of the Ti:sapphire as a pump is 810 nm with a pulse duration of 78.2 fs, the pulse repetition

rate is 80 MHz, thence the pulse energy are 0.4 nJ, 0.8 nJ, 1.5 nJ, and 2.5 nJ (the peak

power are 5 kW, 10 kW, 20 kW, and 32 kW respectively). The nonlinear parameter for

our PCF is reported as 𝛾 = 0.0198𝑊−1𝑚−1, and we use a 25 cm length of PCF. The

chromatics coefficients we adopt for the simulation are

𝛽2 = 1.7893 × 10−2𝑝𝑠2

𝑚,𝛽3 = 3.7755 × 10−5

𝑝𝑠3

𝑚,𝛽4 = 1.5469 × 10−8

𝑝𝑠4

𝑚,𝛽5

= −4.6604 × 10−13𝑝𝑠5

𝑚,

𝛽6 = 1.4041 × 10−17

𝑝𝑠6

𝑚 , 𝛽7 = −4.2303 × 10−22

𝑝𝑠7

𝑚,𝛽8

= 1.2745 × 10−26𝑝𝑠8

𝑚,𝛽9 = −3.8398 × 10−31

𝑝𝑠9

𝑚,𝛽10

= 1.1568 × 10−35𝑝𝑠10

𝑚.

Figure 16 show the input and output simulate spectra for an input field 𝐴 =

√𝑃 sech (𝑇

𝑡0). The input peak power is varied to demonstrate their effect on the output

spectrum. The peak power values correspond to our estimates of the power coupled into

40

the PCF. The spectrum is narrower as the power is reduced, but we still expected to have

a broad output spectrum,

Figure 16. Input and output spectra for different peak powers.

Experimental Results

In this section we will describe de experimental result obtained for the

supercontinuum light source using a normal-dispersion PCF. The mode-locked Ti:sapphire

laser (Tsunami XP) with a 400 mW (average power) pump was already discussed and

modeled in the previous subsection. The autocorrelation of the pulse is shown in Figure 17

41

below. The structure of our autocorrelation function is highly stable and reproducible.

-600 -500 -400 -300 -200 -100 0 100 200 300 400 500 600

0

2

4

6

8

Au

toc

orr

ela

tio

n S

ign

al

(a. u

.)

Time (fs)

Figure 17. Autocorrelation spectrum from Ti:sapphire laser.

Using the results from Figure 17 we extract the pulse duration, which is inferred to

be 78 fs. The laser beam diameter was measured by knife edge experiments as 2.8 mm.

The beam diameter is necessary to choose the correct lens for coupling light into the PCF.

From our experiment we use a 3 mm focal length lens. The normal-dispersion PCF from

NKT Photonics (NL-1050-NEG-1); it has a ~2.3 micron core diameter and ~0.37 NA. The

PCF length for our supercontinuum source is 25 cm. The diagram of the experimental setup

is illustrated in Figure 18. The output from the Ti:sapphire laser is coupled directly into

the PCF fiber. An isolator is placed before the coupling lens to prevent reflection feedback

42

into the laser cavity. Because the fiber core is so small we used a Thorlabs nano-positioning

stage to align the focused beam from the lens with the fiber core. The other end of the PCF

fiber is spliced directly to our fiber sensor leads. While the NA of the PCF fiber is 0.37,

the NA of our visible single-mode fiber is only 0.18. This mismatch results in a power loss

to the sensor, but we still have sufficient power to measure a good signal over the

wavelength region of interest.

Figure 18. 3D diagram of the experimental setup for a supercontinuum generation.

The supercontinuum transmission spectrum through the PCF is captured using an

Ocean Optics spectrometer with 0.2 nm resolution; the data is plotted in Figure 19. The

figure represents the power in a logarithmic scale with arbitrary units for the power scale.

The red curve is a plot of the pump’s transmission spectrum, which has a center wavelength

43

is close to 810 nm. The input laser spectrum is relatively narrow and has a Gaussian shape.

The blue curve is the transmission spectrum of the light after the PCF. It represents a

supercontinuum generation with a bandwidth around 230 nm. The supercontinuum

spectrum extends from the visible to the near infrared, i.e. extends from 710 nm to 940 nm.

The shape of the supercontinuum spectrum is stable over a long time.

Figure 19. The input spectrum and output supercontinuum spectrum using the Ti:sapphire laser.

44

Chapter Summary

We developed a supercontinuum light source using mode-locked Ti;sapphire laser

as a pump. The nonlinear properties of a PCF were used to obtain a broadband transmission

spectrum from the visible wavelengths (710 nm) to the near infrared (940 nm). This stable

supercontinuum design was coupled into a bi-tapered optical fiber sensor to study the

sensor characteristics and especially its sensitivity and resolution.

45

CHAPTER 4

FIBER OPTICS SENSORS

Introduction

The application of optical fibers in sensing systems is a signature development in

manipulation of light based on different physical phenomena in fibers. It has led to optical

fiber design modifications over the past decades to bring out higher sensitivity and with

high reliability at the same time. The researchers discovered that fiber optic sensors are not

only useful for signal transportation, but it also can be developed as sensor elements too.

Some applications of fiber sensors are to measure physical variables, such as acceleration

14, pressure 15, and temperature 16. Now, many optical fiber sensors are developed to

measure how biological and chemical species interact with light present in the surrounding

of the fiber. The light remains inside the fiber and carries the signal response with

information about the measured analyte.

Tapered Fiber Optics

Many fiber sensors base their operation on the interaction of the guided mode with

the surrounding environment. From the earliest research, one of the most developed

techniques is sensor applications incorporating tapered fibers. The tapered fiber can begin

with a standard, single mode optical fiber; the diameter is gradually reduce in a specific

region (see Figure 1) by heating and stretching the fiber to form a waist that maintains its

cross-sectional profile over a specified length 4. Generally, we keep the diameter of the

waist of our bi-tapered fibers between 2 and 15 𝜇m.

46

4.2.1. Tapered Fiber Geometry

We will describe the geometry of our common biconically tapered (or simply bi-

tapered) optical fibers. It is fabricated with three regions (illustrated here in Figure 20). The

bi-tapered fiber is fabricate with three sections in region 2. In the first section the fiber is

transitioned via a down-taper where the diameter of the fiber is gradually decreased until a

fixed, minimum diameter is reached. The second section is the taper waist section, where

its length and the waist diameter can be independently varied. The third section is the up-

taper section, where the diameter is increased back to the original diameter. The middle

section is the sensing part of the bi-tapered fiber, where the optical fiber is multimode and

the fiber modes have different propagation constants. Each mode also has a different

sensitivity to refractive index changes in the environment outside the fiber. The sensitivity

to the local environment surrounding the fiber depends on the extent of each mode’s

evanescent field.

For a sensor based on the tapered fiber, the light is launched into the fundamental

mode of the fiber at one end (region 1 from Figure 1). In the taper region (region 2) the

fundamental mode is gradually changed in the taper-down section and couples energy to

higher-order modes; in the waist section the modes propagate with an evanescent field

extending outside the fiber that can sense the local refractive index. At the up-taper end of

the fiber the higher-order modes are transformed back to the single fundamental mode

again (region 3). The modes generate interference due to the differences in the effective

indices between the modes at each wavelength. For some wavelengths the mode amplitudes

constructively interfere and at other wavelengths they destructively interfere. Therefore,

the total intensity 𝐼𝑇 light transmitted at the output of the fiber is largely determined by the

47

interference among the modes and their propagation constants.

Figure 20. Geometry of biconical tapered (aka bi-tapered) fiber.

Significantly decreasing the fiber diameter in the down-taper pushes the fiber mode

power from the core into the cladding region. Across the waist region of the tapered fiber

optical power is shared among several cladding modes in addition to the fundamental

mode. Due to the reduced size of the fiber in the waist region light has an evanescent

component in the surrounded environment (for example air) and each mode is differently

affected by the environment. After passing through the waist region the light is coupled

back into the single, core mode of the single-mode fiber in the up-taper region 17.

Close to the waist region the energy internal to the fiber decreases and the

evanescent field interacts with the materials near the surface of the fiber. The parameter

that describe the distance between the core-cladding and the evanescent field is well known

as penetration depth 𝑑𝑝 18. It is a parameter to estimate the extent of the evanescent field,

which is a function of the incident wavelength 𝜆, the incident angle 𝜃 of a plane wave

(above the angle for total internal reflection), and the refractive indices (𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔 and

48

𝑛𝑒𝑛𝑣𝑖𝑟 (say, air or water)), so this relation is represent as,

𝑑𝑝 =𝜆

2𝜋√𝑛𝑐𝑙𝑎𝑑𝑑𝑖𝑛𝑔2 𝑠𝑖𝑛2𝜃−𝑛𝑒𝑛𝑣𝑖𝑟

2. ( 4-1)

For angles well removed from the angle for total internal reflection at the boundary, the

penetration depth is much less than the wavelength of the light. For air or water on the

outside the minimum penetration depth is around 14% or 21% of the wavelength in

vacuum. In other words the sensing volume is confined very close to the surface, i.e. within

a couple hundred nanometers.

Sensing Principles

For a tapered fiber based sensor the light that enters the fiber from one end passes

through the previously described three regions in Figure 20. The interference signal

measured at the other end of the fiber due to the small effective index differences between

the coupled modes, which have evanescent fields into the local environment (water or air)

19. We express the total intensity 𝐼𝑇 at the output of the fiber in terms of the phase difference

between the modes. We approximate the propagation constants of each mode by using a

calculation for the waist modes. The intensity of the mth mode in the waist region is 𝐼𝑚

and the total transmitted intensity is related to the modes in the waist of the fiber sensor by

𝐼𝑇 =∑𝐼𝑛 + 2 ∑ √𝐼𝑚𝐼𝑛𝑐𝑜𝑠∆𝜙𝑛𝑚𝑛>𝑚𝑛

. ( 4-2)

The difference in the phase of two modes due to the interference between the modes

can be describe as,

49

∆𝜙𝑛𝑚 = (𝛽𝑛(𝜆) − 𝛽𝑚(𝜆))𝐿, ( 4-3)

where L is the length of the taper, 𝜆 is the wavelength of the incident the light source.

Recall in Chapter 2 we calculate the cladding modes of the fiber; these can be used to

examine the signal structure and even estimate the periodicity of the signal as a function of

wavelength.

4.3.1. Signal Processing

The signal processing is used to extract the amplitude and phase properties from each

data set. In other words we can gain information from the signal to determine small

refractive index changes in the environment. We developed a Fourier analysis technique to

process the signal. This computational tool provides high precision because the dominant

Fourier component is the result of all the data points. In this section we describe the signal

processing method. A discrete Fourier transform (FT) of the signal is discussed in the

following sub-section using the Fast Fourier Transform (FFT) algorithm.

4.3.1.1. Fourier Transform

For many time series signals it is important to convert it from the time domain to

the frequency domain. This is standard practice in many applications and one uses the FFT

to do it. The inverse Fourier transform (IFT) will take the data in the frequency domain

back to its original time domain signal. A complex signal can be obtained from the

frequency domain data by zeroing the negative frequency data before taking the IFT.

Mathematically we can define the Fourier spectrum represented by 𝐹(𝑘) of the wavelength

𝜆 , where 𝑘 is refer to as the wave frequency 20,

50

𝐹(𝑘) = ∫ 𝑓(𝜆)

∞

−∞

𝑒−𝑗𝑘𝜆𝑑𝜆, ( 4-4)

In the same way, we can define the inverse Fourier spectrum of a function 𝑓(𝜆) as,

𝑓(𝜆) =

1

2𝜋∫ 𝐹(𝑘)∞

−∞

𝑒𝑗𝑘𝜆𝑑𝑘, ( 4-5)

Now, if we have a discrete input signal {𝑥[𝑛], 𝑛 = 1,…𝑁} with N samples, the discrete

Fourier transform (DFT) of the sequence 𝑥[𝑛] is defined as 21,

𝑦𝑘 = ∑ 𝑥[𝑛]𝑒−𝑗𝑘𝑛Δ𝜆

𝑁−1

𝑛=0

, ( 4-6)

where Δ𝜆 is the uniform wavelength spacing of the data points and the wave frequency is

defined as 𝑘 = 2𝜋𝑚/Λ where 𝑚 is an integer from 0 to N-1, and the domain size is Λ =

Δ𝜆𝑁. DFT is computed using a FFT algorithm that is implemented in Matlab. The ordering

of the vector components in wave space is {𝑚 = 0,…𝑁

2− 1,−

𝑁

2, … , −1}. The

computational time is not an issue for our data. However, we note that the DFT has a total

of 𝑁2 components that corresponds to 4𝑁2 real multiplications; the FFT algorithm uses

needs only 𝑁𝑙𝑜𝑔2(𝑁) multiplications 22.

Now, let as examine a standard signal represent for a cosine wave such as

cos [(𝛽𝑛(𝜆) − 𝛽𝑚(𝜆))𝐿], where 𝛽𝑚,𝑛 are the modes that we calculated in Chapter 2, and

𝐿 = 12 𝑚𝑚 is the length of the fiber. Figure 21 represent the cosine wave as a function of

the wavelength 𝜆 from 740 nm to 910 nm for the modes 𝛽1 and 𝛽3 with a water as external

refractive index. The slope value for the argument of the cosine function, S13, is taken from

Table 3.

51

Figure 21. Cosine wave for two interfering modes, as a function of wavelength; the fiber

diameter is 10 m and it is immersed in water; the fiber length is 12 mm.

From Figure 21 we have a periodic function, then is possible to estimate the

corresponding frequencies using the computational program discussed above by applying

the FFT algorithm to the signal from Figure 21 with 100 points (𝑁). We calculate the

corresponding spatial frequency spectrum of the signal and plot it in Figure 22.

52

Figure 22. Spatial frequency spectrum of the cosine wave.

In the previous figure we can observe two maximum peaks that are symmetrical

with respect to the zero frequency. They correspond to the positive and negative frequency

components in the original signal. Due to the equal amplitude for the two different

frequencies we select one Fourier component at a peak Fourier spectrum wave frequency

and extract the signal phase from the complex amplitude. For example, the first peak

correspond to a frequency of 0.1471 𝑛𝑚−1.

From the complex Fourier amplitude in Figure 22 we calculate the phase as a function of

the wavelength as show in Figure 23. This corresponds to the phase we used to generate

the cosine data.

53

Figure 23. Phase for the wave frequency 0.55 nm-1 Fourier component versus wavelength.

Bi-Tapered Faber Fabrication

To fabricate tapered optical fiber we utilized a Vytran GPX-3000. This machine

has a high precision for the fabrication of taper devices (see Figure 24). The system has

two holding blocks where each end of the fiber is firmly attached. The holding blocks are

moved by long screws that move the blocks and are computer controlled. There is a

filament in the center of the system that locally heats the fiber and the entire system is

computer controlled. An image of a portion of a typical tapered fiber is shown in Figure

25.

54

Figure 24. Vytran GPX-3000 machine for fabrication of tapered fiber.

The important fiber taper parameters, such as waist length and size, can be fixed

using the computer-controls for the Vytran. For the desired size and uniformity of the fiber

taper system the speed of the fiber holding blocks and the temperature of the filament are

determined by conducting several experiments and analyzing the results to get the desired

taper shape.

Figure 25. Taper fiber image using the Vytran machine.

55

Experimental Sensing in IR Region

The schematic diagram of experimental setup of the tapered fiber sensor used for

study in IR band is illustrated in Figure 26 and the 3D diagram in Figure 27. The

configuration to measure the change in the refractive index is a broadband light source,

which is a tunable laser with a wavelength range from 1480 nm to 1550 nm, that correspond

a bandwidth of 70 nm. The light travels through the fiber and passes through the tapered

fiber region, which is set inside a flow cell. The light continues to the untapered region at

the end of the fiber that is connected to a photodetector on our laser system. The spectral

measurement is done by a virtual interface between the laser and the computer.

Figure 26. Experimental setup of tapered refractive index change.

56

Figure 27. 3D diagram of the tabletop experimental setup with all the devices.

4.5.1. Results in IR Bands

Solutions were made of water-glycerol mixtures with four different weight

concentrations that correspond to different refractive indices at room temperature. The

concentrations of the solutions are between 0% and 0.15%, as shown in Table 4. The

refractive indices were inferred by doing an interpolation of the concentrations from a

published water-glycerol refractive index table 23. For our experiments we chose a taper

waist length (𝐿) 10 mm and the waist diameter (𝑑) 10 𝜇m. We analyzed the transmission

spectrum, i.e. intensity versus wavelength, for each refractive index solution that we

studied; the four data sets are plotted in Figure 28. The difference in the refractive indices

of each solution produce a nearly linear wavelength shift with concentration. The results

57

are attributable to the differences in propagation constants of the modes that are present in

the tapered fiber produce some change in the refractive indices that generate a phase shift.

Table 4. Glycerol concentrations in water and their corresponding refractive indices n.

Concentration (%) 𝒏 0.00 1.33295 0.05 1.33301 0.10 1.33307 0.15 1.33313

Figure 28. Power transmission spectrum of the tapered fiber sensor as a function of

wavelength.

58

We clearly observed on the plots in Figure 28 how the initial points and the final

curves changed with index concentration; it is consistent with work present in a previous

publication 24. In Figure 29 we plot the wave frequency spectrum of the transmission

spectrum by applying the previously described Fourier transform technique. There are 𝑁 =

701 data points from 1480 to 1550 nm and the wavelength span Λ = 70 𝑛𝑚. The Fourier

wave vector increment each wave frequency is Δ𝐾 = 2𝜋

Λ= 0.089 nm-1. The predominant

peak is found for 𝑚 = 7 and the corresponding wave vector value is 0.628 nm-1. The

corresponding period of the signal is around 10 nm.

Figure 29. Spatial frequency spectrum from transmission for tapered fiber with L=10 mm, d=10 μm.

59

Figure 30 displays the corresponding phase change for the predominant Fourier

component with respect to the wavelength. This corresponds to the linear behavior between

phase and wavelength discuss in Chapter 2. Additionally, the values of the phase decrease

according the refractive index increase.

Figure 30. Phase shift versus wavelength for the complex data for a tapered fiber with L=5 mm, d=10 μm.

However, we used a similar signal analysis to that proposed in our earlier

publication 24 and calculated the linear relation between the phase difference Δ𝜙 and the

difference on the refractive indices Δ𝑛, as shown in Figure 31. We observed a clear

60

difference between each refractive index that corresponds to a linear response of the sensor

using the analysis an average of the all data points present in the transmission spectrum.

Additionally, in we had a detected sensitive limited around 0.0001 Refractive Index Unit

(RIU) which agrees with the earlier reference study.

Figure 31. Phase shift sensitivity versus refractive index differences for tapered fiber with L=10 mm,

d=10 μm.

Experimental Sensing from Visible to Near Infrared Region

The experimental setup for a second tapered fiber sensor is illustrated in Figure 32.

The principal elements that compound the experimental setup are: a laser, mirror, lens,

61

fiber optics, and the spectrometer. The objective is to generate a supercontinuum spectrum

that was discussed in Chapter 3. We focus a Ti:sapphire mode-locked laser beam into a

photonics crystal fiber (PCF) that is designed to enhance the nonlinearity of the material

by confining the light to a small effective area. The bi-tapered fiber that we used is designed

using a single-mode fiber (SMF) for the visible region (bought from Thorlabs), then the

PCF fiber is spliced to the specialty SMF. The output from the tapered fiber is coupled into

a spectrometer that records the signal. As illustrated in Figure 32 the laser output beam is

turned using two mirrors (M1 and M2) and an isolator that prevents light coupling back

into the laser oscillator.

Figure 32. (a) . Experimental setup of bi-tapered fiber sensor coupled to a supercontinuum

generating light source. (b) 3D diagram of the experimental setup with all the elements.

62

We fabricated the bi-tapered fiber using the Vytran Glass Processing system that was

previously described. The difference here is that we used a specialty fiber supplied by

Thorlabs that is single mode in the visible regime. Our bi-tapered fiber sample was

fabricated with the following physical parameters: the up-taper and down-tapers lengths

were each 5 mm, the waist length 𝐿 = 10 mm, and fiber waist diameter = 10 𝜇𝑚.

The light exiting the tapered fiber section was coupled into a multimode fiber

leading to the spectrometer (Ocean Optics USB4000-VIS-NIR) with a fiber optic connector

SMA 905. The spectrometer has an array of 3648-element CCD. The resolution of the

spectrometer is 0.2 nm. The data was captured by a computer through a USB port

connection. In the next section we will present the results.

4.6.1. Results from Visible to Near Infrared Region

The output spectrum of the bi-tapered surrounded by air (𝑛 = 1.00) is presented in

Figure 33. As a result, the light from the pump laser, broadened by the nonlinear properties

of the PCF (from VIS to IR), is inserted into the bi-tapered and collected at the fiber output

end. The transmission spectrum with the bi-tapered fiber is normalized by dividing it by

the supercontinuum spectrum after the PCF without the spliced SMF section (the

supercontinuum spectrum in Figure 19). This has the effect of flattening the spectrum over

the entire wavelength range. As observed in our IR data, we again observe a dominant

periodic, sinusoidal change of the intensity due to interference between the multiple modes

in the taper region. The amplitudes are relatively constant over the range with an

anomalously high peak around 900 nm, which is evidence of especially high coupling

efficiency in that region.

63

Figure 33. Transmission spectrum taken from the tapered fiber sensor immersed in air

generated by the supercontinuum light source.

To determine the sensitivity of the fiber sensor for this new configuration the signal

transmitted by the bi-tapered fiber was studied using the same concentrations of water-

glycerol mixtures presented in Table 4 (section 4.5.1). The intensity modulation with

respect to the wavelength for the four aqueous solutions (water/glycerol) are plotted in

Figure 34. The spectrum has periodic intensity changes similar to those observed in Figure

33. We find the dominant period in Figure 33 that is 9.3 nm and the period in Figure 34

that is 10.3 nm. Hence, there is a small shift of the dominant period when we change from

64

air to water in the local environment.

Figure 34. Transmission spectrum of the tapered fiber sensor as a function of the wavelength

for different refractive indices.

In Figure 34 we observe that the intensity modulation amplitude for different

refractive indices shows a clear change as the wavelength is scanned. There is also a

repeatable shift of the phase for each solution that cannot be easily observed in the data.

However, to examine the small phase shift, we implemented the FFT signal analysis

developed and applied in previous sections with the objective to observe in greater detail

using all data points the change of the phase corresponding to changes in the refractive

65

index 25. The data in Figure 34 is subjected to an FFT in order to extract the complex

coefficients of each wave component (as we previously analyzed the IR data). The

wavelengths span around Λ = 193.7 nm, i.e. from around 723.3 nm to 917. There are 𝑁 =

1062 data points in this range. The incremental Fourier wave vector is defined as Δ𝐾 =

2𝜋

Λ= 0.032 nm-1.

The Fourier spectra have a prominent Fourier component that corresponds to the

interference between two modes; the Fourier data corresponding to the results in Figure 34

are plotted in Figure 35. The spectra have a strong and dominant peak at one value of the

wave frequency value 0.65 nm-1. The wave vector corresponds to the period 9.75 nm.

Using the multi-mode model calculations, the period is extracted from Equation 2-47 by

setting the phase equal to 2. For a 1.2 cm sensor length (1 mm on each side to the effective

waist length to account for the up- and down-tapers) we find using Equation 2-47 that the

1-3 mode interference gives a period 11.4 nm. The corresponding period for the same

modes for the fiber/air interface (Figure 33) is 10.3 nm, where the corresponding

experimental result is 9.3. The other modes have period values that substantially differ

from the experimental results. The two cases for the 1-3 modes follow the trend of the

experimental results.

66

Figure 35. Spatial frequency spectrum of the transmission for different refractive indices.

We extract the phase from the complex Fourier amplitudes and use the Fourier

component with the largest amplitude. The phase data is plotted versus wavelength in

Figure 36. Only a small span of wavelengths was used in order to highlight the tiny phase

difference between concentrations. In Figure 36 the relationship between the phase and

refractive index demonstrates that our sensor can detect changes in phase due to changes

in the refractive index on the order of 0.00006 RIU.

67

Figure 36. Phase for the dominant Fourier component respect the wavelength for

different refractive indices.

The difference in phase with respect to the difference in refractive index is plotted

in Figure 37. The data is reproducible to a very high degree; uncertainty in the

concentrations, sample contamination and sensor holder add to variations. We washed the

sensor between measurements with deionized water and the fluid was extracted from a

closed tube using a dedicated suction extractor. The data is fit using a linear function of the

refractive index increment. The slope is 447.33 radians and a phase shifts of 0.01 are

distinguished yielding a detection sensitivity close to 2x10-5 RIU. The unambiguous

68

dynamic range in phase could extend to 2 which corresponds to an index difference of

0.014 using the slope from the curve fit.

Figure 37. Difference phase versus the refractive index difference for a bi-tapered sensor.

We also noted that the amplitudes of the dominant Fourier component in Figure 35

varies with refractive index. The inset in Figure 38, for predominant frequency clearly

displays the amplitude changes for each measured refractive index. The corresponding

peak amplitude values are 48.85 (1.33295), 45.67 (1.33301), 39.23 (1.33307), and 32.10

(1.33313), where the corresponding refractive index is written in parentheses. We

normalize the peak amplitudes with respect to the maximum value (48.85) and plot the

69

results for four different refractive indices in Figure 38. The inset is a view of the spectral

peak as a function of the spatial frequency. Again there is a close relationship between the

data points and the refractive index increment. The data points are fit to the linear curve

shown by the dotted line in Figure 38.

Figure 38. Normalized spectral peak as a function of the refractive index difference for a

taper with 𝐿 = 10 𝑚𝑚 and 𝑑 = 10 𝜇𝑚.

70

Chapter Summary

Our data analysis show a high correlation between the signal phase shift, the Fourier

amplitude, and changes in the refractive index. We used differences in the refractive index

close to 6 × 10−5, the RIU sensitivity using the tunable laser for the IR region is around

2 × 10−4 and using the supercontinuum light source is around 2 × 10−5. Working with a

laser source with a wavelength less that 1𝜇𝑚 we increased the sensitivity of our sensor by

about ten times for the same bi-tapered fiber parameters.

In the VIS to NIR experiment for small refractive index differences the transmission

spectra do not reveal a clear phase shift by visually inspecting the data; however, by

Fourier analyzing our data and using the predominant Fourier component, the phase shift

can be reproducible measured and correlated with changes in the phase down to 0.01

radians. That contrasted with change in phase in IR region experiment that was around

0.15 radians. Additionally, in the VIS to NIR experiments we observed correlated changes

in the Fourier amplitude peak that was not clearly observed in the IR experiments.

In conducting our experiments we developed an optical system to generate a

supercontinuum light source that enables measurements over a wide range of wavelengths

(200 nm) for our transmission spectra. The pump laser wavelengths are smaller than 1𝜇𝑚

and after supercontinuum broadening we were able to develop sensing modalities close to

the visible wavelength range. This is desirable for future work on the bi-tapered fibers

and will be described in Chapter 5.

71

CHAPTER 5

SUMMARY AND FUTURE WORK

Summary

As discussed in Chapters 2 and 4, optical fibers form a cylindrical waveguide made

of silica/doped silica glass used for light transmission over long distances. The index of

refraction of the cladding is slightly lower than the index of the core that is doped with

germanium. The light propagates inside of the fiber following the phenomenon known as

total internal reflection (TIR), due to the difference between core and cladding refractive

indices the light is reflected along the fiber axis. There is a small portion of the light power

that is outside the core surface and propagates in the cladding region. This portion of light

decays exponentially away from the interface and is known as an evanescent field.

However, the evanescent field in a regular fiber is purposely kept small using a cladding

layer that is much thicker than the core’s diameter, so that the evanescent field is well

contained inside the glass and doesn’t directly interact with the surrounding medium.

By gradually reducing the diameter of the fiber the light will couple to the cladding,

as cladding modes, and the field will extend outside the glass into the surrounding medium.

That is the principal used for fabricating a tapered fiber sensor. The tapered fiber

redistributes the field and in the waist region pushes the evanescent field beyond the

cladding boundary. Consequently, if the tapered region is immersed in a liquid, the

evanescent field senses the small changes in the liquid refractive index.

72

The penetration depth, i.e. the distance from the core and cladding boundary where

the light decays to 𝑒−1 its boundary value, give us a measure of how much evanescent field

energy is available for sensing. The penetration depth depends of the wavelength, so when

the range of the wavelength is longer we have a greater penetration depth for a fixed waist

diameter. We initially worked at wavelengths between the 1480 nm to 1550 nm because it

was convenient for coupling light into the fiber, as well as the availability of broad tunable

sources.

Our work in the visible/near IR wavelength region (710 nm to 920 nm) presented

serval challenges that had to be overcome. The major challenge is the small core size of

the specialty fiber with the same NA as standard IR fibers. We were able to overcome this

obstacle by using a supercontinuum source spliced directly to our fiber sensor. With a goal

to optimize the sensitivity of the tapered fiber in the future we can design a set of

experiments for different waist lengths and diameters that allow us to determine the best

taper parameters.

Ongoing Work

5.2.1. Tapered Fiber with Metal Coating

In the future we intend to design a new generation of high sensitivity bi-tapered

optical fiber sensors by adding a thin metal (or dielectric) film on the surface to extend

more optical power into the environment. This will allow further improvements to detect

trace amounts of selected biomolecules in aqueous solutions or in vapors. The optical

sensitivity of the fiber sensors are based on the ability of a metal film to draw the

electromagnetic field to the fiber’s surface where the analytes are captured. The device

73

sensitivity will be increased by depositing a gold metal film or titanium dioxide (TiO2) film

a few nanometers in thickness on the surface. By attaching selected capture molecules to

the surface we can determine the presence of specific analyte biomolecules for label free

detection.

As we mentioned previously we will use TiO2 or Au films to coat the tapered fiber.

The idea is to present a clear protocol on how to deposit and attach the thin film over the

fiber for future research in our lab.

In Chapter 4 we analyzed our data using FFT based signal analysis to measure the

phase and amplitude changes with refractive index changes 24. The dominant Fourier

component for signal was used. However in the future we will explore analyzing the data

using other significant Fourier components with the purpose of determining the effect of

multiple cladding modes on the signal. We will explore if the same analysis can be applied

to signals that may not be sinusoidal with a single, dominant frequency in the data.

Finally, we will study specific chemical and biological species, so the metal film

fiber sensor will function as a label free detection technique. The concept is shown in

Figure 39. The nanofilm in the waist region attracts only specific biomolecules that bind

to the surface. The molecules effectively change the local refractive index, which is

measured by our fiber sensor.

74

Figure 39. Schematic diagram of the tapered fiber coated with AuNPs.

5.2.2. Surface Plasmon Polariton

One important characteristic of metals is related to the free electrons present in the

material. Consequently, the free electrons are responsible of the electric conductivity and

optical reflectivity. Additionally, the free electrons form an electron gas (plasma) that

affects the optical properties of the metal 26. One of the interesting phenomena connected

to the interface between metal and dielectric is the excitation of propagating modes called

surface plasmon polaritons (SPPs). At a metal/dielectric interface the electromagnetic field

couples to the electrons in the conduction band. Generally the SPP at a metal/dielectric

interface consists of an evanescent wave that extend into the dielectric medium and is also

confined near the surface of the metal. Using an electromagnetic wave it is possible to

exciting SPP using the Kretschmann configuration shown in Figure 40 for a p-polarized

wave incident on a planar surface. The transmission is zero due to total internal reflection

at the interface. At a specific angle of incidence the wave couples to the SPP at the outer

surface of the metal film. The phenomenon produces a dip in the reflected light called a

75

surface plasmon resonance (SPR) and the dip sensitively depends on the index of the

ambient medium.

Figure 40. Kretschmann configuration for SPP excitation as a function of incident angle.

In the Kretschmann configuration a metal film placed at the interface of two

mediums, one with higher refractive index (prism) and another with lower refractive index

(air or the surrounding medium). The angle of incidence of the light is changed in order to

achieve a phase matching condition between the incident wave and the SPP 27. By exciting

the SPP at the metal/ambient interface there is a reduced reflection around a specific angle.

The minimum reflection angle is sensitive to the refractive index near the surface.

5.2.3. Optical Properties of Metals

One way to describe the interaction of the electromagnetic field with metal is the

Lorentz-Drude model 28. For this model the metal is consider a gas with a free electrons in

a fixed lattice of positive ions. The equation of motion for an electron in the present of

76

external field is describe as

𝑚𝑑𝒗

𝑑𝑡+𝑚𝛾𝒗 = −𝑒(𝑬 + 𝒗 × 𝑩). ( 5-1)

where 𝑚, 𝒗, and 𝑒 are the mass, velocity and charge of the electron respectively. 𝛾 is the

frequency where the electron damped by interactions with ions. The dipole moment of

typical electron is given by 𝑷 = −𝑒𝑛𝒓, 𝑛 is the density of electron gas per unit volume.

Now, defining the plasma frequency of the electron gas as 𝜔𝑝 = √𝑛𝑒2휀0𝑚⁄ , with can

write the previous equation as,

�̈� + 𝛾�̇� = 휀0𝜔𝑝2𝑬 + �̇� × 𝑩. ( 5-2)

On the other hand, if we ignore the contribution of the magnetic force, the polarization

density can be written in the frequency domain as

𝑷(𝜔) = −𝜀0𝜔𝑝

2

𝜔2+𝑖𝛾𝜔𝑬(𝜔), ( 5-3)

where the term susceptibility for plasmons can be write as 𝜒(𝜔) = − 𝜔𝑝2

𝜔2+𝑖𝛾𝜔. Now, we

have the contribution of two different susceptibilities, one for free electrons (𝜒𝐷𝑟𝑢𝑑𝑒) and

the another for bound electrons (𝜒𝐿𝑜𝑟𝑒𝑛𝑡𝑧), these can be represent as,

𝜒𝐷𝑟𝑢𝑑𝑒(𝜔) = −

𝜔𝑝,𝑓2

𝜔2+𝑖𝛾𝑓𝜔 𝑎𝑛𝑑 𝜒𝐿𝑜𝑟𝑒𝑛𝑡𝑧(𝜔) =

𝜔𝑝,𝑏2

𝜔𝑏2−𝜔2−𝑖𝛾𝑏𝜔

, ( 5-4)

Finally, using the Lorentz-Drude model the permittivity can be modeled by adding both

contributions

휀(𝜔) = 1 + 𝜒𝐷𝑟𝑢𝑑𝑒(𝜔) + ∑ 𝜒𝐿𝑜𝑟𝑒𝑛𝑡𝑧,𝑛(𝜔)𝑛 . ( 5-5)

The sum over Lorentz contributions acknowledges the fact that there are many bound

electronic modes that contribute to the total optical response.

77

5.2.4. Metal Film Selection

The idea in this project is cover the surface of the fiber with a metal film to obtain

a plasmonic fiber. With the tapered fiber sensor we expect to develop a SPR for biological

applications that redistributes the electromagnetic field outside the surface of the fiber

waist. Gold nanoparticles (AuNPs) have some advantages with respect to other metal for

detection and study of biological and chemical elements in tapered fiber sensors 29. The

advantages of AuNPs include: (a) the high stability of the NPs when synthesized, (b) the

favorable dielectric properties for forming surface plasmon resonances, (c) AuNPs can be

attached to the fiber surface using a well-known chemistry and (d) the AuNPs can be grown

to form thin films that support SPPs. On the other hand it is important to analyze the

operational wavelength range that will support the excitation of the surface plasmon on the

fiber tapered covered with an Au film. The change the metal permittivity in gold respect

to the wavelength is significant and can be modeled using the Lorentz-Drude model discuss

in the previous section (see Figure 41) 30. Additionally, TiO2 has a high dielectric constant

and is used in many integrated circuits.

78

Figure 41. Real and Imaginary components of gold permittivity as a function of the wavelength.

From the previous figure of the permittivity of gold we observe that its real part is

negative with a relatively steep slope. It becomes quite negative at the infrared frequencies

of interest, which will limit the excitation of the SPP and the large imaginary part will

increase the absorption of the wave and ultimately limit the propagation distance and

length of the tapered fiber’s waist for a given waist diameter.

5.2.5. Experimental Results with TiO2

We also studied the sensitivity of a bi-tapered fiber by adding a thin of TiO2 film to

the surface of the bi-tapered fiber for a 10 nm and 20 nm of thickness. The electron beam

evaporation tool was used to coat the fiber. The deposition of a TiO2 film on the fiber has

high deposition rates and low cost. One of the advantages of TiO2 is its higher refractive

index compared to SiO2. The high TiO2 index reshapes the electromagnetic field to the

surface of the fiber and should improve the sensitivity of the sensor31. To test this concept

we used the same experimental setup described in Section 4.5 with the tunable IR laser as

79

a source. The length (𝐿) of the taper was 10 mm and the diameter of the waist (𝑑) was 10

𝜇m. The experiment was run for 10 nm and 20 nm thick films of TiO2. Our preliminary

results will be describe below.

5.2.5.1. 10 nm TiO2 Thickness

The corresponding transmission spectrum of the sensor coated and uncoated

surrounded by air for the same bi-tapered fiber is present in Figure 42. The graph show a

clear increase of the power transmission when the fiber is coated with TiO2. Additionally,

we observed the same sinusoidal shape of the signal function in the spectrum due to the

interference between the propagation modes in the taper region.


wavelength coated and uncoated with 10 nm of TiO2.

80

The intensity modulation with respect to the wavelength for four solutions of water-

glycerol describe in Table 4 is shown in Figure 43 for the TiO2 film on the fiber. We

observed a phase shift with respect to the variation of the refractive index. The period of

the graph is around 11 nm. Additionally, the power transmission decreases when the

refractive index increase, i.e. less output power is obtained with the increment of the

concentration of glycerol on the solution.


wavelength for different refractive indices.

81

Using the Fourier analysis we developed earlier the phase of the transmission data

as function of wavelength, is shown in Figure 44. This graph represents the phase shift when

the refractive index surrounding the coated fiber is changed. It shows a linear behavior of

the phase extracted from the predominant Fourier component of the data from Figure 43.

The inset plot in Figure 44 is a result from a the linear regression analysis of the phase

changes with respect to the difference in the refractive index, defined as ∆𝑛 = 𝑛 −

1.33295,. The slope of the plot is -7777.3 rad/nm.

Figure 44. Phase for the dominant Fourier component respect to the wavelength for different

refractive indices. Inset: Phase difference versus refractive index differences.

82

Now, the corresponding Fourier components of the data in Figure 43 was taken.

For the higher order mode (0.54 𝑛𝑚−1) or the predominant peak there is a change in the

amplitude for each refractive index (inset Figure 45). The peaks correspond to 66.5, 58.6,

59.6, and 51.7 for 1.33295, 1.33301, 1.33307, and 1.33313 respectively. The variation of

peak respect to the refractive index difference, ∆𝑛, is plotted in Figure 45.

Figure 45. Normalized spectral peak as a function of the refractive index differences for a bi-

tapered sensor coated with 10 nm of TiO2.

83

5.2.5.2. 20 nm TiO2 Thickness

The same experiment made for 10 nm of thickness was repeated for 20 nm of

thickness. The transmission spectrum as a function of the wavelength in a bi-tapered fiber

taper surrounding by air is show in Figure 46. In this case the power transmission decreased.

The signal still has a periodic component.


wavelength coated with 20 nm of TiO2.

The corresponding Fourier spectrum from data in Figure 46 is show in Figure 47.

The higher order mode is present around 0.56 𝑛𝑚−1 .The Fourier spectrum in this case

84

does not present a significant variation of the peak for each refractive index as we observed

for 10 nm.

Figure 47. Spatial frequency spectrum from transmission for tapered fiber coated with 20 nm of

TiO2.

Finally, the phase of the transmission data as function of wavelength, as shown in

Figure 48. We can observed the phase shift when the refractive index surrounding the

coated fiber is changed. The inset plot in Figure 48 show the linear regression between the

different in phase respect to the different in refractive index. The slope of the plot is -

9792.01 rad/nm. This is larger than the slope for 10 nm thickness TiO2 films, which shows

85

that the thicker 20 nm film has an improved sensitivity performance than the 10 nm films.

The optimal thickness is somewhere in the range around 20 nm, but further experiments

need to be done.

Figure 48. Phase for the dominant Fourier component respect to the wavelength for different

refractive index. Inset: Phase difference versus refractive index difference.

86

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bi-tapered fiber sensor using a supercontinuum light source for a wide measuring range

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