A NOVEL IDEA OF USING SOLITON IN FIBER BRAGG GRATING
HARYANA BINTI MOHD HAIRI
UNIVERSITI TEKNOLOGI MALAYSIA
A NOVEL IDEA OF USING SOLITON IN FIBER BRAGG GRATING
HARYANA BINTI MOHD HAIRI
A thesis submitted in fulfillment of the
requirements for the award of the degree of
Master of Science (Physics)
Faculty of Science
Universiti Teknologi Malaysia
AUGUST 2010
iii
To all the beloved person in life especially
Mom, Dad and My Lovely Siblings
No Love
can cross the path of our destiny without leaving some
mark on it forever.......
To my dearest friends:
There are no limits to our possibilities.
At any moment, we have more possibilities that we can act upon.
When we imagine the possibilities, our vision expands,
We capture our friends and our life is meaningful.
We can reach out and touch the limits of our being.
iv
ACKNOWLEDGEMENTS
First and foremost, I would like to express my deepest gratitude to Allah S.W.T
for giving the strength to complete my research successfully.
Secondly, without their guidance, I would be nowhere. I would like to convey
my deepest appreciation to my supervisors, Prof. Dr. Jalil Ali, Prof. Dr. Rosly Abd.
Rahman, Dr. Saktioto and Prof. Dr. Preecha Yupapin (KMITL, Thailand) for all their
guidance and support throughout the duration of this research and thesis writing. I am
greatly indebted to them for the knowledge imparted and the precious time they
allocated to guide me. Prof. Dr. Jalil Ali provided the overall framework of this studies.
Together with Prof. Dr. Preecha Yupapin, they guided me on how to produce good
results and publish papers. Prof. Dr. Rosly Abdul Rahman provided the FBG research
facilities and Dr. Saktioto assisted in modeling work. I would like to extend my sincere
appreciation to my family especially mom and dad for their tender support, morally and
financially. During the final stage of my thesis writing, my dad had a severe stroke, I am
thankful to my supervisors for being understanding during this point of time. I would
also like to convey many thanks to members of the Institute of Advanced Photonics and
Sciences (APSI) for their assistance. They had provided me with ample information, cooperation and help during the process of conducting my research.
Last but not least, I would like to thanks my constant companions, Asiah,
Nafisah and Hanim who had given me a lot of support as well as fruitful ideas and
comments which had helped me a lot in completing this research.
v
ABSTRACT
With the rapid development in sensing and optical telecommunication, fiber
optic plays an important role in transmission systems as a low-loss and wide-bandwidth
medium.
In this study, three fiber Bragg gratings (FBGs) are fabricated using
conventional method known as the phase mask technique. Bragg’s wavelength of
1551.09 nm, 1551.29 nm and 1551.66 nm and reflectivities values of 30.18%, 78.12%
and 44.73% respectively are obtained. For soliton writing, the equations based on the
coupled mode theory have been derived. A Matlab coding has been developed in order
to solve some of these equations.
The simulation of potential energy distribution
throughout the grating is examined by varying the value of nonlinear parameters of α, β,
γ, and a new element known as θ is added in the equations. The results show that the
nonlinear parameters affect the motion of photon in the FBG and under certain
condition, it is possible to trap the photon and hence obtain the optical soliton. The
fabrication results show that the FBG with reflectivity of 78.12% can be classified as
good FBG compared to the other two FBGs. The simulation studies show that amongst
those nonlinear parameters, α significantly affects the potential well due to its ability of
this parameter in order of photon trapping. This study thus shows that is plausible to use
soliton for FBG writing and the properties of soliton for such purpose can be controlled
by manipulating α, β, γ and θ.
vi
ABSTRAK
Sejajar dengan perkembangan pesat dalam bidang penderia dan teknologi
komunikasi, gentian optik memainkan peranan penting dalam sistem pancaran sebagai
medium yang mempunyai daya kehilangan yang rendah dan jalur lebar yang luas.
Dalam kajian ini, tiga gentian parutan Bragg (FBG) telah berjaya difabrikasi
menggunakan teknik topeng fasa dengan panjang gelombang masing-masing ialah
1551.09 nm, 1551.29 nm dan 1551.66 nm bersama darjah pantulan masing-masing
sebanyak 30.18%, 78.12% dan 44.73%.
Teknik penghasilan parutan Bragg
menggunakan soliton telah diterbitkan dalam beberapa persamaan yang diperolehi
daripada Teori Mod Pengganding.
Kod Matlab juga telah dihasilkan dalam
menyelesaikan persamaan-persamaan yang telah diterbitkan. Simulasi taburan tenaga
keupayaan sepanjang parutan telah dibuat dengan mengubah nilai-nilai parameter tak
linear iaitu nilai-nilai α, β dan γ. Selain itu, satu parameter yang baru telah ditambah
dalam persamaan tenaga keupayaan untuk mengkaji kesannya terhadap taburan tenaga
keupayaan. Hasil keputusan kajian fabrikasi menunjukkan FBG dengan darjah pantulan
78.12% adalah yang terbaik berbanding FBG yang lain dan dari simulasi pula jelas
menunjukkan α memberi impak yang paling besar terhadap pola pergerakan foton dalam
telaga keupayaan berbanding parameter-paramater tak linear yang lain.
Ini
menyumbang terhadap penangkapan foton sekaligus kewujudan elemen yang dikenali
sebagai soliton optik. Ini menunjukkan bahawa adalah mungkin penggunaan soliton
untuk fabrikasi FBG dan ciri-ciri soliton untuk tujuan berkenaan boleh dikawal dengan
memanipulasi α, β, γ dan θ.
vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENT
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF TABLES
x
LIST OF FIGURES
xii
LIST OF SYMBOLS
xiii
LIST OF APPENDICES
xvi
INTRODUCTION
1
1.1 Introduction
1
1.2 Background of the Study
3
1.3 Problem Statement
3
1.4 Aims and Objectives
4
1.5 Scope of the Study
4
1.6 Research Methodology
5
1.7 Significance of the Study
7
1.8 Organization of the Study
7
viii
2
3
4
LITERATURE REVIEW
8
2.1
Optical Soliton
8
2.2
Coupled-Mode Theory for FBG
9
2.3
Soliton in Fiber Bragg Grating
12
2.4
Pulse propagation in FBG
13
2.5
Properties of Fiber Bragg Grating
15
2.5.1 Bragg condition
15
2.5.2 Uniform Bragg grating reflectivity
17
2.6
Photosensitivity in Optical Fiber
19
2.7
Fabrication Technique for Fiber Bragg Grating
21
2.7.1
Internal Inscription of Bragg Gratings
21
2.7.2
External Inscription of Bragg Gratings
23
2.7.3
Point-by-point Writing Technique
25
2.7.4
The Phase Mask Technique
26
EXPERIMENTAL SETUP
3.1
Introduction
3.2
Experimental Setup of Fiber Bragg Grating
28
28
Fabrication
28
3.2.1 KrF Excimer Laser Overview
31
3.2.2 Mask Aligner Overview
34
3.2.3 Phase mask
36
3.2.4 Tunable Laser Source
36
3.2.5 Optical Spectrum Analyzer
37
FIBER BRAGG GRATING MODEL OF POTENTIAL ENERGY
DISTRIBUTION
39
4.1
Coupled Mode Theory
39
4.2
Derivation of Nonlinear Coupled Mode
Equation (NLCM)
4.3
4.4
43
Derivation of Potential Energy Distribution
in Fiber Bragg Grating
49
Modelling of Optical Soliton using NLCM
50
ix
4.5
Modelling of Potential Energy Distribution
in Fiber Bragg Grating structures
4.6
53
Multi Perturbation of Potential Energy
Photon in Fiber Bragg Grating
53
4.6.1 External Perturbation of Potential
Energy
4.7
5
Flowchart for computational modelling
RESULTS AND DISCUSSION
53
55
59
5.1
Introduction
59
5.2
Results of Fiber Bragg Grating Fabrication
59
5.3
Results for Simulation of Soliton in Fiber
Bragg Grating
62
5.3.1 Nonlinear Parametric Studies of
Photon in Fiber Bragg Grating
62
5.3.2 External Disturbance of Potential Energy
Photon in Fiber Bragg Grating
65
5.3.3 Motion of Photon due to External Energy
Perturbation in Potential Well
5.4
6
Summary
68
72
CONCLUSION
73
6.1
Introduction
73
6.2
Conclusions
73
6.3
Future Work
75
7
REFERENCES
76
8
APPENDICES
79
9
PUBLISHED PAPERS
93
x
LIST OF TABLES
TABLE NO.
5.1
TITLE
Summary of the data collected for fabrication
PAGE
62
xi
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
Illustration of Fiber Bragg Grating.
1
1.2
The flow chart for the research methodology on the
6
novel idea of using optical soliton in FBG.
2.1
Cross-section of an optical fiber with the corresponding
11
refractive index profile.
2.2
A basic diagram of Fiber Bragg Grating
16
2.3
Oxygen-deficient germania defects thought to be
20
responsible for the photosensitive effect in
germania-doped silica.
2.4
Schematic of original apparatus used for recording Bragg
22
Gratings in optical fibers. A position sensor monitored the
Amount of strectching of the Bragg gratings as it was
strain-tuned to measure its very narrow-band response.
2.5
Schematic design of the diffraction of an incident beam
27
from a phase mask.
3.1
Schematic diagram of Fiber Bragg Grating fabrication
29
experimental setup.
3.2
KrF Excimer Laser
33
3.3
Functional design of the COMPex laser system
33
3.4
Optical components of mask aligner
35
3.5
Schematic diagram on propagation of light in mask aligner
35
3.6
Phase Mask Holder
36
3.7
Tunable Laser Source Overview
37
xii
3.8
The Optical Spectrum Analyzer
38
4.1
Flow chart in the case where there is no energy disturbance
55
4.2
Flow chart of simulation with potential energy disturbance factor 56
4.3
Flow chart of potential energy under multi-perturbation condition 57
5.1
The transmission spectrum to monitor the growth of fiber
60
grating in FBG1
5.2
Results of fabricated FBG1
61
5.3
The motion of photon in double well for different values of α
63
5.4
The optimized point of the double well potential for
63
different values of α
5.5
Under Bragg resonance condition the system possesses
64
double well potential for γ = 0.13 to 0.53
5.6
The optimized point of the double well potential when
65
γ = 0.1 to 1.0
5.7
The motion of photon in potential well for α = 0.9, β = 0.3,
66
θ = 0.09 and γ is varies from 0.3 to 0.9.
5.8
The effect of theta,θ to γ and shape of the potential well of
67
the photon.
5.9
The disturbance to the potential energy by β factor
68
5.10
The motion of photon in potential well for α = 0.9, β = 0.3,
69
θ = 0.09 and γ is varies from 0.3 to 0.9.
5.11
The disturbance factor that affect the shape of the
potential well of the motion of photon.
71
xiii
LIST OF SYMBOLS
λB
-
Bragg wavelength
Λ
-
Spatial period (or pitch) of the periodic variation
Neff
-
Effective index for light propagating in a single mode fiber
A(z)
-
Forward propagating modes
B(z)
-
Backward propagating modes
ψ (x, y )
-
Transverse modal field distribution
ω
-
Frequency
β
-
Propagation constant of the mode
n g2 (x, y, z )
-
K
-
Spatial frequency of the grating
Δn 2
-
Index modulation of the grating
Γ
-
Coupling coefficient
r
-
Radius of the core of FBG
a
-
Radius of the cladding of FBG
l
-
Length of the grating
R
-
Reflectivity of the grating
n2
-
Kerr coefficient
δng(z)
-
Periodic index variation inside the grating
n2I
-
Nonlinear index change
n
-
Average refractive index of the medium
ε(z)
-
Perturbed permittivity
E f ,b ( z , t )
-
Refractive index variation along the fiber
Forward and backward propagating waves
xiv
κ
-
Coupling between the forward and backward propagating
waves in the FBG
ki
-
Incident wavevector
K
-
Grating wavevector
kf
-
Wavevector of the scattered radiation
neff
-
Effective refractive index of the fiber core at free space center
wavelength
Δn
-
Amplitude of the induced refractive index perturbation formed
in the core of the fiber
z
-
Distance along the fiber in longitudinal axis
R( l,λ)
-
Reflectivity
λ
-
Wavelength
Ω
-
Coupling coefficient
Δk
-
Detuning wavevector
K
-
Propagation constant
Mp
-
Fraction of the fiber mode power contained by the fiber core
V
-
Normalized frequency of the fiber
nco
-
Core radius
ncl
-
Cladding radius
λw
-
Irradiation wavelength
ϕ
-
Intersecting beams
Λg
-
Period of the grating
Λpm
-
Period of the phase mask
Λg
-
Period of fringes
λuv
-
UV wavelength
N
-
Number of grating
Punperturbed
-
Unperturbed polarization
Pgrating
-
Perturbed polarization
μ
-
Transverse mode number
êz
-
Unit vector along the propagation direction z
δ μυ
-
Kronecker’s delta
xv
r
E
r
H
r
D
v
B
-
Electric field vectors
-
Magnetic field vectors
-
Displacement vectors
-
Flux density
c
r
E (z, t )
-
Speed of light
-
Electric field
ω0
-
Central frequency
k0
-
Wavenumber
P0
-
Total power inside the grating
ef
-
Forward propagating modes
eb
-
Backward propagating modes
Γs
-
Self Phase Modulation
Γx
-
Cross-phase modulation effects
C
-
Constant of integration
δˆ
-
Detuning parameter
V(A0)
-
Potential energy distribution in a FBG structures while the
light propagating through the grating structures
xvi
LIST OF APPENDICES
APPENDIX
A
TITLE
PAGE
The transmission spectrum to monitor the growth of
79
fiber grating during FBG fabrication using phase mask
technique
B
Characteristics of fabricated FBGs based on the
81
transmission spectrum
C
MatLab coding of potential energy distribution in
83
Bragg grating
D
MatLab coding for optimizing photon trapping under the
85
effects of nonlinear parameters, α, β, γ and θ in an FBG
E
Matlab coding of potential well insertion of θ factor
87
when soliton propagates in FBG
F
MatLab coding for higher order disturbance factor under
multi-perturbation factor
89
CHAPTER 1
INTRODUCTION
1.1
Introduction
A Fiber Bragg Grating (FBG) is a periodic variation of the refractive index of
the core in the fiber optic along the length of the fiber as shown in Figure 1.1. The
principal property of FBGs is that they reflect light in a narrow bandwidth that is
centered abour the Bragg wavelength, λB which is given as (A. Orthonos and K.
Kalli, 1999)
Figure 1.1: Illustration of Fiber Bragg Grating (R. Kashyap, 1999)
2
λB = 2Neff Λ
(1.1.)
where Λ is the spatial period (or pitch) of the periodic variation and Neff is the
effective index for light propagating in a single mode fiber.
FBGs are simple intrinsic devices that are made in the fibre core by imaging
an interference pattern through the side of the fibre (Meltz et. al, 1989). FBGs have
all the advantages of an optical fibre, such as electrically passive operation,
lightweight, high sensitivity with also unique features for self-referencing and
multiplexing capabilities. This gives them a distinct edge over conventional devices
(Nahar Singh et. al, 2006). Therefore, FBGs in optical fibers have a wide range of
applications, such as for sensors, dispersion compensators, optical fibre filters, and
all-optical switching and routing (T. Sun et. al,2002). An UV laser source is used to
form FBG’s in fiber optics either through internal writing (Hill et. al, 1978) or
external writing technique (A. Orthonos and K. Kalli, 1999). In this study, the novel
idea of using soliton is introduced for FBG.
Solitons are particle-like waves that propagate in dispersive or absorptive
media without changing their pulse shapes and can survive after collisions. Various
types of optical soliton phenomenon have been studied extensively in the area of
nonlinear optical physics. These includes the nonlinear Schr edinger solitons in
dispersive optical fibers, spatial and vortex solitons in photorefractive material,
waveguides and cavity solitons in resonators (Y. S. Kivshar and G. P. Agrawal,
2003).
The first step in this study is to fabricate FBGs using conventional method.
Then the novel method of writing the gratings on FBG using soliton is introduced.
This will be studied numerically. Mathematical modelling is developed through the
first principle of derivation. Simulated result obtain will be able to characterize the
soliton waves and FBG’s. Further details about FBG and soliton history,
3
development, theory, fabrication, simulation, testing and evaluation are expounded in
this thesis.
1.2
Background of the Study
Over the last decade fiber Bragg gratings(FBG) have become the key
components for optical communications systems and sensor applications. They are
used as flexible and low cost in-line components to manipulate any part of the optical
transmission and reflection spectrum. FBG is formed by the periodic variations of
the refractive index in the fiber core. Several techniques have been established to
inscribe them with UV-lasers (R. Kashyap, 1999). However, these technologies are
limited to photosensitive fiber core material, which are unsuitable for high power
applications.
Only recently modifications have been demonstrated in a non
photosensitive fiber but at the expense of longer exposure times (K. W. Chow et. al,
2008).
1.3
Problem Statement
The main motivation of this research is to pursue the novel idea of using
optical soliton writing in Fiber Bragg Gratings. First, the FBGs are fabricated using
the Excimer UV Laser conventional method. For the soliton writing, distribution of
potential energy equations has been derived based on coupled-mode theory.
Simulation has shown the trend of photon movement along the grating in order to
obtained optical soliton.
Current method of using UV laser source could be
enhanced by introducing soliton since we know that lasers are expensive and bulky
in size. Usage of solitons gives less external interference since it only consists of
4
minimal amount of losses along the propagation regarding the properties of soliton
itself. Based on this study, the optimized parameters will be identified for inscribing
grating to fiber optics using optical soliton.
1.4
Aims and Objectives
This research aims to introduce new soliton writing in FBG. The principal
objective of this study is to investigate the novel idea of using soliton in FBG. A
mathematical model on soliton FBG writing will be developed. The equations will
be derived based on the coupled mode theory. A MatLab coding will be developed
to solve these equations.
1.5
Scope of the Study
This research starts with a literature review of FBG’s. Next the FBG’s
principle of operation, and fabrication techniques are discussed. The theory involved
in the modelling of soliton will be developed. It is based on the coupled-mode
theory including the Kerr nonlinearity, group velocity dispersion (GVD) and self
phase modulation (SPM) and simulation on soliton writing of FBG will be
performed. The conventional method of FBG fabrication process will be conducted
using the phase mask technique using Excimer UV laser source at a wavelength of
248 nm.
Results obtained from experiments, modelling and simulation will be
analysed in terms of Bragg wavelength, reflectivity and the bandwidth.
5
1.6
Research Methodology
This study covers two main areas, namely, experimental setup of FBG
fabrication, evaluation, modelling and simulation on the existence of optical soliton
in grating structure in FBG. Phase mask technique is utilized to fabricate the FBGs
in this research. The motion of a particle moving in FBG represents the pulse
propagation in the grating structure of fiber optics exhibiting the existence of optical
fiber. In order to describe the photon motion, the function of potential energy is
depicted via modelling and the simulation. Figure 1.2 shows the flow and steps
undertaken to conduct this research.
6
Literature Review on FBG’s
Fabrication of FBGs by phase mask technique
FBG experiments
¾ The measurements of FBG transmission
spectrum while inscribing the gratings
¾ The measurement of fabricated Fiber
Bragg Gratings
Modelling of optical soliton
¾ Derive equations using the Coupled-Mode
Theory (CMT)
¾ Develop and write the MatLab coding for
solving equations
Run the MatLab coding by setting several
parameters such as the value of α, β, γ and θ
Results, Analysis and Discussion
Conclusions
Figure 1.2: The flow chart for the research methodology on using optical soliton
writing in FBG.
7
1.7
Significance of the Study
This research will contribute towards the research areas of nanophotonics and
optical solitons especially in FBG writing. These lasers are complicated devices, and
additionally their use restricts significantly the possibilities to adjust pulse
parameters like its duration and shape.
Furthermore it may overcome the
disadvantages of the bulky lasers and high power requirements. The novel idea of
using soliton writing in Fiber Bragg Grating will be plausible.
1.8
Organization of the Study
Chapter 1 provides a brief introduction on the overall review of the research
background, work undertaken including the problem statement, objectives,
scope,significance of the study and the research outline. The literature review is
introduced in Chapter 2. Chapter 3 describes the simulations related to the modelling
of FBG according to the certain properties and characteristics. In Chapter 4, the
mathematical modelling of soliton will be shown numerically. Chapter 5 describes
the fabrication technique used and the results of the FBG experiments. Chapter 6
presents the results and discusses the parameters obtained from the fabricated FBG
through experiment and simulation. Finally, the thesis is summed up as Chapter 6
and recommendations for future work are suggested.
CHAPTER 2
LITERATURE REVIEW
2.1
Optical Soliton
Soliton was first discovered by James Scott Russell in 1834, when he first
observed that a heap of water in a canal propagated undistorted over several
kilometres (J. S. Russell, 1834).
Such waves are called solitary waves.
Mathematical models were introduced to explain the properties of solitary waves and
the inverse scattering method was developed in the 1960s (Y. S. Kivshar and G. P.
Agrawal, 2003). The term soliton was coined in 1965 to reflect the particle-like
nature of solitary waves that remained intact even after mutual collisions. In the
context of nonlinear optics, solitons are classified as either being temporal or spatial
soliton depending on whether the confinement of light occurs in time or space during
wave propagation. Temporal soliton are optical pulses that maintain their shapes.
Spatial soliton represents self-guided beams that remained confined in the traverse
direction orthogonal to the direction of propagation. Thus, soliton are pulses that
either maintain their shapes or widths as they propagate over any distance (higher
order solitons) (J. A. Buck, 2004).
9
The existence of optical solitons in lossless fiber was theoretically
demonstrated first by Hasegawa and Tappert in 1973. Bright and dark solitons
appear in anomalous and normal dispersion regime respectively (Hasegawa and
Tappert, 1973). The existence of an optical soliton in fibers is made by deriving the
evolution equation for the complex light wave envelope via the slowly varying
Fourier amplitude by retaining the lowest order of the group dispersion. This lower
order is taken from the variation of the group velocity as a function of light
frequency and the nonlinearity. For a glass fiber it is cubic and originates from the
Kerr effect (K. Porsezian and K. Senthilnathan, 2006). The one soliton solution of
the nonlinear Schrödinger equation is given by a sech T function which is
characterized by four parameters, the amplitude, the pulsewidth, the frequency, time
position and the phase (K. Porsezian and K. Senthilnathan, 2006). In particular, the
soliton speed is a parameter independent of the amplitude unlike the case of Kortweg
de Vries (KdV) soliton. This is important fact in the use of optical soliton as a digital
signal (A. Hasegawa, 2000). Originally in 1980, L. F. Mollenauer and his colleagues
at Bell Laboratories succeeded in observing optical soliton in fiber (L. F. Mollenauer
et. al, 1980). During the 1990’s, many other kinds of optical soliton were discovered
such as spatiotemporal solitons and quadratic solitons (Y. S. Kivshar and G. P.
Agrawal, 2003).
2.2
Coupled-Mode Theory for FBG
Several methods have been adopted to study and analyze the reflection and
transmission properties of FBG (R. Kasyhap, 2004). The pulse propagation in FBG
and its effect on Bragg grating affects the wave propagation in optical fibers can be
examined using the coupled-mode theory (CMT) and Bloch wave technique.
However, in this study we take CMT only into consideration.
10
One of the standard methods of analysis of FBG is using the coupled-mode
theory (A. Ghatak and K. Thyagarajan, 1998). According to this theory, the total
field at any value of z can be written as a superposition of the two interacting modes
and the coupling process results in a z-dependent amplitude of the two coupled
modes. It is assumed that any point along the grating within the single-mode fiber
has a forward propagating mode and a backward propagating mode. Thus the total
field within the core of the fiber is given by
Ψ (x, y, z , t ) = A(z )ψ ( x, y )e i (ωt − βz ) + B(z )ψ (x, y )e i (ωt + βz )
(2.1)
where x, y, z refers to space while t refers to variation of time, A(z) and B(z)
represents the amplitudes of the forward and backward propagating modes (assumed
to be the same order mode), ψ ( x, y ) represents the transverse modal field
distribution, ω refers to frequency and β is the propagation constant of the mode. The
total field given by Equation (2.1) has to satisfy the wave equation given by
∇ 2 Ψ + k 02 n g2 (x, y, z )Ψ = 0
(2.2)
where n g2 ( x, y, z ) represents the refractive index variation along the fiber. For an
FBG it is given by
ng2 ( x, y, z ) = n 2 ( x, y ) + Δn 2 ( x, y )sin(Kz)
(2.3)
where K = 2π / Λ represents the spatial frequency of the grating and Δ n 2 represents
the index modulation of the grating. For a uniform grating K is independent of z;
when K depends on z, such gratings are referred to as chirped gratings. However,
now we further focused on uniform gratings.
Substituting Equation (2.1) and Equation (2.2) into Equation (2.3) and
making some simplifying approximations, we can obtain the following coupledmode equations:
11
dA
= κBe iΓz
dz
and
(2.4)
dB
= κAe −iΓz
dz
where Γ = 2 β − K and κ represents the coupling coefficient given by
κ=
ωε 0
8
∫∫ψ
*
Δn 2 ( x, y )ψdxdy
(2.5)
Figure 2.1: Cross-section of an optical fiber with the corresponding refractive index
profile (R. Kasyhap, 1999)
If the perturbation in the refractive index is constant and finite only within the
core of the fiber, then
Δn 2 ( x , y ) = Δn 2 ,
=0
,
r<a
(2.6)
r>a
and we obtain
κ≈
πΔn
l
λB
(2.7)
where λ B is the Bragg wavelength and
a
l=
∫ψ
2
∫ψ
2
rdr
0
∞
0
(2.8)
rdr
12
The coupled-mode Equations (2.4) can be solved using the boundary conditions of
A (z = 0) = 1
and
B (z = L) = 0
(2.9)
where L is the length of the grating. Equation (2.9) implies that the incident wave
has unit amplitude at z = 0 and the amplitude of the reflected wave at z = L is zero
because there is no reflected wave beyond z = L. We defined the reflectivity of the
FBG by the ratio of the reflected power at z = 0 to the incident power at z = 0.
Solving the coupled-mode equations and using the boundary conditions we obtain
the reflectivity of the grating as follows:
R=
κ 2 sinh 2 (ΩL )
Γ2
Ω cosh (ΩL ) +
sinh 2 (ΩL )
4
2
(2.10)
2
where
Ω2 = κ 2 −
2.3
Γ2
4
Soliton in Fiber Bragg Grating
Soliton in fibers is formed after the exact balancing of group velocity
dispersion (GVD) arising as a combination of material and waveguide dispersion
with that of the self-phase modulation (SPM) due to the Kerr nonlinearity. Due to
this, a similar soliton-type pulse formation in Fiber Bragg Grating where the strong
grating-induced dispersion is exactly counterbalanced by the Kerr nonlinearity
through the SPM and cross-phase modulation (CPM) effects. As a result, there is a
formation of slowly travelling localized envelope in FBG structures known as Bragg
grating solitons. They are often referred to as gap solitons if their spectra lies well
within the frequency of the photonic bandgap if the frequency of incident pulse
matches the Bragg frequency. Thus based on the pulse spectrum with respect to the
13
photonic bandgap, solitons in FBG can be classified into two categories as either
Bragg grating solitons or gap solitons.
There are basically two conditions that one can determine the formation of
solitons in FBG. First is based on high intensity pulse propagation in which the
refractive index modulation is weak in FBG where nonlinear coupled-mode (NCM)
equations are used to describe a coupling between forward and backward
propagating modes.
The other conditions deals with the low intensity pulse
propagation in FBG where the peak intensity of the pulse is assumed to be small
enough so that the nonlinear index change, n2I is much smaller than the maximum
value of δn. Under the low intensity limit, the NCM equations can be reduced to the
nonlinear Schrödinger equation by using multiple scale analysis.
2.4
Pulse Propagation in FBG
Wave propagation in a linear periodic medium has been studied extensively
using coupled-mode theory. In the case of a dispersive nonlinear medium, the
refractive index is given as
n (ω, z, I ) = n (ω ) + n2 I + δn g ( z )
(2.11)
where n2 is the Kerr coefficient and δng(z) accounts for periodic index variation
inside the grating. The coupled-mode theory can be generalized to include the
nonlinear effects if the nonlinear index change, n2I in Equation (2.11) is so small
that it can be treated as a perturbation.
14
The starting point consists of solving Maxwell’s equations with the refractive
index given in Equation (2.11). When the nonlinear effects are relatively weak, we
can work in the frequency domain and solve the Helmholtz equation
~
~
∇ 2 E + n~ 2 (ω , z )ω 2 / c 2 E = 0
(2.12)
The forward and backward propagating modes in FBG due to Bragg reflection can be
described using CMT as been explained by Yariv in the distributed feedback
structure (K. Senthilnathan, 2003). As usual, the governing equations for the pulse
propagation in FBG are derived using Maxwell’s equation. In this study the focus is
on the frequency domain as the nonlinear effects are assumed to be relatively weak.
It can easily be shown that Maxwell’s equation are reduced to the following wave
equation in the form
r
∂ 2 E ε (z ) ∂ 2 E
− 2
=0
dz 2
c ∂t 2
(2.13)
where perturbed permittivity, ε ( z ) = n 2 + ε~ (z ), n 2 is the spatial average of ε~ (z ) ,
and n is the average refractive index of the medium. We consider the term ε~ (z )
with a period Λ and define k0 = π /Λ. Using the Fourier series, ε~ (z ) can be written
as
ε~(z ) = 2ε~ cos(2k 0 z )
(2.14)
This electric field inside the grating can be written as
r
r
r
E ( z, t ) = E f ( z, t )e + i (kb z −ωd t ) + Eb ( z , t )e − i (k a z −ωa t ) + ...
where
(2.15)
E f ,b ( z, t ) represents the forward and backward propagating waves,
respectively, inside the FBG structure. Now, inserting Equation (2.14) and (2.15)
into Equation (2.13) and considering that the fields E f ,b (z , t ) are varying slowly with
respect to ω 0−1 in time and k 0−1 in space, the resulting frequency domain coupled mode
equations can be written as
15
r
∂E f
r
r
n ∂E f
i
+i
+ κE b = 0
∂z
c ∂t
(2.16)
r
r
r
∂E b
n ∂E b
−i
+i
+ κE f = 0
∂z
c ∂t
In the above equations, κ represents the coupling between the forward and
backward propagating waves in the FBG. The set of Equations (2.16) are called
linear coupled-mode (LCM) equations in which the non-phase-matched terms have
been neglected. The LCM equations assume slowly varying amplitudes rather than
the electric field itself. Note that CMT is an approximate description that is valid for
shallow gratings and for wavelength close to the Bragg resonance.
2.5
Properties of Fiber Bragg Grating
2.5.1
Bragg Condition
A simple form of Fiber Bragg Grating (FBG) consists of a periodic
modulation of the refractive index in the core of a single-mode optical fiber. These
types of uniform fiber gratings, where the phase fronts are perpendicular to the fiber
longitudinal axis with grating planes have a constant grating period, Λ.
16
Grating
Broadband
spectrum
Fiber
cladding
λ
Λ
Fiber core
ki
kf
Transmitted
spectrum
Broadband spectrum
Reflected spectrum
K
Bragg Gratings
λ
λ
Reflected
spectrum
Transmitted
spectrum
Figure 2.2: A basic diagram of Fiber Bragg Grating(A. Orthonos and K. Kalli, 1999)
The Bragg condition is a manifestation of both energy and momentum
conservation.
Energy conservation requires that the frequency of the incident
radiation and the reflected radiation is the same, means
hω f = hω i
(2.17)
Momentum conservation requires that the incident wavevector, ki, plus the grating
wavevector, K, equal the wavevector of the scattered radiation, kf. This leads to an
equation in which,
ki + K = k f
(2.18)
17
where the grating wavevector, K, has a direction normal to the grating planes with a
magnitude
2π
. The diffracted wavevector is equal in magnitude, but opposite in
Λ
direction to the incident wavevector. Hence, the momentum conservation becomes
⎛ 2πneff
2⎜⎜
⎝ λB
⎞ 2π
⎟⎟ =
⎠ Λ
(2.19)
Equation (2.19) simplifies to the first-order Bragg condition
λ B = 2neff Λ
(2.20)
λB is the Bragg wavelength. This is the free space center wavelength of the input
light that will be back-reflected from the Bragg grating region). neff is the effective
refractive index of the fiber core at free space center wavelength.
2.5.2
Uniform Bragg grating reflectivity
Consider a uniform Bragg grating formed within the core of an optical fiber
with an average refractive index n0. The refractive index profile can be expressed as
⎛ 2πz ⎞
n( z ) = n0 + Δn cos⎜
⎟
⎝ Λ ⎠
Here;
(2.21)
Δn is the amplitude of the induced refractive index perturbation formed in
the core of the fiber (conventional values 10-5 to 10-3).
z is the distance along the fiber in longitudinal axis.
18
The coupled mode theory of Lam and Garside (1981), describes the reflection
properties of a Bragg grating. The reflectivity of a grating with constant modulation
amplitude and period is given by
R(l , λ ) =
Ω 2 sinh2 (sl )
Δk 2 sinh2 (sl ) + s 2 cosh2 (sl)
(2.22)
where
R( l,λ) is the reflectivity that is a function of the grating length, l and wavelength, λ,
Ω is the coupling coefficient,
π
Δk is the detuning wavevector, with Δk equals to ⎛⎜ k − ⎞⎟ and k is the propagation
λ⎠
⎝
constant. s is related to Ω via the equation (s 2 = Ω 2 − Δk 2 ) .
The coupling coefficient, Ω, for sinusoidal variation of index perturbation along the
fiber axis is given by
Ω=
πΔn
Mp
λ
(2.23)
Mp is the fraction of the fiber mode power contained by the fiber core.
Where
Assuming that the grating is uniformly written through the core, MP can be
approximated by 1-V-2, where V is the normalized frequency of the fiber.
⎛ 2π ⎞
V = ⎜ ⎟a nco2 − ncl2
⎝ λ ⎠
(
)
1
2
(2.24)
where a is the core radius, nco and ncl are the core and the cladding indices,
respectively.
At the Bragg grating center wavelength there is no wavevector
detuning and Δk is equals to zero. Therefore, the expression for the reflectivity
becomes
R(l , λ ) = tanh 2 (Ωl )
(2.25)
19
The reflectivity increases as the induced index of refraction increases. So, it can
be concluded that length of the grating increases too as the resultant reflectivity.
2.6
Photosensitivity in Optical Fiber
The refractive index variations are formed by exposure of the fiber core to an
intense optical interference pattern of ultraviolet light. The capability of light to
induce permanent refractive index changes in the core of an optical fiber has been
known for years as photosensitivity. Photosensitivity has been discovered first by
Hill et. al in 1978 at Communications Research Centre in Canada or best known as
CRC. The discovery of photosensitivity has led to techniques for fabricating Bragg
gratings in the core of optical fiber and a means for manufacturing a wide range of
FBG-based devices that have many applications especially in fiber communication
and optical sensing industries for the past three decades.
Photosensitivity in optical fibers refers to a permanent change in the index of
refraction of the fiber core when exposed to light with characteristics wavelength and
intensity that depends on the core material.
The first observation of index of
refraction changes were noticed in germanosilica fibers and were reported by Hill
and co-workers in 1978 (K. O. Hill, 1978). They described a permanent grating
written in the core of the fibers by the argon ion laser line at 488 nm launched into
the fiber.
This particular grating had very weak index modulation, which was
estimated to be of the order of 10-6 resulting in a narrow-band reflection filter at the
writing wavelength. In 1981, Lam and Garside (Lam and Garside, 1981) showed
that the magnitude of the photo-induced refractive index change depended on the
square of the writing power at the argon ion wavelength (488 nm). This suggested a
two-photon process as the possible mechanism of refractive-index change. The lack
of international interest in fiber photosensitivity at the time was attributed to the
20
effect being viewed as the phenomenon present only in this special fiber. However
in 1989, Meltz et al. showed that a strong index of refraction change occurred when a
germanium-doped fiber was exposed to the UV light close to the absorption peak of
a germania-related defect at a wavelength range of 240-250 nm (Meltz et. al, 1989).
eOxygen
Ge
Ge/ Si
Figure 2.3: Oxygen-deficient germania defects thought to be responsible for the
photosensitive effect in germania-doped silica. An electron is released on breaking
of the bond (A. Orthonos, 1997).
Figure 2.3 shows an oxygen-deficient Germania defect thought to be
responsible for the photosensitivity in germania-doped silica. The peak wavelength
of absorption of the well-known GeO defect is at ~240nm. This absorption has been
shown to bleach when exposed to UV radiation. Hand and Russel has developed a
model to explain the change in the index of refraction by relating by relating it to the
absorption changes via the Kramer-Kronig relationship (Hand and Russel, 1989).
The model proposed the breaking of the GeO defect resulting in a GeΕ′ center with
the release of an electron, which is free to move within the glass matrix until it is
retrapped.
There are also other fibers that exhibit photosensitivity phenomena such as
fibers doped by europium, cerium, and erbium:germanium but the best is fiber doped
21
with germania. The next section will describe the various techniques for including
photosensitivities in a fiber optics.
2.7
Fabrication Technique for Fiber Bragg Grating
There are various techniques used in fabricating standard and complex Bragg
grating structures in optical fibers. Briefly, Bragg gratings may be classified as
internally or externally written which basically referred to the fabrication technique
used (R. Kashyap, 1999).
2.7.1 Internal Inscription of Bragg Gratings
The internal writing technique was first demonstrated in 1978 by Hill et. al
(Hill et. al, 1978). This technique requires the use of single-frequency laser light
which two-photon absorption lies in the UV photosensitivity region of the fiber
which initiates the change in the index of refraction (A. Orthonos and K. Kalli,
1999). This technique is simple and only requires minimal experimental setup.
However, these gratings are limited to operating at a Bragg wavelength coincides
with the laser wavelength. An argon ion laser is used as the source, oscillating on a
single longitudinal mode at 514.5 nm (or 488 nm) and exposing the photosensitive
fiber by coupling light into its core. Isolation of the argon ion laser from the backreflected beam is necessary to avoid instability in the pump laser and usually the
fiber is placed in a tube for thermal isolation. The incident laser light interferes with
the Fresnel reflection (approximately 4% from the cleaved end of the fiber) to
initially form a weak standing wave intensity pattern within the core of the fiber.
The high-intensity points alter the index of refraction in the photosensitive fiber
22
permanently.
Thus, a refractive index perturbation having the same spatial
periodicity as the interference pattern is formed. Reflectivity may only be achieved
for gratings having a long length due to the small index of refraction change induced
via this method. Figure 2.4 basically shows the typical experimental setup for
internal inscription of Bragg gratings.
x 50 objective
Power
meter
Optical fiber
enclosed in quartz
tube
Absorber
50 %
M1
Variable
Attenuator
Position sensor
Power
meter
Rigid Quartz
Clamp
Spring steel
Positioner
Single-mode
Argon laser
Figure 2.4: Schematic of original apparatus used for recording Bragg gratings in
optical fibers. A position sensor monitored the amount of stretching of the Bragg
grating as it was strain-tuned to measure its very narrow-band response (K. O. Hill
and G. Meltz, 1997).
23
2.7.2
External Inscription of Bragg Gratings
Inscribing Bragg gratings in optical fibers is a formidable task.
The
requirements of a submicron periodic pattern make the stability a severe constraint
on the technique able to write Bragg gratings in optical fibers. Due to this, Bragg
gratings are inscribed using external writing techniques which overcome the
fundamental limitation of internally written gratings. These techniques could be
classified into three main groups which are interferometric techniques, point-bypoint techniques and also phase mask techniques.
Meltz and co-workers were the first to demonstrate the interferometric
fabrication technique, which known as external writing approach for inscribing
Bragg gratings into the photosensitive fibers (A. Orthonos and K. Kalli, 1999). It
utilized an interferometer that split the incoming UV light into two beams then
recombined them to form an interference pattern. The fringe pattern was used to
expose a photosensitive fiber, inducing a refractive index modulation in the core.
Bragg gratings in optical fibers have been fabricated using both amplitude splitting
and wave-front-splitting interferometers (A. Orthonos, 1997).
In an amplitude-splitting interferometer, the UV writing laser light is split into
two equal intensity beams and are later recombined after traversing through two
different optical paths.
This forms an interference pattern at the core of a
photosensitive fiber. Cylindrical lenses are normally placed in the interferometer to
focus the interfering beams to a fine line matching the fiber core. The Bragg grating
period, Λ, which is identical to the period of the interference fringe pattern, depends
on both the irradiation wavelength, λw, and the half angle between the intersecting
beams, ϕ. The period of the grating is defined by
Λ=
λw
2 sin ϕ
(2.10)
24
where λw is the UV wavelength and ϕ is the half-angle between the intersection UV
beams. The most important advantage offered by this fabrication technique is the
ability to inscribe Bragg gratings at any wavelength. This is accomplished by simply
changing the intersecting angle between the UV beams. Moreover, this technique
also offers complete flexibility for producing gratings of various length, which
allows the fabrication of wavelength narrowed or broadened gratings and unique
grating geometries such as linearly chirped gratings can be produced by using curved
reflecting surfaces in the beam delivery path. However, the main disadvantage of the
amplitude-splitting technique is its susceptibility to mechanical vibrations.
Displacements as small as submicrons in the position of mirrors, beam splitter, or
mounts in the interferometer can cause the fringe pattern to drift, washing out the
gratings.
Besides, due to long separate optical path lengths involved in the
interferometers, air currents, which affect the refractive index locally, may cause a
problem in the formation of a stable fringe pattern. In addition to the above short
comings, quality gratings can only be produced with a laser source that has good
spatial and temporal coherence with excellent output power stability.
Wave-front splitting interferometers are not as popular as the amplitude
splitting interferometers for grating fabrication. However, they have some useful
advantages over the amplitude splitting interferometer.
Two such wave-front-
splitting interferometers that have been used to fabricate Bragg gratings in optical
fibers are the prism interferometer (B. J Eggleton et. al, 1994) and the Lloyd
interferometer (A. Othonos and X. Lee, 1995). A key advantage of the wave-frontsplitting interferometer is that only one optical component is used. This greatly
reduces the sensitivity to mechanical vibrations. In addition, the short distance
where the UV beams are separated reduces the wave-front distortion induced by air
current and air differences between the two interfering beams. Furthermore, this
setup can be rotated easily to vary the angle of intersection of the two beams for
wavelength tuning. One disadvantage of this system is the limitation on the grating
length, which is restricted to half of the beam width. Another disadvantage is the
range of Bragg wavelength tunability, which is restricted by the physical
arrangement of the interferometers. That is, as the intersection angle increases, the
25
difference between beam path lengths increases.
Therefore, the beam coherence
length limits the Bragg wavelength tunability.
2.7.3 Point-by-point writing technique
The point-by-point technique for fabricating Bragg gratin is accomplished by
inducing a change in the index of refraction a step at a time along the core of the
fiber. Each grating plane is produced separately by a focused single pulse from an
excimer laser. A single pulse of UV light from an excimer laser. A single pulse of
UV light from an excimer laser passes through a phase mask containing a slit. A
focusing lens images the slit onto the core of the optical fiber from the side and the
refractive index of the core in the irradiated fiber section increases locally. The fiber
is then translated through a distance Λ corresponding to the grating pitch in a
direction parallel to the fiber axis and the process is repeated to form the grating
structure in the fiber core. Essential to the point-by-point fabrication technique is a
very stable and precise submicron translational system.
The main advantage of the point-by-point writing technique lies in its
flexibility to alter the Bragg grating parameters. It is because the grating structure is
built up a point at a time, variations in grating length, grating pitch, and spectral
response can easily be incorporated. Chirped gratings can be produced accurately
simply by increasing the amount of fiber translation each time the fiber is irradiated.
The point-by-point method allows for the fabrication of spatial-mode converters and
polarization-mode converters or rocking filters that have gratings periods, Λ, ranging
from tens of micrometers to tens of milimeters. Because the UV pulse energy can be
varied between points of the induced index change, the refractive-index profile of the
grating can be tailored to provide any desired spectral response.
26
One disadvantage of the point-by-point technique is that it is a tedious
process. Because it is a step-by-step procedure, this method requires a relatively
long time process. Errors in the grating spacing due to the thermal effects and/or
small variations in the fiber’s strain can occur. This limits the gratings to a very
short length. Typically, the grating period required for first-order reflection at 1550
nm is approximately 530 nm.
Because of the submicrons translation and tight
focusing required, first-order 1550 nm Bragg gratings have yet to be demonstrated
using the point-by-point technique.
2.7.4
The Phase Mask Technique
One of the techniques commonly used to inscribe Bragg gratings in the core
of optical fibers utilizes a phase mask to spatially modulate and diffract the UV beam
to form an interference pattern (A. Orthonos and X. Lee, 1995). The interference
patterns induces a refractive index modulation in the core of the photosensitive
optical fiber which is placed directly behind the phase mask to form Bragg grating.
Basically the discoveries of phase mask techniques is gaining over the
interferometric and point-point by methods of writing Bragg gratings due to its
simplicity and reduced mechanical sensitivity (A. Orthonos and X. Lee, 1995). The
phase mask is made from flat slab of silica glass which is transparent to ultraviolet
light (Hill et. al, 1997).
Generally, phase masks may be formed either
holographically or by electron beam lithography .
27
θm/2
θm/2
Figure 2.5: Schematic design of the diffraction of an incident beam from a phase
mask
Figure 2.5 shows that the UV radiation at normal incidence to the phase mask
and diffracted radiation is split into m = 0 and ± 1 order. The interference pattern at
the two fiber of two beams of order ± 1 brought together has a period of the grating
Λg related to the diffraction angle θm/2 by
Λ g = λUV / 2 sin (θ m / 2) = Λ pm / 2
(2.11)
where Λpm is the period of the phase mask, Λg is the period of fringes and λuv is the
UV wavelength. The period of grating etched in the mask is determined by the
required Bragg wavelength λB for the grating in the fiber, yielding
Λ g = Nλ B / 2neff = Λ pm / 2
where N > 1 is an integer indicating the number of grating.
(2.12)
CHAPTER 3
EXPERIMENTAL SETUP
3.1
Introduction
This chapter will discuss thoroughly on the research methodology methods
that are involved in this study. Detailed explanations will be given under several
sections such as the experimental setup for FBG fabrication and also the methods
that have been used in executing the mathematical modelling and simulation for the
optical soliton.
3.2
Experimental Setup of Fiber Bragg Grating Fabrication
The FBG fabrication experimental setup consists of a KrF Excimer Laser
(248 nm), mask aligner, tunable laser source (TLS) and an optical spectrum analyzer
(OSA). TLS provides the broadband light source which pass through the optical fiber
while the OSA plays a critical role in the demodulation to detect the fiber grating
growth and obtain the relevant transmission or reflection spectrum. The fabrication
29
system was setup on the vibration isolated table to reduce the mechanical vibration
that will disturb the fabrication process.
EXCIMER LASER
248 nm
TUNABLE LASER
SOURCE
The fiber optic
undergo the writing
process to performs
FBG
Grating Image
MASK ALIGNER
CCD (to show the
diffraction order)
OPTICAL SPECTRUM
ANALYZER
Figure 3.1:Schematic diagram of Fibre Bragg Grating fabrication experimental setup
30
Before the fiber is placed on the platform, the jacket of the section where the
grating is supposed to be written should be removed. For a photosensitive fiber,
which has a standard diameter of 125 micron, the fiber jacket should be removed by
a cleaver or stripper. For other types of fibers, a mixture of 50% dichloromethane
and 50% acetone is used. Alternatively, one could use notrometers as paint stripper
but it took a longer time.
For certain jacket materials, the percentage of
dichlorometane and acetone in the mixture need to be changed. When the fiber is
placed on the platform, a slight strain is applied to ensure that the fiber is straight. It
is worthy to note that the naked photosensitive fiber should be cleaned thoroughly
with acetone or alcohol before placing on the platform. Otherwise, the UV beam
ablate any reminiscence of the jacket and might cause damage to the expensive phase
mask.
The fiber is connected to the OSA and TLS. This real time growth of the
FBG is monitored with an OSA. It is necessary to clean all the optical elements in
the mask aligner such as the reflecting mirror, cylindrical lens, phase mask and
quartz plate with compressed nitrogen gas. Any residual dust could absorb UV light
and thus reduce the efficiency of the fabrication process. Thus rendering the FBG
produced to be inefficient in terms of reflectivity and transmission.
The excimer laser needs about 8 to 10 minutes to warm up. In order to ensure
that the energy status of the excimer laser is in the operating range, several laser
pulses with energies of 100 to 130 mJ at 20 to 30 kV voltage supply are tested in a
closed tube. If the excimer laser output dropped below the operation range, a new
filling and a fine tune on the optical alignment of the laser pulse is required. (Lambda
Physik, 2003)
After the optical alignment has been completed, the excimer laser is tuned to
the pulsed mode. A number of pulses are input into the laser controller and the
grating writing process can then take place. The laser beam from the excimer laser
enters the mask aligner and hit the fiber through the phase mask after passing
31
through some mirrors and lens. The schematic experimental setup to monitor the
growth of fiber Bragg grating is shown in Figure 3.1. The growth of the grating in
terms of the centre wavelength and reflectivity is monitored by using an OSA. The
UV light that passes through the cylindrical lens is focused linearly onto the fiber
core. The growth of the fiber grating in terms of the transmission spectrum is
monitored with a digital stopwatch. Light from a TLS is launched into the core of
the fiber at one end and monitored with the OSA at the other end. The writing
process is stopped when the desired characteristics of the grating are achieved. With
the fabrication of each subsequent gratings the time is recorded at the same point of
reference.
3.2.1
KrF Excimer Laser
Excimer lasers take their name from the excited state dimmers from which
lasing occurs. The excimer gas is a dimeric gas consisting of two phases (Lambda
Physik, 2003). Most important are the excimer gases composition of a rare gas and
halides, such as Argon Fluoride (ArF), Krypton Fluoride (KrF), Xenon Fluoride
(XeF).
The COMPex system uses these excimer gases as the lasing medium.
Depending on their composition these excimer gases produce intense Ultraviolet
(UV) light on distinct spectral lines between 193 nm and 351 nm. COMPex laser
devices are designed to emit laser light pulses (Lambda Physik, 2003).
The
COMPex laser device uses a gaseous material as an active lasing medium, which
contained in its laser tube.
The electrons in this laser-active medium are pumped, to an excited state by
an energy source thus producing the stimulated emission.
The external source
provides the photons to emit the stored energy in the form of photons. The photons
thus emitted travel in step with the stimulating photons and, in turn, impinge on other
excited atoms to release more photons. The optical resonator normally consists of
32
two mirrors which are placed on two sides of the active laser medium.
Light
amplification is achieved as the photons move back and forth between the two
mirrors, triggering further stimulated emissions. Part of the intense, directional, and
monochromatic laser light finally leaves the resonator through one of the mirrors,
which is partially reflective. COMPex laser devices have these two mirrors attached
to the rear and front side of the laser tube.
Stimulation of the active lasing medium for emission of laser light for
population inversion uses an electric discharge which is integrated to the laser tube.
The amount of energy needed for the electrical discharge requires high voltages.
Therefore COMPex laser devices are equipped with a high voltage power supply.
For the control of the laser beam energy COMPex laser devices are equipped with
an energy monitor. The electrical energy for each laser pulse is stored in an array of
capacitors, which are supplied by the high power voltage supply. When a laser pulse
is needed, an electronic switching using a thyratron, enables the capacitors to be
discharged. The electrical energy stored in the capacitors is then transferred to the
laser-active medium via an electrical discharge between a set of electrodes
The internal control of the components in COMPex laser devices is achieved
by a built-in laser control device, the communication interface. KrF excimer laser is
shown in Figure 3.2 and followed by the functional design of the COMPex laser
system in Figure 3.3
33
Figure 3.2: KrF excimer laser device
Communication
interface
Capacitor
array
C
B
Energy
monitor
High voltage
power supply
A
D
Laser Tube
Vacuum
pump
Key:
A – Thyratron
B – Front mirror (partly reflective)
C – Rear mirror (high reflective)
D - Shutter
Figure 3.3: Functional design of the COMPex laser system (Lambda Physic, 2003)
34
3.2.2
Mask Aligner Overview
Mask Aligner in Figure 3.4 is designed to align the laser beam before it
irradiates the photosensitive optical fiber. The system consist of an exposure unit,
which contains a manual beam attenuator, plano-convex cylinder lens, two planoconcave spherical lenses, a mask holder and a CCD camera. The optical system is
designed to transport the beam from the laser onto a phase mask, which is held in
removable holder. The holder can also hold an optical fiber below and in close
proximity to the mask and an aperture above the phase mask for limiting the
exposure length on the fiber.
The optical fiber that is exposed to the laser is viewed for alignment purposes
using a CCD camera based viewing system. Alignment procedure of the mask
aligner is important in order to fabricate the FBG successfully. First, the laser output
key must be switched on. Then, the attenuator unit control is turned to minimum
transmission. Next, the front panel of the exposure unit is removed. Then, lasing is
initiated at a low power with repetition rate of 1Hz and the shutter on the control box
is opened. The laser beam will pass centered through the shutter and hits the first
turning mirror such that the beam is central on the input aperture of the attenuator
unit was adjusted. The beam using the second turning mirror is adjusted to centre the
beam on the third turning mirror. This is then followed by the fiber jig placed on its
three-point mounting. The third turning mirror is used to align the laser beam to be
positioned central on the aperture plate. Finally, the position of the cylinder lens is
adjusted, so that the beam is precisely centred on the aperture plate. This beam is
now aligned and focused. Figure 3.4 shows the optical components of mask aligner.
35
Plano-concave
cylinder
Shutter
Planoconvex
cylinder
lens
Planoconvex
Attenuator
Focusing lens
Phase mask
Figure 3.4: Optical components of mask aligner
Plano-concave cylinder
lens (1st turning mirror)
Shutter
Attenuator
Plano-convex cylinder
lenses (2nd and 3rd turning
mirrors)
Focusing
lens
Plano-convex
spherical lens
Fiber
Phase
Figure 3.5: Schematic diagram on propagation of light in mask aligner
36
3.2.3
Phase Mask
Phase masks are corrugated circular pieces of fused silica. A phase mask is
placed into a phase mask holder as shown in Figure 3.6. Each phase mask has
different pitch known as periodicity on the corrugated ridges on its surface. In this
research, a uniform period rectangular phase mask with a period of 1070.22 nm is
used. This pitch determines the value of wavelength of Bragg grating that will be
fabricated. The phase mask that has been used in this study is applicable for an
operating wavelength of 248 nm.
Phase mask
Figure 3.6: Phase mask holder
3.2.4
Tunable Laser Source Overview
MG9638A wavelength tunable laser sources enable the output of any
wavelength.
It can also sweep out an output wavelength in a specific range.
Furthermore, it enables one to specify a laser output level and select the successive
laser output level and the modulation laser output either via internal or external
modulation.
This laser source is suitable for the measurement of wavelength
37
characteristics of an optical device and perform experiments with specific
wavelengths.
The MG9638A wavelength tunable laser source provides the GPIB and
RS-232C as a remote interface.
Combining them with a computer and other
measuring instruments (optical spectrum analyzer, etc.) enables automatic
measurement and synchronous measurement.
Figure 3.7 shows the MG9638A
tunable laser source that have been used thoroughly in this research.
Optical Connector
for 2nd output
Power switch
Contrast knob
Laser output
ON/OFF key
Front panel
Optical Connector
for main output
Figure 3.7: Tunable Laser Source (MG9638A)
3.2.5
Optical Spectrum Analyzer
An Anritsu Corporation Optical Spectrum Analyzer (OSA) model MS9710B
is chosen throughout the experiment in order to monitor the waveform of the
fabricated fiber Bragg grating. The wavelength range which this particular OSA
38
possessed was from 600 to 1750 nm. Optical levels in this wavelength range can be
modulated with a maximum resolution 0.07 nm. The level measurement range is -90
to +10 dBm and this can be increased to +20 dBm by using the internal attenuator.
The measured data and waveform can be saved to a floppy disk in the
MS9710B data format, MS-DOS text format, or MS-Windows bitmapped format.
The text and bitmapped files can easily incorporated into popular word-processor and
spread sheet applications.
Figure 3.8 shows the Anritsu Corporation Optical
Spectrum Analyzer model MS9710B. Basically, in this study, the wavelength being
consider is range from 1500 nm – 1600 nm.
Figure 3.8: The Optical Spectrum Analyzer
CHAPTER 4
FIBER BRAGG GRATING MODEL OF POTENTIAL ENERGY
DISTRIBUTION
4.1 Coupled-mode Theory
In order to derive the coupled-mode equations, effects of perturbation have to be
included, assuming that the modes of the unperturbed waveguide remain unchanged.
The derivation begins with the wave equation
∇ 2 E = μ 0ε 0
∂2E
∂2P
μ
+
0
∂t 2
∂t 2
(4.1)
Assuming that wave propagation takes place in a perturbed system with a dielectric
grating, the total polarization response of the dielectric medium described in Equation
(4.1) can be separated into two terms, unperturbed and perturbed polarization, as
P = Punperturbe d + Pgrating
where
(4.2)
40
P unperturbe d = ε 0 χ (1) E μ
(4.3)
and χ is the linear susceptibility. Thus, Equation (4.1) becomes,
∇ 2 Eμt = μ0ε 0ε r
∂2
∂2
+
E
μ
Pgrating , μ ,
μt
0
∂t 2
∂t 2
(4.4)
where the subscripts refer to the transverse mode number,μ. For the present, the nature
of the perturbed polarization is driven by the propagating electric field and is due to the
presence of the grating.
Substituting the modes in Equation (4.5) in Equation (4.4) provides the following
relationship:
[
]
ρ =∞
1 μ =1
i (ωt − β μ z )
i (ωt − β ρ z )
Et = ∑ Aμ (z )ξ μt e
dρ
+ cc + ∑ ∫ Aρ (z )ξ ρt e
ρ
=
0
2 μ =0
(4.5)
where ξ μt and ξ ρt are the radial transverse field distribution of the μth guided and ρth
radiation modes, respectively.
[
] ∑∫
⎡ 1 μ =1
i (ω t − β μ z )
∇ 2 ⎢ ∑ A μ (z )ξ μ t e
+ cc +
2
⎣ μ =0
μ 0ε 0ε r
∂2
∂t 2
[
ρ =∞
ρ =0
A ρ (z )ξ ρ t e
] ∑∫
⎡ 1 μ =1
i (ω t − β μ z )
+ cc +
⎢ ∑ A μ (z )ξ μ t e
⎣ 2 μ =0
ρ =∞
ρ =0
(
i ωt − β ρ z
)
⎤
dρ ⎥ −
⎦
A ρ (z )ξ ρ t e
(
i ωt − β ρ z
)
⎤
∂2
d ρ ⎥ = μ 0 2 Pgrating
∂t
⎦
,μ
(4.6)
By ignoring the coupling to the radiation models for the moment allows the left-handed
side of Equation (4.7) to be expanded.
41
∇ 2 E = μ0ε 0ε ij
∂2E
∂t 2
(4.7)
where ε ij is the permittivity tensor and subscripts ij refers to two laboratory frame
polarization.
Further simplification is possible in weak coupling by applying the slowly
varying envelope approximation (SVEA). This requires that the amplitude of the mode
change slowly over a distance of the wavelength of the light as
∂ 2 Aμ
∂z
<< β μ
2
∂Aμ
(4.8)
∂z
so that
∇ 2 Et =
1
2
⎡
∑ ⎢⎣− 2iβ μ
∂Aμ
∂z
ξ μt e
(
i ωt − β μ z
)
− β 2 μ Aμξ μt e
(
i ωt − β μ z
)
⎤
+ cc ⎥
⎦
(4.9)
Expanding the second term in Equation (4.6), noting that ω2μ0ε0ε r = β 2μ and combining
with Equation (4.9), the wave equation simplifies to
∂Aμ
⎡
⎤
∂2
i (ωt − β μ z )
−
+
=
i
β
ξ
e
cc
μ
Pgrating,t
∑ ⎢ μ ∂z μt
0
⎥
∂t 2
⎣
⎦
(4.10)
The t on the polarization Pgrating,t reminds that the grating has a transverse profile.
Multiplying both sides by ξ μ* and integrating over the wave-guide cross-section leads to
+∞ +∞
∂Aμ
⎡
⎤
∂2
i (ωt − β μ z )
*
ξ μt ξ μt e
− iβ μ
+ cc ⎥dxdy = ∫ ∫ μ 0 2 Pgrating ,t ξ μ*t dxdy (4.11)
∑
∫
− ∞ ∫− ∞ ⎢
−∞ −∞
∂t
∂z
μ =0
⎣
⎦
μ =1 + ∞ + ∞
42
1 +∞ +∞
1 ⎡ β μ ⎤ +∞ +∞
eˆ z ⋅ ξ μt × ξυ*t dxdy = ⎢
ξ μt ⋅ξυ*t dxdy = δ μυ
∫
∫
2 −∞ −∞
2 ⎣ ωμ0 ⎥⎦ ∫−∞ ∫−∞
[
]
(4.12)
Where êz is a unit vector along the propagation direction z , δ μυ is the Kronecker’s delta
and is unity for μ = υ but zero for μ ≠ υ .
Applying the orthogonality relationship of Equation (4.12) directly results in
∂Aμ i (ωt − β μ z )
+∞ +∞
⎤
⎡
∂2
+ cc⎥ = ∫ ∫ μ 0 2 Pgrating ,t ξ μ∗t dxdy
∑
⎢ − 2iωμ0 ∂z e
μ =0 ⎣
⎦ −∞ −∞ ∂t
μ =1
(4.13)
Equation (4.13) is fundamentally the wave propagation equation which can be
used to describe a variety of phenomena in the coupling of modes. Equation (4.13)
applies to a set of forward- and backward- propagating modes; it is now easy to see how
mode coupling occurs by introducing forward- and backward- propagating modes. The
total transverse field may be described as a sum of both fields, not necessarily composed
of the same mode order:
(
)
Et =
1
i (ωt + β μ z )
Aυ ξυt ei (ωt − βυ z ) + cc + Bμξμt e
+ cc
2
Ht =
1
i (ωt + β μ z )
Aυ Hυt ei (ωt − βυ z ) + cc − Bμ H μt e
− cc
2
(
(4.14)
)
(4.15)
Here the negative sign in the exponent signifies the forward- and the positive
sign the backward- propagating mode, respectively. The modes of a waveguide form an
orthogonal set, which in an ideal fiber will not couple unless there is a perturbation.
Using Equation (4.14) and (4.15) in Equation (4.13) leads to
⎤
i +∞ +∞ ∂2
⎡ ∂Aυ i (ωt − βυ z )
⎤ ⎡ ∂Bμ i (ωt + β μ z )
e
+
cc
−
e
+
cc
=
+
Pgrating ,tξ μ* ,υt dxdy (4.16)
⎥
∫
⎢ ∂z
⎥ ⎢ ∂z
− ∞ ∫− ∞ ∂t 2
ω
2
⎣
⎦ ⎣
⎦
43
4.2 Derivation of Nonlinear Coupled Mode Equations (NLCM)
To derive the pulse governing equation oppositely signed Kerr coefficient, we start
with Maxwell’s equations
r
∇⋅D = 0
r
∇⋅B = 0
r
r
∂B
∇× E = −
∂t
(4.17)
r
r r ∂D
∇× H = J +
∂t
(
)
r
r r
r
r
where E and H are electric and magnetic field vectors D = ε 0 E + P is the displacement
r
v
vector, and B = μ 0 H is the flux density. It can easily be shown that Maxwell’s
equations are reduced to the following wave equation in the form
(
∂ 2 E n z, E
−
c2
∂z 2
where c =
1
ε 0 μ0
2
) ∂ Er = 0
2
∂t 2
(4.18)
r
is the speed of light and E( z, t ) is the electric field.
The electric field inside the grating can be written as
r
r
r
E ( z , t ) = E f ( z, t )e+ i (k 0 z −ω0t ) + Eb ( z, t )e− i (k 0 z −ω0t ) + ...
(4.19)
44
where E f ,b ( z, t ) represents the forward- and backward propagating waves, respectively,
inside the FBG structure.
nln =
The central frequency is given by ω0 = ck0 / nln
,
n01 + n02
where n01 and n02 are the linear refractive indices of the two different
2
media and the wave number is given by k0 = 2π nln / λ . Note that peak reflectance
occurs at the center of forbidden gap and can be written as λ = 2nln Λ . In other words,
resonance in the first bandgap occurs when k = 2k0 . Now substituting Equation (4.19)
into Equation (4.18), obtained the first term as
r
r
r
2
v
∂E f
∂ 2 E ⎛⎜ ∂ E f
2
=
+
−
2
ik
k
E
f
0
0
∂z 2 ⎜⎝ ∂z 2
∂z
r
r
r ⎞ −i (k 0 z +ω0 t )
⎞ i (k z −ω t ) ⎛ ∂ 2 Eb
∂
E
2
b
⎟e 0 0 + ⎜
⎟
−
−
2
ik
k
E
(4.20)
0
0
b ⎟e
⎜ ∂z 2
⎟
∂
z
⎝
⎠
⎠
Assume that the forward and backward field components E f ,b ( z, t ) are varying slowly
with respect to ω0−1 in time and k0−1 in space, viz
r
∂E f ,b
∂t
v
<< ω0 E f ,b
,
r
∂E f ,b
∂z
v
<< k 0 E f ,b
(4.21)
By applying the slowly varying envelope approximation, Equation (4.20) transforms to
r
r
r
∂E f
∂ 2 E ⎛⎜
2
2
ik
k
E
=
−
0
0
f
∂z 2 ⎜⎝
∂z
r
r ⎞
⎞ i ( k z −ω t ) ⎛
E
∂
⎟e 0 0 + ⎜ − 2ik0 b − k02 Eb ⎟e− i (k 0 z +ω0t )
⎜
⎟
⎟
∂z
⎝
⎠
⎠
(4.22)
Similarly by applying the slowly varying envelope approximation, the second term in
Equation (4.18) can be written as
r
r
⎡⎛ ∂E f
r
∂2E
⎜ 2i
ω
ω
E
=
−
+
⎢
0
0
f
⎜
∂t 2
⎣⎢⎝ ∂t
r
r
⎞ i ( k z −ω ) ⎛ ∂E f
⎟e 0 0 t + ⎜ 2i
+ ω0 E f
⎟
⎜ ∂t
⎠
⎝
⎞ −i (k z +ω ) ⎤
⎟e 0 0 t ⎥ (4.23)
⎟
⎠
⎦⎥
45
(
n z, E
2
)= n
2
ln
2
+ nln E + 2 n0 k cos kz + 2 n2 k E cos kz
(4.24)
where k is defined as wavenumber.
(
n z, E
2
)= a
∞
0
+ 2 ∑ an cos (2πnf 0 z ) + bn sin (2πnf 0 z )
(4.25)
n =1
where f0 is the fundamental frequency.
Using Equations (4.24) and (4.23), the second term in Equation (4.25) can be
written as
( )∂ E =
n zE
c2
2
2
∂t 2
r
2
∂
k0
E
2
2
n 2 + 2nln nnl E + 2nln n0 k + 2nln n2 k E eikz + e −ikz
ω0 nln c ln
∂t 2 (4.26)
[
(
)(
)]
Neglecting all the higher order terms in n 2 k , the solutions becomes
r2
n⎛⎜ z, E ⎞⎟ 2 r
⎝
⎠ ∂ E = k 0 n 2 + 2n E 2 + 2 n + n E 2 + e i 2 k 0 z + e −i 2 k 0 z
ln
0k
2k
nl
2
c
∂t 2 ω0 n ln c
r
r
⎤
⎡⎛ ∂E f
v ⎞ i (k0 z −ω0t ) ⎛ ∂E f
v ⎞
− ω0 ⎢⎜ 2i
+ ω0 E f ⎟e
+ ⎜ 2i
+ ω0 E f ⎟e −i (k0 z +ω0t ) ⎥
⎟
⎜
⎟
∂t
∂t
⎥⎦
⎢⎣⎜⎝
⎠
⎝
⎠
[
(
)(
)]
(4.27)
r2
The intensity term E in the above equation can be expressed in terms of the field
components as
r2 r r
r 2 r 2 r r
r r
E = E ⋅ E ∗ = E f + Eb + E f Eb*ei 2 k0 z + Eb E *f e−i 2 k0 z
(4.28)
46
Using Equation (4.28), Equation (4.27) can be simplified to
(
)
r
r
n z, E 2 ∂ 2 E
k
=− 0
2
2
∂t
c
c
r
r
r
⎡
∂E f
+ 2n0 kω0 Eb +
⎢nlnωo E f + 2inln
∂t
⎢⎣
r 2 r 2
r r v
v
2nnl ⎛⎜ E f + Eb ⎞⎟ω0 E f + 2nnl E f Eb*ω0 Eb
⎝
⎠
r
r
r r
v
v
2
2
+ 2nn2 k ⎛⎜ E f + Eb ⎞⎟ω0 Eb + 2nn2 k Eb E *f ω0 E f
⎝
⎠
r
r r* v
r
k0 ⎡
∂Eb
i ( k 0 z −ω 0 t )
+ 2nn2 k E f Ebω0 E f ]e
− ⎢nlnω0 Eb + 2inln
c ⎣
∂t
r
r 2 r 2
r
r r
v
+ 2n0 kω0 E f + 2nnl ⎛⎜ E f + Eb ⎞⎟ω0 Eb + 2nnl Eb E*f ω0 E f
⎠
⎝
r r v
r 2 r 2
v
+ 2n2 k ⎛⎜ E f + Eb ⎞⎟ω0 E f + 2nn2 k E f Eb*ω0 Eb
⎠
⎝
r
v
+ 2n2 k Eb E *f ω0 Eb ]e − i (k0 z +ω0t )
(4.29)
Now combining Equation (4.20) and Equation (4.29) and using them in Equation (4.18)
and collecting all the terms having coefficient of e i (k 0 z − ω 0 t ) , obtained
r
r
r
∂E f
k0
− k E f + [nln ωo E f + 2inln
2ik0
c
∂t
∂z
r
r 2 r 2
r
r r v
+ 2n0 kω0 Eb + 2nnl ⎛⎜ E f + Eb ⎞⎟ωo E f + 2nnl E f Eb*ω0 Eb
⎝
⎠
r 2 r 2
v
+ 2nn2 k ⎛⎜ E f + Eb ⎞⎟ω0 Eb
⎝
⎠
r r
r r
r
v
+ 2nn2 k Eb E *f ω0 E f + 2n2 k E f Eb*ω0 E f ] = 0
r
∂E f
2
0
(4.30)
After simplification, the above equation becomes
r
r
r
r 2 r 2 r
c ∂E f
nln ∂E f
i
+i
+ n0k Eb + nnl ⎛⎜ E f + Eb ⎞⎟ E f
⎝
⎠
ω0 ∂z
ω0 ∂t
r 2r
r 2 r 2 r
r 2r
+ nnl Eb E f + n2 k ⎛⎜ E f + Eb ⎞⎟ Eb + n2 k E f Eb
⎝
⎠
(4.31)
47
r r
+ n2 k E 2f Eb* = 0
Similarly by grouping all the terms e − i (k 0 z +ω 0 t ) , obtained
r
r
r
r 2 r 2 r
c ∂Eb
nln ∂Eb
−i
+i
+ n0 k E f + nnl ⎛⎜ E f + Eb ⎞⎟ Eb
⎝
⎠
ω0 ∂z
ω0 ∂t
r 2r
r 2 r 2 r
r 2r
+ nnl E f Eb + n2 k ⎛⎜ E f + Eb ⎞⎟ E f + n2 k Eb E f
⎝
⎠
r 2 r*
+ n2 k Eb E f = 0
(4.32)
To normalize Equation (4.31) and Equation (4.32), ξ = ω0 z / c and τ = ω0t / nln . Now
the normalized coupled-mode equations can be written as
i
r
∂E f
∂ξ
+i
r
∂E f
r
r 2
r 2 r
+ n0 k Eb + nnl ⎛⎜ E f + 2 Eb ⎞⎟ E f
⎝
⎠
∂τ
r 2
r 2 r
r r
+ n2 k ⎡⎢⎛⎜ 2 E f + 2 Eb ⎞⎟ Eb + E 2f Eb* ⎤⎥ = 0
⎠
⎣⎝
⎦
r
r
r
r 2
r 2 r
∂E
∂E
− i b + i b + n0 k E f + n nl ⎛⎜ E b + 2 E f ⎞⎟ E b
⎝
⎠
∂ξ
∂τ
(4.33)
r 2
r 2 r
r r
+ n2 k ⎡⎢⎛⎜ 2 Eb + 2 E f ⎞⎟ E f + Eb2 E *f ⎤⎥ = 0
⎠
⎣⎝
⎦
Equation (4.33) represents the nonlinear pulse propagation in a nonlinearity
management system.
If we consider a medium consisting of periodically varying
nonlinear refractive indices in which the Kerr coefficients of two adjacent layers are
oppositely signed, that is nnl1 = nnl 2 , under this condition nnl = 0 and n2k ≠ 0 . Under this
circumstance, Equation (4.33) becomes
i
r
∂E f
∂ξ
+i
r
∂E f
r
r 2 r 2 r
r r
+ n0 k Eb + n2 k ⎡⎢⎛⎜ 2 E f + Eb ⎞⎟ Eb + E 2f Eb* ⎤⎥ = 0
⎠
∂τ
⎣⎝
⎦
48
r
r
r
r 2 r 2 r
r r
∂Eb
∂Eb
−i
+i
+ n0 k E f + n2 k ⎡⎢⎛⎜ 2 Eb + E f ⎞⎟ E f + Eb2 E *f ⎤⎥ = 0
⎠
∂ξ
∂τ
⎣⎝
⎦
(4.34)
By considering the medium having the same Kerr coeffiecients throughout the
periodic structure, then nnl1 = nnl 2 . Based on the following four paramaters, it is clear
that nnl ≠ 0 and n2k = 0 .
nln =
nnl =
(4.35)
nnl1 + nnl 2
,
2
n0k =
n2 k =
n01 + n02
,
2
n01 − n02
π
nnl1 − nnl 2
π
,
.
Under the above mentioned condition. Equation (4.33) reduces to
r
∂E f
r
∂E f
r
r 2
r 2 r
+ n0 k Eb + nnl ⎛⎜ E f + 2 Eb ⎞⎟ E f = 0
⎝
⎠
∂ξ
∂τ
r
r
r
r 2
r 2 r
∂Eb
∂Eb
−i
+i
+ n0 k E f + nnl ⎛⎜ Eb + 2 E f ⎞⎟ Eb = 0
⎝
⎠
∂ξ
∂τ
i
+i
(4.36)
The set of Equation (4.36) are the well known Nonlinear Coupled Mode Equations for
the medium having the same positive Kerr coefficients (G. P. Agrawal, 2001).
49
4.3 Derivation of Potential Energy Distribution in Fiber Bragg Grating
In the presence of Kerr nonlinearity, using CMT, the NLCM equations can be
written as
r
∂E f
r
r
r
r r
n ∂E f
i
+i
+ κEb + Γs E 2f + 2Γx Eb2 E f = 0
∂z
c ∂t
(
)
r
r
r
r
r r
∂E b
n ∂E b
−i
+i
+ κE f + Γs E b2 + 2Γx E 2f E b = 0
∂z
c ∂t
(
)
(4.37)
where Ef and Eb are the slowly varying amplitudes of forward and backward propagating
waves, n is the average refractive index, and Γs and Γx are SPM and Cross-Phase
modulation terms. In Equation (4.37) the material and waveguide dispersive effects are
not included due to the dispersion arising from the periodic structures dominates the rest
near Bragg resonance condition. Noted that the above NLCM equation are valid only
for wavelengths close to the Bragg wavelength.
Now, by substituting the stationary solution to the above coupled-mode
equations by assuming
E ( f ,b ) (z , t ) = e( f ,b ) (z )e − iδct / n
ˆ
(4.38)
where δˆ is the detuning parameter.
Using the stationary solution in Equation (4.37), we obtain
i
de f
−i
dz
(
+ δˆe f + κeb + Γs e f
(
2
+ 2 Γx e b
deb ˆ
2
+ δeb + κe f + Γs eb + 2Γx e f
dz
2
)e
2
=0
f
)e
b
=0
(4.39)
50
Equation (4.39) represents the time-independent light transmission through the grating
structure, and it has been extensively investigated by many researchers. The NLCM
equations are non-integrable in general. But in a few cases, NLCM equations have exact
analytical solutions representing the solitary wave solutions. However, Christoudolides
and Joseph (D. N. Christodoulides and R. I. Joseph, 1989) has obtained the soliton
solution to the NLCM equation, known as slow Bragg soliton, under the integrable
massive Thirring model where the SPM and detuning parameter is set to zero. After
using suitable transformation, it is used in nonlinear optics as a simple model to explain
the self-induced transparency effect. Using the Stokes parameters they derived the
relation of energy density for the stationary solution for the NLCM equation in terms of
the Jacobi elliptic function (C. M. de Sterke adn J. E. Sipe, 1994).
There are some possible interesting soliton-like solutions apart from these
stationary solutions. In the fiber Bragg grating, these soliton-like solution for the NLCM
equations carry a lot of practical importance.
4.4
Modelling of Optical Soliton using NLCM
Wave propagation in optical fibers is analyzed by solving Maxwell’s Equation
with appropriate boundary conditions. In the presence of Kerr nonlinearity, using the
coupled-mode theory, the nonlinear coupled mode equation is defined under the absence
of material and waveguide dispersive effects. The dispersion arising from the periodic
structure dominates near Bragg resonance conditions and it is valid only for wavelengths
close to the Bragg wavelength. By substituting the stationary solution to the coupled
r
mode equation and by assuming E ± ( z , t ) = e± ( z )e −iδct / n , we obtain
51
i
de f
dz
de
i b + δˆeb + κe f
dz
and
(
+ (Γ e
+ δˆe f + κeb + Γs e f
s
2
2
b
+ 2Γx eb
+ 2Γx e f
2
)e = 0
,
)e = 0
f
2
(4.40)
b
Equation (4.40) represents the time-independent light transmission through the
gratings structure where ef and eb are the forward and backward propagating modes κ
n − n02 ⎞
represents n0k , ⎛⎜ n0 k = 01
⎟ where n01 is the core refractive index and n02 is the
π
⎝
⎠
cladding refractive respectively, Γs represents Self Phase Modulation and Γx represents
Cross-phase modulation effects.
In order to explain the formation of Bragg soliton, consider the Stokes parameter
since it will provide useful information about the total energy and energy difference
between the forward and backward propagating modes. In this study, the following
Stokes parameter are considered where
A0 = e f
2
2
+ eb ,
A1 = e f eb* + e *f eb ,
(
A2 = i e f eb* − e *f eb
A3 = e f
2
− eb
)
and
(4.41)
2
with the constraint A02 equals to the sum of A12 + A22 + A32 . In the FBG theory, the
nonlinear
coupled-mode
P0 = A3 = e f
2
(NLCM)
equation
requires
that
the
total
power
− eb inside the grating is constant along the grating structures.
2
Rewriting the NLCM equations in terms of Stokes parameter gives
52
dA0
= −2κA2 ,
dz
dA1
= 2δˆA2 + 3ΓA0 A2 ’
dz
dA2
= −2δˆA1 − 2κA0 − 3ΓA0 A1 ,
dz
dA3
=0
dz
(4.42)
In Equation (4.42), we drop the distinction between the SPM and cross
modulation effects. Hence Equation (4.42) becomes 3Γ = 2Γx + Γs . It can be clearly
shown that the total power, P0 (=A3) inside the grating and is found to be constant
meaning it is conserved along the grating structure. In the derivation of the anharmonic
oscillator type equation, it is necessary to use the conserved quantity. This is obtained in
3 2
the form δˆA0 + ΓA0 + κA1 = C , where C is the constant of integration and δˆ is the
4
detuning parameter. Equation (4.42) can further be simplified to
d 2 A0
− αA0 + β A02 + γA03 = 4δˆC
2
dz
[
(4.43)
]
9 2
where α = 2 2δˆ 2 − 2κ 2 − 3ΓC , β = 9Γδˆ and γ = Γ . Equation (4.43) contains all
4
the physical parameter of the NLCM equation. Physically, α represents the function of
detuning parameters, phase modulation factors (SPM and CPM).
β represents the
function of phase modulation factors (SPM and CPM) and the detuning parameters.
Lastly, γ represents the phase modulation factor (SPM and CPM). In general, α, β and γ
are the oscillation factors.
53
4.5
Modelling of potential energy distribution in Fiber Bragg Grating
structures.
In order to describe the motion of a particle moving within a classical
anharmonic potential, we have the solution in the form of
A02
A03
A04
+β
+γ
V ( A0 ) = −α
2
3
4
(4.44)
It represents the potential energy distribution in a FBG structures while the light is
propagating through the grating structures.
4.6
Multi Perturbation of Potential Energy Photon in Fiber Bragg Grating
4.6.1
External perturbation of potential energy
Consider the case in Equation (4.41) with a set of constraints which is governed by
∞
φ (e ) = ∑ A0 n . The perturbation factor then is
n =0
d 2 A0
″
= φe
2
dz
n =0
If Equation (4.43) is accumulated using the external perturbation then
(4.45)
54
∞
φ ′′ n =0 + ∑ C m A0 n = ψ
n =0
m =1
where ψ is a function of f (δ, C, Cm,) and Cm = [α , β , γ ,...]
The value of m = 2n for n = 1, 2, 3, …, m = 2n + 1 for n = 0, 1, 2, …
C is constant and C = (C1, C2, C3, …, Cm). The value of C is linear to A0 but not to V.
Equation (4.45) can then be modified by
∞
V ( A0 ) = ∑ C m A0n
(4.46)
m =1
n =0
Equation (4.46) represents the complete potential energy distribution in the Fiber Bragg
Grating structure. We believe at this juncture, the potential function is modified from
Conti and Mills (C. Conti and S. Trillo, 2001). Using well-known Duffing oscillator
type equation, analogically it is written as
∞
φe ″ + ∑ C m A0 n = 0
(4.47)
m =1
n =0
For multi perturbation of nonlinear parameters, two major shapes will be simplified in
the series term.
The flow charts in the subsequent chapters describe how the coupled mode
equations are solved under different conditions when soliton is used for FBG writing.
The cases examined are (i) when there is no energy disturbance (ii) effect of potential
energy disturbance factor (iii) potential energy with the highest disturbance factor.
55
4.7
Flowchart for Computational Modelling
Figure 4.1 shows simulation flow chart which considers the case when there is
no energy disturbance.
Figure 4.1: Flow chart in the case where there is no energy disturbance
56
Figure 4.2 shows the simulation flow chart with the addition of theta to the potential
energy disturbance.
difference Figure 4.2: Flow chart of simulation with potential energy disturbance factor.
57
Figure 4.3 shows the simulation flow chart with the higher order of potential energy
disturbance.
difference difference Figure 4.3: Flow chart of potential energy under multi-perturbation condition
58
The results of these computational for single and multi perturbations based on the
nonlinear parameters are presented in Section 5.3 of Chapter 5.
CHAPTER 5
RESULTS AND DISCUSSION
5.1
Introduction
This chapter presents the results of fiber Bragg grating fabrication and
simulation of optical soliton-like pulses in fiber Bragg grating. For the simulation
aspect, the discussion have been divided into three parts which covers the potential
energy distribution in fiber Bragg grating, nonlinear parametric studies of photon in
a fiber Bragg grating, external disturbance of potential energy photon in FBG.
5.2
Results of Fiber Bragg Grating fabrication
In this section, the results of the experiments to fabricate FBGs are presented.
The measurements of FBGs are observed during and after the fabrication process. A
photosensitive optical fiber (Stocker Yale PS-1550-Y3 non-hydrogenated) which has
an outer diameter (OD) ~ 125 μm, a numerical aperture (NA) ~ 0.07, GeO2
construction of 8% mol and cut-off wavelength (λc) ~ 1.36 μm is used in this
6
60
e
experiment.
Krypton fluoride, K
KrF excimeer laser is utilized thhroughout thhe
f
fabrication
process
p
of FBGs
F
using the phase mask
m
techniqque. This lasers
l
deliveers
p
pulses
at 2488 nm with duuration 30 nns, repetition rate 5 Hz, and
a pulse flu
uency (Ip) ~ 40
4
mJ/cm2. Th
m
he uniform reectangular phhase mask with
w a periodd of 1070.22 nm is used in
t study.
this
A staandard FBG
G have a periiodic refracttive index m
modulation along
a
the fibber
c
core
length that is form
med by expossure of the core
c
to an inntense opticaal interferencce
p
pattern.
Thhe forming of
o the perioodic modulaation occurs due to the defect in thhe
p
photosensitiv
ve fiber. Figgure 5.1 shoows the transsmission spectra obtainedd from opticcal
s
spectrum
an
nalyzer for fabricated
f
FB
BGs known as FBG1. For FBG2 and
a FBG3 thhe
r
results
are sh
hown in the Appendix.
Figure 5.1: The transsmission speectrum to moonitor the grrowth of fibeer grating in
FBG1
ws results off the fabricated FBG1. From thee transmissioon
Figurre 5.2 show
s
spectrum,
we
w are able too observe chharacteristicss of the fabrricated FBGss. Results aare
a
attached
in the
t Appendiix for furtheer referencess. A UV puulse energy of 70 mJ annd
6
61
a
after
exposing the fiberr for 20 minnutes, produ
uces an FBG
G with Braggg wavelenggth
1551.09 nm
m, bandwidthh 0.15 nm annd reflectivity of 30.18%
%. The chaaracteristics of
t fabricateed FBG’s aree shown in Table
the
T
5.1.
Bandw
width Dip Po
ower Bragg Wavellength,λB Dip Power
Figure
F
5.2: R
Results of faabricated FB
BG1
T reflectiv
The
vity of the sp
pectrum can be calculateed using Equuation (5.1)
−D
⎛
⎞
⎜
R = ⎜1 − 10 10 ⎟⎟ × 100%
⎝
⎠
w
where
D is the
t dip poweer in decibel..
(5.1)
62
Data obtained from the fabrication experiments are shown in the table below.
Table 5.1: Summary of the data collected for fabrication
Fabricated
UV
UV
Dip
Reflectivity
Bragg
Bandwidth
FBG
Pulse
exposure
Power
(R± 0.01)
wavelength
(Δλ± 0.1)
energy
time (t±1)
(P ±
%
(λB±0.01)
nm
(P ±
minute
0.1) dB
nm
0.1) mJ
FBG1
70
20
1.6
30.18
1551.09
0.15
FBG2
78.1
20
6.6
78.12
1551.29
0.12
FBG3
130
15
2.6
44.73
1551.66
0.14
5.3
Results for Simulation of Soliton in Fiber Bragg Grating
In this section, the results of simulation in FBG are obtained using MatLab
software R700b.
The nonlinear parametric studies on the motion of photon in
grating are described using those simulation results specifically in soliton
perspective.
5.3.1
Nonlinear Parametric Studies of Photon in a Fiber Bragg Grating
In Equation (4.44), β is not considered due to power conservation along the
propagating of this FBG structure. The qualitative aspects of the potential well will
change if the nonlinearity parameter of the wave equation is varied.
63
2
alpha=0.1
alpha=0.3
alpha=0.5
alpha=0.7
alpha=0.9
alpha=1.0
1.5
1
V(1o)
0.5
0
-0.5
-1
-1.5
-2
-4
-2
0
Ao
2
4
Figure 5.3: The motion of photon in double well for different values of α
Figure 5.3 depicts the double-well potential under Bragg resonance condition
where β = 0, γ = 0.23 and α is varied from 0.1 to 1.0. Photon with power of less than
the total power, P0 will only travel inside the well unless their energy exceeds the
energy level. This would allow the photon to move outside the well.
0
-0.2
-0.4
V(1o)
-0.6
-0.8
-1
-1.2
-1.4
0.1
0.2
0.3
0.4
0.5
0.6
alpha
0.7
0.8
0.9
1
Figure 5.4: The optimized point of the double well potential for different values of α
64
Figure 5.4 explains the optimized point for various values of α. The graph
clearly shows that the optimized points decreased exponentially when values of α are
increased. However, when α >>1, the trend of the curve is no longer valid since it
turns into an almost linear relationship.
1.5
gamma=0.13
gamma=0.23
gamma=0.33
gamma=0.43
gamma=0.53
1
V(1o)
0.5
0
-0.5
-1
-4
-2
0
Ao
2
4
Figure 5.5: Under Bragg resonance condition the system possesses double well
potential for γ = 0.13 to 0.53.
Figure 5.5 shows the motion of photon in double well potential under
different values of gamma for the Bragg resonance condition of γ from 0.13 to 0.53.
Note that the increment of gamma which is between 0.53<γ<1 will reduce the double
well potential to a single well potential.
65
-0.05
-0.1
-0.15
V(1o)
-0.2
-0.25
-0.3
-0.35
-0.4
-0.45
-0.5
0.1
0.2
0.3
0.4
0.5
0.6
gamma
0.7
0.8
0.9
1
Figure 5.6: The optimized point of the double well potential when γ = 0.1 to 1.0
Figure 5.6 describes the optimized point for varies of gamma, γ. Parametric
variation of gamma produces a potential energy function which increases
exponentially. However, when γ >> 1, a plateau is observed. This shows that it is not
valid if γ→∞.
5.3.2
External Disturbance of Potential Energy Photon in Fiber Bragg Grating
Figure 5.7 depicts the motion of photon in a potential well which changes
when few nonlinear parameter are taken into account as shown is Equation (4.8).
Photon is trapped by the α parameter which is depicted by legend V. When α is too
large, the potential well produces A0 increases and have a wider of double well. The
γ parameter is shown by X legend. When γ is large, the potential well produces A0
increases. Suppose that the source is imposed to FBG than initial power is used to
generate the particles. It shows that double well potential well is not symmetrical
and the potential energy will decrease at certain region and is shown in Figure 5.7 in
6
66
l
legend
Y. The
T other eff
ffect is the disturbance at
a potential eenergy by legend Z wheere
p
photon
cannnot be trappped symmetrrically. It will
w tend to equilibrium
m but it is not
n
s
stable
wheree the photon leave the pootential curve as a lossess.
metric functiion, we can describe it aas follows. The
T change in
In terrms of param
α will affectt the dip of the potential well. If α is
i approximately too sm
mall, the shappe
o the potenntial well devvelop into a single poten
of
ntial well. T
The occurrennce of β effeect
i the motioon of photonn gives an efffect to the negative
in
n
reggion which means
m
A0 < 0.
T effect of
The
o γ also sh
hows that thhe width off potential w
well will deecrease if we
w
i
increase
the value of γ. Therefore
T
iff we increasee the value oof gamma, we
w can assum
me
t
that
the photon will be localized annd can be traapped. In adddition, anotther nonlineear
f
factor
θ, it will
w change the shape off potential well
w rapidly. We could say that if we
w
i
include
the existence
e
off θ, the shape of potentiaal well becoomes chaoticc. The photoon
d
does
not onlly move witthin a certaiin region thaat is known as the potenntial well annd
m
moving
freely.
X
V
Y
Z
F
Figure
5.7: The motion
n of photon inn potential well
w for α = 00.9, β = 0.3,θ = 0.09 andd
γ is varies frrom 0.3 to 0.9.
67
Figure 5.8 shows the effect of external disturbance, θ. It shows that by
increasing the value of θ, it will also affect the change in γ. In other words, the
negative part of Ao will be influenced it potential energy. The different values of γ
will produce different profiles. By simulating, we assumed that the increased of γ
value from 0.3 to 0.9, the curve will be positioned within the region C. The peak of
V for each γ from 0.3 to 0.9 describes θ increases linearly and large gradient
compare to the initial V. This represents that potential energy cannot maintain
photon to be trapped and equilibrium state if γ is relatively small.
18
gamma = 0.3
gamma = 0.9
16
14
12
V(10)
10
C
8
6
4
2
0
‐2
0
0.05
0.1
0.15
0.2
θ
0.25
0.3
0.35
Figure 5.8: The effect of theta,θ to γ and shape of the potential well of the photon.
In Figure 5.9 it can be shown that by increasing the value of β the potential
energy of the potential well will be reduced. The highest potential drop occurs
within the range of β, 0.2 to 0.3. If the disturbance is large, it requires a high
potential energy to maintain the photon especially for γ = 0.7. In other words,
increasing the γ value will affect the shape of the potential well in terms of the
potential energy. It will affect the equilibrium of the potential well and therefore the
trapped photons are no longer being trapped or localized.
6
68
0.9
0
0.8
0
0.7
0
V‐Vo
0.6
0
0.5
0
0.4
0
gamm
ma = 0.9
0.3
0
gamm
ma = 0.7
0.2
0
0.1
0
0
0
0.1
1
0.2
0.3
0.4
0.5
0.6
β
Figure
F
5.9: The
T disturbaance to the potential enerrgy by β facttor.
5
5.3.3
Motiion of photo
on due to exxternal enerrgy perturbaation in pottential well
Figurre 5.10 deppicts the mootion of photon in potenntial well which
w
changes
w
when
few nonlinear
n
parrameter is taake into acccount as described by Equation
E
(4.88).
T
There
are thheoretically some com
mments in thhis figure. Photon is trapped
t
by α
p
parameter
which
w
is depicted by leggend V. Whhen α is tooo large, the potential
p
weell
p
produces
A0 increases and
a have a wider
w
doublle well. γ pparameter iss shown by X
l
legend.
Wh
hen γ is largee, the potenttial well produces A0 inccreases. Supppose that thhe
s
source
is im
mposed to FBG than initial power is used to ggenerate thee particles. It
s
shows
that double welll potential w
well is not symmetric
s
aand potentiaal energy will
d
decrease
at the certain region in leegend Y. Th
he other efffect is the perturbation of
p
potential
eneergy by legeend Z where photon cannot be trappped symmetrrically. It will
t
tend
to equillibrium but it
i is not stablle where it can
c go for lossses.
69
X V
Y Z Figure 5.10: The motion of photon in potential well for α = 0.9, β = 0.3, θ = 0.09
and γ is varies from 0.3 to 0.9.
The change in the parametric function can be easily described in terms of α, β
and γ. The dip in the potential well will transform with a single potential well when
α is extremely small. β affects the photon motion which in turn will effect to the
negative region of the potential well when A0 < 0.The effect of γ shows that the
width of potential well will decrease if we increased the value of γ. The photon will
be trapped when γ is increased. The shape of the potential well can be controlled by
a nonlinear factor θ. Changes in the value of θ leads to a chaotic behaviour of the
potential well. Under these condition the photon can either move within certain
specific regions or act as a free particle. Thus Figure 5.10 illustrates the single
perturbation as described by the nonlinear parameter, θ .
When multi perturbations are considered then the photon will be trapped and
untrapped for various conditions.
As depicted in Figure 5.11, it explains the
extrapolation of the graph if more factors of perturbation added into Equation (4.10).
The addition of parametric factors by the higher odd number, Figure 5.11 (a) will
allow the photon to move in a well, and Figure 5.11 (b) will lead the photon to be
70
untrapped and higher even number. It is clearly shown in the graphs that as n>>∞ ,
the value of |A0| will remain constant in the range of -2<A0<2. However, when the
value of V(0) is equal to zero, there are many possibilities of A0, meaning the exact
value of intentsity, A0 to trap the photon is difficult to determine in this condition. If
the parametric factor considered is too large then we may conclude that the photon is
in indifferent state part of the equlibrium
71
Figure 5.11 (a)
Figure 5.11 (b)
Figure 5.11: The disturbance factor that affect the shape of the potential well of the
motion of photon.
The stationary solutions of Equation (4.10) are applied neither for bright nor
dark soliton solution since the dominant parametric factor in contributing A0 is
unknown. However, from Equation (4.10) we have
72
A0 = A0 (C m , z )
(9)
Under these conditions, the frequencies with photonic band gap keep forming
an envelope after the exact balancing at grating-induced dispersion with nonlinearity.
Either decay or increase, the forward and backward waves are transferred by Bragg
reflection process. The total energy of the system, potential energy function is equal
to zero having multi perturbation which is -1<A0<1 and if V→∞, A0 = 2.
5.4 Summary
We have discussed thoroughly results of fabricated FBGs.
We have
succesfully fabricate three fiber Bragg gratings which are FBG1 with Bragg
wavelength, 1551.09 nm, FBG2, 1551.29 nm and the third FBG3 with Bragg
wavelength equal 1551.66 nm. In this chapter, we have successfully simulated the
motion of photon in grating structure in terms of the potential energy distribution. It
must be noted that nonlinear parametric effects such as α, β and γ influences the
photon trapping in the Bragg grating. Furthermore, a new nonlinear parameter that is
θ of study the effect to the potential energy distribution when considering multiperturbations for photon trapping. It is clearly shown that these nonlinear parameters
play an important role that governs the existence of optical soliton in fiber Bragg
grating.
CHAPTER 6
CONCLUSION
6.1
Introduction
A series of experiments on fabrication FBG and modelling and simulation on
the existence of Bragg soliton in FBG have been performed. The results of the
experiments are discussed in Chapter 5. This chapter presents the summary and
conclusions that can be made in this study. The recommendations for future work
are also given in this chapter.
6.2
Conclusion
Three FBGs with Bragg wavelength approximately 1551.09 nm, 1551.29 nm
and 1551.66 nm have been successfully produced using the conventional FBG
fabrication technique which consists of the KrF excimer laser, associated optics and
the phase mask aligner. The phase mask technique is first used to inscribe grating
into the fiber’s core.
It is simple, flexible, steady and effective conventional
74
fabrication technique.
Their reflectivities are 30.18%, 78.12% and 44.73%
respectively. In this studies the FBGs are produced using different pulse energies
and durations and are further examined in terms of Bragg wavelength, reflectivity
and wavelength.
It can be concluded from the results that the spectrum of fabricated FBG
broadens and the grating becomes saturated after long exposure to the UV laser light.
Its reflectivity rise rapidly at the beginning and then gradually reaches a plateau
afterwards. However, the reflectivity does not increase when the grating become
saturated even after exposing the fiber to the laser pulses. Any increase in the
average refractive index of the fiber has an effect of the Bragg wavelength shift
towards a longer wavelength. Extended exposure to laser pulses, does not shift the
peak wavelength towards higher wavelength region.
The novel idea of using soliton in FBG writing shows that the motion of a
particle moving in FBG represents the pulse propagation in the grating structure of
FBG. This indicates the existence of optical soliton. It is described in terms of the
photon motion and as a function of potential energy.
Results obtained show that
photon can be trapped by nonlinear parameters of potential energy which are
identified as α, β, γ and θ.
In the first simulation results of nonlinear parametric studies of photon in a
FBG, we have successfully shown that the changes of nonlinearity parameter will
affect the motion in the potential well. This will influence the existence of Bragg
soliton in a fiber Bragg grating.
In the second simulation results, we have added new nonlinear parameter
which is known as θ. We have preset the value of θ, α and β and vary the value of γ
over certain range. From the results, it is depicted that the factor θ will affect the
75
shapes of potential well. If the existence of θ is taken into account, the potential well
profile becomes chaotic.
The simulation data are then expanded on the multi perturbation of potential
energy photon in FBG. It shows that the change of α affect the dip of the potential
well. The occurrence of β effect in the motion will affect the soliton propagation in
the region for A0<0. The effect of γ shows that the width of potential well will
decreased if the value of γ is increased. However, another nonlinear factor, θ will
turns the shape of potential well rapidly which necessities the multi perturbation
studies. When multi perturbations are considered, the photon will be trapped and
entrapped under various conditions. From this, we may conclude the addition of
nonlinear parametric factors by the higher odd number will allow the photon to move
in a well instead to be entrapped with the higher even number. It is found from this
study that the potential well under Bragg resonance condition is not symmetrical and
conserved. The higher perturbation series representing the potential well is much
indifferent of the equilibrium in both odd and even nonlinear parametric factor of n.
As a conclusions, these studies have successfully shown that it is plausible to use
soliton for FBG writing and the solitons can be controlled by manipulating the
parametric effects which are α, β, γ and θ.
6.3
Future Work
The model developed in this study can be further extended by optimizing the
nonlinear parameters in terms of the potential energy, soliton trapping and its
applications as optical tweezers. The model can be tested by developing compact
miniature FBG inscribing system using laser diodes
76
REFERENCES
Andreas Orthonos and Kyriacos Kalli. (1999).“Fiber Bragg Gratings –
Telecommunications and Sensing”. Artech House Boston, London.
D. Boardman and K. Xie. “Spatial bright-dark soliton steering through waveguide
coupling,” Optical and Quantum Electronics 30 (1998) 783-794.
Andreas Orthonos and Xavier Lee. “Novel and Improved Methods of Writing
Bragg Gratings with Phase Masks”. IEEE Photonics Technology Letters, Vol.
7, No. 10, October 1995.
C. Conti and S. Trillo, “Bifurcation of gap solitons through catastrophe theory”.
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7
79
A
APPENDIX
XA
The transm
mission specctrum to moonitor the growth
g
of fib
ber grating during FBG
G
fab
brication ussing phase mask
m
techniique
Fabricated FB
BG1
BG2
Faabricated FB
880
Faabricated FB
BG3
8
81
A
APPENDIX
XB
Characcteristics of fabricated FBGs based
d on the traansmission spectrum
s
Faabricated FB
BG1
Fabricated FB
BG2
882
Faabricated FB
BG3
83
APPENDIX C
MatLab coding of potential energy distribution in Bragg grating
clc
clear
alpha = (0.1:0.1:0.9);
number = length (alpha);
beta = 0;
gamma = 0.23;
gamma1 = (0.13:0.1:0.53);
number2 = length (gamma1);
A = (-2:0.01:2);
Ao = A';
number1 = length(Ao);
for i=1:number;
for j=1:number1;
V(i,j) = (alpha(i)*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
for k = number2;
V1(i,j,k) = (alpha(i)*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma1(k)*((Ao(j)^4)/4
));
end
end
end
hold on
%figure (1)
plot(Ao,V(1,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.1')
xlabel('Ao')
ylabel('V(1o)')
%figure (2)
plot(Ao,V(2,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.2')
xlabel('Ao')
ylabel('V(1o)')
%figure (3)
plot(Ao,V(3,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.3')
xlabel('Ao')
ylabel('V(1o)')
%figure (4)
84
plot(Ao,V(4,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.4')
xlabel('Ao')
ylabel('V(1o)')
%figure (5)
plot(Ao,V(5,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.5')
xlabel('Ao')
ylabel('V(1o)')
%figure (6)
plot(Ao,V(6,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.6')
xlabel('Ao')
ylabel('V(1o)')
%figure (7)
plot(Ao,V(7,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.7')
xlabel('Ao')
ylabel('V(1o)')
%figure (8)
plot(Ao,V(8,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.8')
xlabel('Ao')
ylabel('V(1o)')
%figure (9)
plot(Ao,V(9,:))
axis([-2 2 -0.35 0.35])
title('at alpha = 0.9')
xlabel('Ao')
ylabel('V(1o)')
hold off
85
APPENDIX D
MatLab coding for optimizing photon trapping under the effects of nonlinear
parameters, α, β, γ and θ in an FBG
clc
clear
alpha1 = 0.1;
% alpha2 = 0.2;
alpha3 = 0.3;
% alpha4 = 0.4;
alpha5 = 0.5;
% alpha6 = 0.6;
alpha7 = 0.7;
% alpha8 = 0.8;
alpha9 = 0.9;
alpha10 = 1.0;
beta = 0;
gamma = 0.23;
A = (-6:0.1:6);
Ao = A';
number=length(Ao);
for j=1:number
V1(j) = (alpha1*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
%
V2(j) = (alpha2*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
V3(j) = (alpha3*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
%
V4(j) = (alpha4*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
V5(j) = (alpha5*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
%
V6(j) = (alpha6*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
V7(j) = (alpha7*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
%
V8(j) = (alpha8*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
V9(j) = (alpha9*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
V10(j) = (alpha10*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4));
end
plot(Ao,V1,'.',Ao,V3,'^',Ao,V5,'*',Ao,V7,':',Ao,V9,'+',Ao,V10,'p')
axis([-4 4 -2.0 2.0])
xlabel('Ao')
86
ylabel('V(1o)')
grid on
legend('alpha=0.1','alpha=0.3','alpha=0.5','alpha=0.7','alpha=0.9',
'alpha=1.0',2)
pause
MV=[min(V1) min(V3) min(V5) min(V7) min(V9) min(V10)]
alpha=[alpha1 alpha3 alpha5 alpha7 alpha9 alpha10]
plot(alpha,MV)
xlabel('alpha')
ylabel('V(volt)')
grid on
box on
87
APPENDIX E
Matlab coding of potential well insertion of θ factor when soliton
propagates in FBG
clc;
clear;
alpha = 0.5;
beta = 0;
gamma = 0.23;
theta1 = 0.1;
%theta2 = 0.2;
theta3 = 0.3;
%theta4 = 0.4;
theta5 = 0.5;
%theta6 = 0.6;
theta7 = 0.7;
%theta8 = 0.8;
theta9 = 0.9;
theta10 = 1.0;
A = (-6:0.1:6);
Ao = A';
number=length(Ao);
for j=1:number
V1(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta1*((Ao(j)^5)/5));
%V2(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta2*((Ao(j)^5)/5));
V3(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta3*((Ao(j)^5)/5));
%V4(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta4*((Ao(j)^5)/5));
V5(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta5*((Ao(j)^5)/5));
%V6(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta6*((Ao(j)^5)/5));
V7(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta7*((Ao(j)^5)/5));
%V8(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta8*((Ao(j)^5)/5));
V9(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta9*((Ao(j)^5)/5));
88
V10(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta10*((Ao(j)^5)/5));
end
%plot(Ao,V1,Ao,V2,Ao,V3,Ao,V4,Ao,V5,Ao,V6,Ao,V7,Ao,V8,Ao,V9,Ao,V10);
plot(Ao,V1,Ao,V3,Ao,V5,Ao,V7,Ao,V9,Ao,V10);
axis([-5 5 -10 20])
xlabel('Ao')
ylabel('V(1o)')
legend('theta1=0.1','theta2=0.3','theta3=0.5','theta4=0.7','theta5=0
.9','theta6=1.0',2)
grid on
89
APPENDIX F
MatLab coding for higher order disturbance factor under
multi-perturbation factor
Part A (odd number)
clc;
clear;
alpha = 0.5;
beta = 0;
gamma = 0.23;
theta = 0.09;
b=0.5;
c1 = 0.1;
c2 = 0.2;
c3 = 0.3;
c4 = 0.4;
c5 = 0.5;
c6 = 0.6;
c7 = 0.7;
c8 = 0.8;
c9 = 0.9;
c10 = 1.0;
A = (-6:0.1:6);
Ao = A';
number=length(Ao);
for j=1:number
V1(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c1*((Ao(j)^7)/7));
V2(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c2*((Ao(j)^7)/7));
V3(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c3*((Ao(j)^7)/7));
V4(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c4*((Ao(j)^7)/7));
V5(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c5*((Ao(j)^7)/7));
V6(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c6*((Ao(j)^7)/7));
V7(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c7*((Ao(j)^7)/7));
90
V8(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c8*((Ao(j)^7)/7));
V9(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c9*((Ao(j)^7)/7));
V10(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b*((Ao(j)^6)/6))+ (c10*((Ao(j)^7)/7));
end
plot(Ao,V1,Ao,V2,Ao,V3,Ao,V4,Ao,V5,Ao,V6,Ao,V7,Ao,V8,Ao,V9,Ao,V10);
axis([-5 5 -5 5])
xlabel('Ao')
ylabel('V(1o)')
grid on
91
Part B (even number)
clc;
clear;
alpha = 0;
beta = 0;
gamma = 0.53;
theta = 0;
b1 = 0.01;
b2 = 0.02;
b3 = 0.03;
b4 = 0.04;
b5 = 0.05;
b6 = 0.06;
b7 = 0.07;
b8 = 0.08;
b9 = 0.09;
b10 = 0.1;
A = (-6:0.1:6);
Ao = A';
number=length(Ao);
for j=1:number
V1(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b1*((Ao(j)^6)/6));
V2(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b2*((Ao(j)^6)/6));
V3(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b3*((Ao(j)^6)/6));
V4(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b4*((Ao(j)^6)/6));
V5(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b5*((Ao(j)^6)/6));
V6(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b6*((Ao(j)^6)/6));
V7(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b7*((Ao(j)^6)/6));
V8(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b8*((Ao(j)^6)/6));
V9(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b9*((Ao(j)^6)/6));
V10(j) = (alpha*((Ao(j)^2)/2))+(beta*((Ao(j)^3)/3))+(gamma*((Ao(j)^4)/4))+
(theta*((Ao(j)^5)/5))+ (b10*((Ao(j)^6)/6));
92
end
plot(Ao,V1,Ao,V2,Ao,V3,Ao,V4,Ao,V5,Ao,V6,Ao,V7,Ao,V8,Ao,V9,Ao,V10);
axis([-5 5 -100 100])
xlabel('Ao')
ylabel('V(1o)')
legend('b1=0.1','b2=0.2','b3=0.3','b4=0.4','b5=0.5','b6=0.6','b7=0.7
','b8=0.8','b9=0.9','b10=1.0',2)
grid on
93
PUBLISHED PAPERS
Saktioto, Haryana Mohd Hairi, Mohamed Fadhali, Preecha P. Yupapin and Jalil
Ali. Nonlinear Parametric Study of Photon in a Fibre Bragg Grating. Frontier
Research in Nanoscale Science and Technology, Elsevier, Science Direct,
Physics Procedia. Physics Procedia Vol2(1) pg. 81-85 (2009).
Saktioto, H.M. Hairi, J. Ali, M.Fadhali. Nonlinear Parametric Studies of Photon
in a Fibre Bragg Grating. IEEE Industrial Electronics and Applications (ISIEA
2009) Proceeding, October 4-6, 2009, Kuala Lumpur Malaysia.
N.F.Hanim, S.Nafisah, H.M.Hairi, T.Saktioto, F.H.Suhailin, J.Ali, P.P.Yupapin,
An Experimental Investigation of Multisoliton generation Using Erbium Doped
Fiber Amplifier and a Fiber Optic Ring Resonator, Microwave and optical
Technology Letters, Vol.51, 2009 (Article in press)
Rosly A.Rahman, H.M.Hairi, Saktioto, S.Nafisah, M.Fadhali, P.P.Yupapin, J.Ali.
Motion of Photon Parametric for Bragg Resonance Condition, ESciNano
Annual Symposium 2009, The Zone Johor Bahru.
S.Nafisah, H.M.Hairi, Saktioto, J. Ali and P.P. Yupapin. Novel Design of
Multiplexed Sensors using a Dual FBGs Scheme, Microwave and Optical
Technology Letters, John Wiley & Sons. MOP-09-0967
H. M. Hairi, T. Saktioto, B. Vanishakorn, P. P. Yupapin, M. Fadhali, J.Ali. An
Investigation of Photon Trapping Stability within a Dynamic Potential Well for
Optical Tweezers Transportation Use. International Journal of Light and
Electron Optiks, OPTIK, (Elsevier)
H. M. Hairi, T.Saktioto, M. Fadhali, B. Vanishkorn, P.P. Yupapin, Jalil Ali.
Photon Trapping within a Fiber Bragg Grating using Anharmonic Potential
Energy Distribution Model, International Journal of Light and Electron
Optiks, OPTIK (Elsevier)
94
H. M. Hairi, Saktioto, M.Nafisah, M.Fadhali, B.A. Tahir, P. P. Yupapin and
J.Ali. Multi Photons Squeezing Control within Fiber Bragg Grating for
Quantum Bits Parallel Processing Operation. Journal of Electromagnetic
Waves & Application (JEMWA)
S.Nafisah, H.M.Hairi, Saktioto, J. Ali and P.P. Yupapin. Optical Multiplexed
Sensors using FBG with Classical and Quantum Measurement Self Calibration,
Microwave and Optical Technology Letters.
Haryana Mohd Hairi, Saktioto, P.P.Yupapin, M.Fadhali, J.Ali, External
disturbance of Potential Energy Photon in a fiber Bragg grating, 4th
International Conference on Experimental Mechanics (ICEM2009), Singapore,
18-20 November 2009.
H. M. Hairi, Toto Saktioto , S. Nafisah , M. Fadhali , Rabia Qindeel , Preecha P.
Yupapin , J. Ali. Multi-photons Trapping Stability within a Fiber Bragg
Grating for Quantum Sensor Use, PIERS 2010 Xi’an Progress In
Electromagnetic Research Symposium, March 22–26, 2010, Xi’an, CHINA.