design & simulation of optical fiber bragg grating pressure sensor
Fibre Bragg Grating - University of Johannesburg
Transcript of Fibre Bragg Grating - University of Johannesburg
An Object-oriented Simulation
Program for Fibre Bragg
Gratingsby
JIANFENG ZHAO
Thesis submitted in partial fulfilment of the requirement for the
degree
Master of Engineering
in
Electrical and Electronic Engineering
in the
Faculty of Engineering
at the
Rand Afrikaans University
Johannesburg
Republic of South Africa
Supervisor: Prof. P.L. Swart
October 2001
i
ABSTRACT
In recent years, many research and development projects have focused on
the study of fibre Bragg gratings. Fibre Bragg gratings have been used in the
field of sensors, lasers and communications systems. Commercial products
that use fibre Bragg gratings are available. On the other hand, in the field of
software development, object-oriented programming techniques are also
becoming very popular and powerful. The focus of this work is on solving fibre
Bragg grating problems by a simulation program with object-oriented
programming techniques.
For fibre Bragg grating problems, widely used theories and numerical
methods such as the coupled-mode theory and the transfer matrix method will
be applied in the analysis, modelling and simulation. The coupled-mode
theory is a suitable tool for analysis and for obtaining quantitative information
about the spectrum of a fibre Bragg grating. The transfer matrix can be used
to solve non-uniform fibre Bragg gratings. Two coupled-mode equations can
be obtained and simplified by using the weak waveguide approximation. The
spectrum characteristics can be obtained by solving these coupled-mode
equations.
The optical numerical libraries of fibre Bragg gratings have been built by using
object-oriented techniques. The code was realized by C++ and Object Pascal
language in the Delphi4, C++ Builder4 and Visual C++6 environment. The
compiled binary files and the code of the simulation program are available for
both the end user and program developer. This simulation program can be
used to analyze the performance of sensors and communication systems that
use fibre Bragg gratings.
Uniform, chirped, apodized, discrete phase shifted and sampled Bragg
gratings have already been simulated by using the direct numerical integration
method and the transfer matrix method. The reflected and transmitted
spectra, time delay and dispersion of fibre Bragg gratings can be obtained by
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using this simulation program. At the same time, the maximum reflectivity,
3dB-bandwidth and centre wavelength can also be obtained.
This thesis consists of three parts. The first part introduces a suitable theory
and modelling that have been used to analyze the characteristics of fibre
Bragg gratings. Secondly, the codes of the modelling are realized by the
suitable programming languages in different development environments.
Finally, this simulation program is utilized to analyse real physical problems
with fibre Bragg grating applications.
iii
ACKNOWLEDGMENTS
I would like to thank my supervisor, Professor P.L. SWART, for his patience,
support and guidance throughout this research.
I am also thankful to my parents for giving me full support during my studies.
I also thank Dr A. A. CHTCHERBAKOV for helping with C++ programming
technology.
I am grateful to Ms F. Velosa for her professional language editing of this
thesis.
Finally, I thank Rand Afrikaans University for its financial support.
CONTENTS
I
Contents
CHAPTER 1:INTRODUCTION .............................................................................. 1
1.1 AIM OF THIS PROJECT ................................................................................. 2
1.2 OVERVIEW OF THE FIBRE BRAGG GRATING ................................................... 3
1.3 PROGRAMMING TECHNIQUE......................................................................... 4
1.4 THE APPLICATION OF THE SIMULATION PROGRAM .......................................... 5
1.5 THE SCOPE OF THIS PROJECT...................................................................... 6
1.6 REFERENCES ............................................................................................. 8
CHAPTER 2:THEORY AND FUNDAMENTALS OF FIBRE BRAGG
GRATINGS............................................................................................................. 9
2.1 INTRODUCTION ......................................................................................... 10
2.2 THE COUPLED-MODE THEORY.................................................................... 14
2.3 APPLICATIONS OF FIBRE BRAGG GRATINGS................................................. 17
2.3.1 Fibre Bragg grating sensors ................................................................. 18
2.3.2 Wavelength Division Multiplexing......................................................... 21
2.3.3 Fibre grating lasers............................................................................... 22
2.4 CONCLUSION............................................................................................ 22
2.5 REFERENCES ........................................................................................... 24
CHAPTER 3: APPROACHES TO THE SIMULATION OF FIBRE BRAGG
GRATINGS........................................................................................................... 27
3.1 INTRODUCTION ......................................................................................... 28
3.2 MODELLING OF FIBRE BRAGG GRATINGS .................................................... 28
3.3 UNIFORM BRAGG GRATINGS ...................................................................... 32
3.4 THE DIRECT NUMERICAL INTEGRATION METHOD .......................................... 34
CONTENTS
II
3.4.1 The Runge-Kutta method ..................................................................... 34
3.5 THE TRANSFER MATRIX METHOD FOR UNIFORM GRATINGS ........................... 36
3.5.1 The transfer matrix method for non-uniform gratings........................... 37
3.6 CALCULATION OF THE TIME DELAY AND DISPERSION ................................... 38
3.7 CONCLUSION............................................................................................ 39
3.8 REFERENCES ........................................................................................... 40
CHAPTER 4:PROGRAMMING TECHNIQUE..................................................... 41
4.1 OBJECT-ORIENTED PROGRAMMING TECHNIQUE .......................................... 42
4.1.1 Using the object-oriented programming technique .............................. 42
4.2 PROGRAMMING LANGUAGES...................................................................... 43
4.2.1 MATLAB................................................................................................ 44
4.2.2 Object Pascal ....................................................................................... 46
4.2.3 C++ ....................................................................................................... 47
4.3 THE IMPLEMENTATION OF SIMULATION PROGRAMMING................................. 49
4.3.1 The flow chart of the design of the simulation programming ............... 49
4.3.2 The design of grating class .................................................................. 51
4.3.3 The design of a user-friendly GUI ........................................................ 51
4.3.4 Using the Bragg grating classes library................................................ 52
4.3.5 Using this simulation program.............................................................. 54
4.4 COMPATIBILITY AND PORTABILITY ............................................................... 56
4.4.1 Sharing the code between Delphi and C++ Builder ............................. 56
4.4.2 Sharing the code between C++ Builder and Visual C++...................... 57
4.4.3 Portal source code from Windows to LINUX........................................ 57
4.5 CONCLUSION............................................................................................ 58
4.6 REFERENCES ........................................................................................... 59
CHAPTER 5: CHIRPED FIBRE BRAGG GRATINGS ....................................... 60
5.1 THE PRINCIPLE OF THE CHIRPED BRAGG GRATING ...................................... 61
5.1.1 Direct integration .................................................................................. 62
5.1.2 Transfer matrix method ........................................................................ 63
5.2 THE SIMULATION RESULTS OF THE SPECTRAL RESPONSE............................. 63
CONTENTS
III
5.2.1 Linear chirped gratings with different chirp variables........................... 65
5.2.2 Linear chirped gratings with different lengths....................................... 68
5.2.3 Linear chirped gratings with different refractive index changes........... 69
5.2.4 Arbitrary chirped Bragg grating ............................................................ 70
5.3 RELATIONSHIP BETWEEN THE MAXIMUM REFLECTANCE AND THE CHIRPED
GRATING COEFFICIENTS........................................................................................ 71
5.3.1 The maximum reflectance and the “chirp parameter”.......................... 72
5.3.2 The maximum reflectance and the length of the grating...................... 73
5.3.3 The maximum reflectance and the index change ................................ 74
5.4 RELATIONSHIP BETWEEN THE 3 dB BANDWIDTH AND THE CHIRPED GRATING
COEFFICIENTS...................................................................................................... 75
5.4.1 The reflectance bandwidth and the “chirp parameter” (or chirp variable)
.......................................................................................................................76
5.4.2 The reflectance bandwidth and the length of the grating..................... 78
5.4.3 The reflectance bandwidth and the index change ............................... 79
5.5 RELATIONSHIP BETWEEN THE CENTRE WAVELENGTH AND THE CHIRPED
GRATING COEFFICIENTS........................................................................................ 79
5.5.1 The centre wavelength and the “chirp parameter” (or chirp variable).. 80
5.5.2 The centre wavelength and the index change ..................................... 82
5.5.3 The centre wavelength and the length of the grating........................... 83
5.6 DISPERSION COMPENSATION..................................................................... 84
5.6.1 Simulation results ................................................................................. 88
5.7 SENSOR APPLICATIONS ............................................................................. 94
5.8 CONCLUSION............................................................................................ 98
5.9 REFERENCES ........................................................................................... 99
CHAPTER 6: APODIZATION OF FIBRE BRAGG GRATINGS...................... 100
6.1 THE PRINCIPLE OF APODIZED GRATINGS ................................................... 101
6.1.1 Direct integration method ................................................................... 101
6.1.2 Transfer matrix method ...................................................................... 102
6.1.3 Apodization functions ......................................................................... 102
6.2 SPECTRAL RESPONSE OF APODIZED GRATINGS......................................... 104
CONTENTS
IV
6.2.1 Refractive index and spectral response ............................................. 104
6.2.2 Comparison of the properties of apodized and unapodized uniform
gratings......................................................................................................... 105
6.2.3 The apodization of linear chirped gratings with different Gauss width
parameters ....................................................................................................110
6.2.4 The apodization of linear chirped gratings with different Kaiser window
parameters ....................................................................................................110
6.3 RELATIONSHIP BETWEEN THE MAXIMUM REFLECTANCE AND THE GAUSS WIDTH
PARAMETERS ......................................................................................................112
6.4 RELATIONSHIP BETWEEN THE 3 dB BANDWIDTH AND THE GAUSS WIDTH
PARAMETERS ......................................................................................................113
6.5 DISPERSION COMPENSATION USING A LINEAR CHIRPED GRATING WITH
APODIZATION ......................................................................................................114
6.5.1 Optimization of the Gauss width parameters for dispersion
compensation................................................................................................117
6.6 CONCLUSION.......................................................................................... 121
6.7 REFERENCES ......................................................................................... 122
CHAPTER 7: OTHER APPLICATIONS OF THE SIMULATION PROGRAM. 123
7.1 SIMULATION OF PHASE-SHIFTED BRAGG GRATINGS ................................... 124
7.1.1 Principle .............................................................................................. 124
7.1.2 Direct integration ................................................................................ 124
7.1.3 Transfer matrix method ...................................................................... 125
7.1.4 Simulation results ............................................................................... 125
7.1.5 Applications ........................................................................................ 130
7.2 SIMULATION OF SAMPLED BRAGG GRATINGS............................................. 130
7.2.1 Principle .............................................................................................. 130
7.2.2 Direct integration ................................................................................ 131
7.2.3 Transfer matrix method ...................................................................... 131
7.2.4 Simulation results ............................................................................... 132
7.2.5 Applications ........................................................................................ 137
7.3 CONCLUSION.......................................................................................... 137
CONTENTS
V
7.4 REFERENCES ......................................................................................... 138
CHAPTER 8:CONCLUSION AND FUTURE WORK ........................................ 139
8.1 CONCLUSION.......................................................................................... 140
8.2 FUTURE WORK ....................................................................................... 141
8.2.1 Simulation of long period gratings...................................................... 141
8.2.2 Bragg grating simulation with the Internet.......................................... 141
8.3 REFERENCES ......................................................................................... 143
BIBLIOGRAPHY................................................................................................ 144
REFERENCE MANUAL..................................................................................... 150
INTRODUCTION
1
CHAPTER 1: INTRODUCTION
CHAPTER 1: ............................................................................... INTRODUCTION1
1.1 AIM OF THIS PROJECT ..................................................................................... 2
1.2 OVERVIEW OF THE FIBRE BRAGG GRATING ....................................................... 3
1.3 PROGRAMMING TECHNIQUE............................................................................. 4
1.4 THE APPLICATION OF THE SIMULATION PROGRAM .............................................. 5
1.5 THE SCOPE OF THIS PROJECT.......................................................................... 6
1.6 REFERENCES ................................................................................................. 8
INTRODUCTION
2
1.1 Aim of this project
In the last few years, a growing interest has resulted in increase research
projects on fibre Bragg gratings in the field of fibre optics, both experimentally
and numerically. Fibre Bragg gratings are very useful and powerful optical
devices used in fibre optical communications [1] and for fibre optical sensors [2]
[3]. The potential applications of Bragg gratings are still under development.
In this project, we will use computer-aided design to study the fibre Bragg
grating. To date, several commercial simulation software packages, which are
supplied by Apollo Photonics Inc. and Optiwave Corporation, have already
become available. Although these software packages are suitable for application
to many fibre Bragg Grating problems, like most commercial software, they do
not supply the details of the modelling and the source code of the simulation
program. This will limit the users to use the program on their particular
application. As is well known, optical simulation programs must use some kind of
special models according to the specific application to be simulated. Normally,
approximation theories and methods must be used in the simulation during the
model building of physical problems. Sometimes, the approximation makes the
modelling suitable for one application, but it may not be proper for other
applications. There is no general modelling that is suitable for every application.
Optical fibre grating simulations have the same problem. This means that users
must rewrite the code themselves for their particular demands. It is not possible
to solve their problems by only changing the parameter values and function
types in the simulation software. It will increase efficiency and avoid repeated
development of the code if we can build some basic numerical libraries of fibre
Bragg gratings for optical program developers and the end user. The numerical
libraries can be used directly and extended according to the special applications
set by the programmer. On the other hand, this simulation program should be
easy to use for the end users who will not be modifying the code themselves.
Object-oriented programming (OOP) techniques are widely used for their
INTRODUCTION
3
advantages in the simulation software field. The simulation program using this
programming technique can comply with the above two basic requirements for
the optical fibre program developer and the end user. In this project, we intend to
build optical numerical libraries for fibre Bragg gratings by using object-oriented
programming techniques.
1.2 Overview of the fibre Bragg grating
The fibre Bragg grating is a periodic variation of the refractive index along the
propagation direction in the core of the fibre. It can be fabricated by exposing the
core of the optical fibre to UV radiation. This induces the refractive index change
along the core of the fibre.
The coupled-mode theory is most widely used to analyze light propagation in a
weakly coupled waveguide medium. The fibre Bragg grating is a weakly coupled
waveguide structure.
The coupled-mode equations that describe the light propagation in the grating
can be obtained by using the coupled-mode theory. There are no analytical
solutions for these coupled-mode equations as yet. Numerical methods must be
used to solve these equations.
The transfer matrix method and the direct numerical integration method have
been used to calculate the solution of the coupled-mode equations.
Several techniques have been used to fabricate fibre Bragg gratings: the phase
mask technique, the point-by-point technique and the interferometric technique.
Controlling, combining and routing light are the three main uses of fibre Bragg
gratings in optical communications.
INTRODUCTION
4
For controlling light, fibre Bragg gratings are used in optical signal amplification
to filter out all but a single, precise wavelength from laser pump sources, which
are used to provide optical power to the amplifier [4]. For combining light, fibre
Bragg gratings can be used to combine different wavelengths on a single optical
fibre [5]. This feature of fibre Bragg gratings can be used in wavelength division
multiplexing (WDM) systems. Different wavelengths can be added or dropped in
a WDM system by using the route feature of the fibre Bragg grating [6].
Uniform Bragg gratings cannot satisfy the demand of some kind of applications
alone. New types of grating are being manufactured and studied by researchers.
The chirped, apodized, phase shifted and sampled Bragg gratings are some
examples of modified gratings that will be studied and simulated in this project.
1.3 Programming technique
Suitable programming languages and development environments are very
important in the simulation. There are several factors, such as, code reusability,
development speed, and code compatibility, that should be considered in
choosing the programming language.
Two programming languages, Object Pascal and C++, are used in this project.
C++ and Object Pascal are object-oriented programming languages, which are
in widespread use. Each has its advantages and disadvantages depending on
the developer and the developing period. Both of them use object-oriented
programming techniques even though they are not pure object-oriented
languages. However, JAVA is truly object-oriented [7].
The programming languages rely on their Integration Development Environment
(IDE). Object Pascal must be used in Delphi. C++ can be used in several IDE,
such as C++ Builder, and Visual C++. There is another tool that can be used to
assist the development of this project. This tool is MATLAB, which can be used
rapidly to evaluate the simulation for a prototype. The four development
INTRODUCTION
5
environments are compared with respect to their features, advantages and
disadvantages in Table 1 - 1. It is based on my experience.
Delphi C++ Builder Visual C ++ MATLAB
Programming
language
Object
Pascal
C++ C++ MATLAB
Application
framework
(Class library)
VCL VCL MFC NULL
Flexibility of GUI
design
Best Best Good Normal
Code compatibility Difficult Better Best Difficult
Debugging Fast Slow Slow Faster
Project
development
speed
Faster Fast Slow Fastest
Supply company Borland
Corp.
Borland
Corp.
Microsoft
Corp.
MathWorks
Corp.
Table 1 - 1 The features of the four development environments
Notes: VCL stands for Visual Component Libraries, MFC stands for Microsoft
Foundation Classes, and GUI stands for Graph User Interface.
1.4 The application of the simulation program
A computer simulation program is a very important tool in the optical fibre
research field. The use of expensive and delicate manufacturing systems and
instrumentation can be avoided until the design is optimized. We can use the
computer-aided design and simulation to study fibre optical problems.
Sometimes, environmental and noise factors will be critical during the
experiment. They may have a serious influence on the experimental results. The
INTRODUCTION
6
theoretical results can be obtained by using the simulation program. The factor
that affects the system can be found by studying the differences between the
theoretical values and the experimental results. Optimization and improvement
of the system can be realized by using the simulation results.
For example, the simulation program can be used to analyze the spectral
characteristics of fibre Bragg gratings. Uniform, chirped, apodized, phase shifted
and sampled Bragg gratings have been simulated by this program. The reflection
and transmission spectra, group time delay and dispersion can be obtained. The
value of the maximum reflectivity, sidelobe reflectivity, full-width-at-half maximum
(FWHM) can be obtained too. The relationship between the grating’s variables
(e.g. coupling coefficient, length of grating, Gaussian distribution coefficient, and
chirped values) and the output spectrum values (e.g. the maximum reflectivity,
bandwidth and centre wavelength) can be calculated by this program.
1.5 The scope of this project
There are three parts to this thesis, namely fibre Bragg grating fundamentals and
theory, modelling and coding, and the application of the simulation program.
Fundamentals and theory are covered in Chapter 2. It focuses on the physical
object of the simulation. The waveguide propagation theory, coupled-mode
theory, and fundamentals and applications of fibre Bragg gratings will be
presented in this chapter.
The second part includes Chapter 3 and Chapter 4. Chapter 3 is an overview of
the basic simulation approaches for fibre Bragg gratings. The mathematical
modelling of the grating is built by using the coupled-mode theory. The coupled-
mode equations are solved by direct numerical integration and transfer matrix
method approaches. An object-oriented programming technique will be
introduced in Chapter 4. This chapter demonstrates how object-oriented
programming techniques should be applied to solve the fibre Bragg grating
problem. Flow charts of the simulation design will also be presented in this
INTRODUCTION
7
chapter. Because of the increasing use of MATLAB, many scientists, engineers
and researchers prefer to write functions in MATLAB rather than a compiled
language, such as Object Pascal and C++. MATLAB will also be introduced in
this chapter. The advantages and disadvantages of programming languages and
environments will also be discussed.
The third part includes Chapter 5, Chapter 6 and Chapter 7. The application of
the simulation program will be presented in this part. Chapter 5 is about
simulating the chirped Bragg grating. Dispersion compensation will be discussed
by using a linear chirped grating. Application to sensor systems will also be
studied. The function of the apodization in the Bragg grating is studied in
Chapter 6. The simulation results with different apodization functions will be
demonstrated in this chapter. Chapter 7 introduces the application of phased-
shifted and sampled gratings to lasers and wavelength division multiplexing
systems.
Finally, Chapter 8 is the conclusion of this project and a discussion of future work
that could be done to extend this project.
INTRODUCTION
8
1.6 References
1. A. Othonos and K. Kalli, “Fibre Bragg gratings: fundamentals and
applications in telecommunications and sensing”, (Artech House), 1999.
2. A. D. Kersey, “A review of recent developments in fibre optic sensor
technology”, Optical Fibre Technology, vol.2, 1996, pp. 291-317.
3. A. D. Kersey, M. A. Davis, H. J. Patrick, M. Leblanc, K. P. Koo, C. G. Askins,
M. A. Putnam, and E. J. Friebele, “Fibre grating sensors”, Journal of Lightwave
Technology, vol.15, no.8, 1997, pp. 1442-1463.
4. Y. Tashiro, S. Koyanagi, K. Aiso, and S. Namiki, “1.5 W Erbium doped fiber
amplifier pumped by the wavelength division-multiplexed 1480 nm laser diodes
with fiber Bragg grating”, OAA’98, WC2, 1998.
5. C. R. Giles, “Lightwave applications of fibre Bragg gratings”, Journal of
Lightwave Technology, vol. 15, no.8, 1997, pp. 1391-1404.
6. C. R. Giles and V. Mizrahi, “Low-loss add/drop multiplexer for WDM
lightwave network”, in Tenth International Conference on Integrated Optics and
Optical Fiber Communication, IOOC95 Technical Digest, paper ThC2-1, 1995,
pp. 66-67.
7. J. E. Moreira, S. P. Midkiff, and M. Gupta, “From flop to megaflops: Java for
technical computing”, Proceedings of the 11th International Workshop on
Languages and Compilers for Parallel Computing, LCPC'98. IBM Research
Report 21166, 1998.
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
9
CHAPTER 2: Theory and Fundamentalsof Fibre Bragg Gratings
CHAPTER 2:THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS9
2.1 INTRODUCTION .......................................................................................... 10
2.2 THE COUPLED-MODE THEORY..................................................................... 14
2.3 APPLICATIONS OF FIBRE BRAGG GRATINGS ................................................. 17
2.3.1 Fibre Bragg grating sensors ................................................................ 18
2.3.2 Wavelength Division Multiplexing........................................................ 21
2.3.3 Fibre grating lasers.............................................................................. 22
2.4 CONCLUSION ............................................................................................ 22
2.5 REFERENCES ............................................................................................ 24
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
10
2.1 Introduction
In 1978, at the Canadian Communications Research Center (CRC), Ottawa,
Ontario, Canada [1], K.O. Hill et al first demonstrated refractive index changes in
a germanosilica optical fibre by launching a beam of intense light into a fiber. In
1989, a new writing technology for fibre Bragg gratings, the ultraviolet (UV) light
side-written technology, was demonstrated by Meltz et al. [2]. Fibre Bragg
grating technology developed rapidly after UV light side-written technology was
developed. Since then, much research has been done to improve the quality and
durability of fibre Bragg gratings. Fibre gratings are the keys to modern optical
fibre communications and sensor systems. The commercial products of fibre
Bragg gratings have been available since early 1995.
z refractive index of the core without perturbation
Induced index change
nδ
0n
nδ
0n
Figure 2 - 1 Refractive index change of the fibre Bragg grating
A fibre Bragg grating is a periodic perturbation structure of the refractive index in
a waveguide. Fibre gratings can be manufactured by exposing the core of a
single mode communication fibre to a periodic pattern of intense UV light. The
exposure induces a permanent refractive index change in the core of the fibre.
This fixed index modulation depends on the exposure pattern. Figure 2 - 1
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
11
shows the periodic change in refractive index of the fibre core. This short length
optical fibre with refractive index modulation is called a fibre Bragg grating.
Refractive index modulation can be represented by [3]
)2
cos(),,(),,(),,( zzyxnzyxnzyxnΛ
+=π
δr
2 - 1
where ),,( zyxnr
is the average refractive index of the core, ),,( zyxnδ is the
modulation of the refractive index, and Λ is the Bragg period.
A small amount of incident light is reflected at each periodic refractive index
change. The entire reflected light waves are combined into one large reflection at
a particular wavelength when the strongest mode coupling occurs. This is
referred to as the Bragg condition (2 - 2), and the wavelength at which this
reflection occurs is called the Bragg wavelength. Only those wavelengths that
satisfy the Bragg condition are affected and strongly reflected. The reflectivity of
the input light reaches a peak at the Bragg wavelength. The Bragg grating is
essentially transparent for incident light at wavelengths other than the Bragg
wavelength where phase matching of the incident and reflected beams occurs.
Bragg wavelength Bλ is given by
Λ= effB n2λ 2 - 2
where effn is the effective refractive index and Λ is the grating period. This is the
condition for Bragg resonance. From equation (2 - 2), we can see that the Bragg
wavelength depends on the refractive index and the grating period.
The bandwidth and maximum reflectance will be presented in the next chapter.
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
12
Long gratings with a small refractive index excursion have a high peak
reflectance and narrow bandwidth, as can be seen (Figure 2 - 2).
reflected wave(λB)
incident wave spacing = λB / 2*neff
Λ= effB n2λ
Λ
λB
λ
λ
λB
Pow
er s
pect
rum
Pow
er s
pect
rum
Pow
er s
pect
rum
transmitted wave
λ
Input Wavelength
Figure 2 - 2 Diagram illustrating the properties of the fibre Bragg grating
The effective refractive index effn and Bragg period Λ are constant for the
uniform Bragg grating. Figure 2 - 3 shows the reflectance and transmittance of a
uniform Bragg grating, with the following parameter: 447.1=effn , )(5000 mL µ= ,
0009.0=nδ , )(53559.0 mµ=Λ ,
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
13
Wavelength (micrometre)1.55301.55201.55101.55001.54901.5480
Refle
ctance
(p. u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Wavelength (micrometre)1.55301.55201.55101.55001.54901.5480
Tra
nsm
ittan
ce(p
. u.
)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 2 - 3 The reflectance and transmittance spectrum of a uniform grating
The fibre Bragg grating has the advantages of a simple structure, low insertion
loss, high wavelength selectivity, polarization insensitivity and full compatibility
with general single mode communication optical fibres. Uniform Bragg gratings
are basically a reflectance filter. According to the application, they can have
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
14
bandwidths of less than 0.1nm. It is also possible to make a wide bandwidth filter
that is tens of nanometres wide. Reflectivity at the Bragg wavelength can also be
designed to be as low as 1% or greater than 99.9%. Fibre grating characteristics
such as photosensitization, apodization, dispersion, and bandwidth control,
temperature and strain responses, thermal compensation and reliability issues
have been used in optical communications and sensor systems [4].
2.2 The coupled-mode theory
In general, we are interested in the spectral response of the Bragg grating. The
characteristics of the fibre Bragg grating spectrum can be understood and
modelled by several approaches. The most widely used theory is the coupled-
mode theory [5],[6]. The coupled-mode theory is a suitable tool to describe the
propagation of the optical waves in a waveguide with a slowly varying index
along the length of the waveguide. Fibre Bragg gratings have this type of
structure. The basic idea of the coupled-mode theory is that the electrical field of
the waveguide with a perturbation can be represented by a linear combination of
the modes of the field distribution without perturbations.
The modal fields of the fibre can be represented by
)exp(),(),,( ziyxezyxE jjtj β±= ±± ,...3,2,1=j 2 - 3
where ),( yxe j± is the amplitude of the transverse electric field of the thj
propagation mode and ± represents the propagation direction, and jβ is called
the propagation constant or eigenvalue of the thj mode. Generally, each mode
has a unique value of jβ . In this thesis, we implicitly assume a time dependence
)exp( tiω− for the fields where ω is the angular frequency. The propagation of
light along the optical waveguides in the fibre can be described by Maxwell’s
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
15
equations. Propagation modes are the solutions of the source-free Maxwell
equation [5].
In terms of the coupled-mode theory, the transverse component of the electric
field at position z in the perturbed fibre can be described by a linear
superposition of the ideal guided modes of the unperturbed fibre, which can be
written as
∑ −+=j
jjt tzyxEtzyxEtzyxE )],,,(),,,([),,,(2 - 4
Be substituting the modal field equation (2 - 3) into (2 - 4), the electric field
),,,( tzyxE t can be written as
∑ −−+= −+
jjtjjjjt tiyxezizAzizAtzyxE )exp(),()]exp()()exp()([),,,( ωββ
r2 - 5
where )(zA j+ and )(zA j
− are slowly varying amplitudes of the thj forward and
backward travelling waves respectively; jβ is the propagation constant; and
),( yxe jt
r is the transverse mode field. This electric field distribution ),,,( tzyxE t
can be solved by modal methods. ),,,( tzyxE t is one of the solutions of
Maxwell’s equation.
The index of the grating is z-dependent along the fibre. The refractive index
),,( zyxn in Equation (2 - 1) can be rewritten as
))(2
cos()()(),,( 00 zzznnnznzyxn ϕπ
δδ +Λ
++== 2 - 6
where the average refractive index n is represented as 00 nn δ+ , and 00 nn δ>> ;
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
16
0n is the refractive index of the core without the perturbation; 0nδ is the average
index modulation (DC change); )(znδ is the small amplitude of the index
modulation (AC change); )(zϕ is the phase of the grating; and Λ is the Bragg
period.
The electric field distribution in the grating, ),,,( tzyxE t , satisfies the scalar wave
propagation equation. This follows from a simplification of Maxwell’s equations
under the weak propagation approximation, and is given by
0),,,(}),,({ 2222 =−+∇ tzyxEzyxnk tt
rβ 2 - 7
where λπ /2=k is the free space propagation constant, and λ is the free space
wavelength.
The electric field ),,,( tzyxE t and refractive index ),,( zyxn are substituted into
the wave propagation equation (2 - 7) to yield the following coupled-mode
equations:
])(exp[)(
])(exp[)(
ziKKAi
ziKKAidz
dA
nmzmn
tmn
mm
nmzmn
tmn
mm
n
ββ
ββ
+−−+
−+=
∑
∑−
++
2 - 8
])(exp[)(
])(exp[)(
ziKKAi
ziKKAidz
dA
nmzmn
tmn
mm
nmzmn
tmn
mm
n
ββ
ββ
−−+−
+−−=
∑
∑−
+−
2 - 9
where )(zK tmn is the transverse coupling coefficient between modes n and m ,
)(zK tmn is given by [6]
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
17
∫∫∞
∆= ),(),(),,(4
)( * yxeyxezyxdxdyzK ntmttmn
rrε
ω2 - 10
where ε∆ is the perturbation to the permittivity. Under the weak waveguide
approximation ( nn δ>> ), nnδε 2≅∆ . In general, tmn
zmn KK ⟨⟨ for fibre modes, and
this coefficient is thus usually neglected.
2.3 Applications of fibre Bragg gratings
Table 2 - 1 shows some applications of fibre Bragg gratings.
The application of fibre Bragg gratings
Fibre grating sensors
Temperature, strain, pressure sensors [7],[8]
Distributed fibre Bragg grating sensor systems [9]
Fibre lasers
Fibre grating semiconductor lasers [10]
Stabilization of external cavity semiconductor lasers [11]
Erbium-doped fibre lasers [12]
Fibre optical communications and others
Dispersion compensation [13]
Wavelength division multiplexed networks [14]
Gain flattening for erbium-doped fibre amplifiers [15]
Add/Drop multiplexers [16]
Comb filters [17]
Interference reflectors [9]
Pulse compression [18]
Wavelength tuning [19]
Raman amplifiers [20]
Chirped pulse amplification [21]
Table 2 - 1 The applications of fibre Bragg gratings
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
18
There are a number of applications of fibre gratings in lasers, communications
and sensors. For examples, fibre Bragg gratings can be used as a multiplexer
and a demultiplexer in wavelength division multiplexed systems, and as a
dispersion compensator in communication systems.
Fibre Bragg gratings have a low insertion loss, a low polarization-dependent
loss, and an excellent spectral response profile. This makes them suitable for the
applications of fibre optical sensors.
They can be used for the manufacturing of the fibre lasers on the device
manufacturing.
The three applications of fibre Bragg gratings are introduced briefly in this
chapter.
2.3.1 Fibre Bragg grating sensors
The fibre Bragg grating is one of the most exciting developments in the field of
fibre optical sensing systems in recent years. Uniform, chirped, phase-shifted
and sampled Bragg gratings can be used in the sensing system [22].
Typical measurands that can be measured by fibre Bragg gratings are
temperature and strain. Either temperature or strain can be monitored by fibre
Bragg gratings. It is also possible to monitor them together [23]. The dual
wavelength fibre grating used discrimination between strain and temperature
effects in the sensor system by M.G. Xu et al [24].
Furthermore, fibre gratings exhibit a well-behaved wavelength response to
temperature and strain, which can be exploited for accurate wavelength tuning
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
19
and the development of sensor transducer elements.
A properly manufactured fibre Bragg grating also offers a high reflectivity [1] and
narrow bandwidth at its Bragg wavelength. A typical fibre Bragg grating has a
reflectivity greater than 75% [1]. This high reflectivity offers a sufficient amount of
optical power for detection in photodiodes. This unique characteristic gives fibre
Bragg grating sensors a unique Bragg wavelength that is independent of the
optical intensity used in the system.
Input Wavelength
Am
plitu
de
Reflected Wavelength
Am
plitu
de
Reflected Wavelength
Am
plitu
de
Transmitted Wavelength
Am
plitu
de
Broadbandsource
Detector andProcessing
2 X 2 Coupler
Detector andProcessing
Grating
Measurands
Change temperature,
strain,pressure
Measurands
Figure 2 - 4 Diagram of basic fibre Bragg grating sensors
Figure 2 - 4 is a diagram of the basic uniform fibre Bragg gratings used in the
sensor systems. The wavelength of the light reflected from the Bragg grating
changes when the fibre grating is deformed. Depending on this characteristic,
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
20
fibre Bragg gratings have already been used in the sensor system.
Figure 2 - 5 shows the reflectance of a uniform Bragg grating, with the following
parameter: 447.1=effn , )(10000 mL µ= , 0002.0=nδ ,
)(53628.0:053594:53559.0:53526.0 mµ=Λ ,
Wavelength (micrometre)1.5541.5521.551.5481.5461.5441.542
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 2 - 5 Sensor application of a uniform Bragg grating.
)(53526.0 mµ=Λ ( red solid line), )(53559.0 mµ=Λ ( green dashed line),)(053594 mµ=Λ ( blue dotted line), and )(53628.0 mµ=Λ ( pink dashed and
dotted line)
The physical measurands can be temperature, pressure and strain. The Bragg
grating sensor is based on the property of the fibre Bragg gratings to change the
characteristic wavelength corresponding to the strain and temperature of the
glass fibre. In general, fibre Bragg gratings can easily be multiplexed [4] for
many sensors along an optical fibre. Such a system is found to have high
expandability in which many sensors can be added to the system for more
measurements.
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
21
Fibre Bragg grating sensors have many advantages, depending on their specific
properties, such as: small size, immunity against electromagnetic interference,
dielectric materials, and the possibility of distributed sensing and passive
multiplexing (sensor networks). There are numerous applications for this type of
sensor. There is great interest in using these devices to monitor the health of civil
structures like buildings, bridges and dams.
2.3.2 Wavelength Division Multiplexing
Broadbandsource
Reflected Spectrum
2 X 2Coupler FBG 1 FBG 2 FBG 3 FBG 4
11 2 Λ= effnλ 22 2 Λ= effnλ 33 2 Λ= effnλ 44 2 Λ= effnλ
1λ 2λ 3λ 4λ
Figure 2 - 6 Diagram of principle of one use of fibre Bragg gratings in WDM
Figure 2 - 6 is a diagram of principle of the fibre Bragg gratings used in a WDM
system. The different types of fibre Bragg gratings, which are uniform, phase
shifted and sampled, can be used in WDM systems. Sampled Bragg gratings will
be discussed in Chapter 7.
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
22
2.3.3 Fibre grating lasers
Fibre Bragg gratings have a number of important applications in this optical
device. They can be used as very narrowband reflectors suitable for providing
feedback at a specific wavelength in fibre lasers (both in short pulse and single
frequency lasers) or as filters for multichannel WDM communication systems.
980 nm pump LD
Er Yb fiber
Laser output
R ~ 100% R ~ 70%
FBG1 FBG2
Figure 2 - 7 Schematic diagram of fibre Bragg grating laser with Fabry Perotcavity [25]
2.4 Conclusion
The field distribution of the perturbed fibre can be described by the superposition
of the fields of the complete set of bound and radiation modes of the unperturbed
fibre. This distribution varies with the position along the fibre and is described by
a set of coupled-mode equations, which determine the amplitude of every mode.
The fibre Bragg grating can be viewed as an ideal fibre (as reference) plus a
certain index variation (as perturbations).
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
23
Fibre Bragg gratings have already been commercialized in recent years. It has
become popular to use fibre Bragg gratings in sensor systems for their high
sensitivity and potentially low cost. Fibre Bragg gratings have been used in many
applications, such as wavelength division multiplexing communication systems,
lasers, strain and temperature sensing, and fibre lasers.
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
24
2.5 References
1. K.O. Hill, Y. Fujii, D.C. Johnson, and B.S. Kawasaki, “Photosensitivity in
optical fibre waveguides: application to reflection filter fabrication”, Applied
Physics Letters, vol. 32, no.10, 1978, pp.647-649.
2. G. Meltz, W. W. Morey, and W. H. Glenn, “Formation of Bragg gratings in
optical fibres by a transverse holographic method”, Optics Letters, vol.14, no. 15,
1989, pp.823-825.
3. A. Othonos and K. Kalli, “Fibre Bragg gratings: fundamentals and
applications in telecommunications and sensing”, (Artech House), 1999.
4. C. R. Giles “Lightwave application of fiber Bragg gratings”, ”, Journal of
Lightwave Technology, vol.15, no.8, 1997, pp. 1391-1404.
5. A. W. Snyder and J. D. Love, “Optical waveguide theory”, (Chapman and
Hall, London), 1983, pp542.
6. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology, vol.15,
no.8, 1997, pp. 1277-1294.
7. A. D. Kersey, M. A. Davis, H. J. Patrick, M. Leblanc, K. P. Koo, C. G. Askins,
M. A. Putnam, and E. J. Friebele, “Fibre grating sensors”, Journal of Lightwave
Technology, vol.15, no.8, 1997, pp. 1442-1463.
8. R. Kashyap, “Photosensitive optical fibres: devices and applications”, Optical
Fibre Technology, 1994, pp.17-34
9. S. J. Spammer, P. L. Swart, and A. A. Chtcherbakov, “Merged Sagnac-
Michelson interferometer for distributed disturbance detection”, Journal of
Lightwave Technology, vol.15, no.6, 1997, pp. 972-976.
10. G. A. Ball, W. W. Morey, and W. H. Glenn, “Standing-wave monomode
erbium fiber laser”, IEEE Photonics Technology Letters, vol. 3, 1991, pp. 613-
615.
11. A. Hamakawa, T. Kato, G. Sasaki, and M. Higehara, “Wavelength
stabilization of 1.48 um pump laser by fiber grating,” in Proc. ECOC’96, Oslo,
Norway, paper MoC.3.6,1996.
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
25
12. P. C. Becker, N. A. Olsson, J. R. Simpson, P. C. Becker, and P. Becker,
“Erbium-doped fiber amplifiers : fundamentals and technology (Optics and
Photonics series)”, (Academic Press), 1999.
13. J. A. R. Williams, I. Bennion, K. Sugden, and N. J. Doran, “Fiber dispersion
compensation using a chirped in-fire Bragg grating”, Electronics Letters, vol. 30,
1994, pp.985-987.
14. C. R. Giles and J. M. P Delavaux, “Repeaterless bidirectional transmission
of 10 Gb/s WDM channels”, in ECOC’95, Brussels, 1995, paper PD2.
15. R. Kashyap, R. Wyatt, and R. J. Campbell, “Wideband gain flattened
erbium fiber amplifier using a photosensitive fiber blazed grating”, Electronics
Letters, vol. 29, 1993, pp.154-156.
16. C. R. Giles and V. Mizrahi, “Low-loss add/drop multiplexers for WDM
lightwave networks”, in Proc. IOOC’95, Hong Kong, 1995, paper ThC2-1.
17. B. H. Lee, Y. Chung, and U. Paek, "Fiber comb filters based on fiber
gratings", 5F/13/1999 , COOC99, pp19-20.
18. N. G. R. Broderick, D. Taverner, D. J. Richardson, M. Isben, and R. I.
Laming, “Optical pulse compression in fibre Bragg gratings”, Physical Review
Letters, vol.79, 1997, pp. 4566-4569.
19. Y. Tohmori, F. Kano, H. Ishii, Y. Yoshikuni, and Y Kondo, "Wide tuning with
narrow linewidth in DFB lasers with superstructure grating (SSG)", Electronic
Letters, vol.29, no.15, 1993, pp. 1350-1351.
20. M. Prabhu, N. S. Kim, L. Jianren, J. Xu, K. Ueda, "Highly-efficient ultra-
broadband supercontinuum generation centered at 1484nm using Raman fiber
laser", Photonics West, LASE2001, San Jose, USA, 2001.
21. A. Boskovic, M. J. Guy, S. V. Chernikov, J. R. Taylor And R. Kashyap “All-
fiber diode-pumped, femtosecond chirped pulse amplification system”,
Electronics Letters, vo.31, 1995, pp.877-879.
22. W. Morey, G. Meltz, and W. Glenn, “Fiber-optic Bragg grating sensors”,
Proc SPIE, Fiber Optic and Laser Sensors VII, vol.1169, 1989, pp. 98-107.
23. G. P. Brady, C. Kent, K. Kalli, D. J. Webb, D. A. Jackson, L. Zhang, and I.
Bennion, “Recent developments in optical fibre sensing using fibre Bragg
THEORY AND FUNDAMENTALS OF FIBRE BRAGG GRATINGS
26
gratings”, Proc SPIE. Fibre Optic and Laser Sensors XIV, vol.2839, 1996, pp. 8-
19.
24. M. G. Xu, J. L. Archambault, L. Reekie, “Discrimination between strain and
temperature effects using dual-wavelength fibre grating sensors”, Electronics
Letters, vol.30, no.13, 1994, pp.1085-1087.
25. K. Hsu, W. H. Loh, L. Dong, and C. M. Miller, "Wavelength tuning in efficient
Er/Yb fiber grating lasers," International Conference on Integrated Optics and
Optical Fiber Communication and European Conference on Optical
Communication (IOOC/ECOC'97).
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
27
CHAPTER 3: Approaches to the simulationof fibre Bragg gratings
CHAPTER 3: APPROACHES TO THE SIMULATION OF FIBRE BRAGGGRATINGS.......................................................................................................... 27
3.1 INTRODUCTION .......................................................................................... 28
3.2 MODELLING OF FIBRE BRAGG GRATINGS ..................................................... 28
3.3 UNIFORM BRAGG GRATINGS....................................................................... 32
3.4 THE DIRECT NUMERICAL INTEGRATION METHOD ........................................... 34
3.4.1 The Runge-Kutta method .................................................................... 34
3.5 THE TRANSFER MATRIX METHOD FOR UNIFORM GRATINGS ............................ 36
3.5.1 The transfer matrix method for non-uniform gratings.......................... 37
3.6 CALCULATION OF THE TIME DELAY AND DISPERSION .................................... 38
3.7 CONCLUSION ............................................................................................ 39
3.8 REFERENCES ............................................................................................ 40
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
28
3.1 Introduction
The refractive index of the core is higher than that of the cladding in the optical
fibre. Assuming that there are no waves propagating in the cladding of the single
mode fibre, only basic counter-propagating modes exist in the fibre. Under the
two-mode approximation, the coupled-mode equations of Bragg gratings (2 - 8)
and (2 - 9) can be simplified into two equations (3 - 2) and (3 - 3). The uniform
Bragg grating, as described by these two equations, can be solved by analytical
methods.
For the non-uniform grating, it is difficult to find an analytical solution for these
coupled-mode equations. The coupled-mode equations can only be solved by
numerical methods. There are two suitable methods available currently. Firstly,
the two-mode coupled-mode equations can be solved by direct integration with
the Runge-Kutta method.
The second approach is the use of the transfer matrix method [1], [2], which can
also be used to solve the coupled-mode equations of the non-uniform gratings.
This method was effective in the analysis of the almost periodic grating [3]. For
this analysis, the grating is divided into a number of uniform pieces, each with an
analytical transfer matrix. The transfer matrix for the entire grating can be
obtained by multiplying the individual transfer matrices. This method is easy to
implement with a computer.
Both uniform gratings and non-uniform gratings have been solved by the above
two approaches in this thesis. The spectral response, time delay and dispersion
can also be obtained by these two methods.
3.2 Modelling of fibre Bragg gratings
In most fibre gratings, the induced index change is approximately uniform across
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
29
the core, and there are no propagation modes outside the core of the fibre. In
terms of this supposition, the cladding modes in the fibre are neglected in this
simulation program. If we neglect the cladding modes, the electric field of the
grating can be simplified only to the superposition of the forward and backward
fundamental mode in the core. The electric field distribution (2 - 4) along the core
of the fibre can be expressed in terms of two counter-propagating modes under
the two-mode approximation [4].
),()]exp()()exp()([),,( yxezizAzizAzyxE tββ −+ +−= 3 - 1
where )(zA+ and )(zA− are slowly varying amplitudes of the forward and
backward travelling waves along the core of the fibre respectively. The ),,( zyxE
(3 - 1) can be substituted into coupled-mode equations (2 - 8) and (2 - 9). The
coupled-mode equations can be simplified into two modes, which are described
as
)()()()(ˆ)(
zSzikzRzidz
zdR+= σ 3 - 2
)()()()(ˆ)( * zRzikzSzi
dzzdS
−−= σ 3 - 3
where )]2/(exp[)()( φδ −= + zizAzR and )]2/(exp[)()( φδ +−= − zizAzS [5]; )(zR is
the forward mode and )(zS is the reverse mode, and they represent slowly
varying mode envelope functions. σ̂ is a general “DC” self-coupling coefficient
[1], also called local detuning; and )(zk is the “AC” coupling coefficient[1], also
called local grating strength [6].
The simplified coupled-mode equations (3 - 2) and (3 - 3) are used in the
simulation of the spectral response of the Bragg grating. The coupling coefficient
)(zk and the local detuning )(ˆ zσ are two important parameters in the coupled-
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
30
mode equations (3 - 2) and (3 - 3). They are fundamental parameters in the
calculation of the spectral response of the fibre Bragg gratings. The notations of
these two parameters are different, depending on the different authors in
literature.
The general “DC” self-coupling coefficient σ̂ can be represented by
dz
dφσδσ
21
ˆ −+= 3 - 4
where dz
dφ21
is describes possible chirp of the grating period, and φ is the
grating phase [1]. The detuning δ can be represented by
)11
(2D
eff
D
nλλ
π
ββ
πβδ
−=
−=Λ
−=
3 - 5
where Λ= effD n2λ is the design wavelength for Bragg reflectance by a very
weak grating( 0→effnδ ).
effnδλπ
σ2
= 3 - 6
where effnδ is the background refractive index change.
The coupling coefficient )(zk can be represented by
vzgznzk )()()( δλπ
= 3 - 7
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
31
where )(zg is the function of the apodization, and v is fringe visibility. The
coupling coefficient )(zk is proportional to the modulation depth of the refractive
index )()()( zgznzn δ=∆ .
Incident light
Reflected light
Transmitted light
Length of Bragg grating (L)
R(-L/2) R(+L/2)
S(+L/2)S(-L/2)
Left hand side : R(-L/2) =1
-L/2 +L/20
Right hand side :S(+L/2) =0
Figure 3 - 1 The initial condition and calculation of the grating response toinput field
There is no input signal that is incident from the right-hand side of the grating
0)2/( =+LS , and there is some known signal that is incident from the left side of
the grating 1)2/( =−LR . Depending on these two boundary conditions, the initial
condition of the grating can be written as in equations (3 - 8) and (3 - 9). The
reflection and transmission coefficients of the grating can be derived from the
initial conditions and the coupled-mode equations.
Left side:
=−=−
1)2/(
?)2/(
LR
LS3 - 8
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
32
Right side:
=+=+
0)2/(
?)2/(
LS
LR3 - 9
The amplitude of the reflection coefficient “ρ ”can be written as
)2/(
)2/(
LR
LS
−−
=ρ 3 - 10
The power reflection coefficient “r ” (reflectivity) can be written by
2ρ=r 3 - 11
3.3 Uniform Bragg gratings
The phase matching and coupling coefficient are constant in the case of uniform
Bragg gratings. Equations (3 - 2) and (3 - 3) are first-order ordinary differential
equations with constant coefficients. There are analytical solutions to equations
for (3 - 2) and (3 - 3). The analytical solutions of the coupled-mode equations
can be found with boundary conditions (3 - 8) and (3 - 9).
As the chirp dzd /φ is zero, the local detuning σ̂ equals the detuning δ [1]. The
solution of the complex reflection and transmission coefficient can be expressed
by [7]
)cosh()sinh(ˆ)]2/(sinh[
)(LLi
LzikzA
BBB
B
γγγσγ+
−−=−
3 - 12
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
33
)cosh()sinh(ˆ)]2/(sinh[ˆ)]2/(cosh[
)(LLi
LziLzzA
BBB
BBB
γγγσγσγγ
+−−−
=+
3 - 13
where Bγ is described by
22 σ̂γ −= kB ( 22 σ̂>k ) 3 - 14
22ˆ kiB −= σγ ( 22 σ̂<k ) 3 - 15
The reflected and transmitted spectrum can be obtained and described by
)(cosh)(sinhˆ
)(sinh)(
2222
22
LL
Lkr
BBB
B
γγγσγ
λ+
= 3 - 16
)(cosh)(sinhˆ)(
2222
2
LLt
BBB
B
γγγσγ
λ+
= 3 - 17
It satisfies the law of energy conservation, which is 1)()( =+ λλ tr . The phase of
the reflected light with respect to the incident light can be obtained from
equations (3 - 12) and (3 - 13), and is described by [7]
)]coth(ˆ
[tan)( 1 LBB γ
σγ
λ −=Φ 3 - 18
At the Bragg wavelength, 0ˆ =σ , the grating has the peak reflectivity maxr , which
is
)(tanh)( 2max Lkrr D == λ 3 - 19
It is evident from equation (3 - 19) that the reflectivity of Bragg gratings is close
to 1 when the modulation of the index and grating length are increased.
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
34
The bandwidth λ∆ can be obtained by 2/)()2/( DD rr λλλ =∆+ and equation (3 -
16). The numerical method will be used to solve these equations.
3.4 The direct numerical integration method
The coupled-mode equations in the case of non-uniform gratings can be solved
by several different approaches to calculate the reflection and transmission
spectrum under the two-mode approximation. Two accurate simulation
techniques are available currently. One is direct numerical integration of the
coupled-mode equations by using the fourth-order fixed or adaptive step Runge-
Kutta numerical integration method. Another is by using a transfer matrix
method.
If we are only interested in the reflection spectrum, there is another simple
method to obtain the spectrum, dispersion and time delay of the fibre Bragg
grating. The two mode coupled-mode equations (3 - 2) and (3 - 3) can be
simplified to a single differential equation, known as the Ricatti differential
equation [3]. The time of the simulation can be reduced because only one
equation is necessary for numerical integration.
In this simulation program, the two coupled-mode equations have been solved
simultaneously to obtain both the reflection and the transmission spectra.
3.4.1 The Runge-Kutta method
The Runge-Kutta method [8] achieves a higher order of accuracy of a Taylor
series without having to calculate the higher derivatives of ),(/ yxfdxdy =
explicitly. These methods make use of midpoint quadrature. The general form of
the prediction formula of an thm Runge-Kutta method is
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
35
)...( 322111 mmnnn KaKaKahyy ++++=+ 3 - 20
The fourth-order Runge-Kutta formula with Runge's coefficients is given by
)22(6 43211 KKKKh
yy nn ++++=+ 3 - 21
The Ks in equation (3 - 21) are determined by evaluating the function value, as
follows:
),(1 nn yxfK = 3 - 22
)2
,2
( 12 Kh
yh
xfK nn ++= 3 - 23
)2
,2
( 23 Kh
yh
xfK nn ++= 3 - 24
),( 34 hKyhxfK nn ++= 3 - 25
where h is the step to the next integration point and x is the state variable.
The local truncation error of the above fourth-order Runge-Kutta methods is of
order 5h .
From a computational point of view, an approximate method will be employed,
that is, we can use two-mode coupling approximation in coupled-mode equations
to describe the fibre grating [9]. Typically, fixed step fourth-order Runge-Kutta
numerical integrating or adaptive step-size fifth-order Runge-Kutta integration
can be used. For the fixed step Runge-Kutta, a suitable step should be used to
keep the results both accurate and rapid when we simulate the different lengths
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
36
of fibre gratings. That is, the number of steps should be increased when the
grating length is increased even if it requires more time and calculations. For a
short grating length, this is not necessary because it requires more time, and the
result is not significantly by different.
Meanwhile, accuracy by varying the Integration step-size is very important when
we want to use the Runge-Kutta method to solve ordinary differential equations.
The derivation of algorithms to regulate step-size is important to maintain the
accuracy. The primary objective in regulating the step-size is to gain
computational efficiency by taking as large a step-size as possible while
maintaining accuracy and minimizing the number of function evaluations.
3.5 The transfer matrix method for uniform gratings
The other numerical approach uses the transfer matrix method [10]. The transfer
matrix method was first used by Yamada [11] to analyze optical waveguides.
This method can also be used to analyze the fibre Bragg problem.
The coupled-mode equations (3 - 2) and (3 - 3) can be solved by the transfer
matrix method for both uniform and non-uniform gratings. Figure 3 - 2 is the
basic ideal structure that the transfer matrix method is uses to solve for a uniform
Bragg grating. The refractive index excursion and the period remain constant.
For this case, the 2 x 2 transfer matrix is identical for each period of the grating.
The total transfer matrix is obtained by multiplying the individual transfer
matrices.
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
37
R(-L/2) R(+L/2)
S(+L/2)S(-L/2)
R(-L/2) R(+L/2)
S(+L/2)S(-L/2)
⋅⋅⋅⋅⋅⋅=
+
−−
−
+
2/
2/11
2/
2/ ......L
LiMM
L
L
S
RFFFF
S
R
⋅=
+
−
−
+
2/
2/
2/
2/
L
LM
L
L
S
RF
S
R
Λ
1Λ 2Λ
1F 2F mF
(a)
(b)
1 2 M
Figure 3 - 2 The principle diagram of the transfer matrix method(a) uniform grating (b) non-uniform grating
3.5.1 The transfer matrix method for non-uniform
gratings
The transfer matrix method can be used to solve non-uniform gratings. This
method is effective in the analysis of the almost-periodic grating. A non-uniform
fibre Bragg grating can be divided into many uniform sections along the fibre.
The incident lightwave propagates through each uniform section i that is
described by a transfer matrix iF . For the structure of the fibre Bragg grating, the
matrix iF can be described as [1]
∆+∆∆
∆−∆−∆=
)sinh(ˆ
)cosh()sinh(
)sinh()sinh(ˆ
)cosh(
zrizrzrk
i
zrk
izrizrF
BB
BBB
BB
BB
B
i
γσ
γ
γγσ
3 - 26
where k is described by equation (3 - 7), σ̂ is described by equation (3 - 4) and
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
38
Bγ is equations (3 - 14) and (3 - 15).
The entire grating can be represented by
⋅⋅⋅⋅⋅⋅=
+
−−
−
+
2/
2/11
2/
2/ ......L
LiMM
L
L
S
RFFFF
S
R3 - 27
3.6 Calculation of the time delay and dispersion
The group time delay and dispersion of the grating can be obtained from the
phase information of the reflection and transmission coefficients.
The delay time ρτ for light reflected in a grating is defined as follows [1]:
λ
θ
πλ
ω
θτ ρρ
ρ d
d
cd
d
2
2
−== 3 - 28
2
2
λπ
τλ
θρ
ρ c
d
d−= 3 - 29
The dispersion ρd (in nmps / ) is defined as follows:
2
2
2
2
22
2
2
2
ω
θ
λπ
λ
θ
πλ
λ
τ
λ
τ
ρ
ρρρρ
d
dc
d
d
cd
dd
−=
−==3 - 30
)2
(2
22
2
ρρρ
λ
τ
λπ
λ
θd
c
d
d−= 3 - 31
The output results of time delay and dispersion calculation in gratings can be
compared to optimize the system parameters. This enables us to find which one
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
39
is suitable for a particular application.
3.7 Conclusion
The coupled-mode equations can be solved by two different approaches for
calculating the reflection and transmission spectra under the two-mode
approximation. One is direct numerical integration of the coupled-mode
equations by using the fourth-order fixed or adaptive step Runge-Kutta numerical
integration. The other involves the use of the transfer matrix method.
We will later use the numerical integration methods and transfer matrix method
to solve the coupled-mode equations.
APPROACHES TO THE SIMULATION OF FIBRE BRAGG GRATINGS
40
3.8 References
1. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology, vol.15,
no.8, 1997, pp. 1277-1294.
2. A. Othonos, “Fibre Bragg gratings”, Review of Scientific Instruments, vol.68,
no.12, 1997, pp. 4309-4341.
3. H. Kogelnik, “Filter response on nonuniform almost-periodic structures”, Bell
System Technical Journal, vol.55, no.1, 1976, pp. 109-126.
4. A. W. Snyder and J. D. Love, “Optical waveguide theory”, (Chapman and
Hall, London), 1983, p. 542.
5 J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through
nonuniform grating structures”, Journal of the Optical Society of America A,
vol.11, 1994, pp. 1307-1320.
6. L. R. Chen, S. D. Benjamin, P. W. E. Smith, and J. E. Sipe, “Ultrashort pulse
reflection from fiber gratings: a numerical investigation”, Journal of Lightwave
Technology, vol.15, no.8, 1997, pp. 1503-1512.
7. Y. Chen and S. Jian, “An introduction to lightwave technology”, (China
Railway Publishing), 2000, p. 248.
8. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, “Numerical
recipes in C - the art of scientific computing”, (Cambridge University Press),
1988, pp. 707-753.
9. G. Allodi and R. Coisson, “Reflection and propagation of waves in one-
dimensional quasi-periodic structures”, Computers in Physics, vol.10, 1996, pp.
385-390.
10. H. Kogelnik,"Coupled wave theory for thick hologram gratings", Bell System
Technical Journal, vol.48, no.9, 1969, pp. 2909-2949.
11. M. Yamada and K. Sakuda, "Analysis of almost-periodic distributed
feedback slab waveguide via a fundamental matrix approach", Applied Optics,
v.26, no.16, 1987, pp. 3474-3478.
PROGRAMMING TECHNIQUE
41
CHAPTER 4: Programming Technique
CHAPTER 4:PROGRAMMING TECHNIQUE.................................................... 41
4.1 OBJECT-ORIENTED PROGRAMMING TECHNIQUE ........................................... 42
4.1.1 Using the object-oriented programming technique ............................. 42
4.2 PROGRAMMING LANGUAGES ...................................................................... 43
4.2.1 MATLAB............................................................................................... 44
4.2.2 Object Pascal ...................................................................................... 46
4.2.3 C++ ...................................................................................................... 47
4.3 THE IMPLEMENTATION OF SIMULATION PROGRAMMING.................................. 49
4.3.1 The flow chart of the design of the simulation programming .............. 49
4.3.2 The design of grating class ................................................................. 51
4.3.3 The design of a user-friendly GUI ....................................................... 51
4.3.4 Using the Bragg grating classes library............................................... 52
4.3.5 Using this simulation program............................................................. 54
4.4 COMPATIBILITY AND PORTABILITY................................................................ 56
4.4.1 Sharing the code between Delphi and C++ Builder ............................ 56
4.4.2 Sharing the code between C++ Builder and Visual C++..................... 57
4.4.3 Portal source code from Windows to LINUX....................................... 57
4.5 CONCLUSION ............................................................................................ 58
4.6 REFERENCES ............................................................................................ 59
PROGRAMMING TECHNIQUE
42
4.1 Object-oriented programming technique
Object-oriented programming languages such as C++ and Object Pascal are
similar in many ways to traditional programming languages called “procedural”
languages such as C and Pascal, but their approaches to solving the problems
are different. Because of the shift in viewpoint, object-oriented programming is
effective and useful in many different kinds of applications, but it is particularly
applicable to computer simulations.
The basic difference between them is how to deal with the data. In a traditional
language, you write a series of procedural codes that are applied to a collection
of data; code and data are firmly separated. In object-oriented programming,
however, you can organize a problem into a set of entities, called objects; each
object contains both the data and the code that describe its state and behaviour.
In this simulation project, an object in the program corresponds directly to a fibre
Bragg grating that has been modelled.
4.1.1 Using the object-oriented programming technique
Taking advantage of the modern programming design technique is important for
developers, particularly in object-oriented programming [1], [2]. There are four
major elements in an object model: abstraction, encapsulation, modularity and
hierarchy [3]. The object-oriented programming technique is proving to be more
powerful than the traditional one [4].
In object-oriented programming, one develops objects that represent certain
physical models in the real world and then embodies these abstractions into
computer code. A class is a structure that defines the data and the methods to
work on that data. An object is an instance of a class. Classes serve as
templates for the creation of objects. Each of these objects consists of both the
data and the methods (member functions). Encapsulation is a technique in which
PROGRAMMING TECHNIQUE
43
data is packaged with methods in an object. The state of the data is said to be
encapsulated from the outside world so that the internal data of an object is only
accessible through the message interface for that object. Providing a fixed
interface between objects is convenient for the code modularity and flexibility,
and simplifies the task of building a program in a large developing project, since
program components are naturally separated. Inheritance is the idea that an
object can inherit or acquire traits of other objects by subclassing those other
objects. The superclass is commonly referred to as the parent and the
subclasses are the children.
4.2 Programming languages
It is an important step to realize object modelling in this project. The suitable
programming languages and developing environments will decide the efficiency
and reusability of the code. There are many programming languages: C, Pascal,
Fortran, C++ and Object Pascal, which can be used for scientific and
engineering simulation calculations. Several programming environments, for
example, MATLAB, Delphi, C++ builder and Visual C++, can be used in the
design of the simulation program.
Suitable programming languages and development environments are very
important for implementing the code of the physical models. There are several
factors: portable, reusable code and efficiency, that should be taken into account
in making choices. The programming language should support important object-
oriented programming features as well as some useful non-object-oriented
programming features, and these features can strengthen and simplify the
programming of scientific application codes.
The simulation program has two purposes. One is to supply the visual and
friendly GUI (Graphic User Interface) for the end user. The user can use this
program without any program knowledge. The simulation results can be obtained
by changing the input parameters from the GUI part of the simulation program.
PROGRAMMING TECHNIQUE
44
The other purpose is that this simulation program supplies the numerical library
for the programmers. There are several functions and class libraries that have
already been available for use in other projects.
In terms of the purpose of design, there are two main programming languages,
i.e. Object Pascal and C++, used in this project. Both have their own advantages
and disadvantages, depending on the background of the developers and the
experience of the potential user. The period of the development is also
important.
Object Pascal under the Delphi environment and C++ language under the
Borland C++ Builder and Microsoft Visual C++ are suitable programming
languages for the development of this project. There is another tool that can be
used to assist in the development of this project, that is MATLAB, which can be
used for rapid implementation of the simulation of a physical prototype.
4.2.1 MATLAB
MATLAB [5], [6] is a mathematically-oriented interpreter language. It is used for
simulation calculations. It can be used as a numerical and symbolic calculator, a
visualization system, a programming language, a modelling and data analysis
environment, and more. It uses symbolic expressions to provide a very general
representation of mathematics. The simple syntax of the programming language
makes it easy to learn and use. This also makes the debug faster than other
programming languages and developing environments.
PROGRAMMING TECHNIQUE
45
ActiveX
The collaborative relationship of differentprogramming environments
DDE DDE
ActiveX
Compileto DLL
Call DLL
Compileto DLL
ActiveX& DLL
VCL &,
ActiveX Call DLL
MFC &
ActiveXM functions
& DLL
VCL &ActiveX
EXE
DLL DLL
DLLDLL
Delphi
MATCOMC++ Builder
MATLAB Visual Basic
Third party components
Visual C++
Figure 4 - 1 The relationship of different programming environments
PROGRAMMING TECHNIQUE
46
It is very simple to realize complex and matrix arithmetic, compared with the
other two programming languages, Object Pascal and C++. With MATLAB, it is
not necessary for any additional work to be done when using complex numbers
and matrix algebra. Many useful toolboxes and functions are also built into the
MATLAB environment.
The shortcomings of MATLAB are simulation running speed, the flexibility of the
GUI design, the organizing of a large project, etc. Generally, the calculation
speed of simulation is slower than the binary code that is compiled. The reason
for this is that an interpreted language was used by MATLAB. Depending on the
third party tools, e.g. MATCOM, Visual C++ and C++ Builder, some functions, but
not every function written in MATLAB, can be compiled to a binary file. But, the
MATLAB environment is still necessary in using a compiled code for simulation.
This means that for every computer to run this program, MATLAB must be
installed in that computer. You cannot run it without MATLAB. It is not a real
stand-alone program.
In terms of its features, MATLAB is a good tool in the prototype design of
modelling at the beginning of the development. Figure 4 - 1 shows the
relationship between MATLAB and other programming environments. We can
use it to realize the modelling rapidly, and verify the accuracy of the modelling.
The code can be transferred to Object Pascal and C++ for the final
implementation of the modelling after the accurate models are built.
4.2.2 Object Pascal
The Object Pascal language is used under a Delphi developing environment.
The simplified syntax of the language makes the developing and debugging
PROGRAMMING TECHNIQUE
47
faster than C++. Some complex and confusable programming techniques, such
as pointer, hide inside the developing environment, but the programmer need not
worry about this. The compiler of the language will deal with them. It is faster to
develop the simulation program by using Object Pascal than C++. In this
simulation project, the original simulation program was developed by Object
Pascal.
There are some disadvantages of the Object Pascal language [7], that is, it does
not support operation overload (it only supports function overload) and template
libraries. It will reduce the code readability if the simulation calculation needs to
use complex numbers and matrix operations. For these reasons, most parts of
the code have been transported to C++ code to eliminate these inconveniences
at a later development of this simulation program. Furthermore, there are no
complex and matrix libraries built into Delphi. The developers must build the
complex number and matrix libraries themselves. At present, the major
calculation class libraries are also available in both Object Pascal and C++ code.
This simulation program is organized by two kinds of Object Pascal Units. One of
them is used to support the end user, the Graphic User Interface (GUI) windows.
The other is the class library of the grating components for the major simulation
calculations. For the GUI code, they are realized by VCL (Visual Component
libraries) under Delphi. Delphi is a flexible and rapid-developing environment
compared with MATLAB and C++ builder. With its advantages, at the beginning,
this program was developed with Object Pascal programming language under
the Delphi environment.
4.2.3 C++
There are several developing environments that use the C++ language. C++
Builder and Visual C++ are the most popular among them. A different application
PROGRAMMING TECHNIQUE
48
framework (classes library) is used by them. The Microsoft Foundation Classes
(MFC) are used in Visual C++. The Visual Component Libraries (VCL) are used
in C++ Builder.
C++ is a full-featured, object-oriented language that provides support for
inheritance and polymorphism [8], [9]. We shall see that all the desired object-
oriented features, including the important notion of inheritance, are present in
C++. This makes the extending of the grating class simpler.
A critical feature of C++ is the template, which allows C++ programmers to build
a portable, reusable code and to improve the efficiency of the evaluation of
complex expressions involving user-defined data types.
To build the numerical library by C++, another advantage is that C++ supports
operator overloading, which Object Pascal does not yet support. This makes the
code of complex numbers, vector and matrix calculations more readable and
convenient compared with Object Pascal.
A shortcoming of C++ is that the developing period is longer than that of Object
Pascal. The syntax of C++ is complex and makes the debugging very slow with
the old Pentium central processing unit (CPU). It will take several minutes to
compile a simple project in Visual C++ and C++ Builder. There is no matrix
library built in these two developing environments, as in Delphi, but a complex
library is available.
Depending on the different features, the different languages and developing
environments are used at different periods of the development. At the beginning,
MATLAB implements the prototype of physical models. Then, the code is
transferred to Object Pascal and C++. This will save time in developing the
PROGRAMMING TECHNIQUE
49
program, instead of directly writing the code by using C++. The source code of
the simulation program can be used or called by the different methods between
Object Pascal and C++. For example, it is easy to call a Delphi dynamic link
library (DLL) from the C++ builder or Visual C++ code. The developed code can
easily be reused by other projects and environments without modification.
4.3 The implementation of simulation programming
4.3.1 The flow chart of the design of the simulation
programming
Figure 4 - 2 is the flow chart of the design of the simulation program. At the
beginning, we use MATLAB to implement the prototype of the fibre Bragg grating
modelling. The simulation results are compared with the known theory value and
experimental results. When the expected results are obtained, the code of
MATLAB can be transferred to Object Pascal and C++ classes and functions.
For the supporting GUI, the different application frameworks, VCL and MFC, are
applied under different developing environments, followed by the α , β test of the
simulation program.
PROGRAMMING TECHNIQUE
50
Figure 4 - 2 The flow chart of the process of the software development
wrong
Start
Simulation by MATLAB
Compare thesimulation results
Building the fibre Bragg grating modelling
Build grating classesusing Object Pascal
The design of the GUI
Finish
test of the simulation program
Build grating classesusing C++
The design of the GUI
right
bugsbugsβα ,
PROGRAMMING TECHNIQUE
51
4.3.2 The design of grating class
The fibre Bragg grating modelling is encapsulated into a grating class. The
Bragg grating variables and apodization and chirp functions are placed inside the
grating class. The initial value of the variable is initialized in the constructor part
of the grating class. The position z , Bragg period, coupling coefficient, index
change function, apodization function, chirp function, etc., are in the class’s
constructor. Those variables and functions are declared by protected or private
type variables. They are not available to the other functions and variables, so
only the grating object can handle them. Several member functions are defined
in the grating class to assign the value of those protected or private variables in
the class. The main calculation of the grating is realized on the “step” and “solve”
member functions. The transfer matrix method and Runge-Kutta method are
realized by these member functions.
4.3.3 The design of a user-friendly GUI
The design of a user-friendly Graphic User Interface (GUI) is another important
part of the simulation program. Taking the advantages of the Windows operating
system on GUI, the simulation software should have the same features as a
normal Windows application, regardless of the program. Otherwise, it will limit its
use if users do not know how to use it, or have difficulties learning the software
that you have designed. The user-friendly interface can help users to save time
and concentrate on their physical problems, by not spending more time learning
how to use it.
At the beginning of the Windows application program development,
programmers had to spend a lot of time on the GUI design by using the Windows
SDK (software development kit). The programmers were released from doing
this work when the rapid development tools, such as Visual Basic, Delphi, Visual
PROGRAMMING TECHNIQUE
52
C++, and C++ Builder, became available.
In this program, the graphic interface was implemented by different methods
depending on which programming language and development environment were
used. The GUI is realized by VCL under Delphi and C++ Builder. The VCL is a
more modern application framework than MFC. Many VCL components are
available to supply the user-friendly interface and plots of the simulation results.
Taking advantage of the VCL, the developing period is shorter than for MFC. The
MFC’s application framework was used in Visual C++ to support the visual
interface. MFC is more stable than VCL during the running phase and it is widely
supported by other C++ programming environments, but the design process is
very slow compared to VCL.
4.3.4 Using the Bragg grating classes library
Figure 4 - 3 shows the flow chart of how to use the grating class in the program.
The object of the grating class should be created for every simulation. The
variable value of the grating should be set by using the member function of the
grating class. By detuning the input wavelength, a single value of the reflected
and transmitted power can be obtained. The phase shift information can also be
obtained from the reflected and transmitted power. The time delay and
dispersion of the grating can be obtained from the derivative of the phase shift.
In the end, the grating object should be destroyed to release the memory to the
system.
PROGRAMMING TECHNIQUE
53
Figure 4 - 3 Flow chart of how to use the grating class
Start
Create the gratingobject
Input wavelength{wi:wf}
Calculate reflectionand transmission spectrum
Define variables, refractive indexfunction and default values
Calculate phase
Calculate dispersion
Calculate time delay
Finish
Yes
No
Plot the simulation results
Destroy the gratingobject
PROGRAMMING TECHNIQUE
54
4.3.5 Using this simulation program
For the end users, there are no programming skills necessary for them to use
this simulation software. The source codes of the simulation program have been
compiled to execute binary files by Delphi, C++ Builder and Visual C++. The
binary files are stand-alone programs. It is not necessary to use them with the
developing environment. Unlike MATCOM, the program is compiled by it. You
must obtain a licence and dynamic link library files if you are going to use the
compiled program in another computer. It is not a true stand-alone program.
Figure 4 - 4 shows us how to use this simulation program. Depending on the
user-friendly GUI, the simulation results can be obtained by simply inputing the
values of the variable and choosing the function types. When using this
simulation program first, a Bragg grating type should be chosen. The uniform,
chirped, apodized, phase-shifted and sampled Bragg gratings are available at
present. Then, the value of the grating variable, apodization function and chirped
function should be defined. Finally, the numerical method, the Runge-Kutta
method or the transfer matrix methods can be used to simulate the results. One
can define the precision of the calculation, depending on the practical
application. For example, 100~400 calculation points in the detuning input
wavelength are sufficient for the plot of the reflected and transmitted spectra. For
the plot of time delay and dispersion, a satisfactory plot can be obtained by
calculating 1000 points.
PROGRAMMING TECHNIQUE
55
Figure 4 - 4 The flow chart of the simulation process.
Reflection spectrum
Time delay
Transmission spectrum
Dispersion
Plot results
Centre wavelength
FWHM
Maximum reflectivity
Sidelobe reflectivity
Output values
Output
Input
Sim
ulation
Grating period
Grating length
Refractive index change
Effective index
Chirp parameter
Apodization functions
Phase-shifted parameter
Sampled parameter
Detuning value ofinput wavelength
Define variablevalues
andfunctions
Choosenumericalmethods
andprecision
Transfer matrix method
Runge-Kutta method
Detuning value ofthe input wavelength
Step value along the z
Step value along the z
PROGRAMMING TECHNIQUE
56
4.4 Compatibility and portability
4.4.1 Sharing the code between Delphi and C++ Builder
Both Delphi and the C++ Builder were issued by Borland Corp. The same
Application Framework was used in these programming environments. Their
application framework is the Visual Component Libraries (VCL). They can share
the code both on binary level and code level.
Only the general shared code methods are mentioned in this project. There are
other methods that can be used in these two programming environments.
4.4.1.1 General method
We know that the Windows operating system is composed of hundreds of
dynamic link libraries (DLL). The codes and resources are shared by different
application programs using DLLs. This technique can also be used to share a
code among these development environments: Delphi, C++ Builder and Visual
C++.
4.4.1.2 Using the Delphi unit under C++ Builder
The most simple and convenient method is the use of the Delphi unit directly
under C++ Builder. Both the Delphi unit and the project can be used and
compiled by C++ Builder. They use the same compiler in both Delphi and C++
Builder. This makes the sharing of codes between them simple and reliable. The
compiled Delphi unit can also be used by C++ Builder even if the source code is
not available.
PROGRAMMING TECHNIQUE
57
4.4.1.3 Using the C++ code under Delphi
At present, Delphi cannot use the C++ source code directly. Delphi cannot share
the C++ code on the code level, but the object file that is compiled under C++
Builder can be used by Delphi.
4.4.2 Sharing the code between C++ Builder and Visual
C++
There are two levels of sharing the code between the C++ Builder and Visual
C++. One is from the source code, and the other one is the compiled files. The
project of Visual C++ including Classes and MFC can be directly compiled and
used by C++ Builder because it supports the MFC application framework. But
Visual C++ cannot compile the project with C++ Builder. The reason for this is
that Visual C++ does not support the VCL application framework. Generally, the
grating classes library can be used with slight changes by these two. The GUI
with VCL should be transferred to MFC in Visual C++. On the binary code level,
the work should be simple. The general method of sharing the code is to use the
dynamic link library. Both classes of the numerical library and GUI part can be
incorporated into DLL.
4.4.3 Portal source code from Windows to LINUX
A good programming language must be portable. There are several platforms
available in the computer world, e.g. Windows, Linux, OS/2 warp, and Macintosh
OS. It is important for a code that is realized in one platform to be ported to
another. Time and money can be saved in this way.
The simulation code was originally developed under a Windows environment,
but this code can be ported to another platform. In recent years, the use of Linux
and the application software based on it increased rapidly in the field of
PROGRAMMING TECHNIQUE
58
operating systems and application software. Linux is a free operating system
under General Public Licence (GPL). Most of the source codes of the program
are available in the Linux world. It is convenient for developers to use a third
party code for their project and share their own for public use.
A Delphi for Linux version is called the “Kylix” version and is being developed by
Borland Corp. The grating classes with Object Pascal and the VCL GUI
component portal from the Windows system to the Linux environment should be
simple and direct when “Kylix” is available.
There are two tasks that should be done when the C++ code of the simulation
program is ported from Windows to Linux. The C++ grating classes library can
be used under Linux by the GNU G++ compiler with slight modifications.
Normally, the code of the GUI should be rewritten under the KDE or GNOME
environment under X-Window, because the working style of the two operating
systems is different.
For MATLAB, there is one similar environment called Octave in Linux. Most of
the functions of MATLAB can be used in Octave.
4.5 Conclusion
Depending on their features, the different programming languages and
development environments are used in the different phases of the simulation
program being developed. At the beginning, the prototypes of grating models
are realized with MATLAB to validate the accuracy of the models. The code of
the simulation program is transported to Object Pascal under Delphi after the
suitable models are built and validated. C++ is the most popular programming
language at present in the development of software. In the end, the C++ code of
the simulation program was realized under C++ Builder and Visual C++
environment to make the code more readable and easy to use, and shared in
other development projects and environments.
PROGRAMMING TECHNIQUE
59
4.6 References
1. A. Eliens, “Principles of object-oriented software development”, (Addison-
Wesley), 1994.
2. G. Booch, “Object-oriented development”, IEEE Transactions on Software
Engineering, vol.12, 1986, p. 211.
3. G. Booch, “Object-oriented analysis and design with applications”,
(Benjamin-Cummings), 1994.
4. G. Buzzi-Ferraris, “Scientific C++: building numerical libraries the object-
oriented way”, (Addison-Wesley), 1993.
5. D. M. Etter, “Engineering problem solving with MATLAB”, (Prentice Hall),
1993.
6. V. Loan and F. Charles, “Introduction to scientific computing: a matrix-vector
approach using MATLAB”, (Prentice Hall), 1999.
7. P. Norton “Peter Norton’s guide to Delphi 2 premier title”, (SAMS), 1996.
8. B. Stroustrup, “The C++ programming language”, (Addison-Wesley), 1991.
9. D. A. Young, “Object-oriented programming with C++ and OSF/Motif”,
(Prentice Hall), 1992.
CHIRPED FIBRE BRAGG GRATINGS
60
CHAPTER 5: Chirped Fibre Bragg Gratings
CHAPTER 5: CHIRPED FIBRE BRAGG GRATINGS ..............................................60
5.1 THE PRINCIPLE OF THE CHIRPED BRAGG GRATING .............................................61
5.1.1 Direct integration .......................................................................................62
5.1.2 Transfer matrix method .............................................................................63
5.2 THE SIMULATION RESULTS OF THE SPECTRAL RESPONSE....................................63
5.2.1 Linear chirped gratings with different chirp variables ...............................65
5.2.2 Linear chirped gratings with different lengths ...........................................68
5.2.3 Linear chirped gratings with different refractive index changes ...............69
5.2.4 Arbitrary chirped Bragg grating .................................................................70
5.3 RELATIONSHIP BETWEEN THE MAXIMUM REFLECTANCE AND THE CHIRPED
GRATING COEFFICIENTS...............................................................................................71
5.3.1 The maximum reflectance and the “chirp parameter” ..............................72
5.3.2 The maximum reflectance and the length of the grating ..........................73
5.3.3 The maximum reflectance and the index change.....................................74
5.4 RELATIONSHIP BETWEEN THE 3 dB BANDWIDTH AND THE CHIRPED GRATING
COEFFICIENTS.............................................................................................................75
5.4.1 The reflectance bandwidth and the “chirp parameter” (or chirp
variable).................................................................................................................76
5.4.2 The reflectance bandwidth and the length of the grating..........................78
5.4.3 The reflectance bandwidth and the index change ....................................79
5.5 RELATIONSHIP BETWEEN THE CENTRE WAVELENGTH AND THE CHIRPED
GRATING COEFFICIENTS...............................................................................................79
5.5.1 The centre wavelength and the “chirp parameter” (or chirp variable) ......80
5.5.2 The centre wavelength and the index change..........................................82
5.5.3 The centre wavelength and the length of the grating ...............................83
5.6 DISPERSION COMPENSATION............................................................................84
5.6.1 Simulation results ......................................................................................88
5.7 SENSOR APPLICATIONS ....................................................................................94
5.8 CONCLUSION...................................................................................................98
5.9 REFERENCES ..................................................................................................99
CHIRPED FIBRE BRAGG GRATINGS
61
5.1 The principle of the chirped Bragg grating
A chirped Bragg grating is a grating that has a varying grating period. There are
two variables that can be changed to obtain a chirped grating from equation (2 –
2): one is to change the Bragg period; another is to change the refractive index
along the propagation direction of the fibre.
Long Wavelengths(Lower Frequencies)
Short Wavelengths(Higher Frequencies)
Figure 5 - 1 Linear chirped Bragg grating
Figure 5 - 1 shows a linear chirped Bragg grating. In this case, the period of the
grating varies linearly with the position. This makes the grating reflect different
wavelengths (or frequencies) at different points along its length.
Chirped Bragg gratings can also be modelled by the coupled-mode theory. The
refractive index of the chirped Bragg grating can be expressed by [1]:
∫+Λ
+=z
od
zzyxnzyxnzyxn ))(2
2cos(),,(),,(),,( ξξφ
πδ
r5 - 1
where Λ is the Bragg period, and )(ξφ describes the instantaneous phase of the
chirped grating. There are no analytical solutions for the coupled-mode
equations of chirped gratings. Numerical methods must be used to solve the
equations. Two simulation methods, the direct integration and the transfer matrix
method, are used.
CHIRPED FIBRE BRAGG GRATINGS
62
5.1.1 Direct integration
The period is changed along the z -direction, so that the Bragg wavelength Bλ of
any point is different in the Bragg grating.
Changing the refractive index nδ along the z -direction has the same effect as
changing the period along the z -direction. This means the optical period is
changed even though the physical period of the grating is fixed. So these two
variables can be merged, and described by one variable.
The phase term in equation (3 - 4) for a linear chirped grating [2] is
dzdzn
dzd D
D
eff λλ
πφ2
4
21
−= 5 - 2
where dzd D /λ is a rate of change of the design wavelength with the position in
the grating. By substituting equation (3 - 4) and (5 - 2) into the coupled-mode
equations, the solution for the chirped grating can be obtained.
5.1.1.1 Linear chirped grating
For the linear chirped grating, the chirp variable dzd D /λ is constant. The unit of
the variable can be chosen in nanometre/centimetre. For some applications, the
chirped grating can be modelled by the “chirp parameter” F [3], given by
dzdFWHM
n
zz
FWHMF
D
Deff
λλ
π
φ
2
2
2
2
)(4
)()(
−=
=5 - 3
where F is a measure of the fractional change in the grating period over the
entire length of the grating [2], and FWHM is the full-width-at-half-maximum of
the grating profile.
CHIRPED FIBRE BRAGG GRATINGS
63
From equations (5 - 2) and (5 - 3), the phase term in equation (3 - 4) for a linear
chirp can be represented by “chirp parameter” F , is written by
2)(21
FWHMz
Fdzd
=φ
5 - 4
Both the chirp variable dzd D /λ and “chirp parameter” F can be used to solve
the coupled-mode equations of the linear chirped grating. In this program,
LFWHM = was used for the calculation.
5.1.1.2 Arbitrary chirp function grating
Depending on the real system, the chirped grating does not always have a linear
chirp. An arbitrary chirp function can be used in the simulation program. If
dzd D /λ is a function of z , compared to the linear chirped grating for which it is a
constant, arbitrary chirp functions can be simulated. Further study of the
application of an arbitrary chirp function grating still needs to be done.
5.1.2 Transfer matrix method
By substituting equation (3 - 4) and (5 - 2) or (5 - 4) into the transfer matrix (3 -
26), the solution for the chirped grating can be obtained.
5.2 The simulation results of the spectral response
Both linear and arbitrary chirped Bragg gratings can be simulated by the
simulation program developed for this thesis. We can use this simulation to
optimize the design of chirped Bragg gratings.
CHIRPED FIBRE BRAGG GRATINGS
64
Grating parameters Figure 5 - 2 Figure 5 - 3 Figure 5 - 4 Figure 5 - 5 Figure 5 - 6 Figure 5 - 7
Grating length
( )L)(10000 mµ )(10000 mµ
)(5000 mµ)(7000 mµ)(10000 mµ
)(5000 mµ)(7000 mµ)(10000 mµ
)(10000 mµ )(10000 mµ
Index change
( nδ )0004.0 0004.0 0004.0 0004.0
0002.00004.00006.0
0004.0
Effective index
( effn )447.1 447.1 447.1 447.1 447.1 447.1
Design grating wavelength
( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ )(550.1 mµ )(550.1 mµ )(550.1 mµ
Chirp variable
( dzd D /λ )
)/(1 cmnm−)/(1 cmnm+
)/(1 cmnm−)/(2 cmnm−)/(4 cmnm−
)/(5.2 cmnm−Null )/(5.2 cmnm− Function
“Chirp parameter”
( F )Null Null Null 100− Null Null
Table 5 - 1 Grating parameters of the simulation in this part
CHIRPED FIBRE BRAGG GRATINGS
65
5.2.1 Linear chirped gratings with different chirp
variables
The first simulation of linear chirped gratings is for two gratings with the same
parameters. Only the sign of the chirp variable dzd D /λ is reversed.
Figure 5 - 2 contains the simulation results of the reflectance with changed chirp
variable dzd D /λ and the same refractive index nδ and grating length L . If
dzd D /λ is positive, the period of the linear chirped grating increases along the
propagation direction. On the other hand, if dzd D /λ is negative, the period of the
linear chirped grating reduces along the propagation direction. In the simulation
program, the value of dzd D /λ can be positive or negative. In the plot of Figure 5
- 2(a), we can see that the spectral responses of these two linear chirped
gratings are shifted from the designed Bragg wavelength. If dzd D /λ is negative,
the centre wavelength of the grating moves to the left hand side (shorter λ ). If
dzd D /λ is positive, the centre wavelength of the grating moves to the right hand
side (longer λ ). Both of them have the same 3 dB bandwidth, which is shown in
Figure 5 - 2(a). Figure 5 - 2(b) contains the time delay plots of these two linear
chirped gratings. The time delay can be obtained from the equations (3 - 28) and
(3 - 29).
CHIRPED FIBRE BRAGG GRATINGS
66
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5541.5521.551.5481.546
Re
flect
an
ce (
p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5541.5521.551.5481.546
Tim
e de
lay
(ps)
140
120
100
80
60
40
20
0
-20
-40
-60
(b)
Figure 5 - 2 The reflectance spectrum of two chirped gratings with an equal
chirp of opposite signs. The values of the chirp variables are
)/(1/ cmnmdzd D −=λ (red solid line) and )/(1/ cmnmdzd D =λ (green dashed
line), and the following parameters: )(10000 mL µ= , 447.1=effn , 0004.0=nδ ,
)(550.1 mD µλ = . (a) is the linear scale, (b) is the time delay
CHIRPED FIBRE BRAGG GRATINGS
67
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5541.5521.551.5481.5461.5441.542
Re
flect
an
ce (
p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5 - 3 The reflectance spectrum of three linear chirped gratings with
different values of the chirp variable: )/(1/ cmnmdzd D −=λ (red solid line),
)/(2/ cmnmdzd D −=λ (green dashed line) and )/(4/ cmnmdzd D −=λ (blue
dotted line), and the following parameters: )(10000 mL µ= , 447.1=effn ,
0004.0=nδ , )(550.1 mD µλ =
Figure 5 - 3 shows that the 3dB bandwidth of the reflectance spectrum is
increased when the value of the chirp variable dzd D /λ is increased, whereas the
reflectance is reduced. This is not what we expected intuitively. So we can see
that the increased bandwidth results in the reduced reflectance at the same time.
Figure 5 - 3 also shows that the centre wavelength is shifted with different the
values of the chirp variable. This feature can be utilized in sensor systems, and
will be discussed later in this chapter.
CHIRPED FIBRE BRAGG GRATINGS
68
5.2.2 Linear chirped gratings with different lengths
Figure 5 - 4 shows the reflectance spectrum of linear chirped gratings with
different lengths where the chirp variable dzd D /λ is the same.
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5541.5521.551.5481.5461.5441.542
Re
flect
an
ce (
p.
u.)
0.750.7
0.650.6
0.550.5
0.450.4
0.350.3
0.250.2
0.150.1
0.05
Figure 5 - 4 The reflectance spectrum of three linear chirped gratings with
different lengths: )(5000 mL µ= (red solid line), )(7000 mL µ= (green dashed
line), )(10000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0004.0=nδ , )(550.1 mD µλ = , )/(5.2/ cmnmdzd D −=λ
In Figure 5 - 4, we used the same value of the chirp variable dzd D /λ in the
simulation. The maximum reflectance is almost the same whereas the length of
the chirped grating is increased. The bandwidth increase is proportional to the
length.
CHIRPED FIBRE BRAGG GRATINGS
69
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5541.5521.551.5481.5461.5441.542
Re
flect
an
ce (
p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5 - 5 The reflectance spectrum of three linear chirped gratings with
different lengths: )(5000 mL µ= (red solid line), )(7000 mL µ= (green dashed
line), )(10000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0004.0=nδ , )(550.1 mD µλ = 100−=F .
Figure 5 - 5 shows the reflectance spectrum of linear chirped gratings with
different lengths, and with the same “chirp parameter” F . The reflectance is
increased when the length of the grating is increased. At the same time, the
bandwidth of the spectrum is reduced. The different behaviour comes from the
equation (5 - 3) where 22)( LFWHMF ∝∝ . Only for the same length of the linear
chirped grating, the “chirp parameter” ./ dzdF Dλ∝
5.2.3 Linear chirped gratings with different refractive
index changes
Figure 5 - 6 shows the reflectance spectrum of linear chirped gratings with
different values of the index changed nδ .
CHIRPED FIBRE BRAGG GRATINGS
70
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5541.5521.551.5481.5461.5441.542
Re
flect
an
ce (
p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5 - 6 The reflectance spectrum of three linear chirped gratings with
different values of the index change: 0002.0=nδ (red solid
line), 0004.0=nδ (green dashed line), 0006.0=nδ (blue dotted line), and the
following parameters: )(10000 mL µ= , 447.1=effn ,
)(550.1 mD µλ = , )/(5.2/ cmnmdzd D −=λ
The reflectance is increased with increasing nδ values. At the same time, the
3dB bandwidth of the reflectance is increased slightly. In this simulation, the
value of chirp variable dzd D /λ is kept constant. The increase in the index value
nδ is limited by the fabrication technology used for the grating. The index
change nδ can only be changed in certain limited range. A flat reflectance
spectrum can also be obtained by increasing the length of the grating as shown
in Figure 5 - 4.
5.2.4 Arbitrary chirped Bragg grating
This simulation program can also simulate chirped Bragg gratings with arbitrary
chirp functions.
CHIRPED FIBRE BRAGG GRATINGS
71
Ref-
dn4
L10
000
c10
--
10.
wmf
Wavelength (micrometre)1.5511.55051.551.54951.549
Re
flect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5 - 7 The reflectance spectrum of chirped grating with arbitrary chirpfunction, and the following parameters: )(10000 mL µ= , 447.1=effn ,
)(550.1 mD µλ = , 0004.0=nδ .
Figure 5 - 7 shows a reflectance spectrum of fibre gratings with an arbitrary chirp
function, which is )/]()3.0
2/(exp[5.2 2 cmnm
LLz −
−− in this simulation. Further
research on the different kinds of chirp functions can be processed by this
simulation program. Depending on the different applications, a grating with
various chirp functions can be simulated to optimize the characteristics. The
analysis of arbitrary chirp function gratings will not be discussed in this thesis.
5.3 Relationship between the maximum reflectance
and the chirped grating coefficients
The maximum reflectance is an important variable for the chirped Bragg grating.
It is determined by the chirp variable or “chirp parameter”, the Gauss width
CHIRPED FIBRE BRAGG GRATINGS
72
parameter, which it is a parameter of the Gauss apodization function, and will be
discussed in the next chapter, the index change and the grating length. The
following simulation results will show the relationship between the maximum
reflectance and the chirped grating coefficients in the grating. The relationship
between the maximum reflectance and the chirped grating coefficients can be
used to analyse the potential application of the grating in sensors and
communication systems.
Grating parameters Figure 5 - 8 Figure 5 - 9 Figure 5 - 10
Grating length
( )L
)(10000 mµ)(15000 mµ)(20000 mµ
x axis )(10000 mµ
Index change
( nδ )0002.0 0004.0 x axis
Effective index
( effn )447.1 447.1 447.1
Design grating
wavelength ( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ
“Chirp parameter”
( F )x axis
100=F150=F200=F
100=F150=F200=F
Table 5 - 2 Grating parameters of the simulation in this part
5.3.1 The maximum reflectance and the “chirp
parameter”
Figure 5 - 8 is a plot of the relationship between the maximum reflectance and
the “chirp parameter” F . It shows that the maximum reflectance is reduced if the
“chirped parameter” F is increased. The speed of the reduction is slower if the
grating length is increased. Because the longer grating can obtain a larger
CHIRPED FIBRE BRAGG GRATINGS
73
optical power of reflectance than that of a short length grating. The simulation
results satisfy the theory.
Chirp parameter180160140120100806040200
Max
imum
re
flect
an
ce (
p.
u.)
0.950.9
0.850.8
0.750.7
0.650.6
0.550.5
0.450.4
0.35
Figure 5 - 8 The maximum reflectance vs. the “chirp parameter” F with
different chirped lengths: )(10000 mL µ= (red solid line), )(15000 mL µ= (green
dashed line), )(20000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0002.0=nδ , )(550.1 mD µλ =
5.3.2 The maximum reflectance and the length of the
grating
Figure 5 - 9 is a plot of the relationship between the maximum reflectance and
the grating length. It shows that the maximum reflectance is increased if the
grating length is increased, until it reaches saturation. It is faster to reach
saturation value of the maximum reflectance when the “chirp parameter” F is
smaller.
CHIRPED FIBRE BRAGG GRATINGS
74
Grating length (micrometre)15,00010,0005,000
Max
imum
re
flect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 5 - 9 The maximum reflectance vs. the grating length with different
values of the “chirped parameter”: 0=F (red solid line), 100=F (green
dashed line), 200=F (blue dotted line), and the following parameters:
)(10000 mL µ= , 447.1=effn , 0004.0=nδ , )(550.1 mD µλ =
5.3.3 The maximum reflectance and the index change
Figure 5 - 10 is a plot of the relationship between the maximum reflectance and
the index change. It shows that the maximum reflectance is increased while the
index change is increased, until it reaches saturation. It is faster to reach the
saturation value of the maximum reflectance when the “chirp parameter” F is
smaller.
The relationship between the maximum reflectance and the Gauss distribution
parameter will be discussed in the next chapter.
CHIRPED FIBRE BRAGG GRATINGS
75
Index change0.00090.00080.00070.00060.00050.00040.0003
Max
imum
re
flect
an
ce (
p.
u.)
0.950.9
0.850.8
0.750.7
0.650.6
0.550.5
0.450.4
0.350.3
Figure 5 - 10 The maximum reflectance vs. the index change with different
values of the “chirp parameter”: 100=F (red solid line), 150=F (green dashed
line), 200=F (blue dotted line), and the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ =
5.4 Relationship between the 3 dB bandwidth and the
chirped grating coefficients
The 3dB bandwidth is another important property of the chirped Bragg grating. It
also depends on the “chirp parameter” or the chirp variable, the Gauss
distribution parameter, the index change and the grating length. The simulation
results will show the relationship between the 3 dB bandwidth and the chirped
grating coefficients in the grating.
CHIRPED FIBRE BRAGG GRATINGS
76
Grating parameters Figure 5 - 11 Figure 5 - 12 Figure 5 - 13 Figure 5 - 14
Grating length
( )L
)(10000 mµ)(15000 mµ)(20000 mµ
)(10000 mµ)(15000 mµ)(20000 mµ
x axis )(10000 mµ
Index change
( nδ )0004.0 0004.0 0004.0 x axis
Effective index
( effn )447.1 447.1 447.1 447.1
Design grating
wavelength ( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ )(550.1 mµ
Chirp variable
( dzd D /λ )x axis Null Null Null
“Chirp parameter”
( F )Null x axis
100=F150=F200=F
100=F150=F200=F
Table 5 - 3 Grating parameters of the simulation in this part
5.4.1 The reflectance bandwidth and the “chirp
parameter” (or chirp variable)
Figure 5 - 11 and Figure 5 - 12 are the plots of the relationship between the 3 dB
bandwidth and the chirp variable dzd D /λ and “chirp parameter” F respectively.
The 3 dB bandwidth is increased if the chirp variable (or “chirp parameter”) is
increased. They have a linear relationship between them. The bandwidth is more
sensitive to the chirp for the shorter grating.
CHIRPED FIBRE BRAGG GRATINGS
77
Chirp variable (nanometre/centimetre)210
3 d
B b
andw
idth
(nanom
etr
e) 8
7
6
5
4
3
2
1
Figure 5 - 11 The 3 dB bandwidth vs. the chirp variable with different chirped
lengths: )(10000 mL µ= (red solid line), )(15000 mL µ= (green dashed line),
)(20000 mL µ= (blue dotted line), and the following parameters: 447.1=effn ,
0004.0=nδ , )(550.1 mD µλ = .
Chirp parameter180160140120100806040200
3 d
B b
andw
idth
(nanom
etr
e)
4
3
2
1
Figure 5 - 12 The 3 dB bandwidth vs. the “chirp parameter” F with different
grating lengths: )(10000 mL µ= (red solid line), )(15000 mL µ= (green dashed
line), )(20000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0004.0=nδ , )(550.1 mD µλ = .
CHIRPED FIBRE BRAGG GRATINGS
78
5.4.2 The reflectance bandwidth and the length of the
grating
Grating length (micrometre)14,00012,00010,0008,0006,000
3 dB
ban
dwid
th (
nano
met
re)
9
8
7
6
5
4
3
Figure 5 - 13 The 3 dB bandwidth vs. the grating length with different values
of the “chirp parameter”: 100=F (red solid line, 150=F (green dashed line),
200=F (blue dotted line), and the following parameters: 447.1=effn ,
0004.0=nδ , )(550.1 mD µλ =
Figure 5 - 13 is a plot of the relationship between the 3 dB bandwidth and the
grating length. It shows that the 3 dB bandwidth is reduced if the grating length is
increased. The rate of reduction is slower if the “chirp parameter” is increased.
CHIRPED FIBRE BRAGG GRATINGS
79
5.4.3 The reflectance bandwidth and the index change
Index change0.00090.00080.00070.00060.00050.00040.0003
3 dB
ban
dwid
th (
nano
met
re)
4
3
2
Figure 5 - 14 The 3 dB bandwidth vs. the index change with a different
“chirped parameter”: 100=F (red solid line), 150=F (green dashed line),
200=F (blue dotted line), and the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ =
Figure 5 - 14 is a plot of the relationship between the 3 dB bandwidth and the
index change. It shows that the 3 dB bandwidth is increased if the index change
is increased. They have approximately a linear relationship. The rate of the
increase is the same for the different “chirp parameter” F .
The relationship between the 3 dB bandwidth and the Gauss distribution
parameter will be discussed in the next chapter.
5.5 Relationship between the centre wavelength and
the chirped grating coefficients
The centre wavelength is an important variable in the chirped Bragg grating. It is
dependent on the “chirp parameters” F or the chirp variable, the Gauss
CHIRPED FIBRE BRAGG GRATINGS
80
distribution parameter, the index change and the grating length. The simulation
results will show the relationship between the centre wavelength and the chirped
grating coefficient.
Grating parameters Figure 5 - 15 Figure 5 - 16 Figure 5 - 17Figure 5 - 18
Figure 5 - 19
Grating length
( )L
)(10000 mµ)(15000 mµ)(20000 mµ
)(10000 mµ)(15000 mµ)(20000 mµ
)(10000 mµ x axis
Index change
( nδ )0004.0 0002.0 x axis 0004.0
Effective index
( effn )447.1 447.1 447.1 447.1
Design grating
wavelength ( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ )(550.1 mµ
Chirp variable
( dzd D /λ )x axis Null Null Null
“Chirp parameter”
( F )Null x axis
100=F150=F200=F
100±=F150±=F200±=F
Table 5 - 4 Grating parameters of the simulation in this part
5.5.1 The centre wavelength and the “chirp parameter”
(or chirp variable)
Figure 5 - 15 and Figure 5 - 16 are the plots of the centre wavelength versus the
chirped variable dzd D /λ in Figure 5 - 15 or versus “chirped parameter” F in
Figure 5 - 16. They show that the centre wavelength is increased if the chirp
variable dzd D /λ or “chirp parameter” F is increased. They have a linear
dependence on them, with a steeper slope. It is faster for the short grating. This
feature can be used in the design of fibre sensor systems.
CHIRPED FIBRE BRAGG GRATINGS
81
Chirp variable (nanometre/centimetre)210
Centr
e w
ave
length
(nanom
etr
e)
1,554
1,553
1,552
1,551
Figure 5 - 15 The centre wavelength vs. the chirp variable with different
grating lengths: )(10000 mL µ= (red solid line), )(15000 mL µ= (green dashed
line), )(20000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0004.0=nδ , )(550.1 mD µλ =
Chirp parameter180160140120100806040200
Centr
e w
ave
length
(nanom
etr
e)
1,552
1,551
Figure 5 - 16 The centre wavelength vs. the “chirp parameter” F with different
grating Lengths: )(10000 mL µ= (red solid line), )(15000 mL µ= (green dashed
line), )(20000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0004.0=nδ , )(550.1 mD µλ =
CHIRPED FIBRE BRAGG GRATINGS
82
5.5.2 The centre wavelength and the index change
Index change0.00090.00080.00070.00060.00050.00040.0003
Cen
tre
wav
elen
gth
(nan
omet
re)
1,553
1,552
Figure 5 - 17 The centre wavelength vs. the index change with different
values of the “chirp parameter”: 100=F (red solid line), 150=F (green dashed
line), 200=F (blue dotted line), and the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ =
Figure 5 - 17 is a plot of the relationship between the centre wavelength and the
index change. It shows that the centre wavelength is increased if the index
change is increased. They have a linear relationship. The rate of the increase is
the same for the different “chirp parameter”. This feature can be used in the
design of fibre sensor systems.
CHIRPED FIBRE BRAGG GRATINGS
83
5.5.3 The centre wavelength and the length of the
grating
Grating length (micrometre)16,00014,00012,00010,0008,0006,000
Cen
tre
wav
elen
gth
(nan
omet
re)
1,555
1,554
1,553
1,552
Figure 5 - 18 The centre wavelength vs. the grating length with different
values of the “chirp parameter”: 100=F (red solid line), 150=F (green dashed
line), 200=F (blue dotted line), and the following parameters: 447.1=effn ,
0004.0=nδ , )(550.1 mD µλ =
Figure 5 - 18 and Figure 5 - 19 are the plots of the relationship between the
centre wavelength and the grating length. The “chirp parameter” is positive in
Figure 5 - 18, and negative in Figure 5 - 19. The centre wavelength is shifted
with respect to the design value, depending on the sign of the “chirp parameter”
F .
CHIRPED FIBRE BRAGG GRATINGS
84
Grating length (micrometre)16,00014,00012,00010,0008,0006,000
Cen
tre
wav
elen
gth
(nan
omet
re)
1,549
1,548
1,547
1,546
Figure 5 - 19 The centre wavelength vs. the grating length with different
values of the “chirp parameter”: 100−=F (red solid line), 150−=F (green
dashed line), 200−=F (blue dotted line), and the following parameters:
447.1=effn , 0004.0=nδ , )(550.1 mD µλ =
The relationship between the centre wavelength and the Gauss width parameter
will be discussed in the next chapter.
5.6 Dispersion compensation
The loss limitation in the long-haul high-bit-rate communication systems was
solved by using an erbium-doped fibre amplifier [4]. The main limitation now in
transmitting over a long distance is the pulse broadening due to dispersion [5].
There are two popular methods to eliminate the dispersion: one is by using a
dispersion compensating fibre, which is better suited to compensate over a wide
CHIRPED FIBRE BRAGG GRATINGS
85
range of wavelengths. Another solution is the insertion of chirped Bragg grating
dispersion compensators. The use of chirped gratings for dispersion
compensation in optical communication systems was first proposed by F.
Ouellette [6]. A linearly chirped fibre Bragg grating can be used as a dispersion
compensator [7]. Chirped gratings are ideally suited to compensate for individual
wavelengths. In contrast, a dispersion compensating fibre is better suited to
compensate over a wide range of wavelengths. However, it introduces higher
loss and additional penalties due to increased non-linearities, compared to
chirped gratings.
In a chirped grating, the resonant frequency is a linear function of the axial
position z along the grating so that different wavelengths acquire different delay
times. Chirped gratings can be used as dispersion compensators to compress
temporally broadened pulses. This is illustrated in Figure 5 - 20. A chirped Bragg
grating effectively introduces different delays at different frequencies.
λ1 λ2 ... λn
Compensated pulse
Uncompensated pulse
λnλ1 λ2 ...
Figure 5 - 20 Diagram for illustrating the principle of using a chirped grating for
dispersion compensation
Figure 5 - 21 is a diagram of optical fibre dispersion compensation with a chirped
Bragg grating. Figure 5 - 22 is a diagram of chirped fibre Bragg gratings for
compensating three wavelengths in a WDM system.
CHIRPED FIBRE BRAGG GRATINGS
86
Tx
SMF SMF
Rx
CirculatorEDFA
Input pulse Broadened pulse Recompressed pulse
Linear chirped grating
Figure 5 - 21 Diagram of optical fibre dispersion compensation with a chirped
Bragg grating
output
input1λ
2λ 3λ
Figure 5 - 22 Chirped fibre Bragg gratings for compensating three
wavelengths in a WDM system
CHIRPED FIBRE BRAGG GRATINGS
87
Wavelength (micrometre)1.5521.55151.5511.55051.551.54951.549
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Wavelength (micrometre)1.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.750.7
0.650.6
0.550.5
0.450.4
0.350.3
0.250.2
0.150.1
0.05
(a)
Wavelength (micrometre)1.5521.55151.5511.55051.551.5495
Tim
e de
lay
(ps)
130
120
110
100
90
80
70
60
50
40
30
20
10
Wavelength (micrometre)1.5521.551.5481.5461.544
Tim
e de
lay
(ps)
120110100
908070605040302010
0-10-20
(b)
Wavelength (micrometre)1.5521.55151.5511.55051.551.5495
Dis
pers
ion
(ps/
nm)
2,000
1,500
1,000
500
0
-500
-1,000
-1,500
-2,000
Wavelength (micrometre)1.5521.551.5481.5461.544
Dis
pers
ion
(ps/
nm)
200
150
100
50
0
-50
-100
-150
-200
(c)
Figure 5 - 23 Frequency response of uniform and linear chirped grating, and
the following parameters: )(10000 mL µ= , 447.1=effn , 0004.0=nδ ,
)(550.1 mD µλ = , )/(5.2/ cmnmdzd D −=λ for chirped grating.
(a) is reflectance (b) is time delay (c) is dispersion.
Figure 5 - 23 contains the simulation results for the reflectance spectrum, time
delay, and dispersion of a uniform (left hand side) and linearly chirped Bragg
CHIRPED FIBRE BRAGG GRATINGS
88
grating (right hand side). The dispersion can be obtained by equations (3 – 30)
and (3 - 31).
This figure shows that the reflection bandwidth of the chirped grating increased
compared with the uniform grating, but the reflectance is reduced at the same
time. Time delay changes almost linearly with wavelength. There are ripples in
both spectra and time delays [8],[9]. This problem can be eliminated by using
apodization of the refractive index. This will be discussed in the next chapter.
5.6.1 Simulation results
The time delay and dispersion of chirped gratings with the different chirp variable
dzd D /λ , length and index change were obtained by simulations, Figure 5 - 24,
Figure 5 - 25 and Figure 5 - 26. Suitable parameters for linear chirped gratings
may be optimized by these simulation results.
Grating parameters Figure 5 - 24 Figure 5 - 25 Figure 5 - 26
Grating length
( )L)(10000 mµ
)(6000 mµ)(8000 mµ)(10000 mµ
)(10000 mµ
Index change
( nδ )0004.0 0002.0
0002.0
0004.0
0006.0
Effective index
( effn )447.1 447.1 447.1
Design grating
wavelength ( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ
Chirp variable
( dzd D /λ )
)/(5.1 cmnm−)/(5.2 cmnm−)/(5.3 cmnm−
)/(5.2 cmnm− )/(5.2 cmnm−
Table 5 - 5 The simulation results of the spectral response
CHIRPED FIBRE BRAGG GRATINGS
89
Wavelength (micrometre)1.5521.551.5481.5461.5441.542
Re
flect
ance
(dB
)
-10
-20
-30
-40
-50
-60
-70
-80
-90
(a)
Wavelength (micrometre)1.5521.551.5481.5461.544
Tim
e de
lay
(ps)
110100
908070605040302010
0-10
(b)
CHIRPED FIBRE BRAGG GRATINGS
90
Wavelength (micrometre)1.5521.551.5481.5461.544
Dis
pe
rsio
n (p
s/nm
)250
200
150
100
50
0
-50
-100
-150
-200
-250
(c)
Figure 5 - 24 The simulation results of three linear chirped gratings with
different values of chirp variable: )/(5.1/ cmnmdzd D −=λ (red solid
line), )/(5.2/ cmnmdzd D −=λ (green dashed line), )/(5.3/ cmnmdzd D −=λ (blue
dotted line), and the following parameters: )(10000 mL µ= , 447.1=effn ,
0004.0=nδ , )(550.1 mD µλ = . (a) is reflectance, (b) is time delay, and (c) is
dispersion
CHIRPED FIBRE BRAGG GRATINGS
91
Wavelength (micrometre)1.5521.5511.551.5491.5481.5471.5461.5451.544
Re
flect
ance
(dB
)-5
-10-15-20-25-30-35-40-45-50-55-60-65-70-75
(a)
Wavelength (micrometre)1.5521.5511.551.5491.5481.5471.5461.545
Tim
e de
lay
(ps)
1009080706050
40302010
0-10-20
(b)
CHIRPED FIBRE BRAGG GRATINGS
92
Wavelength (micrometre)1.5521.5511.551.5491.5481.5471.5461.545
Dis
pe
rsio
n (p
s/nm
)150
100
50
0
-50
-100
-150
-200
(c)
Figure 5 - 25 The simulation results of three linear chirped gratings with
different chirped lengths: )(6000 mL µ= (red solid line), )(8000 mL µ= (green
dashed line), )(10000 mL µ= (blue dotted line), and the following parameters:
447.1=effn , 0002.0=nδ , )(550.1 mD µλ = , )/(5.2/ cmnmdzd D −=λ .
(a) reflectance, (b) time delay, and (c) dispersion
Wavelength (micrometre)1.5521.5511.551.5491.5481.5471.5461.5451.544
Re
flect
ance
(dB
)
-10
-20
-30
-40
-50
-60
-70
-80
(a)
CHIRPED FIBRE BRAGG GRATINGS
93
Wavelength (micrometre)1.5521.5511.551.5491.5481.5471.5461.5451.544
Tim
e de
lay
(ps)
110
100
90
80
70
60
50
40
30
20
10
0
-10
(b)
Wavelength (micrometre)1.5521.551.5481.5461.544
Dis
pe
rsio
n (p
s/nm
)
140120100
80604020
0-20-40-60-80
-100-120-140
(c)
Figure 5 - 26 The simulation results of three linear chirped gratings with
different index changes: 0002.0=nδ (red solid line), 0004.0=nδ (green
dashed line), 0006.0=nδ (blue dotted line), and the following parameters:
)(10000 mL µ= , 447.1=effn , )(550.1 mD µλ = , )/(5.2/ cmnmdzd D −=λ .
(a) is reflectance, (b) is time delay, and (c) is dispersion
CHIRPED FIBRE BRAGG GRATINGS
94
The main disadvantage of fibre Bragg gratings for the application of optical
communications is the thermal dependence of its centre wavelength. This is a
major factor that needs to be addressed. Reducing the thermal variability of the
fibre Bragg grating is the key to improve the commercial applications in
communication system.
5.7 Sensor applications
Fibre chirped Bragg gratings can also be used for optical fibre sensor
systems[10]. The basic principle of this application is the measurement of the
change of the reflection wavelength. The measurands can be strain,
temperature, pressure, etc.
)()(2)( zznz effB Λ=λ 5 - 5
where )(zneff and )(zΛ may vary along the grating due to the effects of the
measurand.
∆Λ+Λ∆=∆ effneff n22λ 5 - 6
The shift of the reflection wavelength is caused by changes in the refractive
index effn∆ and in the Bragg period ∆Λ .
Table 5 - 6 shows the simulation results of linear chirped gratings in sensor
applications.
Table 5 - 6 The simulation results of linear chirped gratings in sensor applications
CHIRPED FIBRE BRAGG GRATINGS
95
Wavelength (micrometre)1.5521.5511.551.5491.5481.5471.5461.5451.544
Re
flect
ance
(dB
)-10
-20
-30
-40
-50
-60
-70
-80
Figure 5 - 27 The reflectance spectrum of the chirped gratings with the
different values of the chirp variable: )/(5.0/ cmnmdzd D −=λ (red solid line),
)/(1/ cmnmdzd D −=λ (green dashed line), )/(5.1/ cmnmdzd D −=λ (blue dotted
line), )/(2/ cmnmdzd D −=λ (pink dashed and dotted line),
)/(5.2/ cmnmdzd D −=λ (dark blue dashed, dotted, dotted line), and the
following parameters: )(10000 mL µ= , 447.1=effn , 0004.0=nδ , )(550.1 mD µλ =
Figure 5 - 27 and Figure 5 - 28 are reflection spectra of chirped gratings with
different values of the chirp variable. This shows that the centre wavelength is
changed when the values of chirp variable dzd D /λ are changed.
CHIRPED FIBRE BRAGG GRATINGS
96
Wavelength (micrometre)1.5561.5541.5521.551.548
Re
flect
ance
(dB
)-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
Figure 5 - 28 The reflectance spectrum of the chirped grating with the different
values of chirp variable: )/(5.0/ cmnmdzd D =λ (red solid line),
)/(1/ cmnmdzd D =λ (green dashed line), )/(5.1/ cmnmdzd D =λ (blue dotted
line), )/(2/ cmnmdzd D =λ (pink dashed and dotted line), )/(5.2/ cmnmdzd D =λ
(dark blue dashed, dotted, dotted line), and the following parameters:
)(10000 mL µ= , 447.1=effn , 0004.0=nδ , )(550.1 mD µλ =
Figure 5 - 29 shows how a chirped grating apples to sensors [11]. This method
transforms a uniform fibre grating into a chirped one. By glueing a uniform
grating to the cantilever beam, and pressing the right hand side of the cantilever
beam, the period of the grating will be linearly variable. It causes the chirp
variable dzd D /λ to be changed linearly. This means, the pressure that is applied
to the cantilever beam causes a shift in the centre wavelength. By measuring the
shift of the centre wavelength, the values of the pressure can be obtained.
CHIRPED FIBRE BRAGG GRATINGS
97
Uniform Grating
pressure
Figure 5 - 29 Cantilever beam tuning structure
For fibre grating sensor systems, the linear relationship between the measurand
and output value are expected. The following simulation results will show this
property of the linear chirped gratings.
Chirp variable (nanometre/centimetre)21
Cen
tre
wav
elen
gth
(nan
omet
re)
1,552
1,551
Figure 5 - 30 The centre wavelength vs. the chirp variable with different
values of the index change: 0002.0=nδ (red solid line), 0004.0=nδ (green
dashed line), 0006.0=nδ (blue dotted line), and the following parameters:
)(10000 mL µ= , 447.1=effn , )(550.1 mD µλ =
CHIRPED FIBRE BRAGG GRATINGS
98
Figure 5 - 30 shows the relationship between the centre wavelength with the
chirp variable dzd D /λ . The centre wavelength is almost linearly related to the
chirp variable, and the rate of the change is almost the same with different
values of the index change. This feature makes the chirped grating a suitable
component for optical fibre sensor systems.
5.8 Conclusion
The features and applications of the linear chirped Bragg gratings were analyzed
and discussed in this chapter. Chirped gratings have some advantages
compared with the uniform gratings, such as a broader bandwidth, linear time
delay, etc. These properties provide many applications in optical fibre
communications and sensor systems.
CHIRPED FIBRE BRAGG GRATINGS
99
5.9 References
1. Y. Chen and S. Jian, “An introduction to lightwave technology”, (China
Railway Publishing), 2000, p. 249.
2. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology, vol.15,
no.8, 1997, pp. 1277-1294.
3. H. Kogelnik, “Filter response on nonuniform almost-periodic structures”, Bell
System Technical Journal, vol.55, no.1, 1976, pp. 109-126.
4. N. Kikuchi and S. Sasaki, “Analytical evaluation technique of self-phase
modulation effect on the performance of cascaded optical amplifier systems,”
Journal of Lightwave Technology, vol.13, no.5, 1995, pp. 868-878.
5. G. Smith, D. Novak, and Z. Ahmed, "Overcoming chromatic-dispersion
effects in fibre-wireless systems incorporating external modulators", IEEE
Transactions on Microwave Theory and Techniques, vol. 45, no. 8, 1997, pp.
1410-1415.
6. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating
filters in optical waveguides,” Optics Letters, vol.12, no.10, 1987, pp. 847-849.
7. F. Ouellette, “Optical equalization with linearly tapered two-dissimilar-core
fibre”, Electronics Letters, vol.27, 1991, pp. 1668-1669.
8. L. Poladian, "Analysis and modeling of group delay ripple in Bragg gratings".
In Bragg Gratings, Photosensitivity, and Poling in Glass Waveguides of OSA
Technical Digest (1999), pp. 258-260.
9. K. Ennser, M. Ibsen, M. Durkin, M. Zervas, and R. Laming, “Influence of
nonideal chirped fibre grating characteristics on dispersion cancellation”, IEEE
Photonics Technology Letters, vol. 10, 1998, pp. 1476-1478.
10. W. Morey, G. Meltz, and W. Glenn, “Fiber-optic Bragg grating sensors”,
Proc SPIE, Fiber Optic and Laser Sensors VII, vol.1169, 1989, pp. 98-107.
11. Y. Yu, Z. Liu, X. Dong, J. Wang, S. Geng and K. S. Chiang, “Linear tuning of
fiber Bragg grating based on a cantilever”, Acta Optica Sinica, vol.19, no.5,
1999, pp. 621-625.
APODIZATION OF FIBRE BRAGG GRATINGS
100
CHAPTER 6: Apodization of Fibre BraggGratings
CHAPTER 6: APODIZATION OF FIBRE BRAGG GRATINGS 100
6.1 THE PRINCIPLE OF APODIZED GRATINGS .................................................. 101
6.1.1 Direct integration method ................................................................ 101
6.1.2 Transfer matrix method ................................................................... 102
6.1.3 Apodization functions ...................................................................... 102
6.2 SPECTRAL RESPONSE OF APODIZED GRATINGS........................................ 104
6.2.1 Refractive index and spectral response.......................................... 104
6.2.2 Comparison of the properties of apodized and unapodized uniform
gratings........................................................................................................ 105
6.2.3 The apodization of linear chirped gratings with different Gauss width
parameters ...................................................................................................110
6.2.4 The apodization of linear chirped gratings with different Kaiser
window parameters......................................................................................111
6.3 RELATIONSHIP BETWEEN THE MAXIMUM REFLECTANCE AND THE GAUSS WIDTH
PARAMETERS .....................................................................................................112
6.4 RELATIONSHIP BETWEEN THE 3 DB BANDWIDTH AND THE GAUSS WIDTH
PARAMETERS .....................................................................................................113
6.5 DISPERSION COMPENSATION USING A LINEAR CHIRPED GRATING WITH
APODIZATION .....................................................................................................114
6.5.1 Optimization of the Gauss width parameters for dispersion
compensation...............................................................................................117
6.6 CONCLUSION......................................................................................... 121
6.7 REFERENCES ........................................................................................ 122
APODIZATION OF FIBRE BRAGG GRATINGS
101
6.1 The principle of apodized gratings
The refractive index change is constant in uniform fibre Bragg gratings. The
reflectance spectrum of a finite-length Bragg grating with a uniform modulation of
the refractive index is accompanied by a series of sidelobes at the adjacent
wavelengths. It is very important to minimize and, if possible, eliminate the
reflectivity of these sidelobes [1].
Figure 2-3 shows the reflectance and transmittance spectra of a uniform Bragg
grating, which have large sidelobes. These features of the uniform Bragg grating
should be improved for applications in communication systems. One method is
by using apodization [2]. Apodization can be achieved by a contoured exposure
to UV light to reduce the refractive index excursions towards both ends of the
grating.
The apodized fibre Bragg grating can be modelled by the coupled-mode theory.
Two simulation methods can be used to solve the coupled-mode equations.
6.1.1 Direct integration method
The effect of the apodization in the models of the Bragg grating can be
represented by using a z -dependent function )(zg in the refractive index [3].
The refractive index of an apodized Bragg grating can be written as
))(2
cos()()( 00 zzzgnnnzn ϕπ
δδ +Λ
++= 6 - 1
where nδ is the depth of the modulation, and )(zg is the modulation function
(also called the apodization function). Generally, this function can be Gaussian,
raised cosine, etc. The apodization function is 1)( =zg for the (unchirped)
uniform Bragg grating.
APODIZATION OF FIBRE BRAGG GRATINGS
102
The coupling coefficient of the apodized fibre Bragg grating is given in equation
(3 – 7). If we substitute equation (3 - 7) into coupled-mode equations (3 - 2) and
(3 - 3), the spectral response of the apodized grating can be obtained by solving
these equations.
6.1.2 Transfer matrix method
By substituting equation (3 - 7) into the transfer matrix equation (3 - 26), the
spectral response of the apodized grating can be obtained by solving these
equations.
6.1.3 Apodization functions
Several apodization functions were built into this simulation program. The user
can also define arbitrary types of functions.
The apodization functions that can be used in the simulations are listed as
follows [4]:
6.1.3.1 Uniform grating
1)( =zg ; ],0[ Lz ∈ 6 - 2
6.1.3.2 Gaussian profile
}])()2/(2
[2lnexp{)( 2
FWHM
Lzzg
−−= ; ],0[ Lz ∈ 6 - 3
where LFWHM 4.0= can be used for this profile [2].
Another expression for the Gaussian profile is as follows [5]:
APODIZATION OF FIBRE BRAGG GRATINGS
103
])2/
(exp[)( 2
L
Lzazg
−−=
],0[ Lz ∈
6 - 4
where a is the Gauss width parameter.
6.1.3.3 Raised-cosine profile
)])()2/(
cos(1[21
)(FWHM
Lzzg
−+=
π; ],0[ Lz ∈ 6 - 5
where LFWHM = can be used for this profile [2].
6.1.3.4 Sinc profile:
])()2/(
sinc[)(FWHM
Lzzg
−= ; ],0[ Lz ∈ 6 - 6
where )2/( πLFWHM = can be used for this profile [4].
6.1.3.5 Kaiser profile:
)(
))1
2(1(
)(0
20
k
k
IN
nI
zgβ
β−
−= ; ]1,0[ −∈ Nn
6 - 7
where kβ is the Kaiser window parameter [6], and 0I is the zero order Bessel
function of the first kind.
APODIZATION OF FIBRE BRAGG GRATINGS
104
6.2 Spectral response of apodized gratings
6.2.1 Refractive index and spectral response
The spectral responses of Bragg gratings with different types of refractive index
profile functions are shown in this section.
Figure 6 - 1 shows four profile functions of the index change. Figure 6 - 2 shows
four reflectance spectrums of the uniform gratings with four apodization
functions, which are shown in Figure 6 - 1. In this case, the average refractive
index changes along the length of the grating. There is a Fabry-Perot resonance
ripple on the short wavelength side of the reflectance spectrum [7]. Figure 6 - 1
shows that the maximum reflectance is increased while the sidelobes are
increased.
Position (millimetre)108642
Re
fra
ctiv
e in
dex
0.00020
0.00018
0.00016
0.00014
0.00012
0.00010
0.00008
0.00006
0.00004
0.00002
0.00000
Figure 6 - 1 Four profile functions of the index change where the solid red line
is equation (6 - 3), the green dashed line is equation (6 - 5), the blue dotted
line is equation (6 - 6), and the pink dashed and dotted line is equation (6 - 7)
with 5=kβ , and with the following parameters: )(10 mmL = , 0002.0=nδ
APODIZATION OF FIBRE BRAGG GRATINGS
105
Wavelength (micrometre)1.55061.55041.55021.55001.5498
Re
flect
an
ce (
p. u
.)0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 6 - 2 The reflectance spectra of four apodized uniform Bragg gratings
with a Gauss profile (equation (6 - 3)) (red solid line), a raised cosine profile
(equation (6 - 5)) (green dashed line), a sinc profile (equation (6 - 6)) (blue
dotted line), and a Kaiser profile (equation (6 - 7) and 5=kβ ) (pink dashed
and dotted line), and with the following parameters: )(10 mmL = , 447.1=effn ,
)(550.1 mD µλ = , 0002.0=nδ
6.2.2 Comparison of the properties of apodized and
unapodized uniform gratings
Figure 6 - 3 shows the simulation results of a uniform grating with and without
Gaussian apodization (equation (6 - 4)). The maximum reflectance of the
apodized grating is reduced, but the sidelobes are suppressed and the ripple is
reduced. This was caused by the reduction in the index change on both sides of
the grating.
There is much ripple on both sides of the time delay and dispersion (apodized
grating). There are only the numerical calculation errors. Only the results of the
centre range of the graphs are useful.
APODIZATION OF FIBRE BRAGG GRATINGS
106
Wavelength (micrometre)1.55051.55001.5495
Re
flect
an
ce (
p. u
.)0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 6 - 3 (a) The reflectance spectrum
Wavelength (micrometre)1.55051.55001.5495
Tim
e de
lay
(ps)
120
100
80
60
40
20
0
-20
Figure 6 - 3 (b) The time delay
APODIZATION OF FIBRE BRAGG GRATINGS
107
Wavelength (micrometre)1.55051.55001.5495
Dis
pers
ion
(ps/
nm)
800
600
400
200
0
-200
-400
-600
-800
Figure 6 - 3 (c) The dispersion
Figure 6 - 3 (a) The reflectance spectrum, (b) the time delay and (c) the
dispersion of a uniform grating without apodization (red solid line) and a
Gauss apodized uniform grating (green dashed line) where the Gauss width
parameter is 10=a , and with the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ = , 0002.0=nδ .
By keeping the average refractive index constant, a symmetrical spectrum is
obtained. The index profile with preconditioning (average refractive index
correction [7]) can be used to obtain this kind of spectrum. The simulation results
with preconditioning are given in Figure 6 - 4.
The output of the simulation has been compared with another simulation
program, namely, the software supplied by Apollo Photonics Inc. The demo
version of that software gave the same simulation results as my program. The
output has also been verified with the example given by T. Erdogan [2].
APODIZATION OF FIBRE BRAGG GRATINGS
108
Position (millimetre)1086420
Re
fra
ctiv
e in
dex
0.00018
0.00016
0.00014
0.00012
0.00010
0.00008
0.00006
0.00004
0.00002
Figure 6 - 4 (a) The profiles of index change
Wavelength (micrometre)1.55051.55041.55031.55021.55011.5500
Re
flect
an
ce (
p. u
.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 6 - 4 (b) The reflectance spectra
APODIZATION OF FIBRE BRAGG GRATINGS
109
Wavelength (micrometre)1.55051.55041.55031.55021.55011.5500
Tim
e de
lay
(ps)
70
65
60
55
50
45
40
35
30
25
20
Figure 6 - 4 (c) The time delay
Wavelength (micrometre)1.55051.55041.55031.55021.55011.5500
Dis
pers
ion
(ps/
nm)
100
80
60
40
20
0
-20
-40
-60
-80
-100
Figure 6 - 4 (d) The dispersion
Figure 6 - 4 (a) the profiles of index change (b) the reflectance spectra (c) the
time delay (d) the dispersion of a Gauss apodized uniform grating (red solid
line), a raised cosine apodized uniform grating (green dashed line), and a sinc
apodized uniform grating (blue dotted line), with the following parameters:
)(10000 mL µ= , 447.1=effn , )(550.1 mD µλ = , 0002.0=nδ .
APODIZATION OF FIBRE BRAGG GRATINGS
110
6.2.3 The apodization of linear chirped gratings with
different Gauss width parameters
We apodized linear chirped gratings with Gaussian functions equation (6 - 4),
and different Gauss width parameters a .
Wavelength (micrometre)1.5521.5511.5501.5491.5481.5471.546
Re
flect
an
ce (
p. u
.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 6 - 5 The reflectance spectrum of linear chirped gratings (with
preconditioning) with different Gauss width parameters: 5=a (red solid line),
10=a (green dashed line), 15=a (blue dotted line), and with the following
parameters: )(10000 mL µ= , 447.1=effn , )(550.1 mD µλ = , 0009.0=nδ ,
)/(5.1/ cmnmdzd D −=λ .
Figure 6 - 5 shows the spectral response with the different Gaussian width
parameters. The reflectance spectrum is smoother if the Gauss width
parameters are increased. The bandwidth becomes narrow. This means that
when one parameter is improved, another becomes worse. Other parameters
should be optimized to obtain both a wide bandwidth and a smooth reflectance
spectrum. The Gauss width parameters affect the bandwidth of the chirped
APODIZATION OF FIBRE BRAGG GRATINGS
111
grating. This parameter can be used to optimize dispersion compensation by
using a linear chirped grating, as will be discussed at the end of this chapter.
6.2.4 The apodization of linear chirped gratings with
different Kaiser window parameters
We apodized linear chirped gratings with Kaiser window functions equation (6 -
7), and different Kaiser window parameters kβ .
Wavelength (micrometre)1.5521.5511.551.5491.5481.547
Re
flect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 6 - 6 The reflectance spectrum of linear chirped gratings (with
preconditioning) with different Kaiser window parameters: 3=kβ (red solid
line), 5=kβ (green dashed line), 7=kβ (blue dotted line), and with the
following parameters: )(10000 mL µ= , 447.1=effn , )(550.1 mD µλ = , 0009.0=nδ ,
)/(5.1/ cmnmdzd D −=λ .
Figure 6 - 6 shows the spectral response with the different Kaiser window
parameters. The reflectance spectrum is smoother if the Kaiser window
parameters are increased. The bandwidth becomes narrow. This is the same
effect that occurs if the Gauss width parameter is increased. Therefore, we only
used Gauss apodization for the analysis in the next section.
APODIZATION OF FIBRE BRAGG GRATINGS
112
6.3 Relationship between the maximum reflectance
and the Gauss width parameters
Gauss width parameter232221201918171615141312111098765
Maxi
mum
re
flect
an
ce (
p.
u.)
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
(a)
Gauss width parameter232221201918171615141312111098765
Maxi
mum
refle
ctance
(p. u.)
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
(b)
Figure 6 - 7 (a) without preconditioning (b) with preconditioning. The
maximum reflectance vs. the Gauss width parameters with different values of
the “chirp parameter”: 0=F (red solid line), 100−=F (green dashed line),
200−=F (blue dotted line), and with the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ = , 0002.0=nδ .
APODIZATION OF FIBRE BRAGG GRATINGS
113
Figure 6 - 7 shows the relationship between the maximum reflectance and the
Gauss width parameters. The Gauss width parameters do not affect the
maximum reflectance of the linear chirped grating (green dashed line and blue
dotted line) seriously, but they have a large effect on the uniform grating (red
solid line).
6.4 Relationship between the 3 dB bandwidth and the
Gauss width parameters
Figure 6 - 8 shows the relationship between the 3 dB bandwidth and the Gauss
width parameters. The bandwidth is reduced when the Gauss width parameter is
increased. The rate of change is almost the same with different values of the
“chirp parameter” F .
Gauss width parameter232221201918171615141312111098765
3 dB
ban
dwid
th (
nano
met
re)
4
3
2
1
Figure 6 - 8 (a) without preconditioning
APODIZATION OF FIBRE BRAGG GRATINGS
114
Gauss width parameter232221201918171615141312111098765
3 dB
ban
dwid
th (
nano
met
re)
4
3
2
1
Figure 6 - 8 (b) with preconditioning
Figure 6 - 8 (a) without preconditioning (b) with preconditioning. The 3dB
bandwidth vs. the Gauss width parameters with different values of the “chirp
parameter”: 100−=F (red solid line), 200−=F (green dashed line), 300−=F
(blue dotted line), and with the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ = , 0002.0=nδ .
6.5 Dispersion compensation using a linear chirped
grating with apodization
In a previous section, we mentioned that chirped gratings can be used as
dispersion compensation components in optical communication systems. The
simulation results show that the spectrum and time delay are improved by using
a linear chirp function, but this is not good enough. Ripples still exist in the
reflectance spectrum, the time delay and the dispersion. These problems should
be solved by improving the apodization function.
APODIZATION OF FIBRE BRAGG GRATINGS
115
Wavelength (micrometre)1.5521.5501.5481.546
Re
flect
an
ce (
p. u
.)0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
Wavelength (micrometre)1.5521.5511.5501.5491.5481.5471.546
Tim
e de
lay
(ps)
110100
908070
605040302010
0-10-20
(b)
Figure 6 - 9 (a) The reflectance spectrum and (b) the time
delay.
APODIZATION OF FIBRE BRAGG GRATINGS
116
Wavelength (micrometre)1.5521.5511.5501.5491.5481.5471.546
Dis
pers
ion
(ps/
nm)
1,400
1,200
1,000
800
600
400
2000
-200
-400
-600
-800
-1,000
-1,200
(c)
Wavelength (micrometre)1.5521.5511.5501.5491.5481.5471.546
Dis
pers
ion
(ps/
nm)
300
200
100
0
-100
-200
-300
-400
-500
(d)
Figure 6 - 9 (c) The dispersion and (d) the dispersion with the range zoom
Figure 6 - 9 (a) The reflectance spectrum, (b) the time delay, (c) the
dispersion, and (d) the dispersion with the range zoom of a chirped grating
(red solid line) and apodized chirped grating with preconditioning (green
dashed line) where the Gauss width parameter is 20=a , and with the
following parameters: )(10000 mL µ= , 447.1=effn , )(550.1 mD µλ = ,
0009.0=nδ , )/(2/ cmnmdzd D −=λ .
APODIZATION OF FIBRE BRAGG GRATINGS
117
Figure 6 - 9 shows the simulation results of the chirped grating both with and
apodization. We can see that the reflectance spectrum is smooth, and the ripple
is eliminated. The time delay and dispersion are improved as well.
Only the Gauss apodized grating was studied in this thesis. Other apodized
functions, such as raised cosine and sinc, still need to be studied in future.
6.5.1 Optimization of the Gauss width parameters for
dispersion compensation
We require that the components used in dispersion compensation have wider
bandwidths and linear time delay characteristics. But this is difficult to obtain
simultaneously by using an apodized chirped Bragg grating. The features of the
time delay and dispersion in the grating are improved whereas the bandwidth of
compensation will be reduced. But we can increase the length of the grating to
increase bandwidth, which is shown in Figure 5 – 11. It is necessary to choose
suitable parameters to optimize the system. Several simulation results are
shown here.
Figure 6 - 10 shows the reflectance spectrum of the linear chirped and apodized
gratings with different Gaussian width parameters. The bandwidth becomes
narrow and the reflectance spectrum becomes smooth if the Gaussian width
parameters are increased.
APODIZATION OF FIBRE BRAGG GRATINGS
118
Wavelength (micrometre)1.5541.5521.5501.5481.5461.5441.5421.540
Re
flect
an
ce (
p. u
.)0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Figure 6 - 10 The reflectance spectrum of linear chirped gratings (with
preconditioning) with different Gauss width parameters: 5=a (red solid line),
15=a (green dashed line), 25=a (blue dotted line), and with the following
parameters: )(15000 mL µ= , 447.1=effn , )(550.1 mD µλ = , 001.0=nδ ,
)/(5.2/ cmnmdzd D −=λ .
Figure 6 - 11 shows the time delay of the linear chirped and apodized gratings
with different Gaussian width parameters. The linearity is improved if the
Gaussian width parameter is increased. The ripple is reduced.
APODIZATION OF FIBRE BRAGG GRATINGS
119
Wavelength (micrometre)1.5521.5501.5481.5461.544
Tim
e de
lay
(ps)
140
120
100
80
60
40
20
0
-20
(a)
Wavelength (micrometre)1.54711.5471.54691.54681.54671.54661.5465
Tim
e de
lay
(ps)
70
68
66
64
62
60
58
56
54
52
(b)
Figure 6 - 11 The time delay of linear chirped and apodized gratings with
different Gauss width parameters: (a) full range plot, (b) the range zoom in.
The same parameters as in Figure 6 - 10 were used.
APODIZATION OF FIBRE BRAGG GRATINGS
120
Wavelength (micrometre)1.5521.5501.5481.5461.544
Dis
pers
ion
(ps/
nm)
2,000
1,500
1,000
500
0
-500
-1,000
-1,500
-2,000
(a)
Wavelength (micrometre)1.5471.54651.546
Dis
pe
rsio
n (p
s/nm
)
250
200
150
100
50
0
-50
-100
-150
(b)
Figure 6 - 12 The dispersion of linear chirped and apodized gratings with
different Gauss width parameters: (a) full range plot, (b) the range zoom in.
The same parameters as in Figure 6 - 10 were used.
Figure 6 - 12 shows the dispersion of the linear chirped and apodized gratings
with different Gaussian width parameters. The linearity is improved if the
Gaussian width parameter is increased. The ripple is reduced. From this
simulation, it is demonstrated that the wide bandwidth and the linearity of the
APODIZATION OF FIBRE BRAGG GRATINGS
121
time delay always oppose each other. Depending on the demand of the
dispersion compensation, optimized parameters should be used.
6.6 Conclusion
In this chapter, apodization of fibre Bragg gratings was discussed and simulated.
The ripple of the spectral response can be reduced and the sidelobes can be
suppressed at the same time. The linearity of time delay and dispersion of the
chirped grating can be improved by using apodization of fibre Bragg gratings.
APODIZATION OF FIBRE BRAGG GRATINGS
122
6.7 References
1. A. Othonos, “Fibre Bragg gratings”, Review of Scientific Instruments, vol.68,
no.12, 1997, pp. 4309-4341.
2. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology, vol.15,
no.8, 1997, pp. 1277-1294.
3. C. Martinez, P. Jougla, S. Magne, and P. Ferdinand, “Phase plate process for
advanced fibre Bragg gratings devices manufacturing”, IEICE Transactions on
Electronics, vol. E83-C, no.3, 2000, pp. 435-439.
4. V. Tzolov, D. Feng, S. Tanev, and Z. Jakubczyk, "Modeling tools for
integrated and fiber optical devices", Integrated Optics Devices III, Photonics
West 99, San Jose, CA, 1999, pp. 23-29.
5. Y. Sun, C. Yun, J. Lin, Y. Qian, B. Bai, Y. Yang and W. Qiu, “Study on the
apodized function of chirped fibre grating for dispersion compensation”, Journal
of Optoelectronics Laser, vol.10, no.3, 1999, pp. 228-231.
6. J.F. Kaiser, "Nonrecursive digital filter design using the 0I - sinh window
function", IEEE Symp. Circuits and Systems, April 1974, pp. 20-23.
7. A. Inoue, T. Iwashima, T. Enomoto, S. Ishikawa, and H. Kanamori,
“Optimization of fibre Bragg grating for dense WDM transmission system”, IEICE
Transactions on Electronics, vol. E81-C, no.8, 1998, pp. 1209-1218.
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
123
CHAPTER 7: Other Applications of theSimulation Program
CHAPTER 7: OTHER APPLICATIONS OF THE SIMULATION PROGRAM 123
7.1 SIMULATION OF PHASE-SHIFTED BRAGG GRATINGS ....................................... 124
7.1.1 Principle........................................................................................... 124
7.1.2 Direct integration ............................................................................. 124
7.1.3 Transfer matrix method ................................................................... 125
7.1.4 Simulation results ............................................................................ 125
7.1.5 Applications ..................................................................................... 130
7.2 SIMULATION OF SAMPLED BRAGG GRATINGS................................................. 130
7.2.1 Principle........................................................................................... 130
7.2.2 Direct integration ............................................................................. 131
7.2.3 Transfer matrix method ................................................................... 131
7.2.4 Simulation results ............................................................................ 132
7.2.5 Applications ..................................................................................... 137
7.3 CONCLUSION.............................................................................................. 137
7.4 REFERENCES ............................................................................................. 138
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
124
7.1 Simulation of phase-shifted Bragg gratings
7.1.1 Principle
The phase-shifted Bragg grating is obtained when the refractive index is
changed in such a way that the phase is not continuous. The schematic
representation of a phase-shifted grating structure is shown in Figure 7 - 1. The
half Bragg period phase-shift φ was placed in the centre of a uniform grating in
this example.
Phase-shifted grating
Uniform grating
Phase shift φ
Figure 7 - 1 Schematic representation of a phase-shifted grating structure
7.1.2 Direct integration
The phase-shifted Bragg grating can be modelled by the coupled-mode theory
and simulated by the direct integration method. A constant phase shift is added
in equation (2-1) for the calculation of the integration. In the coupled-mode
equations (3-2) and (3-3), we multiply the current value of )(zk by )exp( φi where
φ is the shift in the grating phase [1].
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
125
7.1.3 Transfer matrix method
The phase-shifted grating can be calculated by the transfer matrix method too.
We can also use one 2 X 2 matrix to represent this phase shift in the grating.
The transfer matrix can be written as [1],
)2
exp(0
0)2
exp(
i
i
Bpi i
i
F φ
φ−
= 7 - 1
where iφ is the value of the phase-shift. If matrix (7 - 1) is substituted into
equation (3 –27), the spectral response of the phase-shifted grating can be
obtained.
7.1.4 Simulation results
Figure 7 - 2 Figure 7 - 3 Figure 7 - 4Grating parameters
Solid line Dashed line (a) (b) (a) (b)
Grating length
( )L)(10000 mµ )(10000 mµ )(10000 mµ
Index change
( nδ )0002.0 0001.0 0002.0 0002.0
Effective index
( effn )447.1 447.1 447.1
Design grating
wavelength ( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ
Phase-shift
(φ )π Null π π
21
π23
Table 7 - 1 Simulation parameters of phase-shifted gratings
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
126
Wavelength (micrometre)1.55101.55051.55001.54951.5490
Re
flect
an
ce (
p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
Wavelength (micrometre)1.55101.55051.55001.54951.5490
Re
flect
an
ce (
p.
u.)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
Figure 7 - 2 (a) Without apodization (b) with apodization (equation (6 - 5)).
Reflectance spectrum of a phase-shifted grating (red solid line, πφ = ), a
uniform grating (green dashed line), a phase-shifted grating with apodization
(blue dotted line, πφ = ) and a phase-shifted grating with preconditioning (pink
dashed and dotted line, πφ = ), and the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ = , 0002.0=nδ .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
127
Figure 7 - 2 is the reflectance spectrum of the π phase-shifted grating compared
with the uniform grating without a phase shift. In terms of the simulation results,
we can see that the bandwidth of the phase-shifted grating is broader than the
one without a phase shift. The peak reflectance is also reduced. The sidelobe
reflectance is also larger than the sidelobe reflectance of the uniform grating.
There is a very narrow transmission band in the centre of the spectrum of the
phase-shifted grating.
7.1.4.1 Phase-shifted gratings with different refractive index
excursions
Figure 7 - 3 (a) and (b) are the transmittance spectra of π phase-shifted gratings
with different refractive index excursions. The transmittance spectrum is
symmetrical for the π phase-shifted grating. The transmission band occurs at
the centre of the stop band. Both the reflectance spectrum and the transmittance
spectrum are symmetrical. The width of the transmission band depends on the
refractive index change. FWHM bandwidth is reduced from )(038.0 nm to
)(008.0 nm when the index change is increased from 0001.0 to 0002.0 .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
128
Wavelength (micrometre)1.55101.55051.55001.54951.5490
Tra
nsm
ittan
ce(p
. u.
)0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
(a)
Wavelength (micrometre)1.55101.55051.55001.54951.5490
Tra
nsm
ittan
ce(p
. u.
)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
Figure 7 - 3 Transmittance spectrum of phase-shifted Bragg gratings with
different values of the refractive index excursions: (a) 0001.0=nδ (b)
0002.0=nδ , and the following parameters: )(10000 mL µ= , 447.1=effn ,
)(550.1 mD µλ = , πφ = .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
129
7.1.4.2 Phase-shifted gratings with different phase shifts
Wavelength (micrometre)1.55101.55051.55001.54951.5490
Tra
nsm
ittan
ce(p
. u.
)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
Wavelength (micrometre)1.55101.55051.55001.54951.5490
Tra
nsm
ittan
ce(p
. u.
)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
Figure 7 - 4 Transmittance spectrum of phase-shifted Bragg gratings with
different values of phase shift: (a) πφ21
= (b) πφ23
= , and the following
parameters: )(10000 mL µ= , 447.1=effn , 0002.0=nδ , )(550.1 mD µλ = .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
130
Figure 7 - 4 (a) and (b) are the transmittance spectra of phase-shifted gratings
with different phase shift values, (a) πφ21
= (b) πφ23
= , respectively. Unlike the
π phase-shifted grating, these are not symmetrical. Other phase shifts were
introduced and the results were similar. The spectrum response is symmetrical
only when the phase shift is π . The wavelength of the resonance was shifted
from )(134.1550 nm=λ to )(296.1550 nm=λ for a phase shift from πφ21
= to
πφ23
= . We can obtain the required wavelength of the resonance by changing
the phase shift value.
7.1.5 Applications
In terms of the simulation results of the spectral response, these types of the
grating have a very sharp narrow passband inside the reflectance spectrum. This
characteristic of the phase-shifted grating can be used for fibre lasers [2], filters
for communication application [3] and sensor systems [4]. For example, the
phase-shifted grating can be used to obtain single mode operation of distributed
feedback (DFB) fibre lasers [5].
7.2 Simulation of sampled Bragg gratings
7.2.1 Principle
Figure 7 - 5 (b) is a schematic representation of a sampled grating structure
compared with the uniform grating in Figure 7 - 5 (a). In Figure 7 - 5, AL is the
sampling period of the sampled grating, and BL is the range that was exposed
by UV light. The range BA LL − is without exposure to the UV light. Another
parameter in the sampled grating is the duty cycle, which is defined as
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
131
AB LLR /= . L is the total length of the sampled grating. These variables will be
used to analyse the spectral characteristics of sampled Bragg gratings in the
simulations.
Grating length
(b) Sampled Bragg grating
(a) Uniform Bragg grating
ALL
BL
Figure 7 - 5 Schematic representation of a sampled grating structure
7.2.2 Direct integration
The spectral response of sampled Bragg gratings can be modelled by the
coupled-mode theory and simulated by the direct integration method. In this
case, we simply set 0=k in the range ( BA LL − ) between the gratings [1], which
is not exposed to UV light. The spectral response can be obtained by solving the
coupled-mode equations (3-2) and (3-3).
7.2.3 Transfer matrix method
There are two implementations of the models of the sampled Bragg gratings with
the transfer matrix method. First, we can obtain a new transfer matrix by
multiplying matrices (7 - 1) and (3-26), resulting in,
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
132
∆+∆−
∆
∆−−
∆−∆=
∗=
)2
exp()]sinh(ˆ
)[cosh()2
exp()sinh(
)2
exp()sinh()2
exp()]sinh(ˆ
)[cosh(
iB
BB
iB
B
iB
B
iB
BB
Bpi
Bi
Bsi
iziz
iz
ki
iz
ki
iziz
FFF
φγ
γσ
γφ
γγ
φγ
γφ
γγσ
γ 7 - 2
where Bγ is equations (3-14) and (3-15).
By using this new transfer matrix (7 - 2) in equation (3-27), we can obtain the
simulation results of sampled Bragg gratings as if dealing with a normal uniform
grating.
Another implementation is simply to calculate matrix (3-27) with different transfer
matrices in the different ranges. This means that the range ( BL ) of the grating is
represented by matrix (3-26), and the range ( BA LL − ) between two grating
sections can be represented by matrix (7 - 1), for which the phase shift is given
by [1],
zneffi ∆=λ
πφ 2
27 - 3
where BA LLz −=∆ is the separation between two grating sections. In this
simulation program, this method was used to obtain the spectral responses of
sampled gratings.
7.2.4 Simulation results
The spectral responses of sampled gratings with the different values of the
grating parameters, such as the length, the sampling period and the duty cycle,
are presented in this section.
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
133
Figure 7 - 6 Figure 7 - 7 Figure 7 - 8Grating parameters
(a) (b) (a) (b) (a) (b)
Grating length
( )L)(10000 mµ )(20000 mµ )(10000 mµ )(10000 mµ
Index change
( nδ )0006.0 0006.0 0006.0
Effective index
( effn )447.1 447.1 447.1
Design grating
wavelength ( )Dλ)(550.1 mµ )(550.1 mµ )(550.1 mµ
Duty cycle
( R )2.0 2.0 1.0 3.0
Sampling period
( AL ))(1000 mµ )(666 mµ )(500 mµ )(1000 mµ
Table 7 - 2 Simulation parameters of sampled gratings
Figure 7 - 6 shows the simulation results of sampled gratings with different
grating lengths L . Figure 7 - 7 shows the simulation results of the sampled
gratings with different sampling periods AL . Figure 7 - 8 shows the simulation
results of the sampled grating with different duty cycles R .
From the simulation results, we can see that the reflectance depends on the
grating length L , sampling period AL , and duty cycle R . The characteristics of
the sampled grating will not be discussed in this thesis; only the simulation
results are given. Further analysis still needs to be done.
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
134
7.2.4.1 Sampled gratings with different grating lengths
Wavelength (micrometre)1.5581.5561.5541.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
Wavelength (micrometre)1.5581.5561.5541.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
Figure 7 - 6 The spectral response of sampled gratings with different grating
lengths: (a) )(10000 mL µ= (b) )(20000 mL µ= , and the following parameters:
447.1=effn , )(550.1 mD µλ = , 0006.0=nδ , 2.0=R , )(1000 mLA µ= .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
135
7.2.4.2 Sampled gratings with different sampling periods
Wavelength (micrometre)1.5581.5561.5541.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(a)
Wavelength (micrometre)1.5581.5561.5541.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
Figure 7 - 7 The spectral response of sampled gratings with different sampling
periods: (a) )(666 mLA µ= , (b) )(500 mLA µ= , and the following parameters:
)(10000 mL µ= , 447.1=effn , )(550.1 mD µλ = , 0004.0=nδ , 2.0=R .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
136
7.2.4.3 Sampled gratings with different duty cycles
Wavelength (micrometre)1.5581.5561.5541.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.650.6
0.550.5
0.450.4
0.350.3
0.250.2
0.150.1
0.05
(a)
Wavelength (micrometre)1.5581.5561.5541.5521.551.5481.5461.544
Ref
lect
ance
(p.
u.)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
(b)
Figure 7 - 8 The spectral response of the sampled grating with different duty
cycles: (a) 1.0=R (b) 3.0=R , and the following parameters: )(10000 mL µ= ,
447.1=effn , )(550.1 mD µλ = , 0004.0=nδ , )(1000 mLA µ= .
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
137
7.2.5 Applications
The sampled gratings [6] can be used in optical fibre communication systems
[7],[8], depending on the characteristics of the spectral responses. From the
simulations of the sampled gratings, we can see that they can be used as comb
filters for signal processing and wavelength division multiplexed (WDM)
communication systems [9].
7.3 Conclusion
In this chapter, two types of non-uniform Bragg gratings were simulated. The
coupled-mode theory is a suitable tool to analyse them. Two simulation methods,
direct integration and the transfer matrix method, can be used to obtain the
spectra. The spectra of phase-shifted and sampled gratings with different
parameter values were simulated and discussed. Their applications were
introduced briefly. The simulation results that were given in this chapter were
solved by the transfer matrix method.
OTHER APPLCIATIONS OF THE SIMULATION PROGRAM
138
7.4 References
1. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology, vol.15,
no.8, 1997, pp. 1277-1294.
2. M. J. Guy, J. R. Taylor, and R. Kashyap, “Single-frequency erbium fibre ring
laser with intracavity phase-shifted fibre Bragg grating narrow-band-filter”,
Electronics Letters, vol.31, 1995, pp. 1924-1925.
3. G. P. Agrawal and S. Radic, “Phase-shifted fibre Bragg gratings and their
application for wavelength demultiplexing,” IEEE Photonics Technology Letters,
vol.6, no.8, 1994, pp. 995-997.
4. A. D. Kersey, M. A. Davis, H. J. Patrick, M. Leblanc, K. P. Koo, C. G. Askins,
M. A. Putnam, and E. J. Friebele, “Fibre grating sensors”, Journal of Lightwave
Technology, vol.15, no.8, 1997, pp. 1442-1463.
5. J.T. Kringlebotn, J.-L. Archambault, L. Reekie, and D.N.Payne, “1.5 m Er :Yb
-doped fiber DFB laser”, Conference on Lasers and Electro-Optics (CLEO-94),
Anaheim, CA, paper CWP2, 1994.
6. B. S. Kim, Y Chung, and S. H. Kim, “A study on the characteristics of a
widely tunable sampled grating DBR laser diode integrated with an external
modulator”, Journal of Korea Electronics Engineers-A, vol. 33, 1996, pp. 174-
185.
7. M. Ibsen, B.J. Eggleton, M.G. Sceats, and F. Ouellette, “Broadly tunable
DBR fibre laser using sampled fibre Bragg gratings”, Electronics Letters, vol.31,
no.1, 1995, pp. 37-38.
8. V. Jayaraman, Z. M. Chuang, and L.A. Coldren, “Theory, design, and
performance of extended tuning range semiconductor lasers with sampled
gratings”, IEEE Journal of Quantum Electronics, vol.29, no.6, 1993, pp. 1824-
1834.
9. J. Hubner, D. Zauner, and M. Kristensen, “Strong sampled Bragg gratings for
WDM applications”, IEEE Photonics Technology Letters, vol.10, no.4, 1998, pp.
552-554.
CONCLUSION AND FUTURE WORK
139
CHAPTER 8: Conclusion and FutureWork
CHAPTER 8: CONCLUSION AND FUTURE WORK........................................................... 139
8.1 CONCLUSION .......................................................................................... 140
8.2 FUTURE WORK ........................................................................................ 141
8.2.1 Simulation of long period gratings..................................................... 141
8.2.2 Bragg grating simulation with the Internet......................................... 141
8.3 REFERENCES .......................................................................................... 143
CONCLUSION AND FUTURE WORK
140
8.1 Conclusion
Fibre optical component modelling is a vital step towards the design of fibre
optical sensors and communication systems. Simulation software is a very rapid,
efficient and economical way to design and analyze fibre optical systems.
Object-oriented programming techniques have been used to develop the
simulation software in this project. It is easily extendible and reusable according
to varying circumstances. It is technically feasible to write a simulation program
by using object-oriented programming.
Fibre Bragg gratings have a very important role in the field of communication and
sensor systems. The coupled-mode theory and two mode approximation are
also simple and accurate theories to analyse the fibre grating. In this project, the
fibre Bragg grating was modelled, simulated and discussed. The uniform,
chirped, apodized, phase-shifted and sampled gratings have been simulated by
using the transfer matrix method and the direct integration method. In this thesis,
complete simulation results were given by using the transfer matrix method, and
the software is coded by using C++ under C++ Builder 4 environment. However,
the program with the direct integration method has been coded by using Object
Pascal under Delphi 4 environment. Because these are simple results, they are
not given in this thesis.
The simulation results of linear chirped Bragg gratings were provided in Chapter
5. The spectral responses of the linear chirped grating with different grating
parameters, such as chirp variable, grating length and index change, were
provided and discussed. The relationship between the maximum reflectance, 3
dB bandwidth and centre wavelength with grating parameters were also given
and discussed. Two applications of linear chirped Bragg gratings have been
analysed by using this program.
Depending on the ripple of the reflectance spectrum and the time delay of the
linear chirped grating, the apodized grating was introduced in Chapter 6. Better
CONCLUSION AND FUTURE WORK
141
simulation results can be obtained by using the apodization of the grating to get
rid of the ripple in the reflectance spectra and the time delay of the linear chirped
grating, making them more suitable for the dispersion compensation on
communication systems. Phase-shifted and sampled gratings were briefly
introduced and simulated in Chapter 7. From the simulation results, we can see
that this program can be used to analyse the problems of fibre Bragg gratings.
8.2 Future work
Other types of grating, such as the long period grating [1], [2], have also been
studied in recent years. It is necessary to simulate these gratings as well. On the
other hand, the application of the Internet in many fields has been developed, or
is under development. There should be many potential applications if the
simulation software could be run on the Internet.
8.2.1 Simulation of long period gratings
Long period gratings, also known as transmission gratings, are periodic
structures, in which coupling occurs between modes travelling in the same
direction. Long period Bragg gratings are of interest to optical researchers for
their higher sensitivity to sensing and some other applications. The modelling of
the long period Bragg grating can also be done by using the coupled-mode
theory. It is therefore important that the software be expanded in future to cover
the simulation of these components as well.
8.2.2 Bragg grating simulation with the Internet
The Internet has become very popular in the last ten years. It is possible to use
the Internet to supply online simulations of physical systems for the engineer,
researcher, student and scientist. It is a convenient method to share simulation
CONCLUSION AND FUTURE WORK
142
programs by many users.
JAVA is a suitable network programming language for realizing these types of
project. It is a pure object-oriented programming language. There are several of
these types of project under development. JAVA is portable and can be
understood and supported by most web browsers. Using the JAVA computing
language enables interactive program execution from a web page.
JAVA and C++ are similar programming languages. The transfer of simulation
codes of C++ to JAVA should be fairly simple. Some libraries could even be
shared. If the current simulation code could be portable to the JAVA code, the
possibility to do simulations on the Internet would increase the use of this
software.
CONCLUSION AND FUTURE WORK
143
8.3 References
1. T. Erdogan, “Fibre grating spectra”, Journal of Lightwave Technology, vol.15,
no.8, 1997, pp. 1277-1294.
2. Y. Zhao and J. Palais, “Simulation and characteristics of long-period fiber
Bragg grating coherence spectrum”, Journal of Lightwave Technology, vol.16,
no.4, 1998, pp. 554-561.
BIBLIOGRAPHY
144
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MANUAL
150
Reference Manual
FBGSFibre Bragg Gratings Simulation
Version 0.9
for Windows® 95, 98, 2000 and Windows NTTM
Tel.: (+27-11) 489-2352
E-mail: [email protected]
Web: http://eng.rau.ac.za/sensors/
Disclaimer
THIS IS PROVIDED "AS IS", WITHOUT ANY EXPRESS OR IMPLIEDWARRANTIES.
MANUAL
151
1. Introduction
The code was realized through C++ in the C++ Builder4 environment. Uniform,
chirped, apodized, phase-shifted and sampled Bragg gratings have already been
simulated by using the transfer matrix method. The reflected and transmitted
spectra, time delay and dispersion of fibre Bragg gratings can be obtained by
using this simulation program.
2. Quick Start
This simulation program can be run directly without installation and other third
party library support. Run “FBGS.exe” file to start the software.
Figure 1 Steps for quick start
There are four steps to use this software to simulate the problems of fibre Bragg
gratings
MANUAL
152
• Step1: specifies the grating parameters (Region A in Figure 1).
• Step2: specifies the apodization function for apodized gratings (Region C in
Figure 1).
• Step3: specifies the line styles of the plot (Region D in Figure 1).
• Step4: specifies the types of fibre Bragg grating: uniform, chirped, apodized
phase-shifted or sampled grating. Click on the button that corresponds to the
type of grating (Region B in Figure 1) to start the simulation.
3. Graphic user interface
3.1 Input interface
Figure 2 The GUI of the simulation program
• Region A: specifies general grating parameters for all simulations.
• Region B: specifies the step of the transfer matrices.
• Region C: “Clear All” is used to clear all the graphs.
MANUAL
153
• Region D: specifies the parameter of the linear chirped grating.
• Region E: specifies the parameter of the phase-shifted grating.
• Region F: specifies the parameters of the sampled grating.
• Region G: specifies the apodization function for the apodized grating.
• Region H: specifies the line styles of the plot.
• Region I: switches the output results and the simulation option.
1: Wavelength step of the simulation
2: Ending wavelength of the simulation
3: Starting Wavelength of the
simulation
4: Refractive index change
5: Effective index
6: Designed Bragg wavelength
7: Grating length
8: Gauss width parameter
or Kaiser windows parameter
9: The step of the transfer matrices
10: Clear the output graphs
11: “chirp parameter” F
12: Chirp variable: dzd D /λ
13: Phase shift
14: Duty cycle
15: Sampling number
Figure 3 The grating parameters
MANUAL
154
3.2 Output interface
Figure 4 Save the results of the simulation
The output graphs can be saved to “wmf” (item 1) and “bmp” (item 2) formats.
Single click on the right button of the mouse inside the graph, and the pop-up
menu will be displayed. Those two formats can be saved to a file.
The graphs can be cleared by clicking on the “Clear All” button.
The line colours of the graphs are changed automatically for each simulation. It
is better to click on the “Clear All” button for each simulation first, because it
makes the line colours of the graphs switch in the following order (red, green,
yellow, blue … ).
MANUAL
155
There is a known bug with this program. Depending on the “Windows” operation
system, the styles (solid, dashed, dotted, etc) of the line can only be displayed
when the width of the line is one pixel. The styles will not be displayed on the
screen if the width of the line is set at more than one. This problem was not
resolved in C++ Builder either. If the line width is one pixel, the line of the output
graphs cannot be seen clearly. The default value of the line width is 4 pixels in
this program. This causes the line style (solid, dashed, dotted, etc) not to be
seen directly in the simulation program. One way resolving this bug is to use
“Acrobat” software to print the output “wmf” format file in the “pdf” format file. The
line styles can be displayed in the file with the “pdf” format.
4. Using FBGS
4.1 Uniform gratings
Simply set item 11, 12 or 13 in Figure 3 at zero, and click on the “Chirped” or
“Phase shift” button to start the simulation. The uniform gratings can be
simulated.
4.2 Linear chirped gratings
Two chirp parameters can be used to simulate linear chirped gratings. One uses
“chirp parameter” F (item 11 in Figure 3) and the other uses chirp variable
dzd D /λ (item 12 in Figure 3). Click on the “chirped” button to start the
simulation.
4.3 Phase-shifted gratings
Specify the value of the phase shift (item 11 in Figure 3). The value is a multiple
π . For example, if it equals 2, the value of the phase shift is π2 . Click on the
“phase shift” button to start the simulation.
MANUAL
156
4.4 Sampled gratings
Specify the value of the duty cycle and the sampling number (items 14 and 15 in
Figure 3). Click on the “Sampled” button to start the simulation.
4.5 Apodized gratings
Figure 5 The steps for the simulation of the apodized grating
There are three steps to simulate the apodized gratings.
• Step1: specifies the apodization function for the apodized grating (Region B
in Figure 5).
• Step2: specifies the parameter of the apodization function if the Gaussian 2
or Kaiser function was used (Region C in Figure 5).
• Step3: Click on the “Apodization” button to start the simulation for the
apodized grating without a precondition. Click on the “Precondition” button to
MANUAL
157
start the simulation for the apodized grating with a precondition (Region A in
Figure 5).
The predefined apodized function was used in this program, as follows
4.5.1 Gaussian profile
}])()2/(2
[2lnexp{)( 2
FWHM
Lzzg
−−= ; ],0[ Lz ∈ 1
where LFWHM 4.0= can be used for this profile.
Another expression for the Gaussian profile is as follows:
])2/
(exp[)( 2
L
Lzazg
−−=
],0[ Lz ∈
2
where a is the Gauss width parameter (Region C in Figure 5).
4.5.2 Raised-cosine profile
)])()2/(
cos(1[21
)(FWHM
Lzzg
−+=
π; ],0[ Lz ∈ 3
where LFWHM = can be used for this profile.
4.5.3 Sinc profile
])()2/(
sinc[)(FWHM
Lzzg
−= ; ],0[ Lz ∈ 4
MANUAL
158
where )2/( πLFWHM = can be used for this profile.
4.5.4 Kaiser profile
)(
))1
2(1(
)(0
20
k
k
IN
nI
zgβ
β−
−= ; ]1,0[ −∈ Nn 5
where kβ is the Kaiser window parameter (Region C in Figure 5), and 0I is the
zero order Bessel function of the first kind.
5. System requirements
FBGS V.0.9 requires the following minimum system configuration:
• Microsoft Windows 95, 98, 2000 or Windows NT 4.0 operating system
• Personal computer 486/33 or higher processor
• Graphic resolution of 800x600, minimum 256 colours